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Theorem List for Intuitionistic Logic Explorer - 7701-7800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdivcanap2zi 7701 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐵 # 0 → (𝐵 · (𝐴 / 𝐵)) = 𝐴)

Theoremdivrecapzi 7702 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐵 # 0 → (𝐴 / 𝐵) = (𝐴 · (1 / 𝐵)))

Theoremdivcanap3zi 7703 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐵 # 0 → ((𝐵 · 𝐴) / 𝐵) = 𝐴)

Theoremdivcanap4zi 7704 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐵 # 0 → ((𝐴 · 𝐵) / 𝐵) = 𝐴)

Theoremrec11api 7705 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))

Theoremdivclapi 7706 Closure law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐵 # 0       (𝐴 / 𝐵) ∈ ℂ

Theoremdivcanap2i 7707 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐵 # 0       (𝐵 · (𝐴 / 𝐵)) = 𝐴

Theoremdivcanap1i 7708 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐵 # 0       ((𝐴 / 𝐵) · 𝐵) = 𝐴

Theoremdivrecapi 7709 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐵 # 0       (𝐴 / 𝐵) = (𝐴 · (1 / 𝐵))

Theoremdivcanap3i 7710 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐵 # 0       ((𝐵 · 𝐴) / 𝐵) = 𝐴

Theoremdivcanap4i 7711 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐵 # 0       ((𝐴 · 𝐵) / 𝐵) = 𝐴

Theoremdivap0i 7712 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

Theoremrec11apii 7713 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)

Theoremdivassapzi 7714 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)))

Theoremdivmulapzi 7715 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))

Theoremdivdirapzi 7716 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)))

Theoremdivdiv23apzi 7717 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))

Theoremdivmulapi 7718 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)

Theoremdivdiv32api 7719 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)

Theoremdivassapi 7720 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))

Theoremdivdirapi 7721 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))

Theoremdiv23api 7722 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)

Theoremdiv11api 7723 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)

Theoremdivmuldivapi 7724 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))

Theoremdivmul13api 7725 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))

𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐷 ∈ ℂ    &   𝐵 # 0    &   𝐷 # 0       ((𝐴 / 𝐵) + (𝐶 / 𝐷)) = (((𝐴 · 𝐷) + (𝐶 · 𝐵)) / (𝐵 · 𝐷))

Theoremdivdivdivapi 7727 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))

Theoremrerecclapzi 7728 Closure law for reciprocal. (Contributed by Jim Kingdon, 9-Mar-2020.)
𝐴 ∈ ℝ       (𝐴 # 0 → (1 / 𝐴) ∈ ℝ)

Theoremrerecclapi 7729 Closure law for reciprocal. (Contributed by Jim Kingdon, 9-Mar-2020.)
𝐴 ∈ ℝ    &   𝐴 # 0       (1 / 𝐴) ∈ ℝ

Theoremredivclapzi 7730 Closure law for division of reals. (Contributed by Jim Kingdon, 9-Mar-2020.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐵 # 0 → (𝐴 / 𝐵) ∈ ℝ)

Theoremredivclapi 7731 Closure law for division of reals. (Contributed by Jim Kingdon, 9-Mar-2020.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐵 # 0       (𝐴 / 𝐵) ∈ ℝ

Theoremdiv1d 7732 A number divided by 1 is itself. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 / 1) = 𝐴)

Theoremrecclapd 7733 Closure law for reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 # 0)       (𝜑 → (1 / 𝐴) ∈ ℂ)

Theoremrecap0d 7734 The reciprocal of a number apart from zero is apart from zero. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 # 0)       (𝜑 → (1 / 𝐴) # 0)

Theoremrecidapd 7735 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 # 0)       (𝜑 → (𝐴 · (1 / 𝐴)) = 1)

Theoremrecidap2d 7736 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 # 0)       (𝜑 → ((1 / 𝐴) · 𝐴) = 1)

Theoremrecrecapd 7737 A number is equal to the reciprocal of its reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 # 0)       (𝜑 → (1 / (1 / 𝐴)) = 𝐴)

Theoremdividapd 7738 A number divided by itself is one. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 # 0)       (𝜑 → (𝐴 / 𝐴) = 1)

Theoremdiv0apd 7739 Division into zero is zero. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 # 0)       (𝜑 → (0 / 𝐴) = 0)

Theoremapmul1 7740 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)))

Theoremdivclapd 7741 Closure law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → (𝐴 / 𝐵) ∈ ℂ)

