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Date | Label | Description | ||||||||||||
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Theorem | ||||||||||||||
24-Jun-2020 | nqprlu 6530 | The canonical embedding of the rationals into the reals. (Contributed by Jim Kingdon, 24-Jun-2020.) | ||||||||||||
⊢ (A ∈ Q → ⟨{𝑙 ∣ 𝑙 <_{Q} A}, {u ∣ A <_{Q} u}⟩ ∈ P) | ||||||||||||||
15-Jun-2020 | imdivapd 9162 | Imaginary part of a division. Related to remul2 9061. (Contributed by Jim Kingdon, 15-Jun-2020.) | ||||||||||||
⊢ (φ → A ∈ ℝ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → A # 0) ⇒ ⊢ (φ → (ℑ‘(B / A)) = ((ℑ‘B) / A)) | ||||||||||||||
15-Jun-2020 | redivapd 9161 | Real part of a division. Related to remul2 9061. (Contributed by Jim Kingdon, 15-Jun-2020.) | ||||||||||||
⊢ (φ → A ∈ ℝ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → A # 0) ⇒ ⊢ (φ → (ℜ‘(B / A)) = ((ℜ‘B) / A)) | ||||||||||||||
15-Jun-2020 | cjdivapd 9155 | Complex conjugate distributes over division. (Contributed by Jim Kingdon, 15-Jun-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → B # 0) ⇒ ⊢ (φ → (∗‘(A / B)) = ((∗‘A) / (∗‘B))) | ||||||||||||||
15-Jun-2020 | riotaexg 5415 | Restricted iota is a set. (Contributed by Jim Kingdon, 15-Jun-2020.) | ||||||||||||
⊢ (A ∈ 𝑉 → (℩x ∈ A ψ) ∈ V) | ||||||||||||||
14-Jun-2020 | cjdivapi 9123 | Complex conjugate distributes over division. (Contributed by Jim Kingdon, 14-Jun-2020.) | ||||||||||||
⊢ A ∈ ℂ & ⊢ B ∈ ℂ ⇒ ⊢ (B # 0 → (∗‘(A / B)) = ((∗‘A) / (∗‘B))) | ||||||||||||||
14-Jun-2020 | cjdivap 9097 | Complex conjugate distributes over division. (Contributed by Jim Kingdon, 14-Jun-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ B # 0) → (∗‘(A / B)) = ((∗‘A) / (∗‘B))) | ||||||||||||||
14-Jun-2020 | cjap0 9095 | A number is apart from zero iff its complex conjugate is apart from zero. (Contributed by Jim Kingdon, 14-Jun-2020.) | ||||||||||||
⊢ (A ∈ ℂ → (A # 0 ↔ (∗‘A) # 0)) | ||||||||||||||
14-Jun-2020 | cjap 9094 | Complex conjugate and apartness. (Contributed by Jim Kingdon, 14-Jun-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ B ∈ ℂ) → ((∗‘A) # (∗‘B) ↔ A # B)) | ||||||||||||||
14-Jun-2020 | imdivap 9069 | Imaginary part of a division. Related to immul2 9068. (Contributed by Jim Kingdon, 14-Jun-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ B ∈ ℝ ∧ B # 0) → (ℑ‘(A / B)) = ((ℑ‘A) / B)) | ||||||||||||||
14-Jun-2020 | redivap 9062 | Real part of a division. Related to remul2 9061. (Contributed by Jim Kingdon, 14-Jun-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ B ∈ ℝ ∧ B # 0) → (ℜ‘(A / B)) = ((ℜ‘A) / B)) | ||||||||||||||
14-Jun-2020 | mulreap 9052 | A product with a real multiplier apart from zero is real iff the multiplicand is real. (Contributed by Jim Kingdon, 14-Jun-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ B ∈ ℝ ∧ B # 0) → (A ∈ ℝ ↔ (B · A) ∈ ℝ)) | ||||||||||||||
13-Jun-2020 | sqgt0apd 9021 | The square of a real apart from zero is positive. (Contributed by Jim Kingdon, 13-Jun-2020.) | ||||||||||||
⊢ (φ → A ∈ ℝ) & ⊢ (φ → A # 0) ⇒ ⊢ (φ → 0 < (A↑2)) | ||||||||||||||
13-Jun-2020 | reexpclzapd 9018 | Closure of exponentiation of reals. (Contributed by Jim Kingdon, 13-Jun-2020.) | ||||||||||||
⊢ (φ → A ∈ ℝ) & ⊢ (φ → A # 0) & ⊢ (φ → 𝑁 ∈ ℤ) ⇒ ⊢ (φ → (A↑𝑁) ∈ ℝ) | ||||||||||||||
13-Jun-2020 | expdivapd 9008 | Nonnegative integer exponentiation of a quotient. (Contributed by Jim Kingdon, 13-Jun-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → B # 0) & ⊢ (φ → 𝑁 ∈ ℕ_{0}) ⇒ ⊢ (φ → ((A / B)↑𝑁) = ((A↑𝑁) / (B↑𝑁))) | ||||||||||||||
13-Jun-2020 | sqdivapd 9007 | Distribution of square over division. (Contributed by Jim Kingdon, 13-Jun-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → B # 0) ⇒ ⊢ (φ → ((A / B)↑2) = ((A↑2) / (B↑2))) | ||||||||||||||
12-Jun-2020 | expsubapd 9005 | Exponent subtraction law for nonnegative integer exponentiation. (Contributed by Jim Kingdon, 12-Jun-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → A # 0) & ⊢ (φ → 𝑁 ∈ ℤ) & ⊢ (φ → 𝑀 ∈ ℤ) ⇒ ⊢ (φ → (A↑(𝑀 − 𝑁)) = ((A↑𝑀) / (A↑𝑁))) | ||||||||||||||
12-Jun-2020 | expm1apd 9004 | Value of a complex number raised to an integer power minus one. (Contributed by Jim Kingdon, 12-Jun-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → A # 0) & ⊢ (φ → 𝑁 ∈ ℤ) ⇒ ⊢ (φ → (A↑(𝑁 − 1)) = ((A↑𝑁) / A)) | ||||||||||||||
12-Jun-2020 | expp1zapd 9003 | Value of a nonzero complex number raised to an integer power plus one. (Contributed by Jim Kingdon, 12-Jun-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → A # 0) & ⊢ (φ → 𝑁 ∈ ℤ) ⇒ ⊢ (φ → (A↑(𝑁 + 1)) = ((A↑𝑁) · A)) | ||||||||||||||
12-Jun-2020 | exprecapd 9002 | Nonnegative integer exponentiation of a reciprocal. (Contributed by Jim Kingdon, 12-Jun-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → A # 0) & ⊢ (φ → 𝑁 ∈ ℤ) ⇒ ⊢ (φ → ((1 / A)↑𝑁) = (1 / (A↑𝑁))) | ||||||||||||||
12-Jun-2020 | expnegapd 9001 | Value of a complex number raised to a negative power. (Contributed by Jim Kingdon, 12-Jun-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → A # 0) & ⊢ (φ → 𝑁 ∈ ℤ) ⇒ ⊢ (φ → (A↑-𝑁) = (1 / (A↑𝑁))) | ||||||||||||||
12-Jun-2020 | expap0d 9000 | Nonnegative integer exponentiation is nonzero if its mantissa is nonzero. (Contributed by Jim Kingdon, 12-Jun-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → A # 0) & ⊢ (φ → 𝑁 ∈ ℤ) ⇒ ⊢ (φ → (A↑𝑁) # 0) | ||||||||||||||
12-Jun-2020 | expclzapd 8999 | Closure law for integer exponentiation. (Contributed by Jim Kingdon, 12-Jun-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → A # 0) & ⊢ (φ → 𝑁 ∈ ℤ) ⇒ ⊢ (φ → (A↑𝑁) ∈ ℂ) | ||||||||||||||
12-Jun-2020 | sqrecapd 8998 | Square of reciprocal. (Contributed by Jim Kingdon, 12-Jun-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → A # 0) ⇒ ⊢ (φ → ((1 / A)↑2) = (1 / (A↑2))) | ||||||||||||||
12-Jun-2020 | sqgt0api 8952 | The square of a nonzero real is positive. (Contributed by Jim Kingdon, 12-Jun-2020.) | ||||||||||||
⊢ A ∈ ℝ ⇒ ⊢ (A # 0 → 0 < (A↑2)) | ||||||||||||||
12-Jun-2020 | sqdivapi 8950 | Distribution of square over division. (Contributed by Jim Kingdon, 12-Jun-2020.) | ||||||||||||
⊢ A ∈ ℂ & ⊢ B ∈ ℂ & ⊢ B # 0 ⇒ ⊢ ((A / B)↑2) = ((A↑2) / (B↑2)) | ||||||||||||||
11-Jun-2020 | sqgt0ap 8935 | The square of a nonzero real is positive. (Contributed by Jim Kingdon, 11-Jun-2020.) | ||||||||||||
⊢ ((A ∈ ℝ ∧ A # 0) → 0 < (A↑2)) | ||||||||||||||
11-Jun-2020 | sqdivap 8932 | Distribution of square over division. (Contributed by Jim Kingdon, 11-Jun-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ B # 0) → ((A / B)↑2) = ((A↑2) / (B↑2))) | ||||||||||||||
11-Jun-2020 | expdivap 8919 | Nonnegative integer exponentiation of a quotient. (Contributed by Jim Kingdon, 11-Jun-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ (B ∈ ℂ ∧ B # 0) ∧ 𝑁 ∈ ℕ_{0}) → ((A / B)↑𝑁) = ((A↑𝑁) / (B↑𝑁))) | ||||||||||||||
11-Jun-2020 | expm1ap 8918 | Value of a complex number raised to an integer power minus one. (Contributed by Jim Kingdon, 11-Jun-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ A # 0 ∧ 𝑁 ∈ ℤ) → (A↑(𝑁 − 1)) = ((A↑𝑁) / A)) | ||||||||||||||
11-Jun-2020 | expp1zap 8917 | Value of a nonzero complex number raised to an integer power plus one. (Contributed by Jim Kingdon, 11-Jun-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ A # 0 ∧ 𝑁 ∈ ℤ) → (A↑(𝑁 + 1)) = ((A↑𝑁) · A)) | ||||||||||||||
11-Jun-2020 | expsubap 8916 | Exponent subtraction law for nonnegative integer exponentiation. (Contributed by Jim Kingdon, 11-Jun-2020.) | ||||||||||||
⊢ (((A ∈ ℂ ∧ A # 0) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (A↑(𝑀 − 𝑁)) = ((A↑𝑀) / (A↑𝑁))) | ||||||||||||||
11-Jun-2020 | expmulzap 8915 | Product of exponents law for integer exponentiation. (Contributed by Jim Kingdon, 11-Jun-2020.) | ||||||||||||
⊢ (((A ∈ ℂ ∧ A # 0) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (A↑(𝑀 · 𝑁)) = ((A↑𝑀)↑𝑁)) | ||||||||||||||
10-Jun-2020 | expaddzap 8913 | Sum of exponents law for integer exponentiation. (Contributed by Jim Kingdon, 10-Jun-2020.) | ||||||||||||
⊢ (((A ∈ ℂ ∧ A # 0) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (A↑(𝑀 + 𝑁)) = ((A↑𝑀) · (A↑𝑁))) | ||||||||||||||
10-Jun-2020 | expaddzaplem 8912 | Lemma for expaddzap 8913. (Contributed by Jim Kingdon, 10-Jun-2020.) | ||||||||||||
⊢ (((A ∈ ℂ ∧ A # 0) ∧ (𝑀 ∈ ℝ ∧ -𝑀 ∈ ℕ) ∧ 𝑁 ∈ ℕ_{0}) → (A↑(𝑀 + 𝑁)) = ((A↑𝑀) · (A↑𝑁))) | ||||||||||||||
10-Jun-2020 | exprecap 8910 | Nonnegative integer exponentiation of a reciprocal. (Contributed by Jim Kingdon, 10-Jun-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ A # 0 ∧ 𝑁 ∈ ℤ) → ((1 / A)↑𝑁) = (1 / (A↑𝑁))) | ||||||||||||||
10-Jun-2020 | mulexpzap 8909 | Integer exponentiation of a product. (Contributed by Jim Kingdon, 10-Jun-2020.) | ||||||||||||
⊢ (((A ∈ ℂ ∧ A # 0) ∧ (B ∈ ℂ ∧ B # 0) ∧ 𝑁 ∈ ℤ) → ((A · B)↑𝑁) = ((A↑𝑁) · (B↑𝑁))) | ||||||||||||||
10-Jun-2020 | expap0i 8901 | Integer exponentiation is apart from zero if its mantissa is apart from zero. (Contributed by Jim Kingdon, 10-Jun-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ A # 0 ∧ 𝑁 ∈ ℤ) → (A↑𝑁) # 0) | ||||||||||||||
10-Jun-2020 | expap0 8899 | Positive integer exponentiation is apart from zero iff its mantissa is apart from zero. That it is easier to prove this first, and then prove expeq0 8900 in terms of it, rather than the other way around, is perhaps an illustration of the maxim "In constructive analysis, the apartness is more basic [ than ] equality." ([Geuvers], p. 1). (Contributed by Jim Kingdon, 10-Jun-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((A↑𝑁) # 0 ↔ A # 0)) | ||||||||||||||
10-Jun-2020 | mvllmulapd 7550 | Move LHS left multiplication to RHS. (Contributed by Jim Kingdon, 10-Jun-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → A # 0) & ⊢ (φ → (A · B) = 𝐶) ⇒ ⊢ (φ → B = (𝐶 / A)) | ||||||||||||||
9-Jun-2020 | expclzap 8894 | Closure law for integer exponentiation. (Contributed by Jim Kingdon, 9-Jun-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ A # 0 ∧ 𝑁 ∈ ℤ) → (A↑𝑁) ∈ ℂ) | ||||||||||||||
9-Jun-2020 | expclzaplem 8893 | Closure law for integer exponentiation. Lemma for expclzap 8894 and expap0i 8901. (Contributed by Jim Kingdon, 9-Jun-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ A # 0 ∧ 𝑁 ∈ ℤ) → (A↑𝑁) ∈ {z ∈ ℂ ∣ z # 0}) | ||||||||||||||
9-Jun-2020 | reexpclzap 8889 | Closure of exponentiation of reals. (Contributed by Jim Kingdon, 9-Jun-2020.) | ||||||||||||
⊢ ((A ∈ ℝ ∧ A # 0 ∧ 𝑁 ∈ ℤ) → (A↑𝑁) ∈ ℝ) | ||||||||||||||
9-Jun-2020 | neg1ap0 7764 | -1 is apart from zero. (Contributed by Jim Kingdon, 9-Jun-2020.) | ||||||||||||
⊢ -1 # 0 | ||||||||||||||
8-Jun-2020 | expcl2lemap 8881 | Lemma for proving integer exponentiation closure laws. (Contributed by Jim Kingdon, 8-Jun-2020.) | ||||||||||||
⊢ 𝐹 ⊆ ℂ & ⊢ ((x ∈ 𝐹 ∧ y ∈ 𝐹) → (x · y) ∈ 𝐹) & ⊢ 1 ∈ 𝐹 & ⊢ ((x ∈ 𝐹 ∧ x # 0) → (1 / x) ∈ 𝐹) ⇒ ⊢ ((A ∈ 𝐹 ∧ A # 0 ∧ B ∈ ℤ) → (A↑B) ∈ 𝐹) | ||||||||||||||
8-Jun-2020 | expn1ap0 8879 | A number to the negative one power is the reciprocal. (Contributed by Jim Kingdon, 8-Jun-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ A # 0) → (A↑-1) = (1 / A)) | ||||||||||||||
8-Jun-2020 | expineg2 8878 | Value of a complex number raised to a negative integer power. (Contributed by Jim Kingdon, 8-Jun-2020.) | ||||||||||||
⊢ (((A ∈ ℂ ∧ A # 0) ∧ (𝑁 ∈ ℂ ∧ -𝑁 ∈ ℕ_{0})) → (A↑𝑁) = (1 / (A↑-𝑁))) | ||||||||||||||
8-Jun-2020 | expnegap0 8877 | Value of a complex number raised to a negative integer power. (Contributed by Jim Kingdon, 8-Jun-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ A # 0 ∧ 𝑁 ∈ ℕ_{0}) → (A↑-𝑁) = (1 / (A↑𝑁))) | ||||||||||||||
8-Jun-2020 | expinnval 8872 | Value of exponentiation to positive integer powers. (Contributed by Jim Kingdon, 8-Jun-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ 𝑁 ∈ ℕ) → (A↑𝑁) = (seq1( · , (ℕ × {A}), ℂ)‘𝑁)) | ||||||||||||||
7-Jun-2020 | expival 8871 | Value of exponentiation to integer powers. (Contributed by Jim Kingdon, 7-Jun-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (A # 0 ∨ 0 ≤ 𝑁)) → (A↑𝑁) = if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {A}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {A}), ℂ)‘-𝑁))))) | ||||||||||||||
7-Jun-2020 | expivallem 8870 | Lemma for expival 8871. If we take a complex number apart from zero and raise it to a positive integer power, the result is apart from zero. (Contributed by Jim Kingodon, 7-Jun-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ A # 0 ∧ 𝑁 ∈ ℕ) → (seq1( · , (ℕ × {A}), ℂ)‘𝑁) # 0) | ||||||||||||||
7-Jun-2020 | df-iexp 8869 | Define exponentiation to nonnegative integer powers. This definition is not meant to be used directly; instead, exp0 8873 and expp1 8876 provide the standard recursive definition. The up-arrow notation is used by Donald Knuth for iterated exponentiation (Science 194, 1235-1242, 1976) and is convenient for us since we don't have superscripts. 10-Jun-2005: The definition was extended to include zero exponents, so that 0↑0 = 1 per the convention of Definition 10-4.1 of [Gleason] p. 134. 4-Jun-2014: The definition was extended to include negative integer exponents. The case x = 0, y < 0 gives the value (1 / 0), so we will avoid this case in our theorems. (Contributed by Jim Kingodon, 7-Jun-2020.) | ||||||||||||
⊢ ↑ = (x ∈ ℂ, y ∈ ℤ ↦ if(y = 0, 1, if(0 < y, (seq1( · , (ℕ × {x}), ℂ)‘y), (1 / (seq1( · , (ℕ × {x}), ℂ)‘-y))))) | ||||||||||||||
4-Jun-2020 | iseqfveq 8867 | Equality of sequences. (Contributed by Jim Kingdon, 4-Jun-2020.) | ||||||||||||
⊢ (φ → 𝑁 ∈ (ℤ_{≥}‘𝑀)) & ⊢ ((φ ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) = (𝐺‘𝑘)) & ⊢ (φ → 𝑆 ∈ 𝑉) & ⊢ ((φ ∧ x ∈ (ℤ_{≥}‘𝑀)) → (𝐹‘x) ∈ 𝑆) & ⊢ ((φ ∧ x ∈ (ℤ_{≥}‘𝑀)) → (𝐺‘x) ∈ 𝑆) & ⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆)) → (x + y) ∈ 𝑆) ⇒ ⊢ (φ → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq𝑀( + , 𝐺, 𝑆)‘𝑁)) | ||||||||||||||
3-Jun-2020 | iseqfeq2 8866 | Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.) | ||||||||||||
⊢ (φ → 𝐾 ∈ (ℤ_{≥}‘𝑀)) & ⊢ (φ → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (𝐺‘𝐾)) & ⊢ (φ → 𝑆 ∈ 𝑉) & ⊢ ((φ ∧ x ∈ (ℤ_{≥}‘𝑀)) → (𝐹‘x) ∈ 𝑆) & ⊢ ((φ ∧ x ∈ (ℤ_{≥}‘𝐾)) → (𝐺‘x) ∈ 𝑆) & ⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆)) → (x + y) ∈ 𝑆) & ⊢ ((φ ∧ 𝑘 ∈ (ℤ_{≥}‘(𝐾 + 1))) → (𝐹‘𝑘) = (𝐺‘𝑘)) ⇒ ⊢ (φ → (seq𝑀( + , 𝐹, 𝑆) ↾ (ℤ_{≥}‘𝐾)) = seq𝐾( + , 𝐺, 𝑆)) | ||||||||||||||
3-Jun-2020 | iseqfveq2 8865 | Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.) | ||||||||||||
⊢ (φ → 𝐾 ∈ (ℤ_{≥}‘𝑀)) & ⊢ (φ → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (𝐺‘𝐾)) & ⊢ (φ → 𝑆 ∈ 𝑉) & ⊢ ((φ ∧ x ∈ (ℤ_{≥}‘𝑀)) → (𝐹‘x) ∈ 𝑆) & ⊢ ((φ ∧ x ∈ (ℤ_{≥}‘𝐾)) → (𝐺‘x) ∈ 𝑆) & ⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆)) → (x + y) ∈ 𝑆) & ⊢ (φ → 𝑁 ∈ (ℤ_{≥}‘𝐾)) & ⊢ ((φ ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → (𝐹‘𝑘) = (𝐺‘𝑘)) ⇒ ⊢ (φ → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq𝐾( + , 𝐺, 𝑆)‘𝑁)) | ||||||||||||||
1-Jun-2020 | iseqcl 8863 | Closure properties of the recursive sequence builder. (Contributed by Jim Kingdon, 1-Jun-2020.) | ||||||||||||
⊢ (φ → 𝑁 ∈ (ℤ_{≥}‘𝑀)) & ⊢ (φ → 𝑆 ∈ 𝑉) & ⊢ ((φ ∧ x ∈ (ℤ_{≥}‘𝑀)) → (𝐹‘x) ∈ 𝑆) & ⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆)) → (x + y) ∈ 𝑆) ⇒ ⊢ (φ → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) ∈ 𝑆) | ||||||||||||||
1-Jun-2020 | fzdcel 8634 | Decidability of membership in a finite interval of integers. (Contributed by Jim Kingdon, 1-Jun-2020.) | ||||||||||||
⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝐾 ∈ (𝑀...𝑁)) | ||||||||||||||
1-Jun-2020 | fztri3or 8633 | Trichotomy in terms of a finite interval of integers. (Contributed by Jim Kingdon, 1-Jun-2020.) | ||||||||||||
⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 < 𝑀 ∨ 𝐾 ∈ (𝑀...𝑁) ∨ 𝑁 < 𝐾)) | ||||||||||||||
1-Jun-2020 | zdclt 8054 | Integer < is decidable. (Contributed by Jim Kingdon, 1-Jun-2020.) | ||||||||||||
⊢ ((A ∈ ℤ ∧ B ∈ ℤ) → DECID A < B) | ||||||||||||||
31-May-2020 | iseqp1 8864 | Value of the sequence builder function at a successor. (Contributed by Jim Kingdon, 31-May-2020.) | ||||||||||||
⊢ (φ → 𝑁 ∈ (ℤ_{≥}‘𝑀)) & ⊢ (φ → 𝑆 ∈ 𝑉) & ⊢ ((φ ∧ x ∈ (ℤ_{≥}‘𝑀)) → (𝐹‘x) ∈ 𝑆) & ⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆)) → (x + y) ∈ 𝑆) ⇒ ⊢ (φ → (seq𝑀( + , 𝐹, 𝑆)‘(𝑁 + 1)) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑁) + (𝐹‘(𝑁 + 1)))) | ||||||||||||||
31-May-2020 | iseq1 8862 | Value of the sequence builder function at its initial value. (Contributed by Jim Kingdon, 31-May-2020.) | ||||||||||||
⊢ (φ → 𝑀 ∈ ℤ) & ⊢ (φ → 𝑆 ∈ 𝑉) & ⊢ ((φ ∧ x ∈ (ℤ_{≥}‘𝑀)) → (𝐹‘x) ∈ 𝑆) & ⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆)) → (x + y) ∈ 𝑆) ⇒ ⊢ (φ → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (𝐹‘𝑀)) | ||||||||||||||
31-May-2020 | iseqovex 8859 | Closure of a function used in proving sequence builder theorems. This can be thought of as a lemma for the small number of sequence builder theorems which need it. (Contributed by Jim Kingdon, 31-May-2020.) | ||||||||||||
⊢ ((φ ∧ x ∈ (ℤ_{≥}‘𝑀)) → (𝐹‘x) ∈ 𝑆) & ⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆)) → (x + y) ∈ 𝑆) ⇒ ⊢ ((φ ∧ (x ∈ (ℤ_{≥}‘𝑀) ∧ y ∈ 𝑆)) → (x(z ∈ (ℤ_{≥}‘𝑀), w ∈ 𝑆 ↦ (w + (𝐹‘(z + 1))))y) ∈ 𝑆) | ||||||||||||||
31-May-2020 | frecuzrdgcl 8840 | Closure law for the recursive definition generator on upper integers. See comment in frec2uz0d 8826 for the description of 𝐺 as the mapping from 𝜔 to (ℤ_{≥}‘𝐶). (Contributed by Jim Kingdon, 31-May-2020.) | ||||||||||||
⊢ (φ → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((x ∈ ℤ ↦ (x + 1)), 𝐶) & ⊢ (φ → 𝑆 ∈ 𝑉) & ⊢ (φ → A ∈ 𝑆) & ⊢ ((φ ∧ (x ∈ (ℤ_{≥}‘𝐶) ∧ y ∈ 𝑆)) → (x𝐹y) ∈ 𝑆) & ⊢ 𝑅 = frec((x ∈ (ℤ_{≥}‘𝐶), y ∈ 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩), ⟨𝐶, A⟩) & ⊢ (φ → 𝑇 = ran 𝑅) ⇒ ⊢ ((φ ∧ B ∈ (ℤ_{≥}‘𝐶)) → (𝑇‘B) ∈ 𝑆) | ||||||||||||||
30-May-2020 | iseqfn 8861 | The sequence builder function is a function. (Contributed by Jim Kingdon, 30-May-2020.) | ||||||||||||
⊢ (φ → 𝑀 ∈ ℤ) & ⊢ (φ → 𝑆 ∈ 𝑉) & ⊢ ((φ ∧ x ∈ (ℤ_{≥}‘𝑀)) → (𝐹‘x) ∈ 𝑆) & ⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆)) → (x + y) ∈ 𝑆) ⇒ ⊢ (φ → seq𝑀( + , 𝐹, 𝑆) Fn (ℤ_{≥}‘𝑀)) | ||||||||||||||
30-May-2020 | iseqval 8860 | Value of the sequence builder function. (Contributed by Jim Kingdon, 30-May-2020.) | ||||||||||||
⊢ 𝑅 = frec((x ∈ (ℤ_{≥}‘𝑀), y ∈ 𝑆 ↦ ⟨(x + 1), (x(z ∈ (ℤ_{≥}‘𝑀), w ∈ 𝑆 ↦ (w + (𝐹‘(z + 1))))y)⟩), ⟨𝑀, (𝐹‘𝑀)⟩) & ⊢ ((φ ∧ x ∈ (ℤ_{≥}‘𝑀)) → (𝐹‘x) ∈ 𝑆) & ⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆)) → (x + y) ∈ 𝑆) ⇒ ⊢ (φ → seq𝑀( + , 𝐹, 𝑆) = ran 𝑅) | ||||||||||||||
30-May-2020 | nfiseq 8858 | Hypothesis builder for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.) | ||||||||||||
⊢ Ⅎx𝑀 & ⊢ Ⅎx + & ⊢ Ⅎx𝐹 & ⊢ Ⅎx𝑆 ⇒ ⊢ Ⅎxseq𝑀( + , 𝐹, 𝑆) | ||||||||||||||
30-May-2020 | iseqeq4 8857 | Equality theorem for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.) | ||||||||||||
⊢ (𝑆 = 𝑇 → seq𝑀( + , 𝐹, 𝑆) = seq𝑀( + , 𝐹, 𝑇)) | ||||||||||||||
30-May-2020 | iseqeq3 8856 | Equality theorem for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.) | ||||||||||||
⊢ (𝐹 = 𝐺 → seq𝑀( + , 𝐹, 𝑆) = seq𝑀( + , 𝐺, 𝑆)) | ||||||||||||||
30-May-2020 | iseqeq2 8855 | Equality theorem for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.) | ||||||||||||
⊢ ( + = 𝑄 → seq𝑀( + , 𝐹, 𝑆) = seq𝑀(𝑄, 𝐹, 𝑆)) | ||||||||||||||
30-May-2020 | iseqeq1 8854 | Equality theorem for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.) | ||||||||||||
⊢ (𝑀 = 𝑁 → seq𝑀( + , 𝐹, 𝑆) = seq𝑁( + , 𝐹, 𝑆)) | ||||||||||||||
30-May-2020 | nffrec 5921 | Bound-variable hypothesis builder for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.) | ||||||||||||
⊢ Ⅎx𝐹 & ⊢ ℲxA ⇒ ⊢ Ⅎxfrec(𝐹, A) | ||||||||||||||
30-May-2020 | freceq2 5920 | Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.) | ||||||||||||
⊢ (A = B → frec(𝐹, A) = frec(𝐹, B)) | ||||||||||||||
30-May-2020 | freceq1 5919 | Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.) | ||||||||||||
⊢ (𝐹 = 𝐺 → frec(𝐹, A) = frec(𝐺, A)) | ||||||||||||||
29-May-2020 | df-iseq 8853 |
Define a general-purpose operation that builds a recursive sequence
(i.e. a function on the positive integers ℕ or some other upper
integer set) whose value at an index is a function of its previous value
and the value of an input sequence at that index. This definition is
complicated, but fortunately it is not intended to be used directly.
Instead, the only purpose of this definition is to provide us with an
object that has the properties expressed by iseq1 8862 and at successors.
Typically, those are the main theorems that would be used in practice.
The first operand in the parentheses is the operation that is applied to the previous value and the value of the input sequence (second operand). The operand to the left of the parenthesis is the integer to start from. For example, for the operation +, an input sequence 𝐹 with values 1, 1/2, 1/4, 1/8,... would be transformed into the output sequence seq1( + , 𝐹) with values 1, 3/2, 7/4, 15/8,.., so that (seq1( + , 𝐹)‘1) = 1, (seq1( + , 𝐹)‘2) = 3/2, etc. In other words, seq𝑀( + , 𝐹) transforms a sequence 𝐹 into an infinite series. Internally, the frec function generates as its values a set of ordered pairs starting at ⟨𝑀, (𝐹‘𝑀)⟩, with the first member of each pair incremented by one in each successive value. So, the range of frec is exactly the sequence we want, and we just extract the range and throw away the domain. (Contributed by Jim Kingdon, 29-May-2020.) | ||||||||||||
⊢ seq𝑀( + , 𝐹, 𝑆) = ran frec((x ∈ (ℤ_{≥}‘𝑀), y ∈ 𝑆 ↦ ⟨(x + 1), (y + (𝐹‘(x + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) | ||||||||||||||
28-May-2020 | frecuzrdgsuc 8842 | Successor value of a recursive definition generator on upper integers. See comment in frec2uz0d 8826 for the description of 𝐺 as the mapping from 𝜔 to (ℤ_{≥}‘𝐶). (Contributed by Jim Kingdon, 28-May-2020.) | ||||||||||||
⊢ (φ → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((x ∈ ℤ ↦ (x + 1)), 𝐶) & ⊢ (φ → 𝑆 ∈ 𝑉) & ⊢ (φ → A ∈ 𝑆) & ⊢ ((φ ∧ (x ∈ (ℤ_{≥}‘𝐶) ∧ y ∈ 𝑆)) → (x𝐹y) ∈ 𝑆) & ⊢ 𝑅 = frec((x ∈ (ℤ_{≥}‘𝐶), y ∈ 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩), ⟨𝐶, A⟩) & ⊢ (φ → 𝑇 = ran 𝑅) ⇒ ⊢ ((φ ∧ B ∈ (ℤ_{≥}‘𝐶)) → (𝑇‘(B + 1)) = (B𝐹(𝑇‘B))) | ||||||||||||||
27-May-2020 | frecuzrdg0 8841 | Initial value of a recursive definition generator on upper integers. See comment in frec2uz0d 8826 for the description of 𝐺 as the mapping from 𝜔 to (ℤ_{≥}‘𝐶). (Contributed by Jim Kingdon, 27-May-2020.) | ||||||||||||
⊢ (φ → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((x ∈ ℤ ↦ (x + 1)), 𝐶) & ⊢ (φ → 𝑆 ∈ 𝑉) & ⊢ (φ → A ∈ 𝑆) & ⊢ ((φ ∧ (x ∈ (ℤ_{≥}‘𝐶) ∧ y ∈ 𝑆)) → (x𝐹y) ∈ 𝑆) & ⊢ 𝑅 = frec((x ∈ (ℤ_{≥}‘𝐶), y ∈ 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩), ⟨𝐶, A⟩) & ⊢ (φ → 𝑇 = ran 𝑅) ⇒ ⊢ (φ → (𝑇‘𝐶) = A) | ||||||||||||||
27-May-2020 | frecuzrdgrrn 8835 | The function 𝑅 (used in the definition of the recursive definition generator on upper integers) yields ordered pairs of integers and elements of 𝑆. (Contributed by Jim Kingdon, 27-May-2020.) | ||||||||||||
⊢ (φ → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((x ∈ ℤ ↦ (x + 1)), 𝐶) & ⊢ (φ → 𝑆 ∈ 𝑉) & ⊢ (φ → A ∈ 𝑆) & ⊢ ((φ ∧ (x ∈ (ℤ_{≥}‘𝐶) ∧ y ∈ 𝑆)) → (x𝐹y) ∈ 𝑆) & ⊢ 𝑅 = frec((x ∈ (ℤ_{≥}‘𝐶), y ∈ 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩), ⟨𝐶, A⟩) ⇒ ⊢ ((φ ∧ 𝐷 ∈ 𝜔) → (𝑅‘𝐷) ∈ ((ℤ_{≥}‘𝐶) × 𝑆)) | ||||||||||||||
27-May-2020 | dffun5r 4857 | A way of proving a relation is a function, analogous to mo2r 1949. (Contributed by Jim Kingdon, 27-May-2020.) | ||||||||||||
⊢ ((Rel A ∧ ∀x∃z∀y(⟨x, y⟩ ∈ A → y = z)) → Fun A) | ||||||||||||||
26-May-2020 | frecuzrdgfn 8839 | The recursive definition generator on upper integers is a function. See comment in frec2uz0d 8826 for the description of 𝐺 as the mapping from 𝜔 to (ℤ_{≥}‘𝐶). (Contributed by Jim Kingdon, 26-May-2020.) | ||||||||||||
⊢ (φ → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((x ∈ ℤ ↦ (x + 1)), 𝐶) & ⊢ (φ → 𝑆 ∈ 𝑉) & ⊢ (φ → A ∈ 𝑆) & ⊢ ((φ ∧ (x ∈ (ℤ_{≥}‘𝐶) ∧ y ∈ 𝑆)) → (x𝐹y) ∈ 𝑆) & ⊢ 𝑅 = frec((x ∈ (ℤ_{≥}‘𝐶), y ∈ 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩), ⟨𝐶, A⟩) & ⊢ (φ → 𝑇 = ran 𝑅) ⇒ ⊢ (φ → 𝑇 Fn (ℤ_{≥}‘𝐶)) | ||||||||||||||
26-May-2020 | frecuzrdglem 8838 | A helper lemma for the value of a recursive definition generator on upper integers. (Contributed by Jim Kingdon, 26-May-2020.) | ||||||||||||
⊢ (φ → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((x ∈ ℤ ↦ (x + 1)), 𝐶) & ⊢ (φ → 𝑆 ∈ 𝑉) & ⊢ (φ → A ∈ 𝑆) & ⊢ ((φ ∧ (x ∈ (ℤ_{≥}‘𝐶) ∧ y ∈ 𝑆)) → (x𝐹y) ∈ 𝑆) & ⊢ 𝑅 = frec((x ∈ (ℤ_{≥}‘𝐶), y ∈ 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩), ⟨𝐶, A⟩) & ⊢ (φ → B ∈ (ℤ_{≥}‘𝐶)) ⇒ ⊢ (φ → ⟨B, (2^{nd} ‘(𝑅‘(^{◡}𝐺‘B)))⟩ ∈ ran 𝑅) | ||||||||||||||
26-May-2020 | frecuzrdgrom 8837 | The function 𝑅 (used in the definition of the recursive definition generator on upper integers) is a function defined for all natural numbers. (Contributed by Jim Kingdon, 26-May-2020.) | ||||||||||||
⊢ (φ → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((x ∈ ℤ ↦ (x + 1)), 𝐶) & ⊢ (φ → 𝑆 ∈ 𝑉) & ⊢ (φ → A ∈ 𝑆) & ⊢ ((φ ∧ (x ∈ (ℤ_{≥}‘𝐶) ∧ y ∈ 𝑆)) → (x𝐹y) ∈ 𝑆) & ⊢ 𝑅 = frec((x ∈ (ℤ_{≥}‘𝐶), y ∈ 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩), ⟨𝐶, A⟩) ⇒ ⊢ (φ → 𝑅 Fn 𝜔) | ||||||||||||||
25-May-2020 | freccl 5932 | Closure for finite recursion. (Contributed by Jim Kingdon, 25-May-2020.) | ||||||||||||
⊢ (φ → ∀z(𝐹‘z) ∈ V) & ⊢ (φ → A ∈ 𝑆) & ⊢ ((φ ∧ z ∈ 𝑆) → (𝐹‘z) ∈ 𝑆) & ⊢ (φ → B ∈ 𝜔) ⇒ ⊢ (φ → (frec(𝐹, A)‘B) ∈ 𝑆) | ||||||||||||||
24-May-2020 | frec2uzrdg 8836 | A helper lemma for the value of a recursive definition generator on upper integers (typically either ℕ or ℕ_{0}) with characteristic function 𝐹(x, y) and initial value A. This lemma shows that evaluating 𝑅 at an element of 𝜔 gives an ordered pair whose first element is the index (translated from 𝜔 to (ℤ_{≥}‘𝐶)). See comment in frec2uz0d 8826 which describes 𝐺 and the index translation. (Contributed by Jim Kingdon, 24-May-2020.) | ||||||||||||
⊢ (φ → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((x ∈ ℤ ↦ (x + 1)), 𝐶) & ⊢ (φ → 𝑆 ∈ 𝑉) & ⊢ (φ → A ∈ 𝑆) & ⊢ ((φ ∧ (x ∈ (ℤ_{≥}‘𝐶) ∧ y ∈ 𝑆)) → (x𝐹y) ∈ 𝑆) & ⊢ 𝑅 = frec((x ∈ (ℤ_{≥}‘𝐶), y ∈ 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩), ⟨𝐶, A⟩) & ⊢ (φ → B ∈ 𝜔) ⇒ ⊢ (φ → (𝑅‘B) = ⟨(𝐺‘B), (2^{nd} ‘(𝑅‘B))⟩) | ||||||||||||||
21-May-2020 | fzofig 8849 | Half-open integer sets are finite. (Contributed by Jim Kingdon, 21-May-2020.) | ||||||||||||
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀..^𝑁) ∈ Fin) | ||||||||||||||
21-May-2020 | fzfigd 8848 | Deduction form of fzfig 8847. (Contributed by Jim Kingdon, 21-May-2020.) | ||||||||||||
⊢ (φ → 𝑀 ∈ ℤ) & ⊢ (φ → 𝑁 ∈ ℤ) ⇒ ⊢ (φ → (𝑀...𝑁) ∈ Fin) | ||||||||||||||
19-May-2020 | fzfig 8847 | A finite interval of integers is finite. (Contributed by Jim Kingdon, 19-May-2020.) | ||||||||||||
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) ∈ Fin) | ||||||||||||||
19-May-2020 | frechashgf1o 8846 | 𝐺 maps 𝜔 one-to-one onto ℕ_{0}. (Contributed by Jim Kingdon, 19-May-2020.) | ||||||||||||
⊢ 𝐺 = frec((x ∈ ℤ ↦ (x + 1)), 0) ⇒ ⊢ 𝐺:𝜔–1-1-onto→ℕ_{0} | ||||||||||||||
19-May-2020 | ssfiexmid 6254 | If any subset of a finite set is finite, excluded middle follows. One direction of Theorem 2.1 of [Bauer], p. 485. (Contributed by Jim Kingdon, 19-May-2020.) | ||||||||||||
⊢ ∀x∀y((x ∈ Fin ∧ y ⊆ x) → y ∈ Fin) ⇒ ⊢ (φ ∨ ¬ φ) | ||||||||||||||
19-May-2020 | enm 6230 | A set equinumerous to an inhabited set is inhabited. (Contributed by Jim Kingdon, 19-May-2020.) | ||||||||||||
⊢ ((A ≈ B ∧ ∃x x ∈ A) → ∃y y ∈ B) | ||||||||||||||
18-May-2020 | frecfzen2 8845 | The cardinality of a finite set of sequential integers with arbitrary endpoints. (Contributed by Jim Kingdon, 18-May-2020.) | ||||||||||||
⊢ 𝐺 = frec((x ∈ ℤ ↦ (x + 1)), 0) ⇒ ⊢ (𝑁 ∈ (ℤ_{≥}‘𝑀) → (𝑀...𝑁) ≈ (^{◡}𝐺‘((𝑁 + 1) − 𝑀))) | ||||||||||||||
18-May-2020 | frecfzennn 8844 | The cardinality of a finite set of sequential integers. (See frec2uz0d 8826 for a description of the hypothesis.) (Contributed by Jim Kingdon, 18-May-2020.) | ||||||||||||
⊢ 𝐺 = frec((x ∈ ℤ ↦ (x + 1)), 0) ⇒ ⊢ (𝑁 ∈ ℕ_{0} → (1...𝑁) ≈ (^{◡}𝐺‘𝑁)) | ||||||||||||||
17-May-2020 | frec2uzisod 8834 | 𝐺 (see frec2uz0d 8826) is an isomorphism from natural ordinals to upper integers. (Contributed by Jim Kingdon, 17-May-2020.) | ||||||||||||
⊢ (φ → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((x ∈ ℤ ↦ (x + 1)), 𝐶) ⇒ ⊢ (φ → 𝐺 Isom E , < (𝜔, (ℤ_{≥}‘𝐶))) | ||||||||||||||
17-May-2020 | frec2uzf1od 8833 | 𝐺 (see frec2uz0d 8826) is a one-to-one onto mapping. (Contributed by Jim Kingdon, 17-May-2020.) | ||||||||||||
⊢ (φ → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((x ∈ ℤ ↦ (x + 1)), 𝐶) ⇒ ⊢ (φ → 𝐺:𝜔–1-1-onto→(ℤ_{≥}‘𝐶)) | ||||||||||||||
17-May-2020 | frec2uzrand 8832 | Range of 𝐺 (see frec2uz0d 8826). (Contributed by Jim Kingdon, 17-May-2020.) | ||||||||||||
⊢ (φ → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((x ∈ ℤ ↦ (x + 1)), 𝐶) ⇒ ⊢ (φ → ran 𝐺 = (ℤ_{≥}‘𝐶)) | ||||||||||||||
16-May-2020 | frec2uzlt2d 8831 | The mapping 𝐺 (see frec2uz0d 8826) preserves order. (Contributed by Jim Kingdon, 16-May-2020.) | ||||||||||||
⊢ (φ → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((x ∈ ℤ ↦ (x + 1)), 𝐶) & ⊢ (φ → A ∈ 𝜔) & ⊢ (φ → B ∈ 𝜔) ⇒ ⊢ (φ → (A ∈ B ↔ (𝐺‘A) < (𝐺‘B))) | ||||||||||||||
16-May-2020 | frec2uzltd 8830 | Less-than relation for 𝐺 (see frec2uz0d 8826). (Contributed by Jim Kingdon, 16-May-2020.) | ||||||||||||
⊢ (φ → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((x ∈ ℤ ↦ (x + 1)), 𝐶) & ⊢ (φ → A ∈ 𝜔) & ⊢ (φ → B ∈ 𝜔) ⇒ ⊢ (φ → (A ∈ B → (𝐺‘A) < (𝐺‘B))) | ||||||||||||||
16-May-2020 | frec2uzuzd 8829 | The value 𝐺 (see frec2uz0d 8826) at an ordinal natural number is in the upper integers. (Contributed by Jim Kingdon, 16-May-2020.) | ||||||||||||
⊢ (φ → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((x ∈ ℤ ↦ (x + 1)), 𝐶) & ⊢ (φ → A ∈ 𝜔) ⇒ ⊢ (φ → (𝐺‘A) ∈ (ℤ_{≥}‘𝐶)) | ||||||||||||||
16-May-2020 | frec2uzsucd 8828 | The value of 𝐺 (see frec2uz0d 8826) at a successor. (Contributed by Jim Kingdon, 16-May-2020.) | ||||||||||||
⊢ (φ → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((x ∈ ℤ ↦ (x + 1)), 𝐶) & ⊢ (φ → A ∈ 𝜔) ⇒ ⊢ (φ → (𝐺‘suc A) = ((𝐺‘A) + 1)) | ||||||||||||||
16-May-2020 | frec2uzzd 8827 | The value of 𝐺 (see frec2uz0d 8826) is an integer. (Contributed by Jim Kingdon, 16-May-2020.) | ||||||||||||
⊢ (φ → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((x ∈ ℤ ↦ (x + 1)), 𝐶) & ⊢ (φ → A ∈ 𝜔) ⇒ ⊢ (φ → (𝐺‘A) ∈ ℤ) | ||||||||||||||
16-May-2020 | frec2uz0d 8826 | The mapping 𝐺 is a one-to-one mapping from 𝜔 onto upper integers that will be used to construct a recursive definition generator. Ordinal natural number 0 maps to complex number 𝐶 (normally 0 for the upper integers ℕ_{0} or 1 for the upper integers ℕ), 1 maps to 𝐶 + 1, etc. This theorem shows the value of 𝐺 at ordinal natural number zero. (Contributed by Jim Kingdon, 16-May-2020.) | ||||||||||||
⊢ (φ → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((x ∈ ℤ ↦ (x + 1)), 𝐶) ⇒ ⊢ (φ → (𝐺‘∅) = 𝐶) | ||||||||||||||
15-May-2020 | nntri3 6014 | A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 15-May-2020.) | ||||||||||||
⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔) → (A = B ↔ (¬ A ∈ B ∧ ¬ B ∈ A))) | ||||||||||||||
14-May-2020 | rdgifnon2 5907 | The recursive definition generator is a function on ordinal numbers. (Contributed by Jim Kingdon, 14-May-2020.) | ||||||||||||
⊢ ((∀z(𝐹‘z) ∈ V ∧ A ∈ 𝑉) → rec(𝐹, A) Fn On) | ||||||||||||||
14-May-2020 | rdgtfr 5901 | The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 14-May-2020.) | ||||||||||||
⊢ ((∀z(𝐹‘z) ∈ V ∧ A ∈ 𝑉) → (Fun (g ∈ V ↦ (A ∪ ∪ x ∈ dom g(𝐹‘(g‘x)))) ∧ ((g ∈ V ↦ (A ∪ ∪ x ∈ dom g(𝐹‘(g‘x))))‘f) ∈ V)) | ||||||||||||||
13-May-2020 | frecfnom 5925 | The function generated by finite recursive definition generation is a function on omega. (Contributed by Jim Kingdon, 13-May-2020.) | ||||||||||||
⊢ ((∀z(𝐹‘z) ∈ V ∧ A ∈ 𝑉) → frec(𝐹, A) Fn 𝜔) | ||||||||||||||
13-May-2020 | frecabex 5923 | The class abstraction from df-frec 5918 exists. This is a lemma for other finite recursion proofs. (Contributed by Jim Kingdon, 13-May-2020.) | ||||||||||||
⊢ (φ → 𝑆 ∈ 𝑉) & ⊢ (φ → ∀y(𝐹‘y) ∈ V) & ⊢ (φ → A ∈ 𝑊) ⇒ ⊢ (φ → {x ∣ (∃𝑚 ∈ 𝜔 (dom 𝑆 = suc 𝑚 ∧ x ∈ (𝐹‘(𝑆‘𝑚))) ∨ (dom 𝑆 = ∅ ∧ x ∈ A))} ∈ V) | ||||||||||||||
8-May-2020 | tfr0 5878 | Transfinite recursion at the empty set. (Contributed by Jim Kingdon, 8-May-2020.) | ||||||||||||
⊢ 𝐹 = recs(𝐺) ⇒ ⊢ ((𝐺‘∅) ∈ 𝑉 → (𝐹‘∅) = (𝐺‘∅)) | ||||||||||||||
7-May-2020 | frec0g 5922 | The initial value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 7-May-2020.) | ||||||||||||
⊢ (A ∈ 𝑉 → (frec(𝐹, A)‘∅) = A) | ||||||||||||||
3-May-2020 | dcned 2209 | Decidable equality implies decidable negated equality. (Contributed by Jim Kingdon, 3-May-2020.) | ||||||||||||
⊢ (φ → DECID A = B) ⇒ ⊢ (φ → DECID A ≠ B) | ||||||||||||||
2-May-2020 | ax-arch 6762 |
Archimedean axiom. Definition 3.1(2) of [Geuvers], p. 9. Axiom for
real and complex numbers, justified by theorem axarch 6733.
This axiom should not be used directly; instead use arch 7914 (which is the same, but stated in terms of ℕ and <). (Contributed by Jim Kingdon, 2-May-2020.) (New usage is discouraged.) | ||||||||||||
⊢ (A ∈ ℝ → ∃𝑛 ∈ ∩ {x ∣ (1 ∈ x ∧ ∀y ∈ x (y + 1) ∈ x)}A <_{ℝ} 𝑛) | ||||||||||||||
30-Apr-2020 | ltexnqi 6392 | Ordering on positive fractions in terms of existence of sum. (Contributed by Jim Kingdon, 30-Apr-2020.) | ||||||||||||
⊢ (A <_{Q} B → ∃x ∈ Q (A +_{Q} x) = B) | ||||||||||||||
28-Apr-2020 | addnqpr1lemiu 6540 | Lemma for addnqpr1 6541. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 28-Apr-2020.) | ||||||||||||
⊢ (A ∈ Q → (2^{nd} ‘⟨{𝑙 ∣ 𝑙 <_{Q} (A +_{Q} 1_{Q})}, {u ∣ (A +_{Q} 1_{Q}) <_{Q} u}⟩) ⊆ (2^{nd} ‘(⟨{𝑙 ∣ 𝑙 <_{Q} A}, {u ∣ A <_{Q} u}⟩ +_{P} 1_{P}))) | ||||||||||||||
28-Apr-2020 | addnqpr1lemil 6539 | Lemma for addnqpr1 6541. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 28-Apr-2020.) | ||||||||||||
⊢ (A ∈ Q → (1^{st} ‘⟨{𝑙 ∣ 𝑙 <_{Q} (A +_{Q} 1_{Q})}, {u ∣ (A +_{Q} 1_{Q}) <_{Q} u}⟩) ⊆ (1^{st} ‘(⟨{𝑙 ∣ 𝑙 <_{Q} A}, {u ∣ A <_{Q} u}⟩ +_{P} 1_{P}))) | ||||||||||||||
28-Apr-2020 | addnqpr1lemru 6538 | Lemma for addnqpr1 6541. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 28-Apr-2020.) | ||||||||||||
⊢ (A ∈ Q → (2^{nd} ‘(⟨{𝑙 ∣ 𝑙 <_{Q} A}, {u ∣ A <_{Q} u}⟩ +_{P} 1_{P})) ⊆ (2^{nd} ‘⟨{𝑙 ∣ 𝑙 <_{Q} (A +_{Q} 1_{Q})}, {u ∣ (A +_{Q} 1_{Q}) <_{Q} u}⟩)) | ||||||||||||||
28-Apr-2020 | addnqpr1lemrl 6537 | Lemma for addnqpr1 6541. The reverse subset relationship for the lower cut. (Contributed by Jim Kingdon, 28-Apr-2020.) | ||||||||||||
⊢ (A ∈ Q → (1^{st} ‘(⟨{𝑙 ∣ 𝑙 <_{Q} A}, {u ∣ A <_{Q} u}⟩ +_{P} 1_{P})) ⊆ (1^{st} ‘⟨{𝑙 ∣ 𝑙 <_{Q} (A +_{Q} 1_{Q})}, {u ∣ (A +_{Q} 1_{Q}) <_{Q} u}⟩)) | ||||||||||||||
26-Apr-2020 | pitonnlem1p1 6702 | Lemma for pitonn 6704. Simplifying an expression involving signed reals. (Contributed by Jim Kingdon, 26-Apr-2020.) | ||||||||||||
⊢ (A ∈ P → [⟨(A +_{P} (1_{P} +_{P} 1_{P})), (1_{P} +_{P} 1_{P})⟩] ~_{R} = [⟨(A +_{P} 1_{P}), 1_{P}⟩] ~_{R} ) | ||||||||||||||
26-Apr-2020 | addnqpr1 6541 | Addition of one to a fraction embedded into a positive real. One can either add the fraction one to the fraction, or the positive real one to the positive real, and get the same result. (Contributed by Jim Kingdon, 26-Apr-2020.) | ||||||||||||
⊢ (A ∈ Q → ⟨{𝑙 ∣ 𝑙 <_{Q} (A +_{Q} 1_{Q})}, {u ∣ (A +_{Q} 1_{Q}) <_{Q} u}⟩ = (⟨{𝑙 ∣ 𝑙 <_{Q} A}, {u ∣ A <_{Q} u}⟩ +_{P} 1_{P})) | ||||||||||||||
26-Apr-2020 | addpinq1 6446 | Addition of one to the numerator of a fraction whose denominator is one. (Contributed by Jim Kingdon, 26-Apr-2020.) | ||||||||||||
⊢ (A ∈ N → [⟨(A +_{N} 1_{𝑜}), 1_{𝑜}⟩] ~_{Q} = ([⟨A, 1_{𝑜}⟩] ~_{Q} +_{Q} 1_{Q})) | ||||||||||||||
26-Apr-2020 | nnnq 6405 | The canonical embedding of positive integers into positive fractions. (Contributed by Jim Kingdon, 26-Apr-2020.) | ||||||||||||
⊢ (A ∈ N → [⟨A, 1_{𝑜}⟩] ~_{Q} ∈ Q) | ||||||||||||||
24-Apr-2020 | pitonnlem2 6703 | Lemma for pitonn 6704. Two ways to add one to a number. (Contributed by Jim Kingdon, 24-Apr-2020.) | ||||||||||||
⊢ (𝐾 ∈ N → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <_{Q} [⟨𝐾, 1_{𝑜}⟩] ~_{Q} }, {u ∣ [⟨𝐾, 1_{𝑜}⟩] ~_{Q} <_{Q} u}⟩ +_{P} 1_{P}), 1_{P}⟩] ~_{R} , 0_{R}⟩ + 1) = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <_{Q} [⟨(𝐾 +_{N} 1_{𝑜}), 1_{𝑜}⟩] ~_{Q} }, {u ∣ [⟨(𝐾 +_{N} 1_{𝑜}), 1_{𝑜}⟩] ~_{Q} <_{Q} u}⟩ +_{P} 1_{P}), 1_{P}⟩] ~_{R} , 0_{R}⟩) | ||||||||||||||
24-Apr-2020 | pitonnlem1 6701 | Lemma for pitonn 6704. Two ways to write the number one. (Contributed by Jim Kingdon, 24-Apr-2020.) | ||||||||||||
⊢ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <_{Q} [⟨1_{𝑜}, 1_{𝑜}⟩] ~_{Q} }, {u ∣ [⟨1_{𝑜}, 1_{𝑜}⟩] ~_{Q} <_{Q} u}⟩ +_{P} 1_{P}), 1_{P}⟩] ~_{R} , 0_{R}⟩ = 1 | ||||||||||||||
23-Apr-2020 | archsr 6668 | For any signed real, there is an integer that is greater than it. This is also known as the "archimedean property". The expression [⟨(⟨{𝑙 ∣ 𝑙 <_{Q} [⟨x, 1_{𝑜}⟩] ~_{Q} }, {u ∣ [⟨x, 1_{𝑜}⟩] ~_{Q} <_{Q} u}⟩ +_{P} 1_{P}), 1_{P}⟩] ~_{R} is the embedding of the positive integer x into the signed reals. (Contributed by Jim Kingdon, 23-Apr-2020.) | ||||||||||||
⊢ (A ∈ R → ∃x ∈ N A <_{R} [⟨(⟨{𝑙 ∣ 𝑙 <_{Q} [⟨x, 1_{𝑜}⟩] ~_{Q} }, {u ∣ [⟨x, 1_{𝑜}⟩] ~_{Q} <_{Q} u}⟩ +_{P} 1_{P}), 1_{P}⟩] ~_{R} ) | ||||||||||||||
23-Apr-2020 | nnprlu 6533 | The canonical embedding of positive integers into the positive reals. (Contributed by Jim Kingdon, 23-Apr-2020.) | ||||||||||||
⊢ (A ∈ N → ⟨{𝑙 ∣ 𝑙 <_{Q} [⟨A, 1_{𝑜}⟩] ~_{Q} }, {u ∣ [⟨A, 1_{𝑜}⟩] ~_{Q} <_{Q} u}⟩ ∈ P) | ||||||||||||||
22-Apr-2020 | axarch 6733 |
Archimedean axiom. The Archimedean property is more naturally stated
once we have defined ℕ. Unless we find
another way to state it,
we'll just use the right hand side of dfnn2 7657 in stating what we mean by
"natural number" in the context of this axiom.