Theoremdivcanap1d 7742 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → ((𝐴 / 𝐵) · 𝐵) = 𝐴)

Theoremdivcanap2d 7743 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → (𝐵 · (𝐴 / 𝐵)) = 𝐴)

Theoremdivrecapd 7744 Relationship between division and reciprocal. Theorem I.9 of [Apostol] p. 18. (Contributed by Jim Kingdon, 29-Feb-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → (𝐴 / 𝐵) = (𝐴 · (1 / 𝐵)))

Theoremdivrecap2d 7745 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 29-Feb-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → (𝐴 / 𝐵) = ((1 / 𝐵) · 𝐴))

Theoremdivcanap3d 7746 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → ((𝐵 · 𝐴) / 𝐵) = 𝐴)

Theoremdivcanap4d 7747 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → ((𝐴 · 𝐵) / 𝐵) = 𝐴)

Theoremdiveqap0d 7748 If a ratio is zero, the numerator is zero. (Contributed by Jim Kingdon, 19-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)    &   (𝜑 → (𝐴 / 𝐵) = 0)       (𝜑𝐴 = 0)

Theoremdiveqap1d 7749 Equality in terms of unit ratio. (Contributed by Jim Kingdon, 19-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)    &   (𝜑 → (𝐴 / 𝐵) = 1)       (𝜑𝐴 = 𝐵)

Theoremdiveqap1ad 7750 The quotient of two complex numbers is one iff they are equal. Deduction form of diveqap1 7658. Generalization of diveqap1d 7749. (Contributed by Jim Kingdon, 19-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → ((𝐴 / 𝐵) = 1 ↔ 𝐴 = 𝐵))

Theoremdiveqap0ad 7751 A fraction of complex numbers is zero iff its numerator is. Deduction form of diveqap0 7637. (Contributed by Jim Kingdon, 19-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → ((𝐴 / 𝐵) = 0 ↔ 𝐴 = 0))

Theoremdivap1d 7752 If two complex numbers are apart, their quotient is apart from one. (Contributed by Jim Kingdon, 20-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)    &   (𝜑𝐴 # 𝐵)       (𝜑 → (𝐴 / 𝐵) # 1)

Theoremdivap0bd 7753 A ratio is zero iff the numerator is zero. (Contributed by Jim Kingdon, 19-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → (𝐴 # 0 ↔ (𝐴 / 𝐵) # 0))

Theoremdivnegapd 7754 Move negative sign inside of a division. (Contributed by Jim Kingdon, 19-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → -(𝐴 / 𝐵) = (-𝐴 / 𝐵))

Theoremdivneg2apd 7755 Move negative sign inside of a division. (Contributed by Jim Kingdon, 19-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → -(𝐴 / 𝐵) = (𝐴 / -𝐵))

Theoremdiv2negapd 7756 Quotient of two negatives. (Contributed by Jim Kingdon, 19-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → (-𝐴 / -𝐵) = (𝐴 / 𝐵))

Theoremdivap0d 7757 The ratio of numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐴 # 0)    &   (𝜑𝐵 # 0)       (𝜑 → (𝐴 / 𝐵) # 0)

Theoremrecdivapd 7758 The reciprocal of a ratio. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐴 # 0)    &   (𝜑𝐵 # 0)       (𝜑 → (1 / (𝐴 / 𝐵)) = (𝐵 / 𝐴))

Theoremrecdivap2d 7759 Division into a reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐴 # 0)    &   (𝜑𝐵 # 0)       (𝜑 → ((1 / 𝐴) / 𝐵) = (1 / (𝐴 · 𝐵)))

Theoremdivcanap6d 7760 Cancellation of inverted fractions. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐴 # 0)    &   (𝜑𝐵 # 0)       (𝜑 → ((𝐴 / 𝐵) · (𝐵 / 𝐴)) = 1)

Theoremddcanapd 7761 Cancellation in a double division. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐴 # 0)    &   (𝜑𝐵 # 0)       (𝜑 → (𝐴 / (𝐴 / 𝐵)) = 𝐵)

Theoremrec11apd 7762 Reciprocal is one-to-one. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐴 # 0)    &   (𝜑𝐵 # 0)    &   (𝜑 → (1 / 𝐴) = (1 / 𝐵))       (𝜑𝐴 = 𝐵)