This construction-dependent theorem should not be referenced directly; instead, use ax-arch 6762. (Contributed by Jim Kingdon, 22-Apr-2020.) (New usage is discouraged.) | ||||||||||||
⊢ (A ∈ ℝ → ∃𝑛 ∈ ∩ {x ∣ (1 ∈ x ∧ ∀y ∈ x (y + 1) ∈ x)}A <_{ℝ} 𝑛) | ||||||||||||||
22-Apr-2020 | pitonn 6704 | Mapping from N to ℕ. (Contributed by Jim Kingdon, 22-Apr-2020.) | ||||||||||||
⊢ (𝑛 ∈ N → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <_{Q} [⟨𝑛, 1_{𝑜}⟩] ~_{Q} }, {u ∣ [⟨𝑛, 1_{𝑜}⟩] ~_{Q} <_{Q} u}⟩ +_{P} 1_{P}), 1_{P}⟩] ~_{R} , 0_{R}⟩ ∈ ∩ {x ∣ (1 ∈ x ∧ ∀y ∈ x (y + 1) ∈ x)}) | ||||||||||||||
22-Apr-2020 | archpr 6613 | For any positive real, there is an integer that is greater than it. This is also known as the "archimedean property". The integer x is embedded into the reals as described at nnprlu 6533. (Contributed by Jim Kingdon, 22-Apr-2020.) | ||||||||||||
⊢ (A ∈ P → ∃x ∈ N A<_{P} ⟨{𝑙 ∣ 𝑙 <_{Q} [⟨x, 1_{𝑜}⟩] ~_{Q} }, {u ∣ [⟨x, 1_{𝑜}⟩] ~_{Q} <_{Q} u}⟩) | ||||||||||||||
20-Apr-2020 | fzo0m 8777 | A half-open integer range based at 0 is inhabited precisely if the upper bound is a positive integer. (Contributed by Jim Kingdon, 20-Apr-2020.) | ||||||||||||
⊢ (∃x x ∈ (0..^A) ↔ A ∈ ℕ) | ||||||||||||||
20-Apr-2020 | fzom 8750 | A half-open integer interval is inhabited iff it contains its left endpoint. (Contributed by Jim Kingdon, 20-Apr-2020.) | ||||||||||||
⊢ (∃x x ∈ (𝑀..^𝑁) ↔ 𝑀 ∈ (𝑀..^𝑁)) | ||||||||||||||
18-Apr-2020 | eluzdc 8283 | Membership of an integer in an upper set of integers is decidable. (Contributed by Jim Kingdon, 18-Apr-2020.) | ||||||||||||
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝑁 ∈ (ℤ_{≥}‘𝑀)) | ||||||||||||||
17-Apr-2020 | zlelttric 8026 | Trichotomy law. (Contributed by Jim Kingdon, 17-Apr-2020.) | ||||||||||||
⊢ ((A ∈ ℤ ∧ B ∈ ℤ) → (A ≤ B ∨ B < A)) | ||||||||||||||
16-Apr-2020 | fznlem 8635 | A finite set of sequential integers is empty if the bounds are reversed. (Contributed by Jim Kingdon, 16-Apr-2020.) | ||||||||||||
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < 𝑀 → (𝑀...𝑁) = ∅)) | ||||||||||||||
15-Apr-2020 | fzm 8632 | Properties of a finite interval of integers which is inhabited. (Contributed by Jim Kingdon, 15-Apr-2020.) | ||||||||||||
⊢ (∃x x ∈ (𝑀...𝑁) ↔ 𝑁 ∈ (ℤ_{≥}‘𝑀)) | ||||||||||||||
15-Apr-2020 | xpdom3m 6244 | A set is dominated by its Cartesian product with an inhabited set. Exercise 6 of [Suppes] p. 98. (Contributed by Jim Kingdon, 15-Apr-2020.) | ||||||||||||
⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊 ∧ ∃x x ∈ B) → A ≼ (A × B)) | ||||||||||||||
13-Apr-2020 | snfig 6227 | A singleton is finite. (Contributed by Jim Kingdon, 13-Apr-2020.) | ||||||||||||
⊢ (A ∈ 𝑉 → {A} ∈ Fin) | ||||||||||||||
13-Apr-2020 | en1bg 6216 | A set is equinumerous to ordinal one iff it is a singleton. (Contributed by Jim Kingdon, 13-Apr-2020.) | ||||||||||||
⊢ (A ∈ 𝑉 → (A ≈ 1_{𝑜} ↔ A = {∪ A})) | ||||||||||||||
10-Apr-2020 | negm 8286 | The image under negation of an inhabited set of reals is inhabited. (Contributed by Jim Kingdon, 10-Apr-2020.) | ||||||||||||
⊢ ((A ⊆ ℝ ∧ ∃x x ∈ A) → ∃y y ∈ {z ∈ ℝ ∣ -z ∈ A}) | ||||||||||||||
8-Apr-2020 | zleloe 8028 | Integer 'Less than or equal to' expressed in terms of 'less than' or 'equals'. (Contributed by Jim Kingdon, 8-Apr-2020.) | ||||||||||||
⊢ ((A ∈ ℤ ∧ B ∈ ℤ) → (A ≤ B ↔ (A < B ∨ A = B))) | ||||||||||||||
7-Apr-2020 | zdcle 8053 | Integer ≤ is decidable. (Contributed by Jim Kingdon, 7-Apr-2020.) | ||||||||||||
⊢ ((A ∈ ℤ ∧ B ∈ ℤ) → DECID A ≤ B) | ||||||||||||||
4-Apr-2020 | ioorebasg 8574 | Open intervals are elements of the set of all open intervals. (Contributed by Jim Kingdon, 4-Apr-2020.) | ||||||||||||
⊢ ((A ∈ ℝ^{*} ∧ B ∈ ℝ^{*}) → (A(,)B) ∈ ran (,)) | ||||||||||||||
30-Mar-2020 | icc0r 8525 | An empty closed interval of extended reals. (Contributed by Jim Kingdon, 30-Mar-2020.) | ||||||||||||
⊢ ((A ∈ ℝ^{*} ∧ B ∈ ℝ^{*}) → (B < A → (A[,]B) = ∅)) | ||||||||||||||
30-Mar-2020 | ubioog 8513 | An open interval does not contain its right endpoint. (Contributed by Jim Kingdon, 30-Mar-2020.) | ||||||||||||
⊢ ((A ∈ ℝ^{*} ∧ B ∈ ℝ^{*}) → ¬ B ∈ (A(,)B)) | ||||||||||||||
30-Mar-2020 | lbioog 8512 | An open interval does not contain its left endpoint. (Contributed by Jim Kingdon, 30-Mar-2020.) | ||||||||||||
⊢ ((A ∈ ℝ^{*} ∧ B ∈ ℝ^{*}) → ¬ A ∈ (A(,)B)) | ||||||||||||||
29-Mar-2020 | iooidg 8508 | An open interval with identical lower and upper bounds is empty. (Contributed by Jim Kingdon, 29-Mar-2020.) | ||||||||||||
⊢ (A ∈ ℝ^{*} → (A(,)A) = ∅) | ||||||||||||||
27-Mar-2020 | zletric 8025 | Trichotomy law. (Contributed by Jim Kingdon, 27-Mar-2020.) | ||||||||||||
⊢ ((A ∈ ℤ ∧ B ∈ ℤ) → (A ≤ B ∨ B ≤ A)) | ||||||||||||||
26-Mar-2020 | 4z 8011 | 4 is an integer. (Contributed by BJ, 26-Mar-2020.) | ||||||||||||
⊢ 4 ∈ ℤ | ||||||||||||||
25-Mar-2020 | elfzmlbm 8718 | Subtracting the lower bound of a finite set of sequential integers from an element of this set. (Contributed by Alexander van der Vekens, 29-Mar-2018.) (Proof shortened by OpenAI, 25-Mar-2020.) | ||||||||||||
⊢ (𝐾 ∈ (𝑀...𝑁) → (𝐾 − 𝑀) ∈ (0...(𝑁 − 𝑀))) | ||||||||||||||
25-Mar-2020 | elfz0add 8709 | An element of a finite set of sequential nonnegative integers is an element of an extended finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 28-Mar-2018.) (Proof shortened by OpenAI, 25-Mar-2020.) | ||||||||||||
⊢ ((A ∈ ℕ_{0} ∧ B ∈ ℕ_{0}) → (𝑁 ∈ (0...A) → 𝑁 ∈ (0...(A + B)))) | ||||||||||||||
25-Mar-2020 | 2eluzge0 8253 | 2 is an integer greater than or equal to 0. (Contributed by Alexander van der Vekens, 8-Jun-2018.) (Proof shortened by OpenAI, 25-Mar-2020.) | ||||||||||||
⊢ 2 ∈ (ℤ_{≥}‘0) | ||||||||||||||
23-Mar-2020 | rpnegap 8350 | Either a real apart from zero or its negation is a positive real, but not both. (Contributed by Jim Kingdon, 23-Mar-2020.) | ||||||||||||
⊢ ((A ∈ ℝ ∧ A # 0) → (A ∈ ℝ^{+} ⊻ -A ∈ ℝ^{+})) | ||||||||||||||
23-Mar-2020 | reapltxor 7333 | Real apartness in terms of less than (exclusive-or version). (Contributed by Jim Kingdon, 23-Mar-2020.) | ||||||||||||
⊢ ((A ∈ ℝ ∧ B ∈ ℝ) → (A # B ↔ (A < B ⊻ B < A))) | ||||||||||||||
22-Mar-2020 | rpcnap0 8338 | A positive real is a complex number apart from zero. (Contributed by Jim Kingdon, 22-Mar-2020.) | ||||||||||||
⊢ (A ∈ ℝ^{+} → (A ∈ ℂ ∧ A # 0)) | ||||||||||||||
22-Mar-2020 | rpreap0 8336 | A positive real is a real number apart from zero. (Contributed by Jim Kingdon, 22-Mar-2020.) | ||||||||||||
⊢ (A ∈ ℝ^{+} → (A ∈ ℝ ∧ A # 0)) | ||||||||||||||
22-Mar-2020 | rpap0 8334 | A positive real is apart from zero. (Contributed by Jim Kingdon, 22-Mar-2020.) | ||||||||||||
⊢ (A ∈ ℝ^{+} → A # 0) | ||||||||||||||
20-Mar-2020 | qapne 8310 | Apartness is equivalent to not equal for rationals. (Contributed by Jim Kingdon, 20-Mar-2020.) | ||||||||||||
⊢ ((A ∈ ℚ ∧ B ∈ ℚ) → (A # B ↔ A ≠ B)) | ||||||||||||||
20-Mar-2020 | divap1d 7518 | If two complex numbers are apart, their quotient is apart from one. (Contributed by Jim Kingdon, 20-Mar-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → B # 0) & ⊢ (φ → A # B) ⇒ ⊢ (φ → (A / B) # 1) | ||||||||||||||
20-Mar-2020 | apmul1 7506 | Multiplication of both sides of complex apartness by a complex number apart from zero. (Contributed by Jim Kingdon, 20-Mar-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → (A # B ↔ (A · 𝐶) # (B · 𝐶))) | ||||||||||||||
19-Mar-2020 | divfnzn 8292 | Division restricted to ℤ × ℕ is a function. Given excluded middle, it would be easy to prove this for ℂ × (ℂ ∖ {0}). The key difference is that an element of ℕ is apart from zero, whereas being an element of ℂ ∖ {0} implies being not equal to zero. (Contributed by Jim Kingdon, 19-Mar-2020.) | ||||||||||||
⊢ ( / ↾ (ℤ × ℕ)) Fn (ℤ × ℕ) | ||||||||||||||
19-Mar-2020 | div2negapd 7522 | Quotient of two negatives. (Contributed by Jim Kingdon, 19-Mar-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → B # 0) ⇒ ⊢ (φ → (-A / -B) = (A / B)) | ||||||||||||||
19-Mar-2020 | divneg2apd 7521 | Move negative sign inside of a division. (Contributed by Jim Kingdon, 19-Mar-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → B # 0) ⇒ ⊢ (φ → -(A / B) = (A / -B)) | ||||||||||||||
19-Mar-2020 | divnegapd 7520 | Move negative sign inside of a division. (Contributed by Jim Kingdon, 19-Mar-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → B # 0) ⇒ ⊢ (φ → -(A / B) = (-A / B)) | ||||||||||||||
19-Mar-2020 | divap0bd 7519 | A ratio is zero iff the numerator is zero. (Contributed by Jim Kingdon, 19-Mar-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → B # 0) ⇒ ⊢ (φ → (A # 0 ↔ (A / B) # 0)) | ||||||||||||||
19-Mar-2020 | diveqap0ad 7517 | A fraction of complex numbers is zero iff its numerator is. Deduction form of diveqap0 7403. (Contributed by Jim Kingdon, 19-Mar-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → B # 0) ⇒ ⊢ (φ → ((A / B) = 0 ↔ A = 0)) | ||||||||||||||
19-Mar-2020 | diveqap1ad 7516 | The quotient of two complex numbers is one iff they are equal. Deduction form of diveqap1 7424. Generalization of diveqap1d 7515. (Contributed by Jim Kingdon, 19-Mar-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → B # 0) ⇒ ⊢ (φ → ((A / B) = 1 ↔ A = B)) | ||||||||||||||
19-Mar-2020 | diveqap1d 7515 | Equality in terms of unit ratio. (Contributed by Jim Kingdon, 19-Mar-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → B # 0) & ⊢ (φ → (A / B) = 1) ⇒ ⊢ (φ → A = B) | ||||||||||||||
19-Mar-2020 | diveqap0d 7514 | If a ratio is zero, the numerator is zero. (Contributed by Jim Kingdon, 19-Mar-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → B # 0) & ⊢ (φ → (A / B) = 0) ⇒ ⊢ (φ → A = 0) | ||||||||||||||
15-Mar-2020 | nneoor 8076 | A positive integer is even or odd. (Contributed by Jim Kingdon, 15-Mar-2020.) | ||||||||||||
⊢ (𝑁 ∈ ℕ → ((𝑁 / 2) ∈ ℕ ∨ ((𝑁 + 1) / 2) ∈ ℕ)) | ||||||||||||||
14-Mar-2020 | zltlen 8055 | Integer 'Less than' expressed in terms of 'less than or equal to'. Also see ltleap 7373 which is a similar result for complex numbers. (Contributed by Jim Kingdon, 14-Mar-2020.) | ||||||||||||
⊢ ((A ∈ ℤ ∧ B ∈ ℤ) → (A < B ↔ (A ≤ B ∧ B ≠ A))) | ||||||||||||||
14-Mar-2020 | zdceq 8052 | Equality of integers is decidable. (Contributed by Jim Kingdon, 14-Mar-2020.) | ||||||||||||
⊢ ((A ∈ ℤ ∧ B ∈ ℤ) → DECID A = B) | ||||||||||||||
14-Mar-2020 | zapne 8051 | Apartness is equivalent to not equal for integers. (Contributed by Jim Kingdon, 14-Mar-2020.) | ||||||||||||
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 # 𝑁 ↔ 𝑀 ≠ 𝑁)) | ||||||||||||||
14-Mar-2020 | zltnle 8027 | 'Less than' expressed in terms of 'less than or equal to'. (Contributed by Jim Kingdon, 14-Mar-2020.) | ||||||||||||
⊢ ((A ∈ ℤ ∧ B ∈ ℤ) → (A < B ↔ ¬ B ≤ A)) | ||||||||||||||
14-Mar-2020 | ztri3or 8024 | Integer trichotomy. (Contributed by Jim Kingdon, 14-Mar-2020.) | ||||||||||||
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ∨ 𝑀 = 𝑁 ∨ 𝑁 < 𝑀)) | ||||||||||||||
14-Mar-2020 | ztri3or0 8023 | Integer trichotomy (with zero). (Contributed by Jim Kingdon, 14-Mar-2020.) | ||||||||||||
⊢ (𝑁 ∈ ℤ → (𝑁 < 0 ∨ 𝑁 = 0 ∨ 0 < 𝑁)) | ||||||||||||||
14-Mar-2020 | zaddcllemneg 8020 | Lemma for zaddcl 8021. Special case in which -𝑁 is a positive integer. (Contributed by Jim Kingdon, 14-Mar-2020.) | ||||||||||||
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℤ) | ||||||||||||||
14-Mar-2020 | zaddcllempos 8018 | Lemma for zaddcl 8021. Special case in which 𝑁 is a positive integer. (Contributed by Jim Kingdon, 14-Mar-2020.) | ||||||||||||
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℤ) | ||||||||||||||
14-Mar-2020 | dcne 2211 | Decidable equality expressed in terms of ≠. Basically the same as df-dc 742. (Contributed by Jim Kingdon, 14-Mar-2020.) | ||||||||||||
⊢ (DECID A = B ↔ (A = B ∨ A ≠ B)) | ||||||||||||||
9-Mar-2020 | 2muliap0 7886 | 2 · i is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.) | ||||||||||||
⊢ (2 · i) # 0 | ||||||||||||||
9-Mar-2020 | iap0 7885 | The imaginary unit i is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.) | ||||||||||||
⊢ i # 0 | ||||||||||||||
9-Mar-2020 | 2ap0 7749 | The number 2 is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.) | ||||||||||||
⊢ 2 # 0 | ||||||||||||||
9-Mar-2020 | 1ne0 7723 | 1 ≠ 0. See aso 1ap0 7334. (Contributed by Jim Kingdon, 9-Mar-2020.) | ||||||||||||
⊢ 1 ≠ 0 | ||||||||||||||
9-Mar-2020 | redivclapi 7497 | Closure law for division of reals. (Contributed by Jim Kingdon, 9-Mar-2020.) | ||||||||||||
⊢ A ∈ ℝ & ⊢ B ∈ ℝ & ⊢ B # 0 ⇒ ⊢ (A / B) ∈ ℝ | ||||||||||||||
9-Mar-2020 | redivclapzi 7496 | Closure law for division of reals. (Contributed by Jim Kingdon, 9-Mar-2020.) | ||||||||||||
⊢ A ∈ ℝ & ⊢ B ∈ ℝ ⇒ ⊢ (B # 0 → (A / B) ∈ ℝ) | ||||||||||||||
9-Mar-2020 | rerecclapi 7495 | Closure law for reciprocal. (Contributed by Jim Kingdon, 9-Mar-2020.) | ||||||||||||
⊢ A ∈ ℝ & ⊢ A # 0 ⇒ ⊢ (1 / A) ∈ ℝ | ||||||||||||||
9-Mar-2020 | rerecclapzi 7494 | Closure law for reciprocal. (Contributed by Jim Kingdon, 9-Mar-2020.) | ||||||||||||
⊢ A ∈ ℝ ⇒ ⊢ (A # 0 → (1 / A) ∈ ℝ) | ||||||||||||||
9-Mar-2020 | divdivdivapi 7493 | Division of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.) | ||||||||||||
⊢ A ∈ ℂ & ⊢ B ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐷 ∈ ℂ & ⊢ B # 0 & ⊢ 𝐷 # 0 & ⊢ 𝐶 # 0 ⇒ ⊢ ((A / B) / (𝐶 / 𝐷)) = ((A · 𝐷) / (B · 𝐶)) | ||||||||||||||
9-Mar-2020 | divadddivapi 7492 | Addition of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.) | ||||||||||||
⊢ A ∈ ℂ & ⊢ B ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐷 ∈ ℂ & ⊢ B # 0 & ⊢ 𝐷 # 0 ⇒ ⊢ ((A / B) + (𝐶 / 𝐷)) = (((A · 𝐷) + (𝐶 · B)) / (B · 𝐷)) | ||||||||||||||
9-Mar-2020 | divmul13api 7491 | Swap denominators of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.) | ||||||||||||
⊢ A ∈ ℂ & ⊢ B ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐷 ∈ ℂ & ⊢ B # 0 & ⊢ 𝐷 # 0 ⇒ ⊢ ((A / B) · (𝐶 / 𝐷)) = ((𝐶 / B) · (A / 𝐷)) | ||||||||||||||
9-Mar-2020 | divmuldivapi 7490 | Multiplication of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.) | ||||||||||||
⊢ A ∈ ℂ & ⊢ B ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐷 ∈ ℂ & ⊢ B # 0 & ⊢ 𝐷 # 0 ⇒ ⊢ ((A / B) · (𝐶 / 𝐷)) = ((A · 𝐶) / (B · 𝐷)) | ||||||||||||||
9-Mar-2020 | div11api 7489 | One-to-one relationship for division. (Contributed by Jim Kingdon, 9-Mar-2020.) | ||||||||||||
⊢ A ∈ ℂ & ⊢ B ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐶 # 0 ⇒ ⊢ ((A / 𝐶) = (B / 𝐶) ↔ A = B) | ||||||||||||||
9-Mar-2020 | div23api 7488 | A commutative/associative law for division. (Contributed by Jim Kingdon, 9-Mar-2020.) | ||||||||||||
⊢ A ∈ ℂ & ⊢ B ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐶 # 0 ⇒ ⊢ ((A · B) / 𝐶) = ((A / 𝐶) · B) | ||||||||||||||
9-Mar-2020 | divdirapi 7487 | Distribution of division over addition. (Contributed by Jim Kingdon, 9-Mar-2020.) | ||||||||||||
⊢ A ∈ ℂ & ⊢ B ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐶 # 0 ⇒ ⊢ ((A + B) / 𝐶) = ((A / 𝐶) + (B / 𝐶)) | ||||||||||||||
9-Mar-2020 | divassapi 7486 | An associative law for division. (Contributed by Jim Kingdon, 9-Mar-2020.) | ||||||||||||
⊢ A ∈ ℂ & ⊢ B ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐶 # 0 ⇒ ⊢ ((A · B) / 𝐶) = (A · (B / 𝐶)) | ||||||||||||||
8-Mar-2020 | nnap0 7684 | A positive integer is apart from zero. (Contributed by Jim Kingdon, 8-Mar-2020.) | ||||||||||||
⊢ (A ∈ ℕ → A # 0) | ||||||||||||||
8-Mar-2020 | divdivap2d 7539 | Division by a fraction. (Contributed by Jim Kingdon, 8-Mar-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → 𝐶 ∈ ℂ) & ⊢ (φ → B # 0) & ⊢ (φ → 𝐶 # 0) ⇒ ⊢ (φ → (A / (B / 𝐶)) = ((A · 𝐶) / B)) | ||||||||||||||
8-Mar-2020 | divdivap1d 7538 | Division into a fraction. (Contributed by Jim Kingdon, 8-Mar-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → 𝐶 ∈ ℂ) & ⊢ (φ → B # 0) & ⊢ (φ → 𝐶 # 0) ⇒ ⊢ (φ → ((A / B) / 𝐶) = (A / (B · 𝐶))) | ||||||||||||||
8-Mar-2020 | dmdcanap2d 7537 | Cancellation law for division and multiplication. (Contributed by Jim Kingdon, 8-Mar-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → 𝐶 ∈ ℂ) & ⊢ (φ → B # 0) & ⊢ (φ → 𝐶 # 0) ⇒ ⊢ (φ → ((A / B) · (B / 𝐶)) = (A / 𝐶)) | ||||||||||||||
8-Mar-2020 | dmdcanapd 7536 | Cancellation law for division and multiplication. (Contributed by Jim Kingdon, 8-Mar-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → 𝐶 ∈ ℂ) & ⊢ (φ → B # 0) & ⊢ (φ → 𝐶 # 0) ⇒ ⊢ (φ → ((B / 𝐶) · (A / B)) = (A / 𝐶)) | ||||||||||||||
8-Mar-2020 | divcanap7d 7535 | Cancel equal divisors in a division. (Contributed by Jim Kingdon, 8-Mar-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → 𝐶 ∈ ℂ) & ⊢ (φ → B # 0) & ⊢ (φ → 𝐶 # 0) ⇒ ⊢ (φ → ((A / 𝐶) / (B / 𝐶)) = (A / B)) | ||||||||||||||
8-Mar-2020 | divcanap5rd 7534 | Cancellation of common factor in a ratio. (Contributed by Jim Kingdon, 8-Mar-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → 𝐶 ∈ ℂ) & ⊢ (φ → B # 0) & ⊢ (φ → 𝐶 # 0) ⇒ ⊢ (φ → ((A · 𝐶) / (B · 𝐶)) = (A / B)) | ||||||||||||||
8-Mar-2020 | divcanap5d 7533 | Cancellation of common factor in a ratio. (Contributed by Jim Kingdon, 8-Mar-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → 𝐶 ∈ ℂ) & ⊢ (φ → B # 0) & ⊢ (φ → 𝐶 # 0) ⇒ ⊢ (φ → ((𝐶 · A) / (𝐶 · B)) = (A / B)) | ||||||||||||||
8-Mar-2020 | divdiv32apd 7532 | Swap denominators in a division. (Contributed by Jim Kingdon, 8-Mar-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → 𝐶 ∈ ℂ) & ⊢ (φ → B # 0) & ⊢ (φ → 𝐶 # 0) ⇒ ⊢ (φ → ((A / B) / 𝐶) = ((A / 𝐶) / B)) | ||||||||||||||
8-Mar-2020 | div13apd 7531 | A commutative/associative law for division. (Contributed by Jim Kingdon, 8-Mar-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → 𝐶 ∈ ℂ) & ⊢ (φ → B # 0) ⇒ ⊢ (φ → ((A / B) · 𝐶) = ((𝐶 / B) · A)) | ||||||||||||||
8-Mar-2020 | div32apd 7530 | A commutative/associative law for division. (Contributed by Jim Kingdon, 8-Mar-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → 𝐶 ∈ ℂ) & ⊢ (φ → B # 0) ⇒ ⊢ (φ → ((A / B) · 𝐶) = (A · (𝐶 / B))) | ||||||||||||||
8-Mar-2020 | divmulapd 7529 | Relationship between division and multiplication. (Contributed by Jim Kingdon, 8-Mar-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → 𝐶 ∈ ℂ) & ⊢ (φ → B # 0) ⇒ ⊢ (φ → ((A / B) = 𝐶 ↔ (B · 𝐶) = A)) | ||||||||||||||
7-Mar-2020 | nn1gt1 7688 | A positive integer is either one or greater than one. This is for ℕ; 0elnn 4283 is a similar theorem for 𝜔 (the natural numbers as ordinals). (Contributed by Jim Kingdon, 7-Mar-2020.) | ||||||||||||
⊢ (A ∈ ℕ → (A = 1 ∨ 1 < A)) | ||||||||||||||
5-Mar-2020 | crap0 7651 | The real representation of complex numbers is apart from zero iff one of its terms is apart from zero. (Contributed by Jim Kingdon, 5-Mar-2020.) | ||||||||||||
⊢ ((A ∈ ℝ ∧ B ∈ ℝ) → ((A # 0 ∨ B # 0) ↔ (A + (i · B)) # 0)) | ||||||||||||||
3-Mar-2020 | rec11apd 7528 | Reciprocal is one-to-one. (Contributed by Jim Kingdon, 3-Mar-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → A # 0) & ⊢ (φ → B # 0) & ⊢ (φ → (1 / A) = (1 / B)) ⇒ ⊢ (φ → A = B) | ||||||||||||||
3-Mar-2020 | ddcanapd 7527 | Cancellation in a double division. (Contributed by Jim Kingdon, 3-Mar-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → A # 0) & ⊢ (φ → B # 0) ⇒ ⊢ (φ → (A / (A / B)) = B) | ||||||||||||||
3-Mar-2020 | divcanap6d 7526 | Cancellation of inverted fractions. (Contributed by Jim Kingdon, 3-Mar-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → A # 0) & ⊢ (φ → B # 0) ⇒ ⊢ (φ → ((A / B) · (B / A)) = 1) | ||||||||||||||
3-Mar-2020 | recdivap2d 7525 | Division into a reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → A # 0) & ⊢ (φ → B # 0) ⇒ ⊢ (φ → ((1 / A) / B) = (1 / (A · B))) | ||||||||||||||
3-Mar-2020 | recdivapd 7524 | The reciprocal of a ratio. (Contributed by Jim Kingdon, 3-Mar-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → A # 0) & ⊢ (φ → B # 0) ⇒ ⊢ (φ → (1 / (A / B)) = (B / A)) | ||||||||||||||
3-Mar-2020 | divap0d 7523 | The ratio of numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 3-Mar-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → A # 0) & ⊢ (φ → B # 0) ⇒ ⊢ (φ → (A / B) # 0) | ||||||||||||||
3-Mar-2020 | div0apd 7505 | Division into zero is zero. (Contributed by Jim Kingdon, 3-Mar-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → A # 0) ⇒ ⊢ (φ → (0 / A) = 0) | ||||||||||||||
3-Mar-2020 | dividapd 7504 | A number divided by itself is one. (Contributed by Jim Kingdon, 3-Mar-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → A # 0) ⇒ ⊢ (φ → (A / A) = 1) | ||||||||||||||
3-Mar-2020 | recrecapd 7503 | A number is equal to the reciprocal of its reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → A # 0) ⇒ ⊢ (φ → (1 / (1 / A)) = A) | ||||||||||||||
3-Mar-2020 | recidap2d 7502 | Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → A # 0) ⇒ ⊢ (φ → ((1 / A) · A) = 1) | ||||||||||||||
3-Mar-2020 | recidapd 7501 | Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → A # 0) ⇒ ⊢ (φ → (A · (1 / A)) = 1) | ||||||||||||||
3-Mar-2020 | recap0d 7500 | The reciprocal of a number apart from zero is apart from zero. (Contributed by Jim Kingdon, 3-Mar-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → A # 0) ⇒ ⊢ (φ → (1 / A) # 0) | ||||||||||||||
3-Mar-2020 | recclapd 7499 | Closure law for reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → A # 0) ⇒ ⊢ (φ → (1 / A) ∈ ℂ) | ||||||||||||||
2-Mar-2020 | div11apd 7547 | One-to-one relationship for division. (Contributed by Jim Kingdon, 2-Mar-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → 𝐶 ∈ ℂ) & ⊢ (φ → 𝐶 # 0) & ⊢ (φ → (A / 𝐶) = (B / 𝐶)) ⇒ ⊢ (φ → A = B) | ||||||||||||||
2-Mar-2020 | divsubdirapd 7546 | Distribution of division over subtraction. (Contributed by Jim Kingdon, 2-Mar-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → 𝐶 ∈ ℂ) & ⊢ (φ → 𝐶 # 0) ⇒ ⊢ (φ → ((A − B) / 𝐶) = ((A / 𝐶) − (B / 𝐶))) | ||||||||||||||
2-Mar-2020 | divdirapd 7545 | Distribution of division over addition. (Contributed by Jim Kingdon, 2-Mar-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → 𝐶 ∈ ℂ) & ⊢ (φ → 𝐶 # 0) ⇒ ⊢ (φ → ((A + B) / 𝐶) = ((A / 𝐶) + (B / 𝐶))) | ||||||||||||||
2-Mar-2020 | div23apd 7544 | A commutative/associative law for division. (Contributed by Jim Kingdon, 2-Mar-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → 𝐶 ∈ ℂ) & ⊢ (φ → 𝐶 # 0) ⇒ ⊢ (φ → ((A · B) / 𝐶) = ((A / 𝐶) · B)) | ||||||||||||||
2-Mar-2020 | div12apd 7543 | A commutative/associative law for division. (Contributed by Jim Kingdon, 2-Mar-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → 𝐶 ∈ ℂ) & ⊢ (φ → 𝐶 # 0) ⇒ ⊢ (φ → (A · (B / 𝐶)) = (B · (A / 𝐶))) | ||||||||||||||
2-Mar-2020 | divassapd 7542 | An associative law for division. (Contributed by Jim Kingdon, 2-Mar-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → 𝐶 ∈ ℂ) & ⊢ (φ → 𝐶 # 0) ⇒ ⊢ (φ → ((A · B) / 𝐶) = (A · (B / 𝐶))) | ||||||||||||||
2-Mar-2020 | divmulap3d 7541 | Relationship between division and multiplication. (Contributed by Jim Kingdon, 2-Mar-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → 𝐶 ∈ ℂ) & ⊢ (φ → 𝐶 # 0) ⇒ ⊢ (φ → ((A / 𝐶) = B ↔ A = (B · 𝐶))) | ||||||||||||||
2-Mar-2020 | divmulap2d 7540 | Relationship between division and multiplication. (Contributed by Jim Kingdon, 2-Mar-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → 𝐶 ∈ ℂ) & ⊢ (φ → 𝐶 # 0) ⇒ ⊢ (φ → ((A / 𝐶) = B ↔ A = (𝐶 · B))) | ||||||||||||||
29-Feb-2020 | prodgt0gt0 7558 | Infer that a multiplicand is positive from a positive multiplier and positive product. See prodgt0 7559 for the same theorem with 0 < A replaced by the weaker condition 0 ≤ A. (Contributed by Jim Kingdon, 29-Feb-2020.) | ||||||||||||
⊢ (((A ∈ ℝ ∧ B ∈ ℝ) ∧ (0 < A ∧ 0 < (A · B))) → 0 < B) | ||||||||||||||
29-Feb-2020 | redivclapd 7549 | Closure law for division of reals. (Contributed by Jim Kingdon, 29-Feb-2020.) | ||||||||||||
⊢ (φ → A ∈ ℝ) & ⊢ (φ → B ∈ ℝ) & ⊢ (φ → B # 0) ⇒ ⊢ (φ → (A / B) ∈ ℝ) | ||||||||||||||
29-Feb-2020 | rerecclapd 7548 | Closure law for reciprocal. (Contributed by Jim Kingdon, 29-Feb-2020.) | ||||||||||||
⊢ (φ → A ∈ ℝ) & ⊢ (φ → A # 0) ⇒ ⊢ (φ → (1 / A) ∈ ℝ) | ||||||||||||||
29-Feb-2020 | divcanap4d 7513 | A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → B # 0) ⇒ ⊢ (φ → ((A · B) / B) = A) | ||||||||||||||
29-Feb-2020 | divcanap3d 7512 | A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → B # 0) ⇒ ⊢ (φ → ((B · A) / B) = A) | ||||||||||||||
29-Feb-2020 | divrecap2d 7511 | Relationship between division and reciprocal. (Contributed by Jim Kingdon, 29-Feb-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → B # 0) ⇒ ⊢ (φ → (A / B) = ((1 / B) · A)) | ||||||||||||||
29-Feb-2020 | divrecapd 7510 | Relationship between division and reciprocal. Theorem I.9 of [Apostol] p. 18. (Contributed by Jim Kingdon, 29-Feb-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → B # 0) ⇒ ⊢ (φ → (A / B) = (A · (1 / B))) | ||||||||||||||
29-Feb-2020 | divcanap2d 7509 | A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → B # 0) ⇒ ⊢ (φ → (B · (A / B)) = A) | ||||||||||||||
29-Feb-2020 | divcanap1d 7508 | A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → B # 0) ⇒ ⊢ (φ → ((A / B) · B) = A) | ||||||||||||||
29-Feb-2020 | divclapd 7507 | Closure law for division. (Contributed by Jim Kingdon, 29-Feb-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → B # 0) ⇒ ⊢ (φ → (A / B) ∈ ℂ) | ||||||||||||||
29-Feb-2020 | divdiv32api 7485 | Swap denominators in a division. (Contributed by Jim Kingdon, 29-Feb-2020.) | ||||||||||||
⊢ A ∈ ℂ & ⊢ B ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ B # 0 & ⊢ 𝐶 # 0 ⇒ ⊢ ((A / B) / 𝐶) = ((A / 𝐶) / B) | ||||||||||||||
29-Feb-2020 | divmulapi 7484 | Relationship between division and multiplication. (Contributed by Jim Kingdon, 29-Feb-2020.) | ||||||||||||
⊢ A ∈ ℂ & ⊢ B ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ B # 0 ⇒ ⊢ ((A / B) = 𝐶 ↔ (B · 𝐶) = A) | ||||||||||||||
28-Feb-2020 | divdiv23apzi 7483 | Swap denominators in a division. (Contributed by Jim Kingdon, 28-Feb-2020.) | ||||||||||||
⊢ A ∈ ℂ & ⊢ B ∈ ℂ & ⊢ 𝐶 ∈ ℂ ⇒ ⊢ ((B # 0 ∧ 𝐶 # 0) → ((A / B) / 𝐶) = ((A / 𝐶) / B)) | ||||||||||||||
28-Feb-2020 | divdirapzi 7482 | Distribution of division over addition. (Contributed by Jim Kingdon, 28-Feb-2020.) | ||||||||||||
⊢ A ∈ ℂ & ⊢ B ∈ ℂ & ⊢ 𝐶 ∈ ℂ ⇒ ⊢ (𝐶 # 0 → ((A + B) / 𝐶) = ((A / 𝐶) + (B / 𝐶))) | ||||||||||||||
28-Feb-2020 | divmulapzi 7481 | Relationship between division and multiplication. (Contributed by Jim Kingdon, 28-Feb-2020.) | ||||||||||||
⊢ A ∈ ℂ & ⊢ B ∈ ℂ & ⊢ 𝐶 ∈ ℂ ⇒ ⊢ (B # 0 → ((A / B) = 𝐶 ↔ (B · 𝐶) = A)) | ||||||||||||||
28-Feb-2020 | divassapzi 7480 | An associative law for division. (Contributed by Jim Kingdon, 28-Feb-2020.) | ||||||||||||
⊢ A ∈ ℂ & ⊢ B ∈ ℂ & ⊢ 𝐶 ∈ ℂ ⇒ ⊢ (𝐶 # 0 → ((A · B) / 𝐶) = (A · (B / 𝐶))) | ||||||||||||||
28-Feb-2020 | rec11apii 7479 | Reciprocal is one-to-one. (Contributed by Jim Kingdon, 28-Feb-2020.) | ||||||||||||
⊢ A ∈ ℂ & ⊢ B ∈ ℂ & ⊢ A # 0 & ⊢ B # 0 ⇒ ⊢ ((1 / A) = (1 / B) ↔ A = B) | ||||||||||||||
28-Feb-2020 | divap0i 7478 | The ratio of numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 28-Feb-2020.) | ||||||||||||
⊢ A ∈ ℂ & ⊢ B ∈ ℂ & ⊢ A # 0 & ⊢ B # 0 ⇒ ⊢ (A / B) # 0 | ||||||||||||||
28-Feb-2020 | divcanap4i 7477 | A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.) | ||||||||||||
⊢ A ∈ ℂ & ⊢ B ∈ ℂ & ⊢ B # 0 ⇒ ⊢ ((A · B) / B) = A | ||||||||||||||
28-Feb-2020 | divcanap3i 7476 | A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.) | ||||||||||||
⊢ A ∈ ℂ & ⊢ B ∈ ℂ & ⊢ B # 0 ⇒ ⊢ ((B · A) / B) = A | ||||||||||||||
28-Feb-2020 | divrecapi 7475 | Relationship between division and reciprocal. (Contributed by Jim Kingdon, 28-Feb-2020.) | ||||||||||||
⊢ A ∈ ℂ & ⊢ B ∈ ℂ & ⊢ B # 0 ⇒ ⊢ (A / B) = (A · (1 / B)) | ||||||||||||||
28-Feb-2020 | divcanap1i 7474 | A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.) | ||||||||||||
⊢ A ∈ ℂ & ⊢ B ∈ ℂ & ⊢ B # 0 ⇒ ⊢ ((A / B) · B) = A | ||||||||||||||
28-Feb-2020 | divcanap2i 7473 | A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.) | ||||||||||||
⊢ A ∈ ℂ & ⊢ B ∈ ℂ & ⊢ B # 0 ⇒ ⊢ (B · (A / B)) = A | ||||||||||||||
28-Feb-2020 | divclapi 7472 | Closure law for division. (Contributed by Jim Kingdon, 28-Feb-2020.) | ||||||||||||
⊢ A ∈ ℂ & ⊢ B ∈ ℂ & ⊢ B # 0 ⇒ ⊢ (A / B) ∈ ℂ | ||||||||||||||
28-Feb-2020 | rec11api 7471 | Reciprocal is one-to-one. (Contributed by Jim Kingdon, 28-Feb-2020.) | ||||||||||||
⊢ A ∈ ℂ & ⊢ B ∈ ℂ ⇒ ⊢ ((A # 0 ∧ B # 0) → ((1 / A) = (1 / B) ↔ A = B)) | ||||||||||||||
28-Feb-2020 | ltleap 7373 | Less than in terms of non-strict order and apartness. (Contributed by Jim Kingdon, 28-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℝ ∧ B ∈ ℝ) → (A < B ↔ (A ≤ B ∧ A # B))) | ||||||||||||||
27-Feb-2020 | divcanap4zi 7470 | A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.) | ||||||||||||
⊢ A ∈ ℂ & ⊢ B ∈ ℂ ⇒ ⊢ (B # 0 → ((A · B) / B) = A) | ||||||||||||||
27-Feb-2020 | divcanap3zi 7469 | A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.) | ||||||||||||
⊢ A ∈ ℂ & ⊢ B ∈ ℂ ⇒ ⊢ (B # 0 → ((B · A) / B) = A) | ||||||||||||||
27-Feb-2020 | divrecapzi 7468 | Relationship between division and reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.) | ||||||||||||
⊢ A ∈ ℂ & ⊢ B ∈ ℂ ⇒ ⊢ (B # 0 → (A / B) = (A · (1 / B))) | ||||||||||||||
27-Feb-2020 | divcanap2zi 7467 | A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.) | ||||||||||||
⊢ A ∈ ℂ & ⊢ B ∈ ℂ ⇒ ⊢ (B # 0 → (B · (A / B)) = A) | ||||||||||||||
27-Feb-2020 | divcanap1zi 7466 | A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.) | ||||||||||||
⊢ A ∈ ℂ & ⊢ B ∈ ℂ ⇒ ⊢ (B # 0 → ((A / B) · B) = A) | ||||||||||||||
27-Feb-2020 | divclapzi 7465 | Closure law for division. (Contributed by Jim Kingdon, 27-Feb-2020.) | ||||||||||||
⊢ A ∈ ℂ & ⊢ B ∈ ℂ ⇒ ⊢ (B # 0 → (A / B) ∈ ℂ) | ||||||||||||||
27-Feb-2020 | recidapzi 7457 | Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.) | ||||||||||||
⊢ A ∈ ℂ ⇒ ⊢ (A # 0 → (A · (1 / A)) = 1) | ||||||||||||||
27-Feb-2020 | recap0apzi 7456 | The reciprocal of a number apart from zero is apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.) | ||||||||||||
⊢ A ∈ ℂ ⇒ ⊢ (A # 0 → (1 / A) # 0) | ||||||||||||||
27-Feb-2020 | recclapzi 7455 | Closure law for reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.) | ||||||||||||
⊢ A ∈ ℂ ⇒ ⊢ (A # 0 → (1 / A) ∈ ℂ) | ||||||||||||||
27-Feb-2020 | divneg2ap 7454 | Move negative sign inside of a division. (Contributed by Jim Kingdon, 27-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ B # 0) → -(A / B) = (A / -B)) | ||||||||||||||
27-Feb-2020 | div2negap 7453 | Quotient of two negatives. (Contributed by Jim Kingdon, 27-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ B # 0) → (-A / -B) = (A / B)) | ||||||||||||||
27-Feb-2020 | negap0 7372 | A number is apart from zero iff its negative is apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.) | ||||||||||||
⊢ (A ∈ ℂ → (A # 0 ↔ -A # 0)) | ||||||||||||||
27-Feb-2020 | gt0ap0d 7371 | Positive implies apart from zero. Because of the way we define #, A must be an element of ℝ, not just ℝ^{*}. (Contributed by Jim Kingdon, 27-Feb-2020.) | ||||||||||||
⊢ (φ → A ∈ ℝ) & ⊢ (φ → 0 < A) ⇒ ⊢ (φ → A # 0) | ||||||||||||||
27-Feb-2020 | gt0ap0ii 7370 | Positive implies apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.) | ||||||||||||
⊢ A ∈ ℝ & ⊢ 0 < A ⇒ ⊢ A # 0 | ||||||||||||||
27-Feb-2020 | gt0ap0i 7369 | Positive means apart from zero (useful for ordering theorems involving division). (Contributed by Jim Kingdon, 27-Feb-2020.) | ||||||||||||
⊢ A ∈ ℝ ⇒ ⊢ (0 < A → A # 0) | ||||||||||||||
27-Feb-2020 | gt0ap0 7368 | Positive implies apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℝ ∧ 0 < A) → A # 0) | ||||||||||||||
26-Feb-2020 | 2times 7776 | Two times a number. (Contributed by NM, 10-Oct-2004.) (Revised by Mario Carneiro, 27-May-2016.) (Proof shortened by AV, 26-Feb-2020.) | ||||||||||||
⊢ (A ∈ ℂ → (2 · A) = (A + A)) | ||||||||||||||
26-Feb-2020 | redivclap 7449 | Closure law for division of reals. (Contributed by Jim Kingdon, 26-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ B # 0) → (A / B) ∈ ℝ) | ||||||||||||||
26-Feb-2020 | rerecclap 7448 | Closure law for reciprocal. (Contributed by Jim Kingdon, 26-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℝ ∧ A # 0) → (1 / A) ∈ ℝ) | ||||||||||||||
26-Feb-2020 | conjmulap 7447 | Two numbers whose reciprocals sum to 1 are called "conjugates" and satisfy this relationship. (Contributed by Jim Kingdon, 26-Feb-2020.) | ||||||||||||
⊢ (((𝑃 ∈ ℂ ∧ 𝑃 # 0) ∧ (𝑄 ∈ ℂ ∧ 𝑄 # 0)) → (((1 / 𝑃) + (1 / 𝑄)) = 1 ↔ ((𝑃 − 1) · (𝑄 − 1)) = 1)) | ||||||||||||||
26-Feb-2020 | divsubdivap 7446 | Subtraction of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.) | ||||||||||||
⊢ (((A ∈ ℂ ∧ B ∈ ℂ) ∧ ((𝐶 ∈ ℂ ∧ 𝐶 # 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 # 0))) → ((A / 𝐶) − (B / 𝐷)) = (((A · 𝐷) − (B · 𝐶)) / (𝐶 · 𝐷))) | ||||||||||||||
26-Feb-2020 | divadddivap 7445 | Addition of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.) | ||||||||||||
⊢ (((A ∈ ℂ ∧ B ∈ ℂ) ∧ ((𝐶 ∈ ℂ ∧ 𝐶 # 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 # 0))) → ((A / 𝐶) + (B / 𝐷)) = (((A · 𝐷) + (B · 𝐶)) / (𝐶 · 𝐷))) | ||||||||||||||
26-Feb-2020 | ddcanap 7444 | Cancellation in a double division. (Contributed by Jim Kingdon, 26-Feb-2020.) | ||||||||||||
⊢ (((A ∈ ℂ ∧ A # 0) ∧ (B ∈ ℂ ∧ B # 0)) → (A / (A / B)) = B) | ||||||||||||||
26-Feb-2020 | recdivap2 7443 | Division into a reciprocal. (Contributed by Jim Kingdon, 26-Feb-2020.) | ||||||||||||
⊢ (((A ∈ ℂ ∧ A # 0) ∧ (B ∈ ℂ ∧ B # 0)) → ((1 / A) / B) = (1 / (A · B))) | ||||||||||||||
26-Feb-2020 | divdivap2 7442 | Division by a fraction. (Contributed by Jim Kingdon, 26-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ (B ∈ ℂ ∧ B # 0) ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → (A / (B / 𝐶)) = ((A · 𝐶) / B)) | ||||||||||||||
26-Feb-2020 | divdivap1 7441 | Division into a fraction. (Contributed by Jim Kingdon, 26-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ (B ∈ ℂ ∧ B # 0) ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((A / B) / 𝐶) = (A / (B · 𝐶))) | ||||||||||||||
26-Feb-2020 | dmdcanap 7440 | Cancellation law for division and multiplication. (Contributed by Jim Kingdon, 26-Feb-2020.) | ||||||||||||
⊢ (((A ∈ ℂ ∧ A # 0) ∧ (B ∈ ℂ ∧ B # 0) ∧ 𝐶 ∈ ℂ) → ((A / B) · (𝐶 / A)) = (𝐶 / B)) | ||||||||||||||
26-Feb-2020 | divcanap7 7439 | Cancel equal divisors in a division. (Contributed by Jim Kingdon, 26-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ (B ∈ ℂ ∧ B # 0) ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((A / 𝐶) / (B / 𝐶)) = (A / B)) | ||||||||||||||
26-Feb-2020 | divdiv32ap 7438 | Swap denominators in a division. (Contributed by Jim Kingdon, 26-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ (B ∈ ℂ ∧ B # 0) ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((A / B) / 𝐶) = ((A / 𝐶) / B)) | ||||||||||||||
26-Feb-2020 | divcanap6 7437 | Cancellation of inverted fractions. (Contributed by Jim Kingdon, 26-Feb-2020.) | ||||||||||||
⊢ (((A ∈ ℂ ∧ A # 0) ∧ (B ∈ ℂ ∧ B # 0)) → ((A / B) · (B / A)) = 1) | ||||||||||||||
26-Feb-2020 | recdivap 7436 | The reciprocal of a ratio. (Contributed by Jim Kingdon, 26-Feb-2020.) | ||||||||||||
⊢ (((A ∈ ℂ ∧ A # 0) ∧ (B ∈ ℂ ∧ B # 0)) → (1 / (A / B)) = (B / A)) | ||||||||||||||
26-Feb-2020 | divmuleqap 7435 | Cross-multiply in an equality of ratios. (Contributed by Jim Kingdon, 26-Feb-2020.) | ||||||||||||
⊢ (((A ∈ ℂ ∧ B ∈ ℂ) ∧ ((𝐶 ∈ ℂ ∧ 𝐶 # 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 # 0))) → ((A / 𝐶) = (B / 𝐷) ↔ (A · 𝐷) = (B · 𝐶))) | ||||||||||||||
26-Feb-2020 | divmul24ap 7434 | Swap the numerators in the product of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.) | ||||||||||||
⊢ (((A ∈ ℂ ∧ B ∈ ℂ) ∧ ((𝐶 ∈ ℂ ∧ 𝐶 # 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 # 0))) → ((A / 𝐶) · (B / 𝐷)) = ((A / 𝐷) · (B / 𝐶))) | ||||||||||||||
26-Feb-2020 | divmul13ap 7433 | Swap the denominators in the product of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.) | ||||||||||||
⊢ (((A ∈ ℂ ∧ B ∈ ℂ) ∧ ((𝐶 ∈ ℂ ∧ 𝐶 # 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 # 0))) → ((A / 𝐶) · (B / 𝐷)) = ((B / 𝐶) · (A / 𝐷))) | ||||||||||||||
25-Feb-2020 | divcanap5 7432 | Cancellation of common factor in a ratio. (Contributed by Jim Kingdon, 25-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ (B ∈ ℂ ∧ B # 0) ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐶 · A) / (𝐶 · B)) = (A / B)) | ||||||||||||||
25-Feb-2020 | divdivdivap 7431 | Division of two ratios. Theorem I.15 of [Apostol] p. 18. (Contributed by Jim Kingdon, 25-Feb-2020.) | ||||||||||||
⊢ (((A ∈ ℂ ∧ (B ∈ ℂ ∧ B # 0)) ∧ ((𝐶 ∈ ℂ ∧ 𝐶 # 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 # 0))) → ((A / B) / (𝐶 / 𝐷)) = ((A · 𝐷) / (B · 𝐶))) | ||||||||||||||
25-Feb-2020 | divmuldivap 7430 | Multiplication of two ratios. (Contributed by Jim Kingdon, 25-Feb-2020.) | ||||||||||||
⊢ (((A ∈ ℂ ∧ B ∈ ℂ) ∧ ((𝐶 ∈ ℂ ∧ 𝐶 # 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 # 0))) → ((A / 𝐶) · (B / 𝐷)) = ((A · B) / (𝐶 · 𝐷))) | ||||||||||||||
25-Feb-2020 | rec11rap 7429 | Mutual reciprocals. (Contributed by Jim Kingdon, 25-Feb-2020.) | ||||||||||||
⊢ (((A ∈ ℂ ∧ A # 0) ∧ (B ∈ ℂ ∧ B # 0)) → ((1 / A) = B ↔ (1 / B) = A)) | ||||||||||||||
25-Feb-2020 | rec11ap 7428 | Reciprocal is one-to-one. (Contributed by Jim Kingdon, 25-Feb-2020.) | ||||||||||||
⊢ (((A ∈ ℂ ∧ A # 0) ∧ (B ∈ ℂ ∧ B # 0)) → ((1 / A) = (1 / B) ↔ A = B)) | ||||||||||||||
25-Feb-2020 | recrecap 7427 | A number is equal to the reciprocal of its reciprocal. (Contributed by Jim Kingdon, 25-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ A # 0) → (1 / (1 / A)) = A) | ||||||||||||||
25-Feb-2020 | divnegap 7425 | Move negative sign inside of a division. (Contributed by Jim Kingdon, 25-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ B # 0) → -(A / B) = (-A / B)) | ||||||||||||||
25-Feb-2020 | diveqap1 7424 | Equality in terms of unit ratio. (Contributed by Jim Kingdon, 25-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ B # 0) → ((A / B) = 1 ↔ A = B)) | ||||||||||||||
25-Feb-2020 | div0ap 7421 | Division into zero is zero. (Contributed by Jim Kingdon, 25-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ A # 0) → (0 / A) = 0) | ||||||||||||||
25-Feb-2020 | dividap 7420 | A number divided by itself is one. (Contributed by Jim Kingdon, 25-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ A # 0) → (A / A) = 1) | ||||||||||||||
25-Feb-2020 | div11ap 7419 | One-to-one relationship for division. (Contributed by Jim Kingdon, 25-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((A / 𝐶) = (B / 𝐶) ↔ A = B)) | ||||||||||||||
25-Feb-2020 | divcanap4 7418 | A cancellation law for division. (Contributed by Jim Kingdon, 25-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ B # 0) → ((A · B) / B) = A) | ||||||||||||||
25-Feb-2020 | divcanap3 7417 | A cancellation law for division. (Contributed by Jim Kingdon, 25-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ B # 0) → ((B · A) / B) = A) | ||||||||||||||
25-Feb-2020 | divdirap 7416 | Distribution of division over addition. (Contributed by Jim Kingdon, 25-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((A + B) / 𝐶) = ((A / 𝐶) + (B / 𝐶))) | ||||||||||||||
25-Feb-2020 | div12ap 7415 | A commutative/associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → (A · (B / 𝐶)) = (B · (A / 𝐶))) | ||||||||||||||
25-Feb-2020 | div13ap 7414 | A commutative/associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ (B ∈ ℂ ∧ B # 0) ∧ 𝐶 ∈ ℂ) → ((A / B) · 𝐶) = ((𝐶 / B) · A)) | ||||||||||||||
25-Feb-2020 | div32ap 7413 | A commutative/associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ (B ∈ ℂ ∧ B # 0) ∧ 𝐶 ∈ ℂ) → ((A / B) · 𝐶) = (A · (𝐶 / B))) | ||||||||||||||
25-Feb-2020 | div23ap 7412 | A commutative/associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((A · B) / 𝐶) = ((A / 𝐶) · B)) | ||||||||||||||
25-Feb-2020 | divassap 7411 | An associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((A · B) / 𝐶) = (A · (B / 𝐶))) | ||||||||||||||
25-Feb-2020 | divrecap2 7410 | Relationship between division and reciprocal. (Contributed by Jim Kingdon, 25-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ B # 0) → (A / B) = ((1 / B) · A)) | ||||||||||||||
24-Feb-2020 | conventions 9165 |
Unless there is a reason to diverge, we follow the conventions of
the Metamath Proof Explorer (aka "set.mm"). This list of
conventions is intended to be read in conjunction with the
corresponding conventions in the Metamath Proof Explorer, and
only the differences are described below.
Label naming conventions Here are a few of the label naming conventions:
The following table shows some commonly-used abbreviations in labels which are not found in the Metamath Proof Explorer, in alphabetical order. For each abbreviation we provide a mnenomic to help you remember it, the source theorem/assumption defining it, an expression showing what it looks like, whether or not it is a "syntax fragment" (an abbreviation that indicates a particular kind of syntax), and hyperlinks to label examples that use the abbreviation. The abbreviation is bolded if there is a df-NAME definition but the label fragment is not NAME.
(Contributed by Jim Kingdon, 24-Feb-2020.) | ||||||||||||
⊢ φ ⇒ ⊢ φ | ||||||||||||||
24-Feb-2020 | divrecap 7409 | Relationship between division and reciprocal. (Contributed by Jim Kingdon, 24-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ B # 0) → (A / B) = (A · (1 / B))) | ||||||||||||||
24-Feb-2020 | recidap2 7408 | Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 24-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ A # 0) → ((1 / A) · A) = 1) | ||||||||||||||
24-Feb-2020 | recidap 7407 | Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 24-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ A # 0) → (A · (1 / A)) = 1) | ||||||||||||||
24-Feb-2020 | recap0 7406 | The reciprocal of a number apart from zero is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ A # 0) → (1 / A) # 0) | ||||||||||||||
24-Feb-2020 | mulap0bbd 7383 | A factor of a complex number apart from zero is apart from zero. Partial converse of mulap0d 7381 and consequence of mulap0bd 7380. (Contributed by Jim Kingdon, 24-Feb-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → (A · B) # 0) ⇒ ⊢ (φ → B # 0) | ||||||||||||||
24-Feb-2020 | mulap0bad 7382 | A factor of a complex number apart from zero is apart from zero. Partial converse of mulap0d 7381 and consequence of mulap0bd 7380. (Contributed by Jim Kingdon, 24-Feb-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → (A · B) # 0) ⇒ ⊢ (φ → A # 0) | ||||||||||||||
24-Feb-2020 | mulap0bd 7380 | The product of two numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) ⇒ ⊢ (φ → ((A # 0 ∧ B # 0) ↔ (A · B) # 0)) | ||||||||||||||
24-Feb-2020 | mulap0b 7378 | The product of two numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ B ∈ ℂ) → ((A # 0 ∧ B # 0) ↔ (A · B) # 0)) | ||||||||||||||
24-Feb-2020 | mulap0r 7359 | A product apart from zero. Lemma 2.13 of [Geuvers], p. 6. (Contributed by Jim Kingdon, 24-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ (A · B) # 0) → (A # 0 ∧ B # 0)) | ||||||||||||||
24-Feb-2020 | 1ap0 7334 | One is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.) | ||||||||||||
⊢ 1 # 0 | ||||||||||||||
23-Feb-2020 | mulap0d 7381 | The product of two numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 23-Feb-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → A # 0) & ⊢ (φ → B # 0) ⇒ ⊢ (φ → (A · B) # 0) | ||||||||||||||
23-Feb-2020 | mulap0i 7379 | The product of two numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 23-Feb-2020.) | ||||||||||||
⊢ A ∈ ℂ & ⊢ B ∈ ℂ & ⊢ A # 0 & ⊢ B # 0 ⇒ ⊢ (A · B) # 0 | ||||||||||||||
23-Feb-2020 | mulext 7358 | Strong extensionality for multiplication. Given excluded middle, apartness would be equivalent to negated equality and this would follow readily (for all operations) from oveq12 5464. For us, it is proved a different way. (Contributed by Jim Kingdon, 23-Feb-2020.) | ||||||||||||
⊢ (((A ∈ ℂ ∧ B ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((A · B) # (𝐶 · 𝐷) → (A # 𝐶 ∨ B # 𝐷))) | ||||||||||||||
23-Feb-2020 | mulreim 7348 | Complex multiplication in terms of real and imaginary parts. (Contributed by Jim Kingdon, 23-Feb-2020.) | ||||||||||||
⊢ (((A ∈ ℝ ∧ B ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((A + (i · B)) · (𝐶 + (i · 𝐷))) = (((A · 𝐶) + -(B · 𝐷)) + (i · ((𝐶 · B) + (𝐷 · A))))) | ||||||||||||||
22-Feb-2020 | divap0 7405 | The ratio of numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 22-Feb-2020.) | ||||||||||||
⊢ (((A ∈ ℂ ∧ A # 0) ∧ (B ∈ ℂ ∧ B # 0)) → (A / B) # 0) | ||||||||||||||
22-Feb-2020 | divap0b 7404 | The ratio of numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 22-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ B # 0) → (A # 0 ↔ (A / B) # 0)) | ||||||||||||||
22-Feb-2020 | diveqap0 7403 | A ratio is zero iff the numerator is zero. (Contributed by Jim Kingdon, 22-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ B # 0) → ((A / B) = 0 ↔ A = 0)) | ||||||||||||||
22-Feb-2020 | divcanap1 7402 | A cancellation law for division. (Contributed by Jim Kingdon, 22-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ B # 0) → ((A / B) · B) = A) | ||||||||||||||
22-Feb-2020 | divcanap2 7401 | A cancellation law for division. (Contributed by Jim Kingdon, 22-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ B # 0) → (B · (A / B)) = A) | ||||||||||||||
22-Feb-2020 | recclap 7400 | Closure law for reciprocal. (Contributed by Jim Kingdon, 22-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ A # 0) → (1 / A) ∈ ℂ) | ||||||||||||||
22-Feb-2020 | divclap 7399 | Closure law for division. (Contributed by Jim Kingdon, 22-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ B # 0) → (A / B) ∈ ℂ) | ||||||||||||||
22-Feb-2020 | divmulap3 7398 | Relationship between division and multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((A / 𝐶) = B ↔ A = (B · 𝐶))) | ||||||||||||||
22-Feb-2020 | divmulap2 7397 | Relationship between division and multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((A / 𝐶) = B ↔ A = (𝐶 · B))) | ||||||||||||||
22-Feb-2020 | divmulap 7396 | Relationship between division and multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((A / 𝐶) = B ↔ (𝐶 · B) = A)) | ||||||||||||||
22-Feb-2020 | mulap0 7377 | The product of two numbers apart from zero is apart from zero. Lemma 2.15 of [Geuvers], p. 6. (Contributed by Jim Kingdon, 22-Feb-2020.) | ||||||||||||
⊢ (((A ∈ ℂ ∧ A # 0) ∧ (B ∈ ℂ ∧ B # 0)) → (A · B) # 0) | ||||||||||||||
22-Feb-2020 | mulext2 7357 | Right extensionality for complex multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐶 · A) # (𝐶 · B) → A # B)) | ||||||||||||||
22-Feb-2020 | mulext1 7356 | Left extensionality for complex multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((A · 𝐶) # (B · 𝐶) → A # B)) | ||||||||||||||
22-Feb-2020 | remulext2 7344 | Right extensionality for real multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐶 · A) # (𝐶 · B) → A # B)) | ||||||||||||||
21-Feb-2020 | divvalap 7395 | Value of division: the (unique) element x such that (B · x) = A. This is meaningful only when B is apart from zero. (Contributed by Jim Kingdon, 21-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ B # 0) → (A / B) = (℩x ∈ ℂ (B · x) = A)) | ||||||||||||||
21-Feb-2020 | receuap 7392 | Existential uniqueness of reciprocals. (Contributed by Jim Kingdon, 21-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ B # 0) → ∃!x ∈ ℂ (B · x) = A) | ||||||||||||||
21-Feb-2020 | mulcanapi 7390 | Cancellation law for multiplication. (Contributed by Jim Kingdon, 21-Feb-2020.) | ||||||||||||
⊢ A ∈ ℂ & ⊢ B ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐶 # 0 ⇒ ⊢ ((𝐶 · A) = (𝐶 · B) ↔ A = B) | ||||||||||||||
21-Feb-2020 | mulcanap2 7389 | Cancellation law for multiplication. (Contributed by Jim Kingdon, 21-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((A · 𝐶) = (B · 𝐶) ↔ A = B)) | ||||||||||||||
21-Feb-2020 | mulcanap 7388 | Cancellation law for multiplication (full theorem form). (Contributed by Jim Kingdon, 21-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐶 · A) = (𝐶 · B) ↔ A = B)) | ||||||||||||||
21-Feb-2020 | mulcanap2ad 7387 | Cancellation of a nonzero factor on the right in an equation. One-way deduction form of mulcanap2d 7385. (Contributed by Jim Kingdon, 21-Feb-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → 𝐶 ∈ ℂ) & ⊢ (φ → 𝐶 # 0) & ⊢ (φ → (A · 𝐶) = (B · 𝐶)) ⇒ ⊢ (φ → A = B) | ||||||||||||||
21-Feb-2020 | mulcanapad 7386 | Cancellation of a nonzero factor on the left in an equation. One-way deduction form of mulcanapd 7384. (Contributed by Jim Kingdon, 21-Feb-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → 𝐶 ∈ ℂ) & ⊢ (φ → 𝐶 # 0) & ⊢ (φ → (𝐶 · A) = (𝐶 · B)) ⇒ ⊢ (φ → A = B) | ||||||||||||||
21-Feb-2020 | mulcanap2d 7385 | Cancellation law for multiplication. (Contributed by Jim Kingdon, 21-Feb-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → 𝐶 ∈ ℂ) & ⊢ (φ → 𝐶 # 0) ⇒ ⊢ (φ → ((A · 𝐶) = (B · 𝐶) ↔ A = B)) | ||||||||||||||
21-Feb-2020 | mulcanapd 7384 | Cancellation law for multiplication. (Contributed by Jim Kingdon, 21-Feb-2020.) | ||||||||||||
⊢ (φ → A ∈ ℂ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → 𝐶 ∈ ℂ) & ⊢ (φ → 𝐶 # 0) ⇒ ⊢ (φ → ((𝐶 · A) = (𝐶 · B) ↔ A = B)) | ||||||||||||||
21-Feb-2020 | apne 7367 | Apartness implies negated equality. We cannot in general prove the converse, which is the whole point of having separate notations for apartness and negated equality. (Contributed by Jim Kingdon, 21-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ B ∈ ℂ) → (A # B → A ≠ B)) | ||||||||||||||
21-Feb-2020 | apti 7366 | Complex apartness is tight. (Contributed by Jim Kingdon, 21-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ B ∈ ℂ) → (A = B ↔ ¬ A # B)) | ||||||||||||||
20-Feb-2020 | recexap 7376 | Existence of reciprocal of nonzero complex number. (Contributed by Jim Kingdon, 20-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ A # 0) → ∃x ∈ ℂ (A · x) = 1) | ||||||||||||||
20-Feb-2020 | recexaplem2 7375 | Lemma for recexap 7376. (Contributed by Jim Kingdon, 20-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ (A + (i · B)) # 0) → ((A · A) + (B · B)) # 0) | ||||||||||||||
19-Feb-2020 | remulext1 7343 | Left extensionality for multiplication. (Contributed by Jim Kingdon, 19-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((A · 𝐶) # (B · 𝐶) → A # B)) | ||||||||||||||
18-Feb-2020 | ax-pre-mulext 6761 |
Strong extensionality of multiplication (expressed in terms of <_{ℝ}).
Axiom for real and complex numbers, justified by theorem axpre-mulext 6732
(Contributed by Jim Kingdon, 18-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((A · 𝐶) <_{ℝ} (B · 𝐶) → (A <_{ℝ} B ∨ B <_{ℝ} A))) | ||||||||||||||
18-Feb-2020 | axpre-mulext 6732 |
Strong extensionality of multiplication (expressed in terms of
<_{ℝ}). Axiom for real and
complex numbers, derived from set
theory. This construction-dependent theorem should not be referenced
directly; instead, use ax-pre-mulext 6761.
(Contributed by Jim Kingdon, 18-Feb-2020.) (New usage is discouraged.) | ||||||||||||
⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((A · 𝐶) <_{ℝ} (B · 𝐶) → (A <_{ℝ} B ∨ B <_{ℝ} A))) | ||||||||||||||
18-Feb-2020 | mulextsr1 6667 | Strong extensionality of multiplication of signed reals. (Contributed by Jim Kingdon, 18-Feb-2020.) | ||||||||||||
⊢ ((A ∈ R ∧ B ∈ R ∧ 𝐶 ∈ R) → ((A ·_{R} 𝐶) <_{R} (B ·_{R} 𝐶) → (A <_{R} B ∨ B <_{R} A))) | ||||||||||||||
18-Feb-2020 | ltmprr 6612 | Ordering property of multiplication. (Contributed by Jim Kingdon, 18-Feb-2020.) | ||||||||||||
⊢ ((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) → ((𝐶 ·_{P} A)<_{P} (𝐶 ·_{P} B) → A<_{P} B)) | ||||||||||||||
17-Feb-2020 | mulextsr1lem 6666 | Lemma for mulextsr1 6667. (Contributed by Jim Kingdon, 17-Feb-2020.) | ||||||||||||
⊢ (((𝑋 ∈ P ∧ 𝑌 ∈ P) ∧ (𝑍 ∈ P ∧ 𝑊 ∈ P) ∧ (𝑈 ∈ P ∧ 𝑉 ∈ P)) → ((((𝑋 ·_{P} 𝑈) +_{P} (𝑌 ·_{P} 𝑉)) +_{P} ((𝑍 ·_{P} 𝑉) +_{P} (𝑊 ·_{P} 𝑈)))<_{P} (((𝑋 ·_{P} 𝑉) +_{P} (𝑌 ·_{P} 𝑈)) +_{P} ((𝑍 ·_{P} 𝑈) +_{P} (𝑊 ·_{P} 𝑉))) → ((𝑋 +_{P} 𝑊)<_{P} (𝑌 +_{P} 𝑍) ∨ (𝑍 +_{P} 𝑌)<_{P} (𝑊 +_{P} 𝑋)))) | ||||||||||||||
17-Feb-2020 | addextpr 6591 | Strong extensionality of addition (ordering version). This is similar to addext 7354 but for positive reals and based on less-than rather than apartness. (Contributed by Jim Kingdon, 17-Feb-2020.) | ||||||||||||
⊢ (((A ∈ P ∧ B ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → ((A +_{P} B)<_{P} (𝐶 +_{P} 𝐷) → (A<_{P} 𝐶 ∨ B<_{P} 𝐷))) | ||||||||||||||
16-Feb-2020 | apadd2 7353 | Addition respects apartness. (Contributed by Jim Kingdon, 16-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ 𝐶 ∈ ℂ) → (A # B ↔ (𝐶 + A) # (𝐶 + B))) | ||||||||||||||
16-Feb-2020 | apcotr 7351 | Apartness is cotransitive. (Contributed by Jim Kingdon, 16-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ 𝐶 ∈ ℂ) → (A # B → (A # 𝐶 ∨ B # 𝐶))) | ||||||||||||||
16-Feb-2020 | apsym 7350 | Apartness is symmetric. This theorem for real numbers is part of Definition 11.2.7(v) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ B ∈ ℂ) → (A # B ↔ B # A)) | ||||||||||||||
16-Feb-2020 | apirr 7349 | Apartness is irreflexive. (Contributed by Jim Kingdon, 16-Feb-2020.) | ||||||||||||
⊢ (A ∈ ℂ → ¬ A # A) | ||||||||||||||
16-Feb-2020 | reapcotr 7342 | Real apartness is cotransitive. Part of Definition 11.2.7(v) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ 𝐶 ∈ ℝ) → (A # B → (A # 𝐶 ∨ B # 𝐶))) | ||||||||||||||
15-Feb-2020 | addext 7354 | Strong extensionality for addition. Given excluded middle, apartness would be equivalent to negated equality and this would follow readily (for all operations) from oveq12 5464. For us, it is proved a different way. (Contributed by Jim Kingdon, 15-Feb-2020.) | ||||||||||||
⊢ (((A ∈ ℂ ∧ B ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((A + B) # (𝐶 + 𝐷) → (A # 𝐶 ∨ B # 𝐷))) | ||||||||||||||
14-Feb-2020 | apneg 7355 | Negation respects apartness. (Contributed by Jim Kingdon, 14-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ B ∈ ℂ) → (A # B ↔ -A # -B)) | ||||||||||||||
13-Feb-2020 | apadd1 7352 | Addition respects apartness. Analogue of addcan 6948 for apartness. (Contributed by Jim Kingdon, 13-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ 𝐶 ∈ ℂ) → (A # B ↔ (A + 𝐶) # (B + 𝐶))) | ||||||||||||||
13-Feb-2020 | reapneg 7341 | Real negation respects apartness. (Contributed by Jim Kingdon, 13-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℝ ∧ B ∈ ℝ) → (A # B ↔ -A # -B)) | ||||||||||||||
13-Feb-2020 | reapadd1 7340 | Real addition respects apartness. (Contributed by Jim Kingdon, 13-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ 𝐶 ∈ ℝ) → (A # B ↔ (A + 𝐶) # (B + 𝐶))) | ||||||||||||||
12-Feb-2020 | apreim 7347 | Complex apartness in terms of real and imaginary parts. (Contributed by Jim Kingdon, 12-Feb-2020.) | ||||||||||||
⊢ (((A ∈ ℝ ∧ B ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((A + (i · B)) # (𝐶 + (i · 𝐷)) ↔ (A # 𝐶 ∨ B # 𝐷))) | ||||||||||||||
8-Feb-2020 | reapmul1 7339 | Multiplication of both sides of real apartness by a real number apart from zero. Special case of apmul1 7506. (Contributed by Jim Kingdon, 8-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 𝐶 # 0)) → (A # B ↔ (A · 𝐶) # (B · 𝐶))) | ||||||||||||||
8-Feb-2020 | reapmul1lem 7338 | Lemma for reapmul1 7339. (Contributed by Jim Kingdon, 8-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (A # B ↔ (A · 𝐶) # (B · 𝐶))) | ||||||||||||||
7-Feb-2020 | apsqgt0 7345 | The square of a real number apart from zero is positive. (Contributed by Jim Kingdon, 7-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℝ ∧ A # 0) → 0 < (A · A)) | ||||||||||||||
6-Feb-2020 | recexgt0 7324 | Existence of reciprocal of positive real number. (Contributed by Jim Kingdon, 6-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℝ ∧ 0 < A) → ∃x ∈ ℝ (0 < x ∧ (A · x) = 1)) | ||||||||||||||
6-Feb-2020 | ax-precex 6753 | Existence of reciprocal of positive real number. Axiom for real and complex numbers, justified by theorem axprecex 6724. (Contributed by Jim Kingdon, 6-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℝ ∧ 0 <_{ℝ} A) → ∃x ∈ ℝ (0 <_{ℝ} x ∧ (A · x) = 1)) | ||||||||||||||
6-Feb-2020 | axprecex 6724 |
Existence of positive reciprocal of positive real number. Axiom for
real and complex numbers, derived from set theory. This
construction-dependent theorem should not be referenced directly;
instead, use ax-precex 6753.
In treatments which assume excluded middle, the 0 <_{ℝ} A condition is generally replaced by A ≠ 0, and it may not be necessary to state that the reciproacal is positive. (Contributed by Jim Kingdon, 6-Feb-2020.) (New usage is discouraged.) | ||||||||||||
⊢ ((A ∈ ℝ ∧ 0 <_{ℝ} A) → ∃x ∈ ℝ (0 <_{ℝ} x ∧ (A · x) = 1)) | ||||||||||||||
6-Feb-2020 | recexgt0sr 6661 | The reciprocal of a positive signed real exists and is positive. (Contributed by Jim Kingdon, 6-Feb-2020.) | ||||||||||||
⊢ (0_{R} <_{R} A → ∃x ∈ R (0_{R} <_{R} x ∧ (A ·_{R} x) = 1_{R})) | ||||||||||||||
1-Feb-2020 | reaplt 7332 | Real apartness in terms of less than. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 1-Feb-2020.) | ||||||||||||
⊢ ((A ∈ ℝ ∧ B ∈ ℝ) → (A # B ↔ (A < B ∨ B < A))) | ||||||||||||||
31-Jan-2020 | apreap 7331 | Complex apartness and real apartness agree on the real numbers. (Contributed by Jim Kingdon, 31-Jan-2020.) | ||||||||||||
⊢ ((A ∈ ℝ ∧ B ∈ ℝ) → (A # B ↔ A #_{ℝ} B)) | ||||||||||||||
30-Jan-2020 | rereim 7330 | Decomposition of a real number into real part (itself) and imaginary part (zero). (Contributed by Jim Kingdon, 30-Jan-2020.) | ||||||||||||
⊢ (((A ∈ ℝ ∧ B ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ A = (B + (i · 𝐶)))) → (B = A ∧ 𝐶 = 0)) | ||||||||||||||
30-Jan-2020 | reapti 7323 | Real apartness is tight. Beyond the development of apartness itself, proofs should use apti 7366. (Contributed by Jim Kingdon, 30-Jan-2020.) (New usage is discouraged.) | ||||||||||||
⊢ ((A ∈ ℝ ∧ B ∈ ℝ) → (A = B ↔ ¬ A #_{ℝ} B)) | ||||||||||||||
29-Jan-2020 | recexre 7322 | Existence of reciprocal of real number. (Contributed by Jim Kingdon, 29-Jan-2020.) | ||||||||||||
⊢ ((A ∈ ℝ ∧ A #_{ℝ} 0) → ∃x ∈ ℝ (A · x) = 1) | ||||||||||||||
29-Jan-2020 | reapval 7320 | Real apartness in terms of classes. Beyond the development of # itself, proofs should use reaplt 7332 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 29-Jan-2020.) | ||||||||||||
⊢ ((A ∈ ℝ ∧ B ∈ ℝ) → (A #_{ℝ} B ↔ (A < B ∨ B < A))) | ||||||||||||||
29-Jan-2020 | axapti 6847 | Apartness of reals is tight. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-apti 6758 with ordering on the extended reals.) (Contributed by Jim Kingdon, 29-Jan-2020.) | ||||||||||||
⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ ¬ (A < B ∨ B < A)) → A = B) | ||||||||||||||
29-Jan-2020 | ax-pre-apti 6758 | Apartness of reals is tight. Axiom for real and complex numbers, justified by theorem axpre-apti 6729. (Contributed by Jim Kingdon, 29-Jan-2020.) | ||||||||||||
⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ ¬ (A <_{ℝ} B ∨ B <_{ℝ} A)) → A = B) | ||||||||||||||
29-Jan-2020 | axpre-apti 6729 |
Apartness of reals is tight. Axiom for real and complex numbers,
derived from set theory. This construction-dependent theorem should not
be referenced directly; instead, use ax-pre-apti 6758.
(Contributed by Jim Kingdon, 29-Jan-2020.) (New usage is discouraged.) | ||||||||||||
⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ ¬ (A <_{ℝ} B ∨ B <_{ℝ} A)) → A = B) | ||||||||||||||
29-Jan-2020 | aptisr 6665 | Apartness of signed reals is tight. (Contributed by Jim Kingdon, 29-Jan-2020.) | ||||||||||||
⊢ ((A ∈ R ∧ B ∈ R ∧ ¬ (A <_{R} B ∨ B <_{R} A)) → A = B) | ||||||||||||||
28-Jan-2020 | aptipr 6611 | Apartness of positive reals is tight. (Contributed by Jim Kingdon, 28-Jan-2020.) | ||||||||||||
⊢ ((A ∈ P ∧ B ∈ P ∧ ¬ (A<_{P} B ∨ B<_{P} A)) → A = B) | ||||||||||||||
28-Jan-2020 | aptiprlemu 6610 | Lemma for aptipr 6611. (Contributed by Jim Kingdon, 28-Jan-2020.) | ||||||||||||
⊢ ((A ∈ P ∧ B ∈ P ∧ ¬ B<_{P} A) → (2^{nd} ‘B) ⊆ (2^{nd} ‘A)) | ||||||||||||||
28-Jan-2020 | aptiprleml 6609 | Lemma for aptipr 6611. (Contributed by Jim Kingdon, 28-Jan-2020.) | ||||||||||||
⊢ ((A ∈ P ∧ B ∈ P ∧ ¬ B<_{P} A) → (1^{st} ‘A) ⊆ (1^{st} ‘B)) | ||||||||||||||
27-Jan-2020 | bj-dcbi 9313 | Equivalence property for DECID. TODO: solve conflict with dcbi 843; minimize dcbii 746 and dcbid 747 with it, as well as theorems using those. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.) | ||||||||||||
⊢ ((φ ↔ ψ) → (DECID φ ↔ DECID ψ)) | ||||||||||||||
27-Jan-2020 | bj-notbid 9312 | Deduction form of bj-notbi 9310. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.) | ||||||||||||
⊢ (φ → (ψ ↔ χ)) ⇒ ⊢ (φ → (¬ ψ ↔ ¬ χ)) | ||||||||||||||
27-Jan-2020 | bj-notbii 9311 | Inference associated with bj-notbi 9310. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.) | ||||||||||||
⊢ (φ ↔ ψ) ⇒ ⊢ (¬ φ ↔ ¬ ψ) | ||||||||||||||
27-Jan-2020 | bj-notbi 9310 | Equivalence property for negation. TODO: minimize all theorems using notbid 591 and notbii 593. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.) | ||||||||||||
⊢ ((φ ↔ ψ) → (¬ φ ↔ ¬ ψ)) | ||||||||||||||
26-Jan-2020 | df-ap 7326 |
Define complex apartness. Definition 6.1 of [Geuvers], p. 17.
Two numbers are considered apart if it is possible to separate them. One common usage is that we can divide by a number if it is apart from zero (see for example recclap 7400 which says that a number apart from zero has a reciprocal). The defining characteristics of an apartness are irreflexivity (apirr 7349), symmetry (apsym 7350), and cotransitivity (apcotr 7351). Apartness implies negated equality, as seen at apne 7367, and the converse would also follow if we assumed excluded middle. In addition, apartness of complex numbers is tight, which means that two numbers which are not apart are equal (apti 7366). (Contributed by Jim Kingdon, 26-Jan-2020.) | ||||||||||||
⊢ # = {⟨x, y⟩ ∣ ∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ ∃𝑡 ∈ ℝ ∃u ∈ ℝ ((x = (𝑟 + (i · 𝑠)) ∧ y = (𝑡 + (i · u))) ∧ (𝑟 #_{ℝ} 𝑡 ∨ 𝑠 #_{ℝ} u))} | ||||||||||||||
26-Jan-2020 | reapirr 7321 | Real apartness is irreflexive. Part of Definition 11.2.7(v) of [HoTT], p. (varies). Beyond the development of # itself, proofs should use apirr 7349 instead. (Contributed by Jim Kingdon, 26-Jan-2020.) | ||||||||||||
⊢ (A ∈ ℝ → ¬ A #_{ℝ} A) | ||||||||||||||
26-Jan-2020 | df-reap 7319 | Define real apartness. Definition in Section 11.2.1 of [HoTT], p. (varies). Although #_{ℝ} is an apartness relation on the reals (see df-ap 7326 for more discussion of apartness relations), for our purposes it is just a stepping stone to defining # which is an apartness relation on complex numbers. On the reals, #_{ℝ} and # agree (apreap 7331). (Contributed by Jim Kingdon, 26-Jan-2020.) | ||||||||||||
⊢ #_{ℝ} = {⟨x, y⟩ ∣ ((x ∈ ℝ ∧ y ∈ ℝ) ∧ (x < y ∨ y < x))} | ||||||||||||||
26-Jan-2020 | gt0add 7317 | A positive sum must have a positive addend. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 26-Jan-2020.) | ||||||||||||
⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ 0 < (A + B)) → (0 < A ∨ 0 < B)) | ||||||||||||||
17-Jan-2020 | addcom 6907 | Addition commutes. (Contributed by Jim Kingdon, 17-Jan-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ B ∈ ℂ) → (A + B) = (B + A)) | ||||||||||||||
17-Jan-2020 | ax-addcom 6743 | Addition commutes. Axiom for real and complex numbers, justified by theorem axaddcom 6714. Proofs should normally use addcom 6907 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 17-Jan-2020.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ B ∈ ℂ) → (A + B) = (B + A)) | ||||||||||||||
17-Jan-2020 | axaddcom 6714 |
Addition commutes. Axiom for real and complex numbers, derived from set
theory. This construction-dependent theorem should not be referenced
directly, nor should the proven axiom ax-addcom 6743 be used later.
Instead, use addcom 6907.
In the Metamath Proof Explorer this is not a complex number axiom but is instead proved from other axioms. That proof relies on real number trichotomy and it is not known whether it is possible to prove this from the other axioms without it. (Contributed by Jim Kingdon, 17-Jan-2020.) (New usage is discouraged.) | ||||||||||||
⊢ ((A ∈ ℂ ∧ B ∈ ℂ) → (A + B) = (B + A)) | ||||||||||||||
16-Jan-2020 | addid1 6908 | 0 is an additive identity. (Contributed by Jim Kingdon, 16-Jan-2020.) | ||||||||||||
⊢ (A ∈ ℂ → (A + 0) = A) | ||||||||||||||
16-Jan-2020 | ax-0id 6751 |
0 is an identity element for real addition. Axiom for
real and
complex numbers, justified by theorem ax0id 6722.
Proofs should normally use addid1 6908 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 16-Jan-2020.) | ||||||||||||
⊢ (A ∈ ℂ → (A + 0) = A) | ||||||||||||||
16-Jan-2020 | ax0id 6722 |
0 is an identity element for real addition. Axiom for
real and
complex numbers, derived from set theory. This construction-dependent
theorem should not be referenced directly; instead, use ax-0id 6751.
In the Metamath Proof Explorer this is not a complex number axiom but is instead proved from other axioms. That proof relies on excluded middle and it is not known whether it is possible to prove this from the other axioms without excluded middle. (Contributed by Jim Kingdon, 16-Jan-2020.) (New usage is discouraged.) | ||||||||||||
⊢ (A ∈ ℂ → (A + 0) = A) | ||||||||||||||
15-Jan-2020 | axltwlin 6844 | Real number less-than is weakly linear. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltwlin 6756 with ordering on the extended reals. (Contributed by Jim Kingdon, 15-Jan-2020.) | ||||||||||||
⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ 𝐶 ∈ ℝ) → (A < B → (A < 𝐶 ∨ 𝐶 < B))) | ||||||||||||||
15-Jan-2020 | axltirr 6843 | Real number less-than is irreflexive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltirr 6755 with ordering on the extended reals. New proofs should use ltnr 6852 instead for naming consistency. (New usage is discouraged.) (Contributed by Jim Kingdon, 15-Jan-2020.) | ||||||||||||
⊢ (A ∈ ℝ → ¬ A < A) | ||||||||||||||
14-Jan-2020 | 2pwuninelg 5839 | The power set of the power set of the union of a set does not belong to the set. This theorem provides a way of constructing a new set that doesn't belong to a given set. (Contributed by Jim Kingdon, 14-Jan-2020.) | ||||||||||||
⊢ (A ∈ 𝑉 → ¬ 𝒫 𝒫 ∪ A ∈ A) | ||||||||||||||
13-Jan-2020 | 1re 6784 | 1 is a real number. (Contributed by Jim Kingdon, 13-Jan-2020.) | ||||||||||||
⊢ 1 ∈ ℝ | ||||||||||||||
13-Jan-2020 | ax-1re 6737 | 1 is a real number. Axiom for real and complex numbers, justified by theorem ax1re 6708. Proofs should use 1re 6784 instead. (Contributed by Jim Kingdon, 13-Jan-2020.) (New usage is discouraged.) | ||||||||||||
⊢ 1 ∈ ℝ | ||||||||||||||
13-Jan-2020 | ax1re 6708 |
1 is a real number. Axiom for real and complex numbers, derived from set
theory. This construction-dependent theorem should not be referenced
directly; instead, use ax-1re 6737.
In the Metamath Proof Explorer, this is not a complex number axiom but is proved from ax-1cn 6736 and the other axioms. It is not known whether we can do so here, but the Metamath Proof Explorer proof (accessed 13-Jan-2020) uses excluded middle. (Contributed by Jim Kingdon, 13-Jan-2020.) (New usage is discouraged.) | ||||||||||||
⊢ 1 ∈ ℝ | ||||||||||||||
12-Jan-2020 | ax-pre-ltwlin 6756 | Real number less-than is weakly linear. Axiom for real and complex numbers, justified by theorem axpre-ltwlin 6727. (Contributed by Jim Kingdon, 12-Jan-2020.) | ||||||||||||
⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ 𝐶 ∈ ℝ) → (A <_{ℝ} B → (A <_{ℝ} 𝐶 ∨ 𝐶 <_{ℝ} B))) | ||||||||||||||
12-Jan-2020 | ax-pre-ltirr 6755 | Real number less-than is irreflexive. Axiom for real and complex numbers, justified by theorem ax-pre-ltirr 6755. (Contributed by Jim Kingdon, 12-Jan-2020.) | ||||||||||||
⊢ (A ∈ ℝ → ¬ A <_{ℝ} A) | ||||||||||||||
12-Jan-2020 | ax-0lt1 6749 | 0 is less than 1. Axiom for real and complex numbers, justified by theorem ax0lt1 6720. Proofs should normally use 0lt1 6898 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 12-Jan-2020.) | ||||||||||||
⊢ 0 <_{ℝ} 1 | ||||||||||||||
12-Jan-2020 | axpre-ltwlin 6727 | Real number less-than is weakly linear. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltwlin 6756. (Contributed by Jim Kingdon, 12-Jan-2020.) (New usage is discouraged.) | ||||||||||||
⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ 𝐶 ∈ ℝ) → (A <_{ℝ} B → (A <_{ℝ} 𝐶 ∨ 𝐶 <_{ℝ} B))) | ||||||||||||||
12-Jan-2020 | axpre-ltirr 6726 | Real number less-than is irreflexive. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltirr 6755. (Contributed by Jim Kingdon, 12-Jan-2020.) (New usage is discouraged.) | ||||||||||||
⊢ (A ∈ ℝ → ¬ A <_{ℝ} A) | ||||||||||||||
12-Jan-2020 | ax0lt1 6720 |
0 is less than 1. Axiom for real and complex numbers, derived from set
theory. This construction-dependent theorem should not be referenced
directly; instead, use ax-0lt1 6749.
The version of this axiom in the Metamath Proof Explorer reads 1 ≠ 0; here we change it to 0 <_{ℝ} 1. The proof of 0 <_{ℝ} 1 from 1 ≠ 0 in the Metamath Proof Explorer (accessed 12-Jan-2020) relies on real number trichotomy. (Contributed by Jim Kingdon, 12-Jan-2020.) (New usage is discouraged.) | ||||||||||||
⊢ 0 <_{ℝ} 1 | ||||||||||||||
8-Jan-2020 | ecidg 6106 | A set is equal to its converse epsilon coset. (Note: converse epsilon is not an equivalence relation.) (Contributed by Jim Kingdon, 8-Jan-2020.) | ||||||||||||
⊢ (A ∈ 𝑉 → [A]^{◡} E = A) | ||||||||||||||
5-Jan-2020 | elfzom1elp1fzo 8788 | Membership of an integer incremented by one in a half-open range of nonnegative integers. (Contributed by Alexander van der Vekens, 24-Jun-2018.) (Proof shortened by AV, 5-Jan-2020.) | ||||||||||||
⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ (0..^(𝑁 − 1))) → (𝐼 + 1) ∈ (0..^𝑁)) | ||||||||||||||
5-Jan-2020 | 1idsr 6656 | 1 is an identity element for multiplication. (Contributed by Jim Kingdon, 5-Jan-2020.) | ||||||||||||
⊢ (A ∈ R → (A ·_{R} 1_{R}) = A) | ||||||||||||||
4-Jan-2020 | nn0ge2m1nn 7978 | If a nonnegative integer is greater than or equal to two, the integer decreased by 1 is a positive integer. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (Revised by AV, 4-Jan-2020.) | ||||||||||||
⊢ ((𝑁 ∈ ℕ_{0} ∧ 2 ≤ 𝑁) → (𝑁 − 1) ∈ ℕ) | ||||||||||||||
4-Jan-2020 | distrsrg 6647 | Multiplication of signed reals is distributive. (Contributed by Jim Kingdon, 4-Jan-2020.) | ||||||||||||
⊢ ((A ∈ R ∧ B ∈ R ∧ 𝐶 ∈ R) → (A ·_{R} (B +_{R} 𝐶)) = ((A ·_{R} B) +_{R} (A ·_{R} 𝐶))) | ||||||||||||||
3-Jan-2020 | mulasssrg 6646 | Multiplication of signed reals is associative. (Contributed by Jim Kingdon, 3-Jan-2020.) | ||||||||||||
⊢ ((A ∈ R ∧ B ∈ R ∧ 𝐶 ∈ R) → ((A ·_{R} B) ·_{R} 𝐶) = (A ·_{R} (B ·_{R} 𝐶))) | ||||||||||||||
3-Jan-2020 | mulcomsrg 6645 | Multiplication of signed reals is commutative. (Contributed by Jim Kingdon, 3-Jan-2020.) | ||||||||||||
⊢ ((A ∈ R ∧ B ∈ R) → (A ·_{R} B) = (B ·_{R} A)) | ||||||||||||||
3-Jan-2020 | addasssrg 6644 | Addition of signed reals is associative. (Contributed by Jim Kingdon, 3-Jan-2020.) | ||||||||||||
⊢ ((A ∈ R ∧ B ∈ R ∧ 𝐶 ∈ R) → ((A +_{R} B) +_{R} 𝐶) = (A +_{R} (B +_{R} 𝐶))) | ||||||||||||||
3-Jan-2020 | addcomsrg 6643 | Addition of signed reals is commutative. (Contributed by Jim Kingdon, 3-Jan-2020.) | ||||||||||||
⊢ ((A ∈ R ∧ B ∈ R) → (A +_{R} B) = (B +_{R} A)) | ||||||||||||||
3-Jan-2020 | caovlem2d 5635 | Rearrangement of expression involving multiplication (𝐺) and addition (𝐹). (Contributed by Jim Kingdon, 3-Jan-2020.) | ||||||||||||
⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆)) → (x𝐺y) = (y𝐺x)) & ⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆 ∧ z ∈ 𝑆)) → ((x𝐹y)𝐺z) = ((x𝐺z)𝐹(y𝐺z))) & ⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆 ∧ z ∈ 𝑆)) → ((x𝐺y)𝐺z) = (x𝐺(y𝐺z))) & ⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆)) → (x𝐺y) ∈ 𝑆) & ⊢ (φ → A ∈ 𝑆) & ⊢ (φ → B ∈ 𝑆) & ⊢ (φ → 𝐶 ∈ 𝑆) & ⊢ (φ → 𝐷 ∈ 𝑆) & ⊢ (φ → 𝐻 ∈ 𝑆) & ⊢ (φ → 𝑅 ∈ 𝑆) & ⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆)) → (x𝐹y) = (y𝐹x)) & ⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆 ∧ z ∈ 𝑆)) → ((x𝐹y)𝐹z) = (x𝐹(y𝐹z))) & ⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆)) → (x𝐹y) ∈ 𝑆) ⇒ ⊢ (φ → ((((A𝐺𝐶)𝐹(B𝐺𝐷))𝐺𝐻)𝐹(((A𝐺𝐷)𝐹(B𝐺𝐶))𝐺𝑅)) = ((A𝐺((𝐶𝐺𝐻)𝐹(𝐷𝐺𝑅)))𝐹(B𝐺((𝐶𝐺𝑅)𝐹(𝐷𝐺𝐻))))) | ||||||||||||||
2-Jan-2020 | bj-d0clsepcl 9314 | Δ_{0}-classical logic and separation implies classical logic. (Contributed by BJ, 2-Jan-2020.) (Proof modification is discouraged.) | ||||||||||||
⊢ DECID φ | ||||||||||||||
2-Jan-2020 | ax-bj-d0cl 9309 | Axiom for Δ_{0}-classical logic. (Contributed by BJ, 2-Jan-2020.) | ||||||||||||
⊢ BOUNDED φ ⇒ ⊢ DECID φ | ||||||||||||||
1-Jan-2020 | mulcmpblnrlemg 6628 | Lemma used in lemma showing compatibility of multiplication. (Contributed by Jim Kingdon, 1-Jan-2020.) | ||||||||||||
⊢ ((((A ∈ P ∧ B ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) ∧ ((𝐹 ∈ P ∧ 𝐺 ∈ P) ∧ (𝑅 ∈ P ∧ 𝑆 ∈ P))) → (((A +_{P} 𝐷) = (B +_{P} 𝐶) ∧ (𝐹 +_{P} 𝑆) = (𝐺 +_{P} 𝑅)) → ((𝐷 ·_{P} 𝐹) +_{P} (((A ·_{P} 𝐹) +_{P} (B ·_{P} 𝐺)) +_{P} ((𝐶 ·_{P} 𝑆) +_{P} (𝐷 ·_{P} 𝑅)))) = ((𝐷 ·_{P} 𝐹) +_{P} (((A ·_{P} 𝐺) +_{P} (B ·_{P} 𝐹)) +_{P} ((𝐶 ·_{P} 𝑅) +_{P} (𝐷 ·_{P} 𝑆)))))) | ||||||||||||||
31-Dec-2019 | eceq1d 6078 | Equality theorem for equivalence class (deduction form). (Contributed by Jim Kingdon, 31-Dec-2019.) | ||||||||||||
⊢ (φ → A = B) ⇒ ⊢ (φ → [A]𝐶 = [B]𝐶) | ||||||||||||||
30-Dec-2019 | mulsrmo 6632 | There is at most one result from multiplying signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.) | ||||||||||||
⊢ ((A ∈ ((P × P) / ~_{R} ) ∧ B ∈ ((P × P) / ~_{R} )) → ∃*z∃w∃v∃u∃𝑡((A = [⟨w, v⟩] ~_{R} ∧ B = [⟨u, 𝑡⟩] ~_{R} ) ∧ z = [⟨((w ·_{P} u) +_{P} (v ·_{P} 𝑡)), ((w ·_{P} 𝑡) +_{P} (v ·_{P} u))⟩] ~_{R} )) | ||||||||||||||
30-Dec-2019 | addsrmo 6631 | There is at most one result from adding signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.) | ||||||||||||
⊢ ((A ∈ ((P × P) / ~_{R} ) ∧ B ∈ ((P × P) / ~_{R} )) → ∃*z∃w∃v∃u∃𝑡((A = [⟨w, v⟩] ~_{R} ∧ B = [⟨u, 𝑡⟩] ~_{R} ) ∧ z = [⟨(w +_{P} u), (v +_{P} 𝑡)⟩] ~_{R} )) | ||||||||||||||
30-Dec-2019 | prsrlem1 6630 | Decomposing signed reals into positive reals. Lemma for addsrpr 6633 and mulsrpr 6634. (Contributed by Jim Kingdon, 30-Dec-2019.) | ||||||||||||
⊢ (((A ∈ ((P × P) / ~_{R} ) ∧ B ∈ ((P × P) / ~_{R} )) ∧ ((A = [⟨w, v⟩] ~_{R} ∧ B = [⟨u, 𝑡⟩] ~_{R} ) ∧ (A = [⟨𝑠, f⟩] ~_{R} ∧ B = [⟨g, ℎ⟩] ~_{R} ))) → ((((w ∈ P ∧ v ∈ P) ∧ (𝑠 ∈ P ∧ f ∈ P)) ∧ ((u ∈ P ∧ 𝑡 ∈ P) ∧ (g ∈ P ∧ ℎ ∈ P))) ∧ ((w +_{P} f) = (v +_{P} 𝑠) ∧ (u +_{P} ℎ) = (𝑡 +_{P} g)))) | ||||||||||||||
29-Dec-2019 | bj-omelon 9347 | The set 𝜔 is an ordinal. Constructive proof of omelon 4274. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.) | ||||||||||||
⊢ 𝜔 ∈ On | ||||||||||||||
29-Dec-2019 | bj-omord 9346 | The set 𝜔 is an ordinal. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.) | ||||||||||||
⊢ Ord 𝜔 | ||||||||||||||
29-Dec-2019 | bj-omtrans2 9345 | The set 𝜔 is transitive. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.) | ||||||||||||
⊢ Tr 𝜔 | ||||||||||||||
29-Dec-2019 | bj-omtrans 9344 |
The set 𝜔 is transitive. A natural number is
included in
𝜔.
The idea is to use bounded induction with the formula x ⊆ 𝜔. This formula, in a logic with terms, is bounded. So in our logic without terms, we need to temporarily replace it with x ⊆ 𝑎 and then deduce the original claim. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.) | ||||||||||||
⊢ (A ∈ 𝜔 → A ⊆ 𝜔) | ||||||||||||||
29-Dec-2019 | ltrnqg 6403 | Ordering property of reciprocal for positive fractions. For a simplified version of the forward implication, see ltrnqi 6404. (Contributed by Jim Kingdon, 29-Dec-2019.) | ||||||||||||
⊢ ((A ∈ Q ∧ B ∈ Q) → (A <_{Q} B ↔ (*_{Q}‘B) <_{Q} (*_{Q}‘A))) | ||||||||||||||
29-Dec-2019 | rec1nq 6379 | Reciprocal of positive fraction one. (Contributed by Jim Kingdon, 29-Dec-2019.) | ||||||||||||
⊢ (*_{Q}‘1_{Q}) = 1_{Q} | ||||||||||||||
28-Dec-2019 | recexprlemupu 6598 | The upper cut of B is upper. Lemma for recexpr 6608. (Contributed by Jim Kingdon, 28-Dec-2019.) | ||||||||||||
⊢ B = ⟨{x ∣ ∃y(x <_{Q} y ∧ (*_{Q}‘y) ∈ (2^{nd} ‘A))}, {x ∣ ∃y(y <_{Q} x ∧ (*_{Q}‘y) ∈ (1^{st} ‘A))}⟩ ⇒ ⊢ ((A ∈ P ∧ 𝑟 ∈ Q) → (∃𝑞 ∈ Q (𝑞 <_{Q} 𝑟 ∧ 𝑞 ∈ (2^{nd} ‘B)) → 𝑟 ∈ (2^{nd} ‘B))) | ||||||||||||||
28-Dec-2019 | recexprlemopu 6597 | The upper cut of B is open. Lemma for recexpr 6608. (Contributed by Jim Kingdon, 28-Dec-2019.) | ||||||||||||
⊢ B = ⟨{x ∣ ∃y(x <_{Q} y ∧ (*_{Q}‘y) ∈ (2^{nd} ‘A))}, {x ∣ ∃y(y <_{Q} x ∧ (*_{Q}‘y) ∈ (1^{st} ‘A))}⟩ ⇒ ⊢ ((A ∈ P ∧ 𝑟 ∈ Q ∧ 𝑟 ∈ (2^{nd} ‘B)) → ∃𝑞 ∈ Q (𝑞 <_{Q} 𝑟 ∧ 𝑞 ∈ (2^{nd} ‘B))) | ||||||||||||||
28-Dec-2019 | recexprlemlol 6596 | The lower cut of B is lower. Lemma for recexpr 6608. (Contributed by Jim Kingdon, 28-Dec-2019.) | ||||||||||||
⊢ B = ⟨{x ∣ ∃y(x <_{Q} y ∧ (*_{Q}‘y) ∈ (2^{nd} ‘A))}, {x ∣ ∃y(y <_{Q} x ∧ (*_{Q}‘y) ∈ (1^{st} ‘A))}⟩ ⇒ ⊢ ((A ∈ P ∧ 𝑞 ∈ Q) → (∃𝑟 ∈ Q (𝑞 <_{Q} 𝑟 ∧ 𝑟 ∈ (1^{st} ‘B)) → 𝑞 ∈ (1^{st} ‘B))) | ||||||||||||||
28-Dec-2019 | recexprlemopl 6595 | The lower cut of B is open. Lemma for recexpr 6608. (Contributed by Jim Kingdon, 28-Dec-2019.) | ||||||||||||
⊢ B = ⟨{x ∣ ∃y(x <_{Q} y ∧ (*_{Q}‘y) ∈ (2^{nd} ‘A))}, {x ∣ ∃y(y <_{Q} x ∧ (*_{Q}‘y) ∈ (1^{st} ‘A))}⟩ ⇒ ⊢ ((A ∈ P ∧ 𝑞 ∈ Q ∧ 𝑞 ∈ (1^{st} ‘B)) → ∃𝑟 ∈ Q (𝑞 <_{Q} 𝑟 ∧ 𝑟 ∈ (1^{st} ‘B))) | ||||||||||||||
28-Dec-2019 | prmuloc2 6546 | Positive reals are multiplicatively located. This is a variation of prmuloc 6545 which only constructs one (named) point and is therefore often easier to work with. It states that given a ratio B, there are elements of the lower and upper cut which have exactly that ratio between them. (Contributed by Jim Kingdon, 28-Dec-2019.) | ||||||||||||
⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 1_{Q} <_{Q} B) → ∃x ∈ 𝐿 (x ·_{Q} B) ∈ 𝑈) | ||||||||||||||
28-Dec-2019 | 1pru 6536 | The upper cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.) | ||||||||||||
⊢ (2^{nd} ‘1_{P}) = {x ∣ 1_{Q} <_{Q} x} | ||||||||||||||
28-Dec-2019 | 1prl 6535 | The lower cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.) | ||||||||||||
⊢ (1^{st} ‘1_{P}) = {x ∣ x <_{Q} 1_{Q}} | ||||||||||||||
27-Dec-2019 | recexprlemex 6607 | B is the reciprocal of A. Lemma for recexpr 6608. (Contributed by Jim Kingdon, 27-Dec-2019.) | ||||||||||||
⊢ B = ⟨{x ∣ ∃y(x <_{Q} y ∧ (*_{Q}‘y) ∈ (2^{nd} ‘A))}, {x ∣ ∃y(y <_{Q} x ∧ (*_{Q}‘y) ∈ (1^{st} ‘A))}⟩ ⇒ ⊢ (A ∈ P → (A ·_{P} B) = 1_{P}) | ||||||||||||||
27-Dec-2019 | recexprlemss1u 6606 | The upper cut of A ·_{P} B is a subset of the upper cut of one. Lemma for recexpr 6608. (Contributed by Jim Kingdon, 27-Dec-2019.) | ||||||||||||
⊢ B = ⟨{x ∣ ∃y(x <_{Q} y ∧ (*_{Q}‘y) ∈ (2^{nd} ‘A))}, {x ∣ ∃y(y <_{Q} x ∧ (*_{Q}‘y) ∈ (1^{st} ‘A))}⟩ ⇒ ⊢ (A ∈ P → (2^{nd} ‘(A ·_{P} B)) ⊆ (2^{nd} ‘1_{P})) | ||||||||||||||
27-Dec-2019 | recexprlemss1l 6605 | The lower cut of A ·_{P} B is a subset of the lower cut of one. Lemma for recexpr 6608. (Contributed by Jim Kingdon, 27-Dec-2019.) | ||||||||||||
⊢ B = ⟨{x ∣ ∃y(x <_{Q} y ∧ (*_{Q}‘y) ∈ (2^{nd} ‘A))}, {x ∣ ∃y(y <_{Q} x ∧ (*_{Q}‘y) ∈ (1^{st} ‘A))}⟩ ⇒ ⊢ (A ∈ P → (1^{st} ‘(A ·_{P} B)) ⊆ (1^{st} ‘1_{P})) | ||||||||||||||
27-Dec-2019 | recexprlem1ssu 6604 | The upper cut of one is a subset of the upper cut of A ·_{P} B. Lemma for recexpr 6608. (Contributed by Jim Kingdon, 27-Dec-2019.) | ||||||||||||
⊢ B = ⟨{x ∣ ∃y(x <_{Q} y ∧ (*_{Q}‘y) ∈ (2^{nd} ‘A))}, {x ∣ ∃y(y <_{Q} x ∧ (*_{Q}‘y) ∈ (1^{st} ‘A))}⟩ ⇒ ⊢ (A ∈ P → (2^{nd} ‘1_{P}) ⊆ (2^{nd} ‘(A ·_{P} B))) | ||||||||||||||
27-Dec-2019 | recexprlem1ssl 6603 | The lower cut of one is a subset of the lower cut of A ·_{P} B. Lemma for recexpr 6608. (Contributed by Jim Kingdon, 27-Dec-2019.) | ||||||||||||
⊢ B = ⟨{x ∣ ∃y(x <_{Q} y ∧ (*_{Q}‘y) ∈ (2^{nd} ‘A))}, {x ∣ ∃y(y <_{Q} x ∧ (*_{Q}‘y) ∈ (1^{st} ‘A))}⟩ ⇒ ⊢ (A ∈ P → (1^{st} ‘1_{P}) ⊆ (1^{st} ‘(A ·_{P} B))) | ||||||||||||||
27-Dec-2019 | recexprlempr 6602 | B is a positive real. Lemma for recexpr 6608. (Contributed by Jim Kingdon, 27-Dec-2019.) | ||||||||||||
⊢ B = ⟨{x ∣ ∃y(x <_{Q} y ∧ (*_{Q}‘y) ∈ (2^{nd} ‘A))}, {x ∣ ∃y(y <_{Q} x ∧ (*_{Q}‘y) ∈ (1^{st} ‘A))}⟩ ⇒ ⊢ (A ∈ P → B ∈ P) | ||||||||||||||
27-Dec-2019 | recexprlemloc 6601 | B is located. Lemma for recexpr 6608. (Contributed by Jim Kingdon, 27-Dec-2019.) | ||||||||||||
⊢ B = ⟨{x ∣ ∃y(x <_{Q} y ∧ (*_{Q}‘y) ∈ (2^{nd} ‘A))}, {x ∣ ∃y(y <_{Q} x ∧ (*_{Q}‘y) ∈ (1^{st} ‘A))}⟩ ⇒ ⊢ (A ∈ P → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <_{Q} 𝑟 → (𝑞 ∈ (1^{st} ‘B) ∨ 𝑟 ∈ (2^{nd} ‘B)))) | ||||||||||||||
27-Dec-2019 | recexprlemdisj 6600 | B is disjoint. Lemma for recexpr 6608. (Contributed by Jim Kingdon, 27-Dec-2019.) | ||||||||||||
⊢ B = ⟨{x ∣ ∃y(x <_{Q} y ∧ (*_{Q}‘y) ∈ (2^{nd} ‘A))}, {x ∣ ∃y(y <_{Q} x ∧ (*_{Q}‘y) ∈ (1^{st} ‘A))}⟩ ⇒ ⊢ (A ∈ P → ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1^{st} ‘B) ∧ 𝑞 ∈ (2^{nd} ‘B))) | ||||||||||||||
27-Dec-2019 | recexprlemrnd 6599 | B is rounded. Lemma for recexpr 6608. (Contributed by Jim Kingdon, 27-Dec-2019.) | ||||||||||||
⊢ B = ⟨{x ∣ ∃y(x <_{Q} y ∧ (*_{Q}‘y) ∈ (2^{nd} ‘A))}, {x ∣ ∃y(y <_{Q} x ∧ (*_{Q}‘y) ∈ (1^{st} ‘A))}⟩ ⇒ ⊢ (A ∈ P → (∀𝑞 ∈ Q (𝑞 ∈ (1^{st} ‘B) ↔ ∃𝑟 ∈ Q (𝑞 <_{Q} 𝑟 ∧ 𝑟 ∈ (1^{st} ‘B))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2^{nd} ‘B) ↔ ∃𝑞 ∈ Q (𝑞 <_{Q} 𝑟 ∧ 𝑞 ∈ (2^{nd} ‘B))))) | ||||||||||||||
27-Dec-2019 | recexprlemm 6594 | B is inhabited. Lemma for recexpr 6608. (Contributed by Jim Kingdon, 27-Dec-2019.) | ||||||||||||
⊢ B = ⟨{x ∣ ∃y(x <_{Q} y ∧ (*_{Q}‘y) ∈ (2^{nd} ‘A))}, {x ∣ ∃y(y <_{Q} x ∧ (*_{Q}‘y) ∈ (1^{st} ‘A))}⟩ ⇒ ⊢ (A ∈ P → (∃𝑞 ∈ Q 𝑞 ∈ (1^{st} ‘B) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2^{nd} ‘B))) | ||||||||||||||
27-Dec-2019 | recexprlemelu 6593 | Membership in the upper cut of B. Lemma for recexpr 6608. (Contributed by Jim Kingdon, 27-Dec-2019.) | ||||||||||||
⊢ B = ⟨{x ∣ ∃y(x <_{Q} y ∧ (*_{Q}‘y) ∈ (2^{nd} ‘A))}, {x ∣ ∃y(y <_{Q} x ∧ (*_{Q}‘y) ∈ (1^{st} ‘A))}⟩ ⇒ ⊢ (𝐶 ∈ (2^{nd} ‘B) ↔ ∃y(y <_{Q} 𝐶 ∧ (*_{Q}‘y) ∈ (1^{st} ‘A))) | ||||||||||||||
27-Dec-2019 | recexprlemell 6592 | Membership in the lower cut of B. Lemma for recexpr 6608. (Contributed by Jim Kingdon, 27-Dec-2019.) | ||||||||||||
⊢ B = ⟨{x ∣ ∃y(x <_{Q} y ∧ (*_{Q}‘y) ∈ (2^{nd} ‘A))}, {x ∣ ∃y(y <_{Q} x ∧ (*_{Q}‘y) ∈ (1^{st} ‘A))}⟩ ⇒ ⊢ (𝐶 ∈ (1^{st} ‘B) ↔ ∃y(𝐶 <_{Q} y ∧ (*_{Q}‘y) ∈ (2^{nd} ‘A))) | ||||||||||||||
26-Dec-2019 | ltaprg 6590 | Ordering property of addition. Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by Jim Kingdon, 26-Dec-2019.) | ||||||||||||
⊢ ((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) → (A<_{P} B ↔ (𝐶 +_{P} A)<_{P} (𝐶 +_{P} B))) | ||||||||||||||
26-Dec-2019 | prarloc2 6486 | A Dedekind cut is arithmetically located. This is a variation of prarloc 6485 which only constructs one (named) point and is therefore often easier to work with. It states that given a tolerance 𝑃, there are elements of the lower and upper cut which are exactly that tolerance from each other. (Contributed by Jim Kingdon, 26-Dec-2019.) | ||||||||||||
⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) → ∃𝑎 ∈ 𝐿 (𝑎 +_{Q} 𝑃) ∈ 𝑈) | ||||||||||||||
25-Dec-2019 | addcanprlemu 6587 | Lemma for addcanprg 6588. (Contributed by Jim Kingdon, 25-Dec-2019.) | ||||||||||||
⊢ (((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) ∧ (A +_{P} B) = (A +_{P} 𝐶)) → (2^{nd} ‘B) ⊆ (2^{nd} ‘𝐶)) | ||||||||||||||
25-Dec-2019 | addcanprleml 6586 | Lemma for addcanprg 6588. (Contributed by Jim Kingdon, 25-Dec-2019.) | ||||||||||||
⊢ (((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) ∧ (A +_{P} B) = (A +_{P} 𝐶)) → (1^{st} ‘B) ⊆ (1^{st} ‘𝐶)) | ||||||||||||||
24-Dec-2019 | addcanprg 6588 | Addition cancellation law for positive reals. Proposition 9-3.5(vi) of [Gleason] p. 123. (Contributed by Jim Kingdon, 24-Dec-2019.) | ||||||||||||
⊢ ((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) → ((A +_{P} B) = (A +_{P} 𝐶) → B = 𝐶)) | ||||||||||||||
23-Dec-2019 | ltprordil 6563 | If a positive real is less than a second positive real, its lower cut is a subset of the second's lower cut. (Contributed by Jim Kingdon, 23-Dec-2019.) | ||||||||||||
⊢ (A<_{P} B → (1^{st} ‘A) ⊆ (1^{st} ‘B)) | ||||||||||||||
22-Dec-2019 | bj-findis 9363 | Principle of induction, using implicit substitutions (the biconditional versions of the hypotheses are implicit substitutions, and we have weakened them to implications). Constructive proof (from CZF). See bj-bdfindis 9335 for a bounded version not requiring ax-setind 4220. See finds 4266 for a proof in IZF. From this version, it is easy to prove of finds 4266, finds2 4267, finds1 4268. (Contributed by BJ, 22-Dec-2019.) (Proof modification is discouraged.) | ||||||||||||
⊢ Ⅎxψ & ⊢ Ⅎxχ & ⊢ Ⅎxθ & ⊢ (x = ∅ → (ψ → φ)) & ⊢ (x = y → (φ → χ)) & ⊢ (x = suc y → (θ → φ)) ⇒ ⊢ ((ψ ∧ ∀y ∈ 𝜔 (χ → θ)) → ∀x ∈ 𝜔 φ) | ||||||||||||||
21-Dec-2019 | ltexprlemupu 6576 | The upper cut of our constructed difference is upper. Lemma for ltexpri 6585. (Contributed by Jim Kingdon, 21-Dec-2019.) | ||||||||||||
⊢ 𝐶 = ⟨{x ∈ Q ∣ ∃y(y ∈ (2^{nd} ‘A) ∧ (y +_{Q} x) ∈ (1^{st} ‘B))}, {x ∈ Q ∣ ∃y(y ∈ (1^{st} ‘A) ∧ (y +_{Q} x) ∈ (2^{nd} ‘B))}⟩ ⇒ ⊢ ((A<_{P} B ∧ 𝑟 ∈ Q) → (∃𝑞 ∈ Q (𝑞 <_{Q} 𝑟 ∧ 𝑞 ∈ (2^{nd} ‘𝐶)) → 𝑟 ∈ (2^{nd} ‘𝐶))) | ||||||||||||||
21-Dec-2019 | ltexprlemopu 6575 | The upper cut of our constructed difference is open. Lemma for ltexpri 6585. (Contributed by Jim Kingdon, 21-Dec-2019.) | ||||||||||||
⊢ 𝐶 = ⟨{x ∈ Q ∣ ∃y(y ∈ (2^{nd} ‘A) ∧ (y +_{Q} x) ∈ (1^{st} ‘B))}, {x ∈ Q ∣ ∃y(y ∈ (1^{st} ‘A) ∧ (y +_{Q} x) ∈ (2^{nd} ‘B))}⟩ ⇒ ⊢ ((A<_{P} B ∧ 𝑟 ∈ Q ∧ 𝑟 ∈ (2^{nd} ‘𝐶)) → ∃𝑞 ∈ Q (𝑞 <_{Q} 𝑟 ∧ 𝑞 ∈ (2^{nd} ‘𝐶))) | ||||||||||||||
21-Dec-2019 | ltexprlemlol 6574 | The lower cut of our constructed difference is lower. Lemma for ltexpri 6585. (Contributed by Jim Kingdon, 21-Dec-2019.) | ||||||||||||
⊢ 𝐶 = ⟨{x ∈ Q ∣ ∃y(y ∈ (2^{nd} ‘A) ∧ (y +_{Q} x) ∈ (1^{st} ‘B))}, {x ∈ Q ∣ ∃y(y ∈ (1^{st} ‘A) ∧ (y +_{Q} x) ∈ (2^{nd} ‘B))}⟩ ⇒ ⊢ ((A<_{P} B ∧ 𝑞 ∈ Q) → (∃𝑟 ∈ Q (𝑞 <_{Q} 𝑟 ∧ 𝑟 ∈ (1^{st} ‘𝐶)) → 𝑞 ∈ (1^{st} ‘𝐶))) | ||||||||||||||
21-Dec-2019 | ltexprlemopl 6573 | The lower cut of our constructed difference is open. Lemma for ltexpri 6585. (Contributed by Jim Kingdon, 21-Dec-2019.) | ||||||||||||
⊢ 𝐶 = ⟨{x ∈ Q ∣ ∃y(y ∈ (2^{nd} ‘A) ∧ (y +_{Q} x) ∈ (1^{st} ‘B))}, {x ∈ Q ∣ ∃y(y ∈ (1^{st} ‘A) ∧ (y +_{Q} x) ∈ (2^{nd} ‘B))}⟩ ⇒ ⊢ ((A<_{P} B ∧ 𝑞 ∈ Q ∧ 𝑞 ∈ (1^{st} ‘𝐶)) → ∃𝑟 ∈ Q (𝑞 <_{Q} 𝑟 ∧ 𝑟 ∈ (1^{st} ‘𝐶))) | ||||||||||||||
21-Dec-2019 | ltexprlemelu 6571 | Element in upper cut of the constructed difference. Lemma for ltexpri 6585. (Contributed by Jim Kingdon, 21-Dec-2019.) | ||||||||||||
⊢ 𝐶 = ⟨{x ∈ Q ∣ ∃y(y ∈ (2^{nd} ‘A) ∧ (y +_{Q} x) ∈ (1^{st} ‘B))}, {x ∈ Q ∣ ∃y(y ∈ (1^{st} ‘A) ∧ (y +_{Q} x) ∈ (2^{nd} ‘B))}⟩ ⇒ ⊢ (𝑟 ∈ (2^{nd} ‘𝐶) ↔ (𝑟 ∈ Q ∧ ∃y(y ∈ (1^{st} ‘A) ∧ (y +_{Q} 𝑟) ∈ (2^{nd} ‘B)))) | ||||||||||||||
21-Dec-2019 | ltexprlemell 6570 | Element in lower cut of the constructed difference. Lemma for ltexpri 6585. (Contributed by Jim Kingdon, 21-Dec-2019.) | ||||||||||||
⊢ 𝐶 = ⟨{x ∈ Q ∣ ∃y(y ∈ (2^{nd} ‘A) ∧ (y +_{Q} x) ∈ (1^{st} ‘B))}, {x ∈ Q ∣ ∃y(y ∈ (1^{st} ‘A) ∧ (y +_{Q} x) ∈ (2^{nd} ‘B))}⟩ ⇒ ⊢ (𝑞 ∈ (1^{st} ‘𝐶) ↔ (𝑞 ∈ Q ∧ ∃y(y ∈ (2^{nd} ‘A) ∧ (y +_{Q} 𝑞) ∈ (1^{st} ‘B)))) | ||||||||||||||
17-Dec-2019 | ltexprlemru 6584 | Lemma for ltexpri 6585. One direction of our result for upper cuts. (Contributed by Jim Kingdon, 17-Dec-2019.) | ||||||||||||
⊢ 𝐶 = ⟨{x ∈ Q ∣ ∃y(y ∈ (2^{nd} ‘A) ∧ (y +_{Q} x) ∈ (1^{st} ‘B))}, {x ∈ Q ∣ ∃y(y ∈ (1^{st} ‘A) ∧ (y +_{Q} x) ∈ (2^{nd} ‘B))}⟩ ⇒ ⊢ (A<_{P} B → (2^{nd} ‘B) ⊆ (2^{nd} ‘(A +_{P} 𝐶))) | ||||||||||||||
17-Dec-2019 | ltexprlemfu 6583 | Lemma for ltexpri 6585. One direction of our result for upper cuts. (Contributed by Jim Kingdon, 17-Dec-2019.) | ||||||||||||
⊢ 𝐶 = ⟨{x ∈ Q ∣ ∃y(y ∈ (2^{nd} ‘A) ∧ (y +_{Q} x) ∈ (1^{st} ‘B))}, {x ∈ Q ∣ ∃y(y ∈ (1^{st} ‘A) ∧ (y +_{Q} x) ∈ (2^{nd} ‘B))}⟩ ⇒ ⊢ (A<_{P} B → (2^{nd} ‘(A +_{P} 𝐶)) ⊆ (2^{nd} ‘B)) | ||||||||||||||
17-Dec-2019 | ltexprlemrl 6582 | Lemma for ltexpri 6585. Reverse directon of our result for lower cuts. (Contributed by Jim Kingdon, 17-Dec-2019.) | ||||||||||||
⊢ 𝐶 = ⟨{x ∈ Q ∣ ∃y(y ∈ (2^{nd} ‘A) ∧ (y +_{Q} x) ∈ (1^{st} ‘B))}, {x ∈ Q ∣ ∃y(y ∈ (1^{st} ‘A) ∧ (y +_{Q} x) ∈ (2^{nd} ‘B))}⟩ ⇒ ⊢ (A<_{P} B → (1^{st} ‘B) ⊆ (1^{st} ‘(A +_{P} 𝐶))) | ||||||||||||||
17-Dec-2019 | ltexprlemfl 6581 | Lemma for ltexpri 6585. One directon of our result for lower cuts. (Contributed by Jim Kingdon, 17-Dec-2019.) | ||||||||||||
⊢ 𝐶 = ⟨{x ∈ Q ∣ ∃y(y ∈ (2^{nd} ‘A) ∧ (y +_{Q} x) ∈ (1^{st} ‘B))}, {x ∈ Q ∣ ∃y(y ∈ (1^{st} ‘A) ∧ (y +_{Q} x) ∈ (2^{nd} ‘B))}⟩ ⇒ ⊢ (A<_{P} B → (1^{st} ‘(A +_{P} 𝐶)) ⊆ (1^{st} ‘B)) | ||||||||||||||
17-Dec-2019 | ltexprlempr 6580 | Our constructed difference is a positive real. Lemma for ltexpri 6585. (Contributed by Jim Kingdon, 17-Dec-2019.) | ||||||||||||
⊢ 𝐶 = ⟨{x ∈ Q ∣ ∃y(y ∈ (2^{nd} ‘A) ∧ (y +_{Q} x) ∈ (1^{st} ‘B))}, {x ∈ Q ∣ ∃y(y ∈ (1^{st} ‘A) ∧ (y +_{Q} x) ∈ (2^{nd} ‘B))}⟩ ⇒ ⊢ (A<_{P} B → 𝐶 ∈ P) | ||||||||||||||
17-Dec-2019 | ltexprlemloc 6579 | Our constructed difference is located. Lemma for ltexpri 6585. (Contributed by Jim Kingdon, 17-Dec-2019.) | ||||||||||||
⊢ 𝐶 = ⟨{x ∈ Q ∣ ∃y(y ∈ (2^{nd} ‘A) ∧ (y +_{Q} x) ∈ (1^{st} ‘B))}, {x ∈ Q ∣ ∃y(y ∈ (1^{st} ‘A) ∧ (y +_{Q} x) ∈ (2^{nd} ‘B))}⟩ ⇒ ⊢ (A<_{P} B → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <_{Q} 𝑟 → (𝑞 ∈ (1^{st} ‘𝐶) ∨ 𝑟 ∈ (2^{nd} ‘𝐶)))) | ||||||||||||||
17-Dec-2019 | ltexprlemdisj 6578 | Our constructed difference is disjoint. Lemma for ltexpri 6585. (Contributed by Jim Kingdon, 17-Dec-2019.) | ||||||||||||
⊢ 𝐶 = ⟨{x ∈ Q ∣ ∃y(y ∈ (2^{nd} ‘A) ∧ (y +_{Q} x) ∈ (1^{st} ‘B))}, {x ∈ Q ∣ ∃y(y ∈ (1^{st} ‘A) ∧ (y +_{Q} x) ∈ (2^{nd} ‘B))}⟩ ⇒ ⊢ (A<_{P} B → ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1^{st} ‘𝐶) ∧ 𝑞 ∈ (2^{nd} ‘𝐶))) | ||||||||||||||
17-Dec-2019 | ltexprlemrnd 6577 | Our constructed difference is rounded. Lemma for ltexpri 6585. (Contributed by Jim Kingdon, 17-Dec-2019.) | ||||||||||||
⊢ 𝐶 = ⟨{x ∈ Q ∣ ∃y(y ∈ (2^{nd} ‘A) ∧ (y +_{Q} x) ∈ (1^{st} ‘B))}, {x ∈ Q ∣ ∃y(y ∈ (1^{st} ‘A) ∧ (y +_{Q} x) ∈ (2^{nd} ‘B))}⟩ ⇒ ⊢ (A<_{P} B → (∀𝑞 ∈ Q (𝑞 ∈ (1^{st} ‘𝐶) ↔ ∃𝑟 ∈ Q (𝑞 <_{Q} 𝑟 ∧ 𝑟 ∈ (1^{st} ‘𝐶))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2^{nd} ‘𝐶) ↔ ∃𝑞 ∈ Q (𝑞 <_{Q} 𝑟 ∧ 𝑞 ∈ (2^{nd} ‘𝐶))))) | ||||||||||||||
17-Dec-2019 | ltexprlemm 6572 | Our constructed difference is inhabited. Lemma for ltexpri 6585. (Contributed by Jim Kingdon, 17-Dec-2019.) | ||||||||||||
⊢ 𝐶 = ⟨{x ∈ Q ∣ ∃y(y ∈ (2^{nd} ‘A) ∧ (y +_{Q} x) ∈ (1^{st} ‘B))}, {x ∈ Q ∣ ∃y(y ∈ (1^{st} ‘A) ∧ (y +_{Q} x) ∈ (2^{nd} ‘B))}⟩ ⇒ ⊢ (A<_{P} B → (∃𝑞 ∈ Q 𝑞 ∈ (1^{st} ‘𝐶) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2^{nd} ‘𝐶))) | ||||||||||||||
16-Dec-2019 | bj-sbime 9182 | A strengthening of sbie 1671 (same proof). (Contributed by BJ, 16-Dec-2019.) | ||||||||||||
⊢ Ⅎxψ & ⊢ (x = y → (φ → ψ)) ⇒ ⊢ ([y / x]φ → ψ) | ||||||||||||||
16-Dec-2019 | bj-sbimeh 9181 | A strengthening of sbieh 1670 (same proof). (Contributed by BJ, 16-Dec-2019.) | ||||||||||||
⊢ (ψ → ∀xψ) & ⊢ (x = y → (φ → ψ)) ⇒ ⊢ ([y / x]φ → ψ) | ||||||||||||||
16-Dec-2019 | bj-sbimedh 9180 | A strengthening of sbiedh 1667 (same proof). (Contributed by BJ, 16-Dec-2019.) | ||||||||||||
⊢ (φ → ∀xφ) & ⊢ (φ → (χ → ∀xχ)) & ⊢ (φ → (x = y → (ψ → χ))) ⇒ ⊢ (φ → ([y / x]ψ → χ)) | ||||||||||||||
16-Dec-2019 | ltsopr 6568 | Positive real 'less than' is a weak linear order (in the sense of df-iso 4025). Proposition 11.2.3 of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Dec-2019.) | ||||||||||||
⊢ <_{P} Or P | ||||||||||||||
15-Dec-2019 | ltpopr 6567 | Positive real 'less than' is a partial ordering. Remark ("< is transitive and irreflexive") preceding Proposition 11.2.3 of [HoTT], p. (varies). Lemma for ltsopr 6568. (Contributed by Jim Kingdon, 15-Dec-2019.) | ||||||||||||
⊢ <_{P} Po P | ||||||||||||||
15-Dec-2019 | ltdfpr 6488 | More convenient form of df-iltp 6452. (Contributed by Jim Kingdon, 15-Dec-2019.) | ||||||||||||
⊢ ((A ∈ P ∧ B ∈ P) → (A<_{P} B ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2^{nd} ‘A) ∧ 𝑞 ∈ (1^{st} ‘B)))) | ||||||||||||||
15-Dec-2019 | prdisj 6474 | A Dedekind cut is disjoint. (Contributed by Jim Kingdon, 15-Dec-2019.) | ||||||||||||
⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ A ∈ Q) → ¬ (A ∈ 𝐿 ∧ A ∈ 𝑈)) | ||||||||||||||
13-Dec-2019 | 1idpru 6565 | Lemma for 1idpr 6566. (Contributed by Jim Kingdon, 13-Dec-2019.) | ||||||||||||
⊢ (A ∈ P → (2^{nd} ‘(A ·_{P} 1_{P})) = (2^{nd} ‘A)) | ||||||||||||||
13-Dec-2019 | 1idprl 6564 | Lemma for 1idpr 6566. (Contributed by Jim Kingdon, 13-Dec-2019.) | ||||||||||||
⊢ (A ∈ P → (1^{st} ‘(A ·_{P} 1_{P})) = (1^{st} ‘A)) | ||||||||||||||
13-Dec-2019 | gtnqex 6532 | The class of rationals greater than a given rational is a set. (Contributed by Jim Kingdon, 13-Dec-2019.) | ||||||||||||
⊢ {x ∣ A <_{Q} x} ∈ V | ||||||||||||||
13-Dec-2019 | ltnqex 6531 | The class of rationals less than a given rational is a set. (Contributed by Jim Kingdon, 13-Dec-2019.) | ||||||||||||
⊢ {x ∣ x <_{Q} A} ∈ V | ||||||||||||||
13-Dec-2019 | sotritrieq 4053 | A trichotomy relationship, given a trichotomous order. (Contributed by Jim Kingdon, 13-Dec-2019.) | ||||||||||||
⊢ 𝑅 Or A & ⊢ ((B ∈ A ∧ 𝐶 ∈ A) → (B𝑅𝐶 ∨ B = 𝐶 ∨ 𝐶𝑅B)) ⇒ ⊢ ((B ∈ A ∧ 𝐶 ∈ A) → (B = 𝐶 ↔ ¬ (B𝑅𝐶 ∨ 𝐶𝑅B))) | ||||||||||||||
12-Dec-2019 | distrprg 6562 | Multiplication of positive reals is distributive. Proposition 9-3.7(iii) of [Gleason] p. 124. (Contributed by Jim Kingdon, 12-Dec-2019.) | ||||||||||||
⊢ ((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) → (A ·_{P} (B +_{P} 𝐶)) = ((A ·_{P} B) +_{P} (A ·_{P} 𝐶))) | ||||||||||||||
12-Dec-2019 | distrlem5pru 6561 | Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.) | ||||||||||||
⊢ ((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) → (2^{nd} ‘((A ·_{P} B) +_{P} (A ·_{P} 𝐶))) ⊆ (2^{nd} ‘(A ·_{P} (B +_{P} 𝐶)))) | ||||||||||||||
12-Dec-2019 | distrlem5prl 6560 | Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.) | ||||||||||||
⊢ ((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) → (1^{st} ‘((A ·_{P} B) +_{P} (A ·_{P} 𝐶))) ⊆ (1^{st} ‘(A ·_{P} (B +_{P} 𝐶)))) | ||||||||||||||
12-Dec-2019 | distrlem4pru 6559 | Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.) | ||||||||||||
⊢ (((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) ∧ ((x ∈ (2^{nd} ‘A) ∧ y ∈ (2^{nd} ‘B)) ∧ (f ∈ (2^{nd} ‘A) ∧ z ∈ (2^{nd} ‘𝐶)))) → ((x ·_{Q} y) +_{Q} (f ·_{Q} z)) ∈ (2^{nd} ‘(A ·_{P} (B +_{P} 𝐶)))) | ||||||||||||||
12-Dec-2019 | distrlem4prl 6558 | Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.) | ||||||||||||
⊢ (((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) ∧ ((x ∈ (1^{st} ‘A) ∧ y ∈ (1^{st} ‘B)) ∧ (f ∈ (1^{st} ‘A) ∧ z ∈ (1^{st} ‘𝐶)))) → ((x ·_{Q} y) +_{Q} (f ·_{Q} z)) ∈ (1^{st} ‘(A ·_{P} (B +_{P} 𝐶)))) | ||||||||||||||
12-Dec-2019 | distrlem1pru 6557 | Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.) | ||||||||||||
⊢ ((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) → (2^{nd} ‘(A ·_{P} (B +_{P} 𝐶))) ⊆ (2^{nd} ‘((A ·_{P} B) +_{P} (A ·_{P} 𝐶)))) | ||||||||||||||
12-Dec-2019 | distrlem1prl 6556 | Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.) | ||||||||||||
⊢ ((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) → (1^{st} ‘(A ·_{P} (B +_{P} 𝐶))) ⊆ (1^{st} ‘((A ·_{P} B) +_{P} (A ·_{P} 𝐶)))) | ||||||||||||||
12-Dec-2019 | ltdcnq 6381 | Less-than for positive fractions is decidable. (Contributed by Jim Kingdon, 12-Dec-2019.) | ||||||||||||
⊢ ((A ∈ Q ∧ B ∈ Q) → DECID A <_{Q} B) | ||||||||||||||
12-Dec-2019 | ltdcpi 6307 | Less-than for positive integers is decidable. (Contributed by Jim Kingdon, 12-Dec-2019.) | ||||||||||||
⊢ ((A ∈ N ∧ B ∈ N) → DECID A <_{N} B) | ||||||||||||||
11-Dec-2019 | mulassprg 6555 | Multiplication of positive reals is associative. Proposition 9-3.7(i) of [Gleason] p. 124. (Contributed by Jim Kingdon, 11-Dec-2019.) | ||||||||||||
⊢ ((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) → ((A ·_{P} B) ·_{P} 𝐶) = (A ·_{P} (B ·_{P} 𝐶))) | ||||||||||||||
11-Dec-2019 | mulcomprg 6554 | Multiplication of positive reals is commutative. Proposition 9-3.7(ii) of [Gleason] p. 124. (Contributed by Jim Kingdon, 11-Dec-2019.) | ||||||||||||
⊢ ((A ∈ P ∧ B ∈ P) → (A ·_{P} B) = (B ·_{P} A)) | ||||||||||||||
11-Dec-2019 | addassprg 6553 | Addition of positive reals is associative. Proposition 9-3.5(i) of [Gleason] p. 123. (Contributed by Jim Kingdon, 11-Dec-2019.) | ||||||||||||
⊢ ((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) → ((A +_{P} B) +_{P} 𝐶) = (A +_{P} (B +_{P} 𝐶))) | ||||||||||||||
11-Dec-2019 | addcomprg 6552 | Addition of positive reals is commutative. Proposition 9-3.5(ii) of [Gleason] p. 123. (Contributed by Jim Kingdon, 11-Dec-2019.) | ||||||||||||
⊢ ((A ∈ P ∧ B ∈ P) → (A +_{P} B) = (B +_{P} A)) | ||||||||||||||
11-Dec-2019 | genpassg 6509 | Associativity of an operation on reals. (Contributed by Jim Kingdon, 11-Dec-2019.) | ||||||||||||
⊢ 𝐹 = (w ∈ P, v ∈ P ↦ ⟨{x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (1^{st} ‘w) ∧ z ∈ (1^{st} ‘v) ∧ x = (y𝐺z))}, {x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (2^{nd} ‘w) ∧ z ∈ (2^{nd} ‘v) ∧ x = (y𝐺z))}⟩) & ⊢ ((y ∈ Q ∧ z ∈ Q) → (y𝐺z) ∈ Q) & ⊢ dom 𝐹 = (P × P) & ⊢ ((f ∈ P ∧ g ∈ P) → (f𝐹g) ∈ P) & ⊢ ((f ∈ Q ∧ g ∈ Q ∧ ℎ ∈ Q) → ((f𝐺g)𝐺ℎ) = (f𝐺(g𝐺ℎ))) ⇒ ⊢ ((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) → ((A𝐹B)𝐹𝐶) = (A𝐹(B𝐹𝐶))) | ||||||||||||||
11-Dec-2019 | genpassu 6508 | Associativity of upper cuts. Lemma for genpassg 6509. (Contributed by Jim Kingdon, 11-Dec-2019.) | ||||||||||||
⊢ 𝐹 = (w ∈ P, v ∈ P ↦ ⟨{x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (1^{st} ‘w) ∧ z ∈ (1^{st} ‘v) ∧ x = (y𝐺z))}, {x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (2^{nd} ‘w) ∧ z ∈ (2^{nd} ‘v) ∧ x = (y𝐺z))}⟩) & ⊢ ((y ∈ Q ∧ z ∈ Q) → (y𝐺z) ∈ Q) & ⊢ dom 𝐹 = (P × P) & ⊢ ((f ∈ P ∧ g ∈ P) → (f𝐹g) ∈ P) & ⊢ ((f ∈ Q ∧ g ∈ Q ∧ ℎ ∈ Q) → ((f𝐺g)𝐺ℎ) = (f𝐺(g𝐺ℎ))) ⇒ ⊢ ((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) → (2^{nd} ‘((A𝐹B)𝐹𝐶)) = (2^{nd} ‘(A𝐹(B𝐹𝐶)))) | ||||||||||||||
11-Dec-2019 | genpassl 6507 | Associativity of lower cuts. Lemma for genpassg 6509. (Contributed by Jim Kingdon, 11-Dec-2019.) | ||||||||||||
⊢ 𝐹 = (w ∈ P, v ∈ P ↦ ⟨{x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (1^{st} ‘w) ∧ z ∈ (1^{st} ‘v) ∧ x = (y𝐺z))}, {x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (2^{nd} ‘w) ∧ z ∈ (2^{nd} ‘v) ∧ x = (y𝐺z))}⟩) & ⊢ ((y ∈ Q ∧ z ∈ Q) → (y𝐺z) ∈ Q) & ⊢ dom 𝐹 = (P × P) & ⊢ ((f ∈ P ∧ g ∈ P) → (f𝐹g) ∈ P) & ⊢ ((f ∈ Q ∧ g ∈ Q ∧ ℎ ∈ Q) → ((f𝐺g)𝐺ℎ) = (f𝐺(g𝐺ℎ))) ⇒ ⊢ ((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) → (1^{st} ‘((A𝐹B)𝐹𝐶)) = (1^{st} ‘(A𝐹(B𝐹𝐶)))) | ||||||||||||||
11-Dec-2019 | preqlu 6454 | Two reals are equal if and only if their lower and upper cuts are. (Contributed by Jim Kingdon, 11-Dec-2019.) | ||||||||||||
⊢ ((A ∈ P ∧ B ∈ P) → (A = B ↔ ((1^{st} ‘A) = (1^{st} ‘B) ∧ (2^{nd} ‘A) = (2^{nd} ‘B)))) | ||||||||||||||
11-Dec-2019 | fssd 4998 | Expanding the codomain of a mapping, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.) | ||||||||||||
⊢ (φ → 𝐹:A⟶B) & ⊢ (φ → B ⊆ 𝐶) ⇒ ⊢ (φ → 𝐹:A⟶𝐶) | ||||||||||||||
11-Dec-2019 | iffalsed 3335 | Value of the conditional operator when its first argument is false. (Contributed by Glauco Siliprandi, 11-Dec-2019.) | ||||||||||||
⊢ (φ → ¬ χ) ⇒ ⊢ (φ → if(χ, A, B) = B) | ||||||||||||||
11-Dec-2019 | iftrued 3332 | Value of the conditional operator when its first argument is true. (Contributed by Glauco Siliprandi, 11-Dec-2019.) | ||||||||||||
⊢ (φ → χ) ⇒ ⊢ (φ → if(χ, A, B) = A) | ||||||||||||||
10-Dec-2019 | mullocprlem 6549 | Calculations for mullocpr 6550. (Contributed by Jim Kingdon, 10-Dec-2019.) | ||||||||||||
⊢ (φ → (A ∈ P ∧ B ∈ P)) & ⊢ (φ → (𝑈 ·_{Q} 𝑄) <_{Q} (𝐸 ·_{Q} (𝐷 ·_{Q} 𝑈))) & ⊢ (φ → (𝐸 ·_{Q} (𝐷 ·_{Q} 𝑈)) <_{Q} (𝑇 ·_{Q} (𝐷 ·_{Q} 𝑈))) & ⊢ (φ → (𝑇 ·_{Q} (𝐷 ·_{Q} 𝑈)) <_{Q} (𝐷 ·_{Q} 𝑅)) & ⊢ (φ → (𝑄 ∈ Q ∧ 𝑅 ∈ Q)) & ⊢ (φ → (𝐷 ∈ Q ∧ 𝑈 ∈ Q)) & ⊢ (φ → (𝐷 ∈ (1^{st} ‘A) ∧ 𝑈 ∈ (2^{nd} ‘A))) & ⊢ (φ → (𝐸 ∈ Q ∧ 𝑇 ∈ Q)) ⇒ ⊢ (φ → (𝑄 ∈ (1^{st} ‘(A ·_{P} B)) ∨ 𝑅 ∈ (2^{nd} ‘(A ·_{P} B)))) | ||||||||||||||
10-Dec-2019 | mulnqpru 6548 | Lemma to prove upward closure in positive real multiplication. (Contributed by Jim Kingdon, 10-Dec-2019.) | ||||||||||||
⊢ ((((A ∈ P ∧ 𝐺 ∈ (2^{nd} ‘A)) ∧ (B ∈ P ∧ 𝐻 ∈ (2^{nd} ‘B))) ∧ 𝑋 ∈ Q) → ((𝐺 ·_{Q} 𝐻) <_{Q} 𝑋 → 𝑋 ∈ (2^{nd} ‘(A ·_{P} B)))) | ||||||||||||||
10-Dec-2019 | mulnqprl 6547 | Lemma to prove downward closure in positive real multiplication. (Contributed by Jim Kingdon, 10-Dec-2019.) | ||||||||||||
⊢ ((((A ∈ P ∧ 𝐺 ∈ (1^{st} ‘A)) ∧ (B ∈ P ∧ 𝐻 ∈ (1^{st} ‘B))) ∧ 𝑋 ∈ Q) → (𝑋 <_{Q} (𝐺 ·_{Q} 𝐻) → 𝑋 ∈ (1^{st} ‘(A ·_{P} B)))) | ||||||||||||||
9-Dec-2019 | prmuloclemcalc 6544 | Calculations for prmuloc 6545. (Contributed by Jim Kingdon, 9-Dec-2019.) | ||||||||||||
⊢ (φ → 𝑅 <_{Q} 𝑈) & ⊢ (φ → 𝑈 <_{Q} (𝐷 +_{Q} 𝑃)) & ⊢ (φ → (A +_{Q} 𝑋) = B) & ⊢ (φ → (𝑃 ·_{Q} B) <_{Q} (𝑅 ·_{Q} 𝑋)) & ⊢ (φ → A ∈ Q) & ⊢ (φ → B ∈ Q) & ⊢ (φ → 𝐷 ∈ Q) & ⊢ (φ → 𝑃 ∈ Q) & ⊢ (φ → 𝑋 ∈ Q) ⇒ ⊢ (φ → (𝑈 ·_{Q} A) <_{Q} (𝐷 ·_{Q} B)) | ||||||||||||||
9-Dec-2019 | appdiv0nq 6543 | Approximate division for positive rationals. This can be thought of as a variation of appdivnq 6542 in which A is zero, although it can be stated and proved in terms of positive rationals alone, without zero as such. (Contributed by Jim Kingdon, 9-Dec-2019.) | ||||||||||||
⊢ ((B ∈ Q ∧ 𝐶 ∈ Q) → ∃𝑚 ∈ Q (𝑚 ·_{Q} 𝐶) <_{Q} B) | ||||||||||||||
9-Dec-2019 | ltmnqi 6387 | Ordering property of multiplication for positive fractions. One direction of ltmnqg 6385. (Contributed by Jim Kingdon, 9-Dec-2019.) | ||||||||||||
⊢ ((A <_{Q} B ∧ 𝐶 ∈ Q) → (𝐶 ·_{Q} A) <_{Q} (𝐶 ·_{Q} B)) | ||||||||||||||
9-Dec-2019 | ltanqi 6386 | Ordering property of addition for positive fractions. One direction of ltanqg 6384. (Contributed by Jim Kingdon, 9-Dec-2019.) | ||||||||||||
⊢ ((A <_{Q} B ∧ 𝐶 ∈ Q) → (𝐶 +_{Q} A) <_{Q} (𝐶 +_{Q} B)) | ||||||||||||||
8-Dec-2019 | bj-nn0sucALT 9362 | Alternate proof of bj-nn0suc 9348, also constructive but from ax-inf2 9360, hence requiring ax-bdsetind 9352. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||
⊢ (A ∈ 𝜔 ↔ (A = ∅ ∨ ∃x ∈ 𝜔 A = suc x)) | ||||||||||||||
8-Dec-2019 | bj-omex2 9361 | Using bounded set induction and the strong axiom of infinity, 𝜔 is a set, that is, we recover ax-infvn 9329 (see bj-2inf 9326 for the equivalence of the latter with bj-omex 9330). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||
⊢ 𝜔 ∈ V | ||||||||||||||
8-Dec-2019 | bj-inf2vn2 9359 | A sufficient condition for 𝜔 to be a set; unbounded version of bj-inf2vn 9358. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) | ||||||||||||
⊢ (A ∈ 𝑉 → (∀x(x ∈ A ↔ (x = ∅ ∨ ∃y ∈ A x = suc y)) → A = 𝜔)) | ||||||||||||||
8-Dec-2019 | bj-inf2vn 9358 | A sufficient condition for 𝜔 to be a set. See bj-inf2vn2 9359 for the unbounded version from full set induction. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) | ||||||||||||
⊢ BOUNDED A ⇒ ⊢ (A ∈ 𝑉 → (∀x(x ∈ A ↔ (x = ∅ ∨ ∃y ∈ A x = suc y)) → A = 𝜔)) | ||||||||||||||
8-Dec-2019 | bj-inf2vnlem4 9357 | Lemma for bj-inf2vn2 9359. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) | ||||||||||||
⊢ (∀x ∈ A (x = ∅ ∨ ∃y ∈ A x = suc y) → (Ind 𝑍 → A ⊆ 𝑍)) | ||||||||||||||
8-Dec-2019 | bj-inf2vnlem3 9356 | Lemma for bj-inf2vn 9358. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) | ||||||||||||
⊢ BOUNDED A & ⊢ BOUNDED 𝑍 ⇒ ⊢ (∀x ∈ A (x = ∅ ∨ ∃y ∈ A x = suc y) → (Ind 𝑍 → A ⊆ 𝑍)) | ||||||||||||||
8-Dec-2019 | bj-inf2vnlem2 9355 | Lemma for bj-inf2vnlem3 9356 and bj-inf2vnlem4 9357. Remark: unoptimized proof (have to use more deduction style). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) | ||||||||||||
⊢ (∀x ∈ A (x = ∅ ∨ ∃y ∈ A x = suc y) → (Ind 𝑍 → ∀u(∀𝑡 ∈ u (𝑡 ∈ A → 𝑡 ∈ 𝑍) → (u ∈ A → u ∈ 𝑍)))) | ||||||||||||||
8-Dec-2019 | bj-inf2vnlem1 9354 | Lemma for bj-inf2vn 9358. Remark: unoptimized proof (have to use more deduction style). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) | ||||||||||||
⊢ (∀x(x ∈ A ↔ (x = ∅ ∨ ∃y ∈ A x = suc y)) → Ind A) | ||||||||||||||
8-Dec-2019 | bj-bdcel 9226 | Boundedness of a membership formula. (Contributed by BJ, 8-Dec-2019.) | ||||||||||||
⊢ BOUNDED y = A ⇒ ⊢ BOUNDED A ∈ x | ||||||||||||||
8-Dec-2019 | bj-ex 9171 | Existential generalization. (Contributed by BJ, 8-Dec-2019.) Proof modification is discouraged because there are shorter proofs, but using less basic results (like exlimiv 1486 and 19.9ht 1529 or 19.23ht 1383). (Proof modification is discouraged.) | ||||||||||||
⊢ (∃xφ → φ) | ||||||||||||||
8-Dec-2019 | mullocpr 6550 | Locatedness of multiplication on positive reals. Lemma 12.9 in [BauerTaylor], p. 56 (but where both A and B are positive, not just A). (Contributed by Jim Kingdon, 8-Dec-2019.) | ||||||||||||
⊢ ((A ∈ P ∧ B ∈ P) → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <_{Q} 𝑟 → (𝑞 ∈ (1^{st} ‘(A ·_{P} B)) ∨ 𝑟 ∈ (2^{nd} ‘(A ·_{P} B))))) | ||||||||||||||
8-Dec-2019 | prmuloc 6545 | Positive reals are multiplicatively located. Lemma 12.8 of [BauerTaylor], p. 56. (Contributed by Jim Kingdon, 8-Dec-2019.) | ||||||||||||
⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ A <_{Q} B) → ∃𝑑 ∈ Q ∃u ∈ Q (𝑑 ∈ 𝐿 ∧ u ∈ 𝑈 ∧ (u ·_{Q} A) <_{Q} (𝑑 ·_{Q} B))) | ||||||||||||||
8-Dec-2019 | appdivnq 6542 | Approximate division for positive rationals. Proposition 12.7 of [BauerTaylor], p. 55 (a special case where A and B are positive, as well as 𝐶). Our proof is simpler than the one in BauerTaylor because we have reciprocals. (Contributed by Jim Kingdon, 8-Dec-2019.) | ||||||||||||
⊢ ((A <_{Q} B ∧ 𝐶 ∈ Q) → ∃𝑚 ∈ Q (A <_{Q} (𝑚 ·_{Q} 𝐶) ∧ (𝑚 ·_{Q} 𝐶) <_{Q} B)) | ||||||||||||||
8-Dec-2019 | nqprxx 6529 | The canonical embedding of the rationals into the reals, expressed with the same variable for the lower and upper cuts. (Contributed by Jim Kingdon, 8-Dec-2019.) | ||||||||||||
⊢ (A ∈ Q → ⟨{x ∣ x <_{Q} A}, {x ∣ A <_{Q} x}⟩ ∈ P) | ||||||||||||||
8-Dec-2019 | nqprloc 6528 | A cut produced from a rational is located. Lemma for nqprlu 6530. (Contributed by Jim Kingdon, 8-Dec-2019.) | ||||||||||||
⊢ (A ∈ Q → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <_{Q} 𝑟 → (𝑞 ∈ {x ∣ x <_{Q} A} ∨ 𝑟 ∈ {x ∣ A <_{Q} x}))) | ||||||||||||||
8-Dec-2019 | nqprdisj 6527 | A cut produced from a rational is disjoint. Lemma for nqprlu 6530. (Contributed by Jim Kingdon, 8-Dec-2019.) | ||||||||||||
⊢ (A ∈ Q → ∀𝑞 ∈ Q ¬ (𝑞 ∈ {x ∣ x <_{Q} A} ∧ 𝑞 ∈ {x ∣ A <_{Q} x})) | ||||||||||||||
8-Dec-2019 | nqprrnd 6526 | A cut produced from a rational is rounded. Lemma for nqprlu 6530. (Contributed by Jim Kingdon, 8-Dec-2019.) | ||||||||||||
⊢ (A ∈ Q → (∀𝑞 ∈ Q (𝑞 ∈ {x ∣ x <_{Q} A} ↔ ∃𝑟 ∈ Q (𝑞 <_{Q} 𝑟 ∧ 𝑟 ∈ {x ∣ x <_{Q} A})) ∧ ∀𝑟 ∈ Q (𝑟 ∈ {x ∣ A <_{Q} x} ↔ ∃𝑞 ∈ Q (𝑞 <_{Q} 𝑟 ∧ 𝑞 ∈ {x ∣ A <_{Q} x})))) | ||||||||||||||
8-Dec-2019 | nqprm 6525 | A cut produced from a rational is inhabited. Lemma for nqprlu 6530. (Contributed by Jim Kingdon, 8-Dec-2019.) | ||||||||||||
⊢ (A ∈ Q → (∃𝑞 ∈ Q 𝑞 ∈ {x ∣ x <_{Q} A} ∧ ∃𝑟 ∈ Q 𝑟 ∈ {x ∣ A <_{Q} x})) | ||||||||||||||
8-Dec-2019 | mpvlu 6522 | Value of multiplication on positive reals. (Contributed by Jim Kingdon, 8-Dec-2019.) | ||||||||||||
⊢ ((A ∈ P ∧ B ∈ P) → (A ·_{P} B) = ⟨{x ∈ Q ∣ ∃y ∈ (1^{st} ‘A)∃z ∈ (1^{st} ‘B)x = (y ·_{Q} z)}, {x ∈ Q ∣ ∃y ∈ (2^{nd} ‘A)∃z ∈ (2^{nd} ‘B)x = (y ·_{Q} z)}⟩) | ||||||||||||||
8-Dec-2019 | plpvlu 6521 | Value of addition on positive reals. (Contributed by Jim Kingdon, 8-Dec-2019.) | ||||||||||||
⊢ ((A ∈ P ∧ B ∈ P) → (A +_{P} B) = ⟨{x ∈ Q ∣ ∃y ∈ (1^{st} ‘A)∃z ∈ (1^{st} ‘B)x = (y +_{Q} z)}, {x ∈ Q ∣ ∃y ∈ (2^{nd} ‘A)∃z ∈ (2^{nd} ‘B)x = (y +_{Q} z)}⟩) | ||||||||||||||
7-Dec-2019 | addlocprlemeqgt 6515 | Lemma for addlocpr 6519. This is a step used in both the 𝑄 = (𝐷 +_{Q} 𝐸) and (𝐷 +_{Q} 𝐸) <_{Q} 𝑄 cases. (Contributed by Jim Kingdon, 7-Dec-2019.) | ||||||||||||
⊢ (φ → A ∈ P) & ⊢ (φ → B ∈ P) & ⊢ (φ → 𝑄 <_{Q} 𝑅) & ⊢ (φ → 𝑃 ∈ Q) & ⊢ (φ → (𝑄 +_{Q} (𝑃 +_{Q} 𝑃)) = 𝑅) & ⊢ (φ → 𝐷 ∈ (1^{st} ‘A)) & ⊢ (φ → 𝑈 ∈ (2^{nd} ‘A)) & ⊢ (φ → 𝑈 <_{Q} (𝐷 +_{Q} 𝑃)) & ⊢ (φ → 𝐸 ∈ (1^{st} ‘B)) & ⊢ (φ → 𝑇 ∈ (2^{nd} ‘B)) & ⊢ (φ → 𝑇 <_{Q} (𝐸 +_{Q} 𝑃)) ⇒ ⊢ (φ → (𝑈 +_{Q} 𝑇) <_{Q} ((𝐷 +_{Q} 𝐸) +_{Q} (𝑃 +_{Q} 𝑃))) | ||||||||||||||
7-Dec-2019 | addnqprulem 6511 | Lemma to prove upward closure in positive real addition. (Contributed by Jim Kingdon, 7-Dec-2019.) | ||||||||||||
⊢ (((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐺 ∈ 𝑈) ∧ 𝑋 ∈ Q) → (𝑆 <_{Q} 𝑋 → ((𝑋 ·_{Q} (*_{Q}‘𝑆)) ·_{Q} 𝐺) ∈ 𝑈)) | ||||||||||||||
7-Dec-2019 | addnqprllem 6510 | Lemma to prove downward closure in positive real addition. (Contributed by Jim Kingdon, 7-Dec-2019.) | ||||||||||||
⊢ (((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐺 ∈ 𝐿) ∧ 𝑋 ∈ Q) → (𝑋 <_{Q} 𝑆 → ((𝑋 ·_{Q} (*_{Q}‘𝑆)) ·_{Q} 𝐺) ∈ 𝐿)) | ||||||||||||||
7-Dec-2019 | lt2addnq 6388 | Ordering property of addition for positive fractions. (Contributed by Jim Kingdon, 7-Dec-2019.) | ||||||||||||
⊢ (((A ∈ Q ∧ B ∈ Q) ∧ (𝐶 ∈ Q ∧ 𝐷 ∈ Q)) → ((A <_{Q} B ∧ 𝐶 <_{Q} 𝐷) → (A +_{Q} 𝐶) <_{Q} (B +_{Q} 𝐷))) | ||||||||||||||
6-Dec-2019 | addlocprlem 6518 | Lemma for addlocpr 6519. The result, in deduction form. (Contributed by Jim Kingdon, 6-Dec-2019.) | ||||||||||||
⊢ (φ → A ∈ P) & ⊢ (φ → B ∈ P) & ⊢ (φ → 𝑄 <_{Q} 𝑅) & ⊢ (φ → 𝑃 ∈ Q) & ⊢ (φ → (𝑄 +_{Q} (𝑃 +_{Q} 𝑃)) = 𝑅) & ⊢ (φ → 𝐷 ∈ (1^{st} ‘A)) & ⊢ (φ → 𝑈 ∈ (2^{nd} ‘A)) & ⊢ (φ → 𝑈 <_{Q} (𝐷 +_{Q} 𝑃)) & ⊢ (φ → 𝐸 ∈ (1^{st} ‘B)) & ⊢ (φ → 𝑇 ∈ (2^{nd} ‘B)) & ⊢ (φ → 𝑇 <_{Q} (𝐸 +_{Q} 𝑃)) ⇒ ⊢ (φ → (𝑄 ∈ (1^{st} ‘(A +_{P} B)) ∨ 𝑅 ∈ (2^{nd} ‘(A +_{P} B)))) | ||||||||||||||
6-Dec-2019 | addlocprlemgt 6517 | Lemma for addlocpr 6519. The (𝐷 +_{Q} 𝐸) <_{Q} 𝑄 case. (Contributed by Jim Kingdon, 6-Dec-2019.) | ||||||||||||
⊢ (φ → A ∈ P) & ⊢ (φ → B ∈ P) & ⊢ (φ → 𝑄 <_{Q} 𝑅) & ⊢ (φ → 𝑃 ∈ Q) & ⊢ (φ → (𝑄 +_{Q} (𝑃 +_{Q} 𝑃)) = 𝑅) & ⊢ (φ → 𝐷 ∈ (1^{st} ‘A)) & ⊢ (φ → 𝑈 ∈ (2^{nd} ‘A)) & ⊢ (φ → 𝑈 <_{Q} (𝐷 +_{Q} 𝑃)) & ⊢ (φ → 𝐸 ∈ (1^{st} ‘B)) & ⊢ (φ → 𝑇 ∈ (2^{nd} ‘B)) & ⊢ (φ → 𝑇 <_{Q} (𝐸 +_{Q} 𝑃)) ⇒ ⊢ (φ → ((𝐷 +_{Q} 𝐸) <_{Q} 𝑄 → 𝑅 ∈ (2^{nd} ‘(A +_{P} B)))) | ||||||||||||||
6-Dec-2019 | addlocprlemeq 6516 | Lemma for addlocpr 6519. The 𝑄 = (𝐷 +_{Q} 𝐸) case. (Contributed by Jim Kingdon, 6-Dec-2019.) | ||||||||||||
⊢ (φ → A ∈ P) & ⊢ (φ → B ∈ P) & ⊢ (φ → 𝑄 <_{Q} 𝑅) & ⊢ (φ → 𝑃 ∈ Q) & ⊢ (φ → (𝑄 +_{Q} (𝑃 +_{Q} 𝑃)) = 𝑅) & ⊢ (φ → 𝐷 ∈ (1^{st} ‘A)) & ⊢ (φ → 𝑈 ∈ (2^{nd} ‘A)) & ⊢ (φ → 𝑈 <_{Q} (𝐷 +_{Q} 𝑃)) & ⊢ (φ → 𝐸 ∈ (1^{st} ‘B)) & ⊢ (φ → 𝑇 ∈ (2^{nd} ‘B)) & ⊢ (φ → 𝑇 <_{Q} (𝐸 +_{Q} 𝑃)) ⇒ ⊢ (φ → (𝑄 = (𝐷 +_{Q} 𝐸) → 𝑅 ∈ (2^{nd} ‘(A +_{P} B)))) | ||||||||||||||
6-Dec-2019 | addlocprlemlt 6514 | Lemma for addlocpr 6519. The 𝑄 <_{Q} (𝐷 +_{Q} 𝐸) case. (Contributed by Jim Kingdon, 6-Dec-2019.) | ||||||||||||
⊢ (φ → A ∈ P) & ⊢ (φ → B ∈ P) & ⊢ (φ → 𝑄 <_{Q} 𝑅) & ⊢ (φ → 𝑃 ∈ Q) & ⊢ (φ → (𝑄 +_{Q} (𝑃 +_{Q} 𝑃)) = 𝑅) & ⊢ (φ → 𝐷 ∈ (1^{st} ‘A)) & ⊢ (φ → 𝑈 ∈ (2^{nd} ‘A)) & ⊢ (φ → 𝑈 <_{Q} (𝐷 +_{Q} 𝑃)) & ⊢ (φ → 𝐸 ∈ (1^{st} ‘B)) & ⊢ (φ → 𝑇 ∈ (2^{nd} ‘B)) & ⊢ (φ → 𝑇 <_{Q} (𝐸 +_{Q} 𝑃)) ⇒ ⊢ (φ → (𝑄 <_{Q} (𝐷 +_{Q} 𝐸) → 𝑄 ∈ (1^{st} ‘(A +_{P} B)))) | ||||||||||||||
5-Dec-2019 | addlocpr 6519 | Locatedness of addition on positive reals. Lemma 11.16 in [BauerTaylor], p. 53. The proof in BauerTaylor relies on signed rationals, so we replace it with another proof which applies prarloc 6485 to both A and B, and uses nqtri3or 6380 rather than prloc 6473 to decide whether 𝑞 is too big to be in the lower cut of A +_{P} B (and deduce that if it is, then 𝑟 must be in the upper cut). What the two proofs have in common is that they take the difference between 𝑞 and 𝑟 to determine how tight a range they need around the real numbers. (Contributed by Jim Kingdon, 5-Dec-2019.) | ||||||||||||
⊢ ((A ∈ P ∧ B ∈ P) → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <_{Q} 𝑟 → (𝑞 ∈ (1^{st} ‘(A +_{P} B)) ∨ 𝑟 ∈ (2^{nd} ‘(A +_{P} B))))) | ||||||||||||||
5-Dec-2019 | addnqpru 6513 | Lemma to prove upward closure in positive real addition. (Contributed by Jim Kingdon, 5-Dec-2019.) | ||||||||||||
⊢ ((((A ∈ P ∧ 𝐺 ∈ (2^{nd} ‘A)) ∧ (B ∈ P ∧ 𝐻 ∈ (2^{nd} ‘B))) ∧ 𝑋 ∈ Q) → ((𝐺 +_{Q} 𝐻) <_{Q} 𝑋 → 𝑋 ∈ (2^{nd} ‘(A +_{P} B)))) | ||||||||||||||
5-Dec-2019 | addnqprl 6512 | Lemma to prove downward closure in positive real addition. (Contributed by Jim Kingdon, 5-Dec-2019.) | ||||||||||||
⊢ ((((A ∈ P ∧ 𝐺 ∈ (1^{st} ‘A)) ∧ (B ∈ P ∧ 𝐻 ∈ (1^{st} ‘B))) ∧ 𝑋 ∈ Q) → (𝑋 <_{Q} (𝐺 +_{Q} 𝐻) → 𝑋 ∈ (1^{st} ‘(A +_{P} B)))) | ||||||||||||||
5-Dec-2019 | genpmu 6501 | The upper cut produced by addition or multiplication on positive reals is inhabited. (Contributed by Jim Kingdon, 5-Dec-2019.) | ||||||||||||
⊢ 𝐹 = (w ∈ P, v ∈ P ↦ ⟨{x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (1^{st} ‘w) ∧ z ∈ (1^{st} ‘v) ∧ x = (y𝐺z))}, {x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (2^{nd} ‘w) ∧ z ∈ (2^{nd} ‘v) ∧ x = (y𝐺z))}⟩) & ⊢ ((y ∈ Q ∧ z ∈ Q) → (y𝐺z) ∈ Q) ⇒ ⊢ ((A ∈ P ∧ B ∈ P) → ∃𝑞 ∈ Q 𝑞 ∈ (2^{nd} ‘(A𝐹B))) | ||||||||||||||
5-Dec-2019 | genpelxp 6493 | Set containing the result of adding or multiplying positive reals. (Contributed by Jim Kingdon, 5-Dec-2019.) | ||||||||||||
⊢ 𝐹 = (w ∈ P, v ∈ P ↦ ⟨{x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (1^{st} ‘w) ∧ z ∈ (1^{st} ‘v) ∧ x = (y𝐺z))}, {x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (2^{nd} ‘w) ∧ z ∈ (2^{nd} ‘v) ∧ x = (y𝐺z))}⟩) ⇒ ⊢ ((A ∈ P ∧ B ∈ P) → (A𝐹B) ∈ (𝒫 Q × 𝒫 Q)) | ||||||||||||||
3-Dec-2019 | addassnq0lemcl 6443 | A natural number closure law. Lemma for addassnq0 6444. (Contributed by Jim Kingdon, 3-Dec-2019.) | ||||||||||||
⊢ (((𝐼 ∈ 𝜔 ∧ 𝐽 ∈ N) ∧ (𝐾 ∈ 𝜔 ∧ 𝐿 ∈ N)) → (((𝐼 ·_{𝑜} 𝐿) +_{𝑜} (𝐽 ·_{𝑜} 𝐾)) ∈ 𝜔 ∧ (𝐽 ·_{𝑜} 𝐿) ∈ N)) | ||||||||||||||
3-Dec-2019 | nndir 6008 | Distributive law for natural numbers (right-distributivity). (Contributed by Jim Kingdon, 3-Dec-2019.) | ||||||||||||
⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) → ((A +_{𝑜} B) ·_{𝑜} 𝐶) = ((A ·_{𝑜} 𝐶) +_{𝑜} (B ·_{𝑜} 𝐶))) | ||||||||||||||
1-Dec-2019 | nnanq0 6440 | Addition of non-negative fractions with a common denominator. You can add two fractions with the same denominator by adding their numerators and keeping the same denominator. (Contributed by Jim Kingdon, 1-Dec-2019.) | ||||||||||||
⊢ ((𝑁 ∈ 𝜔 ∧ 𝑀 ∈ 𝜔 ∧ A ∈ N) → [⟨(𝑁 +_{𝑜} 𝑀), A⟩] ~_{Q0} = ([⟨𝑁, A⟩] ~_{Q0} +_{Q0} [⟨𝑀, A⟩] ~_{Q0} )) | ||||||||||||||
1-Dec-2019 | archnqq 6400 | For any fraction, there is an integer that is greater than it. This is also known as the "archimedean property". (Contributed by Jim Kingdon, 1-Dec-2019.) | ||||||||||||
⊢ (A ∈ Q → ∃x ∈ N A <_{Q} [⟨x, 1_{𝑜}⟩] ~_{Q} ) | ||||||||||||||
30-Nov-2019 | bj-2inf 9326 | Two formulations of the axiom of infinity (see ax-infvn 9329 and bj-omex 9330) . (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.) | ||||||||||||
⊢ (𝜔 ∈ V ↔ ∃x(Ind x ∧ ∀y(Ind y → x ⊆ y))) | ||||||||||||||
30-Nov-2019 | bj-om 9325 | A set is equal to 𝜔 if and only if it is the smallest inductive set. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.) | ||||||||||||
⊢ (A ∈ 𝑉 → (A = 𝜔 ↔ (Ind A ∧ ∀x(Ind x → A ⊆ x)))) | ||||||||||||||
30-Nov-2019 | bj-ssom 9324 | A characterization of subclasses of 𝜔. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.) | ||||||||||||
⊢ (∀x(Ind x → A ⊆ x) ↔ A ⊆ 𝜔) | ||||||||||||||
30-Nov-2019 | bj-omssind 9323 | 𝜔 is included in all the inductive sets (but for the moment, we cannot prove that it is included in all the inductive classes). (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.) | ||||||||||||
⊢ (A ∈ 𝑉 → (Ind A → 𝜔 ⊆ A)) | ||||||||||||||
30-Nov-2019 | bj-omind 9322 | 𝜔 is an inductive class. (Contributed by BJ, 30-Nov-2019.) | ||||||||||||
⊢ Ind 𝜔 | ||||||||||||||
30-Nov-2019 | bj-dfom 9321 | Alternate definition of 𝜔, as the intersection of all the inductive sets. Proposal: make this the definition. (Contributed by BJ, 30-Nov-2019.) | ||||||||||||
⊢ 𝜔 = ∩ {x ∣ Ind x} | ||||||||||||||
30-Nov-2019 | bj-indint 9320 | The property of being an inductive class is closed under intersections. (Contributed by BJ, 30-Nov-2019.) | ||||||||||||
⊢ Ind ∩ {x ∈ A ∣ Ind x} | ||||||||||||||
30-Nov-2019 | bj-bdind 9319 | Boundedness of the formula "the setvar x is an inductive class". (Contributed by BJ, 30-Nov-2019.) | ||||||||||||
⊢ BOUNDED Ind x | ||||||||||||||
30-Nov-2019 | bj-indeq 9318 | Equality property for Ind. (Contributed by BJ, 30-Nov-2019.) | ||||||||||||
⊢ (A = B → (Ind A ↔ Ind B)) | ||||||||||||||
30-Nov-2019 | bj-indsuc 9317 | A direct consequence of the definition of Ind. (Contributed by BJ, 30-Nov-2019.) | ||||||||||||
⊢ (Ind A → (B ∈ A → suc B ∈ A)) | ||||||||||||||
30-Nov-2019 | df-bj-ind 9316 | Define the property of being an inductive class. (Contributed by BJ, 30-Nov-2019.) | ||||||||||||
⊢ (Ind A ↔ (∅ ∈ A ∧ ∀x ∈ A suc x ∈ A)) | ||||||||||||||
30-Nov-2019 | bj-bdsucel 9271 | Boundedness of the formula "the successor of the setvar x belongs to the setvar y". (Contributed by BJ, 30-Nov-2019.) | ||||||||||||
⊢ BOUNDED suc x ∈ y | ||||||||||||||
30-Nov-2019 | bj-bd0el 9257 | Boundedness of the formula "the empty set belongs to the setvar x". (Contributed by BJ, 30-Nov-2019.) | ||||||||||||
⊢ BOUNDED ∅ ∈ x | ||||||||||||||
30-Nov-2019 | bj-sseq 9200 | If two converse inclusions are characterized each by a formula, then equality is characterized by the conjunction of these formulas. (Contributed by BJ, 30-Nov-2019.) | ||||||||||||
⊢ (φ → (ψ ↔ A ⊆ B)) & ⊢ (φ → (χ ↔ B ⊆ A)) ⇒ ⊢ (φ → ((ψ ∧ χ) ↔ A = B)) | ||||||||||||||
30-Nov-2019 | nqpnq0nq 6435 | A positive fraction plus a non-negative fraction is a positive fraction. (Contributed by Jim Kingdon, 30-Nov-2019.) | ||||||||||||
⊢ ((A ∈ Q ∧ B ∈ Q_{0}) → (A +_{Q0} B) ∈ Q) | ||||||||||||||
30-Nov-2019 | mulclnq0 6434 | Closure of multiplication on non-negative fractions. (Contributed by Jim Kingdon, 30-Nov-2019.) | ||||||||||||
⊢ ((A ∈ Q_{0} ∧ B ∈ Q_{0}) → (A ·_{Q0} B) ∈ Q_{0}) | ||||||||||||||
30-Nov-2019 | prarloclemarch 6401 | A version of the Archimedean property. This variation is "stronger" than archnqq 6400 in the sense that we provide an integer which is larger than a given rational A even after being multiplied by a second rational B. (Contributed by Jim Kingdon, 30-Nov-2019.) | ||||||||||||
⊢ ((A ∈ Q ∧ B ∈ Q) → ∃x ∈ N A <_{Q} ([⟨x, 1_{𝑜}⟩] ~_{Q} ·_{Q} B)) | ||||||||||||||
29-Nov-2019 | bj-elssuniab 9199 | Version of elssuni 3599 using a class abstraction and explicit substitution. (Contributed by BJ, 29-Nov-2019.) | ||||||||||||
⊢ ℲxA ⇒ ⊢ (A ∈ 𝑉 → ([A / x]φ → A ⊆ ∪ {x ∣ φ})) | ||||||||||||||
29-Nov-2019 | bj-intabssel1 9198 | Version of intss1 3621 using a class abstraction and implicit substitution. Closed form of intmin3 3633. (Contributed by BJ, 29-Nov-2019.) | ||||||||||||
⊢ ℲxA & ⊢ Ⅎxψ & ⊢ (x = A → (ψ → φ)) ⇒ ⊢ (A ∈ 𝑉 → (ψ → ∩ {x ∣ φ} ⊆ A)) | ||||||||||||||
29-Nov-2019 | bj-intabssel 9197 | Version of intss1 3621 using a class abstraction and explicit substitution. (Contributed by BJ, 29-Nov-2019.) | ||||||||||||
⊢ ℲxA ⇒ ⊢ (A ∈ 𝑉 → ([A / x]φ → ∩ {x ∣ φ} ⊆ A)) | ||||||||||||||
29-Nov-2019 | nq02m 6447 | Multiply a non-negative fraction by two. (Contributed by Jim Kingdon, 29-Nov-2019.) | ||||||||||||
⊢ (A ∈ Q_{0} → ([⟨2_{𝑜}, 1_{𝑜}⟩] ~_{Q0} ·_{Q0} A) = (A +_{Q0} A)) | ||||||||||||||
29-Nov-2019 | distnq0r 6445 | Multiplication of non-negative fractions is distributive. Version of distrnq0 6441 with the multiplications commuted. (Contributed by Jim Kingdon, 29-Nov-2019.) | ||||||||||||
⊢ ((A ∈ Q_{0} ∧ B ∈ Q_{0} ∧ 𝐶 ∈ Q_{0}) → ((B +_{Q0} 𝐶) ·_{Q0} A) = ((B ·_{Q0} A) +_{Q0} (𝐶 ·_{Q0} A))) | ||||||||||||||
29-Nov-2019 | addassnq0 6444 | Addition of non-negaative fractions is associative. (Contributed by Jim Kingdon, 29-Nov-2019.) | ||||||||||||
⊢ ((A ∈ Q_{0} ∧ B ∈ Q_{0} ∧ 𝐶 ∈ Q_{0}) → ((A +_{Q0} B) +_{Q0} 𝐶) = (A +_{Q0} (B +_{Q0} 𝐶))) | ||||||||||||||
29-Nov-2019 | addclnq0 6433 | Closure of addition on non-negative fractions. (Contributed by Jim Kingdon, 29-Nov-2019.) | ||||||||||||
⊢ ((A ∈ Q_{0} ∧ B ∈ Q_{0}) → (A +_{Q0} B) ∈ Q_{0}) | ||||||||||||||
29-Nov-2019 | mulcanenq0ec 6427 | Lemma for distributive law: cancellation of common factor. (Contributed by Jim Kingdon, 29-Nov-2019.) | ||||||||||||
⊢ ((A ∈ N ∧ B ∈ 𝜔 ∧ 𝐶 ∈ N) → [⟨(A ·_{𝑜} B), (A ·_{𝑜} 𝐶)⟩] ~_{Q0} = [⟨B, 𝐶⟩] ~_{Q0} ) | ||||||||||||||
28-Nov-2019 | ax-bdsetind 9352 | Axiom of bounded set induction. (Contributed by BJ, 28-Nov-2019.) | ||||||||||||
⊢ BOUNDED φ ⇒ ⊢ (∀𝑎(∀y ∈ 𝑎 [y / 𝑎]φ → φ) → ∀𝑎φ) | ||||||||||||||
28-Nov-2019 | bj-peano4 9343 | Remove from peano4 4263 dependency on ax-setind 4220. Therefore, it only requires core constructive axioms (albeit more of them). (Contributed by BJ, 28-Nov-2019.) (Proof modification is discouraged.) | ||||||||||||
⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔) → (suc A = suc B ↔ A = B)) | ||||||||||||||
28-Nov-2019 | bj-nnen2lp 9342 |
A version of en2lp 4232 for natural numbers, which does not require
ax-setind 4220.
Note: using this theorem and bj-nnelirr 9341, one can remove dependency on ax-setind 4220 from nntri2 6012 and nndcel 6016; one can actually remove more dependencies from these. (Contributed by BJ, 28-Nov-2019.) (Proof modification is discouraged.) | ||||||||||||
⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔) → ¬ (A ∈ B ∧ B ∈ A)) | ||||||||||||||
27-Nov-2019 | mulcomnq0 6442 | Multiplication of non-negative fractions is commutative. (Contributed by Jim Kingdon, 27-Nov-2019.) | ||||||||||||
⊢ ((A ∈ Q_{0} ∧ B ∈ Q_{0}) → (A ·_{Q0} B) = (B ·_{Q0} A)) | ||||||||||||||
27-Nov-2019 | distrnq0 6441 | Multiplication of non-negative fractions is distributive. (Contributed by Jim Kingdon, 27-Nov-2019.) | ||||||||||||
⊢ ((A ∈ Q_{0} ∧ B ∈ Q_{0} ∧ 𝐶 ∈ Q_{0}) → (A ·_{Q0} (B +_{Q0} 𝐶)) = ((A ·_{Q0} B) +_{Q0} (A ·_{Q0} 𝐶))) | ||||||||||||||
25-Nov-2019 | prcunqu 6467 | An upper cut is closed upwards under the positive fractions. (Contributed by Jim Kingdon, 25-Nov-2019.) | ||||||||||||
⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐶 ∈ 𝑈) → (𝐶 <_{Q} B → B ∈ 𝑈)) | ||||||||||||||
25-Nov-2019 | prarloclemarch2 6402 | Like prarloclemarch 6401 but the integer must be at least two, and there is also B added to the right hand side. These details follow straightforwardly but are chosen to be helpful in the proof of prarloc 6485. (Contributed by Jim Kingdon, 25-Nov-2019.) | ||||||||||||
⊢ ((A ∈ Q ∧ B ∈ Q ∧ 𝐶 ∈ Q) → ∃x ∈ N (1_{𝑜} <_{N} x ∧ A <_{Q} (B +_{Q} ([⟨x, 1_{𝑜}⟩] ~_{Q} ·_{Q} 𝐶)))) | ||||||||||||||
25-Nov-2019 | subhalfnqq 6397 | There is a number which is less than half of any positive fraction. The case where A is one is Lemma 11.4 of [BauerTaylor], p. 50, and they use the word "approximate half" for such a number (since there may be constructions, for some structures other than the rationals themselves, which rely on such an approximate half but do not require division by two as seen at halfnqq 6393). (Contributed by Jim Kingdon, 25-Nov-2019.) | ||||||||||||
⊢ (A ∈ Q → ∃x ∈ Q (x +_{Q} x) <_{Q} A) | ||||||||||||||
24-Nov-2019 | bj-nnelirr 9341 | A natural number does not belong to itself. Version of elirr 4224 for natural numbers, which does not require ax-setind 4220. (Contributed by BJ, 24-Nov-2019.) (Proof modification is discouraged.) | ||||||||||||
⊢ (A ∈ 𝜔 → ¬ A ∈ A) | ||||||||||||||
24-Nov-2019 | bdcriota 9272 | A class given by a restricted definition binder is bounded, under the given hypotheses. (Contributed by BJ, 24-Nov-2019.) | ||||||||||||
⊢ BOUNDED φ & ⊢ ∃!x ∈ y φ ⇒ ⊢ BOUNDED (℩x ∈ y φ) | ||||||||||||||
24-Nov-2019 | nq0nn 6424 | Decomposition of a non-negative fraction into numerator and denominator. (Contributed by Jim Kingdon, 24-Nov-2019.) | ||||||||||||
⊢ (A ∈ Q_{0} → ∃w∃v((w ∈ 𝜔 ∧ v ∈ N) ∧ A = [⟨w, v⟩] ~_{Q0} )) | ||||||||||||||
24-Nov-2019 | enq0eceq 6419 | Equivalence class equality of non-negative fractions in terms of natural numbers. (Contributed by Jim Kingdon, 24-Nov-2019.) | ||||||||||||
⊢ (((A ∈ 𝜔 ∧ B ∈ N) ∧ (𝐶 ∈ 𝜔 ∧ 𝐷 ∈ N)) → ([⟨A, B⟩] ~_{Q0} = [⟨𝐶, 𝐷⟩] ~_{Q0} ↔ (A ·_{𝑜} 𝐷) = (B ·_{𝑜} 𝐶))) | ||||||||||||||
24-Nov-2019 | dfplq0qs 6412 | Addition on non-negative fractions. This definition is similar to df-plq0 6409 but expands Q_{0} (Contributed by Jim Kingdon, 24-Nov-2019.) | ||||||||||||
⊢ +_{Q0} = {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ ((𝜔 × N) / ~_{Q0} ) ∧ y ∈ ((𝜔 × N) / ~_{Q0} )) ∧ ∃w∃v∃u∃f((x = [⟨w, v⟩] ~_{Q0} ∧ y = [⟨u, f⟩] ~_{Q0} ) ∧ z = [⟨((w ·_{𝑜} f) +_{𝑜} (v ·_{𝑜} u)), (v ·_{𝑜} f)⟩] ~_{Q0} ))} | ||||||||||||||
23-Nov-2019 | addnq0mo 6429 | There is at most one result from adding non-negative fractions. (Contributed by Jim Kingdon, 23-Nov-2019.) | ||||||||||||
⊢ ((A ∈ ((𝜔 × N) / ~_{Q0} ) ∧ B ∈ ((𝜔 × N) / ~_{Q0} )) → ∃*z∃w∃v∃u∃𝑡((A = [⟨w, v⟩] ~_{Q0} ∧ B = [⟨u, 𝑡⟩] ~_{Q0} ) ∧ z = [⟨((w ·_{𝑜} 𝑡) +_{𝑜} (v ·_{𝑜} u)), (v ·_{𝑜} 𝑡)⟩] ~_{Q0} )) | ||||||||||||||
23-Nov-2019 | nnnq0lem1 6428 | Decomposing non-negative fractions into natural numbers. Lemma for addnnnq0 6431 and mulnnnq0 6432. (Contributed by Jim Kingdon, 23-Nov-2019.) | ||||||||||||
⊢ (((A ∈ ((𝜔 × N) / ~_{Q0} ) ∧ B ∈ ((𝜔 × N) / ~_{Q0} )) ∧ (((A = [⟨w, v⟩] ~_{Q0} ∧ B = [⟨u, 𝑡⟩] ~_{Q0} ) ∧ z = [𝐶] ~_{Q0} ) ∧ ((A = [⟨𝑠, f⟩] ~_{Q0} ∧ B = [⟨g, ℎ⟩] ~_{Q0} ) ∧ 𝑞 = [𝐷] ~_{Q0} ))) → ((((w ∈ 𝜔 ∧ v ∈ N) ∧ (𝑠 ∈ 𝜔 ∧ f ∈ N)) ∧ ((u ∈ 𝜔 ∧ 𝑡 ∈ N) ∧ (g ∈ 𝜔 ∧ ℎ ∈ N))) ∧ ((w ·_{𝑜} f) = (v ·_{𝑜} 𝑠) ∧ (u ·_{𝑜} ℎ) = (𝑡 ·_{𝑜} g)))) | ||||||||||||||
23-Nov-2019 | addcmpblnq0 6425 | Lemma showing compatibility of addition on non-negative fractions. (Contributed by Jim Kingdon, 23-Nov-2019.) | ||||||||||||
⊢ ((((A ∈ 𝜔 ∧ B ∈ N) ∧ (𝐶 ∈ 𝜔 ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ 𝜔 ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ 𝜔 ∧ 𝑆 ∈ N))) → (((A ·_{𝑜} 𝐷) = (B ·_{𝑜} 𝐶) ∧ (𝐹 ·_{𝑜} 𝑆) = (𝐺 ·_{𝑜} 𝑅)) → ⟨((A ·_{𝑜} 𝐺) +_{𝑜} (B ·_{𝑜} 𝐹)), (B ·_{𝑜} 𝐺)⟩ ~_{Q0} ⟨((𝐶 ·_{𝑜} 𝑆) +_{𝑜} (𝐷 ·_{𝑜} 𝑅)), (𝐷 ·_{𝑜} 𝑆)⟩)) | ||||||||||||||
23-Nov-2019 | ee8anv 1807 | Rearrange existential quantifiers. (Contributed by Jim Kingdon, 23-Nov-2019.) | ||||||||||||
⊢ (∃x∃y∃z∃w∃v∃u∃𝑡∃𝑠(φ ∧ ψ) ↔ (∃x∃y∃z∃wφ ∧ ∃v∃u∃𝑡∃𝑠ψ)) | ||||||||||||||
23-Nov-2019 | 19.42vvvv 1787 | Theorem 19.42 of [Margaris] p. 90 with 4 quantifiers. (Contributed by Jim Kingdon, 23-Nov-2019.) | ||||||||||||
⊢ (∃w∃x∃y∃z(φ ∧ ψ) ↔ (φ ∧ ∃w∃x∃y∃zψ)) | ||||||||||||||
22-Nov-2019 | bdsetindis 9353 | Axiom of bounded set induction using implicit substitutions. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.) | ||||||||||||
⊢ BOUNDED φ & ⊢ Ⅎxψ & ⊢ Ⅎxχ & ⊢ Ⅎyφ & ⊢ Ⅎyψ & ⊢ (x = z → (φ → ψ)) & ⊢ (x = y → (χ → φ)) ⇒ ⊢ (∀y(∀z ∈ y ψ → χ) → ∀xφ) | ||||||||||||||
22-Nov-2019 | setindis 9351 | Axiom of set induction using implicit substitutions. (Contributed by BJ, 22-Nov-2019.) | ||||||||||||
⊢ Ⅎxψ & ⊢ Ⅎxχ & ⊢ Ⅎyφ & ⊢ Ⅎyψ & ⊢ (x = z → (φ → ψ)) & ⊢ (x = y → (χ → φ)) ⇒ ⊢ (∀y(∀z ∈ y ψ → χ) → ∀xφ) | ||||||||||||||
22-Nov-2019 | setindf 9350 | Axiom of set-induction with a DV condition replaced with a non-freeness hypothesis (Contributed by BJ, 22-Nov-2019.) | ||||||||||||
⊢ Ⅎyφ ⇒ ⊢ (∀x(∀y ∈ x [y / x]φ → φ) → ∀xφ) | ||||||||||||||
22-Nov-2019 | setindft 9349 | Axiom of set-induction with a DV condition replaced with a non-freeness hypothesis (Contributed by BJ, 22-Nov-2019.) | ||||||||||||
⊢ (∀xℲyφ → (∀x(∀y ∈ x [y / x]φ → φ) → ∀xφ)) | ||||||||||||||
22-Nov-2019 | bj-nntrans2 9340 | A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.) | ||||||||||||
⊢ (A ∈ 𝜔 → Tr A) | ||||||||||||||
22-Nov-2019 | bj-nntrans 9339 | A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.) | ||||||||||||
⊢ (A ∈ 𝜔 → (B ∈ A → B ⊆ A)) | ||||||||||||||
22-Nov-2019 | bdfind 9334 | Bounded induction (principle of induction when A is assumed to be bounded), proved from basic constructive axioms. See find 4265 for a nonconstructive proof of the general case. See findset 9333 for a proof when A is assumed to be a set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.) | ||||||||||||
⊢ BOUNDED A ⇒ ⊢ ((A ⊆ 𝜔 ∧ ∅ ∈ A ∧ ∀x ∈ A suc x ∈ A) → A = 𝜔) | ||||||||||||||
22-Nov-2019 | findset 9333 | Bounded induction (principle of induction when A is assumed to be a set) allowing a proof from basic constructive axioms. See find 4265 for a nonconstructive proof of the general case. See bdfind 9334 for a proof when A is assumed to be bounded. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.) | ||||||||||||
⊢ (A ∈ 𝑉 → ((A ⊆ 𝜔 ∧ ∅ ∈ A ∧ ∀x ∈ A suc x ∈ A) → A = 𝜔)) | ||||||||||||||
22-Nov-2019 | cbvrald 9196 | Rule used to change bound variables, using implicit substitution. (Contributed by BJ, 22-Nov-2019.) | ||||||||||||
⊢ Ⅎxφ & ⊢ Ⅎyφ & ⊢ (φ → Ⅎyψ) & ⊢ (φ → Ⅎxχ) & ⊢ (φ → (x = y → (ψ ↔ χ))) ⇒ ⊢ (φ → (∀x ∈ A ψ ↔ ∀y ∈ A χ)) | ||||||||||||||
22-Nov-2019 | addnnnq0 6431 | Addition of non-negative fractions in terms of natural numbers. (Contributed by Jim Kingdon, 22-Nov-2019.) | ||||||||||||
⊢ (((A ∈ 𝜔 ∧ B ∈ N) ∧ (𝐶 ∈ 𝜔 ∧ 𝐷 ∈ N)) → ([⟨A, B⟩] ~_{Q0} +_{Q0} [⟨𝐶, 𝐷⟩] ~_{Q0} ) = [⟨((A ·_{𝑜} 𝐷) +_{𝑜} (B ·_{𝑜} 𝐶)), (B ·_{𝑜} 𝐷)⟩] ~_{Q0} ) | ||||||||||||||
22-Nov-2019 | dfmq0qs 6411 | Multiplication on non-negative fractions. This definition is similar to df-mq0 6410 but expands Q_{0} (Contributed by Jim Kingdon, 22-Nov-2019.) | ||||||||||||
⊢ ·_{Q0} = {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ ((𝜔 × N) / ~_{Q0} ) ∧ y ∈ ((𝜔 × N) / ~_{Q0} )) ∧ ∃w∃v∃u∃f((x = [⟨w, v⟩] ~_{Q0} ∧ y = [⟨u, f⟩] ~_{Q0} ) ∧ z = [⟨(w ·_{𝑜} u), (v ·_{𝑜} f)⟩] ~_{Q0} ))} | ||||||||||||||
21-Nov-2019 | bj-findes 9365 | Principle of induction, using explicit substitutions. Constructive proof (from CZF). See the comment of bj-findis 9363 for explanations. From this version, it is easy to prove findes 4269. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) | ||||||||||||
⊢ (([∅ / x]φ ∧ ∀x ∈ 𝜔 (φ → [suc x / x]φ)) → ∀x ∈ 𝜔 φ) | ||||||||||||||
21-Nov-2019 | bj-findisg 9364 | Version of bj-findis 9363 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-findis 9363 for explanations. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) | ||||||||||||
⊢ Ⅎxψ & ⊢ Ⅎxχ & ⊢ Ⅎxθ & ⊢ (x = ∅ → (ψ → φ)) & ⊢ (x = y → (φ → χ)) & ⊢ (x = suc y → (θ → φ)) & ⊢ ℲxA & ⊢ Ⅎxτ & ⊢ (x = A → (φ → τ)) ⇒ ⊢ ((ψ ∧ ∀y ∈ 𝜔 (χ → θ)) → (A ∈ 𝜔 → τ)) | ||||||||||||||
21-Nov-2019 | bj-bdfindes 9337 | Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 9335 for explanations. From this version, it is easy to prove the bounded version of findes 4269. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) | ||||||||||||
⊢ BOUNDED φ ⇒ ⊢ (([∅ / x]φ ∧ ∀x ∈ 𝜔 (φ → [suc x / x]φ)) → ∀x ∈ 𝜔 φ) | ||||||||||||||
21-Nov-2019 | bj-bdfindisg 9336 | Version of bj-bdfindis 9335 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 9335 for explanations. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) | ||||||||||||
⊢ BOUNDED φ & ⊢ Ⅎxψ & ⊢ Ⅎxχ & ⊢ Ⅎxθ & ⊢ (x = ∅ → (ψ → φ)) & ⊢ (x = y → (φ → χ)) & ⊢ (x = suc y → (θ → φ)) & ⊢ ℲxA & ⊢ Ⅎxτ & ⊢ (x = A → (φ → τ)) ⇒ ⊢ ((ψ ∧ ∀y ∈ 𝜔 (χ → θ)) → (A ∈ 𝜔 → τ)) | ||||||||||||||
21-Nov-2019 | bj-bdfindis 9335 | Bounded induction (principle of induction for bounded formulas), using implicit substitutions (the biconditional versions of the hypotheses are implicit substitutions, and we have weakened them to implications). Constructive proof (from CZF). See finds 4266 for a proof of full induction in IZF. From this version, it is easy to prove bounded versions of finds 4266, finds2 4267, finds1 4268. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) | ||||||||||||
⊢ BOUNDED φ & ⊢ Ⅎxψ & ⊢ Ⅎxχ & ⊢ Ⅎxθ & ⊢ (x = ∅ → (ψ → φ)) & ⊢ (x = y → (φ → χ)) & ⊢ (x = suc y → (θ → φ)) ⇒ ⊢ ((ψ ∧ ∀y ∈ 𝜔 (χ → θ)) → ∀x ∈ 𝜔 φ) | ||||||||||||||
21-Nov-2019 | bdeqsuc 9270 | Boundedness of the formula expressing that a setvar is equal to the successor of another. (Contributed by BJ, 21-Nov-2019.) | ||||||||||||
⊢ BOUNDED x = suc y | ||||||||||||||
21-Nov-2019 | bdop 9264 | The ordered pair of two setvars is a bounded class. (Contributed by BJ, 21-Nov-2019.) | ||||||||||||
⊢ BOUNDED ⟨x, y⟩ | ||||||||||||||
21-Nov-2019 | bdeq0 9256 | Boundedness of the formula expressing that a setvar is equal to the empty class. (Contributed by BJ, 21-Nov-2019.) | ||||||||||||
⊢ BOUNDED x = ∅ | ||||||||||||||
21-Nov-2019 | bj-rspg 9195 | Restricted specialization, generalized. Weakens a hypothesis of rspccv 2647 and seems to have a shorter proof. (Contributed by BJ, 21-Nov-2019.) | ||||||||||||
⊢ ℲxA & ⊢ ℲxB & ⊢ Ⅎxψ & ⊢ (x = A → (φ → ψ)) ⇒ ⊢ (∀x ∈ B φ → (A ∈ B → ψ)) | ||||||||||||||
21-Nov-2019 | bj-rspgt 9194 | Restricted specialization, generalized. Weakens a hypothesis of rspccv 2647 and seems to have a shorter proof. (Contributed by BJ, 21-Nov-2019.) | ||||||||||||
⊢ ℲxA & ⊢ ℲxB & ⊢ Ⅎxψ ⇒ ⊢ (∀x(x = A → (φ → ψ)) → (∀x ∈ B φ → (A ∈ B → ψ))) | ||||||||||||||
21-Nov-2019 | elabg2 9193 | One implication of elabg 2682. (Contributed by BJ, 21-Nov-2019.) | ||||||||||||
⊢ (x = A → (ψ → φ)) ⇒ ⊢ (A ∈ 𝑉 → (ψ → A ∈ {x ∣ φ})) | ||||||||||||||
21-Nov-2019 | elab2a 9192 | One implication of elab 2681. (Contributed by BJ, 21-Nov-2019.) | ||||||||||||
⊢ A ∈ V & ⊢ (x = A → (ψ → φ)) ⇒ ⊢ (ψ → A ∈ {x ∣ φ}) | ||||||||||||||
21-Nov-2019 | elab1 9191 | One implication of elab 2681. (Contributed by BJ, 21-Nov-2019.) | ||||||||||||
⊢ (x = A → (φ → ψ)) ⇒ ⊢ (A ∈ {x ∣ φ} → ψ) | ||||||||||||||
21-Nov-2019 | elabf2 9190 | One implication of elabf 2680. (Contributed by BJ, 21-Nov-2019.) | ||||||||||||
⊢ Ⅎxψ & ⊢ A ∈ V & ⊢ (x = A → (ψ → φ)) ⇒ ⊢ (ψ → A ∈ {x ∣ φ}) | ||||||||||||||
21-Nov-2019 | elabf1 9189 | One implication of elabf 2680. (Contributed by BJ, 21-Nov-2019.) | ||||||||||||
⊢ Ⅎxψ & ⊢ (x = A → (φ → ψ)) ⇒ ⊢ (A ∈ {x ∣ φ} → ψ) | ||||||||||||||
21-Nov-2019 | elabgf2 9188 | One implication of elabgf 2679. (Contributed by BJ, 21-Nov-2019.) | ||||||||||||
⊢ ℲxA & ⊢ Ⅎxψ & ⊢ (x = A → (ψ → φ)) ⇒ ⊢ (A ∈ B → (ψ → A ∈ {x ∣ φ})) | ||||||||||||||
21-Nov-2019 | elabgf1 9187 | One implication of elabgf 2679. (Contributed by BJ, 21-Nov-2019.) | ||||||||||||
⊢ ℲxA & ⊢ Ⅎxψ & ⊢ (x = A → (φ → ψ)) ⇒ ⊢ (A ∈ {x ∣ φ} → ψ) | ||||||||||||||
21-Nov-2019 | elabgft1 9186 | One implication of elabgf 2679, in closed form. (Contributed by BJ, 21-Nov-2019.) | ||||||||||||
⊢ ℲxA & ⊢ Ⅎxψ ⇒ ⊢ (∀x(x = A → (φ → ψ)) → (A ∈ {x ∣ φ} → ψ)) | ||||||||||||||
21-Nov-2019 | elabgf0 9185 | Lemma for elabgf 2679. (Contributed by BJ, 21-Nov-2019.) | ||||||||||||
⊢ (x = A → (A ∈ {x ∣ φ} ↔ φ)) | ||||||||||||||
21-Nov-2019 | bj-vtoclgf 9184 | Weakening two hypotheses of vtoclgf 2606. (Contributed by BJ, 21-Nov-2019.) | ||||||||||||
⊢ ℲxA & ⊢ Ⅎxψ & ⊢ (x = A → φ) & ⊢ (x = A → (φ → ψ)) ⇒ ⊢ (A ∈ 𝑉 → ψ) | ||||||||||||||
21-Nov-2019 | bj-vtoclgft 9183 | Weakening two hypotheses of vtoclgf 2606. (Contributed by BJ, 21-Nov-2019.) | ||||||||||||
⊢ ℲxA & ⊢ Ⅎxψ & ⊢ (x = A → φ) ⇒ ⊢ (∀x(x = A → (φ → ψ)) → (A ∈ 𝑉 → ψ)) | ||||||||||||||
21-Nov-2019 | bj-exlimmpi 9179 | Lemma for bj-vtoclgf 9184. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) | ||||||||||||
⊢ Ⅎxψ & ⊢ (χ → φ) & ⊢ (χ → (φ → ψ)) ⇒ ⊢ (∃xχ → ψ) | ||||||||||||||
21-Nov-2019 | bj-exlimmp 9178 | Lemma for bj-vtoclgf 9184. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) | ||||||||||||
⊢ Ⅎxψ & ⊢ (χ → φ) ⇒ ⊢ (∀x(χ → (φ → ψ)) → (∃xχ → ψ)) | ||||||||||||||
21-Nov-2019 | 3anim2i 1090 | Add two conjuncts to antecedent and consequent. (Contributed by AV, 21-Nov-2019.) | ||||||||||||
⊢ (φ → ψ) ⇒ ⊢ ((χ ∧ φ ∧ θ) → (χ ∧ ψ ∧ θ)) | ||||||||||||||
20-Nov-2019 | mulnq0mo 6430 | There is at most one result from multiplying non-negative fractions. (Contributed by Jim Kingdon, 20-Nov-2019.) | ||||||||||||
⊢ ((A ∈ ((𝜔 × N) / ~_{Q0} ) ∧ B ∈ ((𝜔 × N) / ~_{Q0} )) → ∃*z∃w∃v∃u∃𝑡((A = [⟨w, v⟩] ~_{Q0} ∧ B = [⟨u, 𝑡⟩] ~_{Q0} ) ∧ z = [⟨(w ·_{𝑜} u), (v ·_{𝑜} 𝑡)⟩] ~_{Q0} )) | ||||||||||||||
20-Nov-2019 | mulcmpblnq0 6426 | Lemma showing compatibility of multiplication on non-negative fractions. (Contributed by Jim Kingdon, 20-Nov-2019.) | ||||||||||||