Theoremdivmulapd 7763 Relationship between division and multiplication. (Contributed by Jim Kingdon, 8-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → ((𝐴 / 𝐵) = 𝐶 ↔ (𝐵 · 𝐶) = 𝐴))

Theoremdiv32apd 7764 A commutative/associative law for division. (Contributed by Jim Kingdon, 8-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → ((𝐴 / 𝐵) · 𝐶) = (𝐴 · (𝐶 / 𝐵)))

Theoremdiv13apd 7765 A commutative/associative law for division. (Contributed by Jim Kingdon, 8-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → ((𝐴 / 𝐵) · 𝐶) = ((𝐶 / 𝐵) · 𝐴))

Theoremdivdiv32apd 7766 Swap denominators in a division. (Contributed by Jim Kingdon, 8-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵 # 0)    &   (𝜑𝐶 # 0)       (𝜑 → ((𝐴 / 𝐵) / 𝐶) = ((𝐴 / 𝐶) / 𝐵))

Theoremdivcanap5d 7767 Cancellation of common factor in a ratio. (Contributed by Jim Kingdon, 8-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵 # 0)    &   (𝜑𝐶 # 0)       (𝜑 → ((𝐶 · 𝐴) / (𝐶 · 𝐵)) = (𝐴 / 𝐵))

Theoremdivcanap5rd 7768 Cancellation of common factor in a ratio. (Contributed by Jim Kingdon, 8-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵 # 0)    &   (𝜑𝐶 # 0)       (𝜑 → ((𝐴 · 𝐶) / (𝐵 · 𝐶)) = (𝐴 / 𝐵))

Theoremdivcanap7d 7769 Cancel equal divisors in a division. (Contributed by Jim Kingdon, 8-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵 # 0)    &   (𝜑𝐶 # 0)       (𝜑 → ((𝐴 / 𝐶) / (𝐵 / 𝐶)) = (𝐴 / 𝐵))

Theoremdmdcanapd 7770 Cancellation law for division and multiplication. (Contributed by Jim Kingdon, 8-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵 # 0)    &   (𝜑𝐶 # 0)       (𝜑 → ((𝐵 / 𝐶) · (𝐴 / 𝐵)) = (𝐴 / 𝐶))

Theoremdmdcanap2d 7771 Cancellation law for division and multiplication. (Contributed by Jim Kingdon, 8-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵 # 0)    &   (𝜑𝐶 # 0)       (𝜑 → ((𝐴 / 𝐵) · (𝐵 / 𝐶)) = (𝐴 / 𝐶))

Theoremdivdivap1d 7772 Division into a fraction. (Contributed by Jim Kingdon, 8-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵 # 0)    &   (𝜑𝐶 # 0)       (𝜑 → ((𝐴 / 𝐵) / 𝐶) = (𝐴 / (𝐵 · 𝐶)))

Theoremdivdivap2d 7773 Division by a fraction. (Contributed by Jim Kingdon, 8-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵 # 0)    &   (𝜑𝐶 # 0)       (𝜑 → (𝐴 / (𝐵 / 𝐶)) = ((𝐴 · 𝐶) / 𝐵))

Theoremdivmulap2d 7774 Relationship between division and multiplication. (Contributed by Jim Kingdon, 2-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐶 # 0)       (𝜑 → ((𝐴 / 𝐶) = 𝐵𝐴 = (𝐶 · 𝐵)))

Theoremdivmulap3d 7775 Relationship between division and multiplication. (Contributed by Jim Kingdon, 2-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐶 # 0)       (𝜑 → ((𝐴 / 𝐶) = 𝐵𝐴 = (𝐵 · 𝐶)))

Theoremdivassapd 7776 An associative law for division. (Contributed by Jim Kingdon, 2-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐶 # 0)       (𝜑 → ((𝐴 · 𝐵) / 𝐶) = (𝐴 · (𝐵 / 𝐶)))

Theoremdiv12apd 7777 A commutative/associative law for division. (Contributed by Jim Kingdon, 2-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐶 # 0)       (𝜑 → (𝐴 · (𝐵 / 𝐶)) = (𝐵 · (𝐴 / 𝐶)))

Theoremdiv23apd 7778 A commutative/associative law for division. (Contributed by Jim Kingdon, 2-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐶 # 0)       (𝜑 → ((𝐴 · 𝐵) / 𝐶) = ((𝐴 / 𝐶) · 𝐵))

Theoremdivdirapd 7779 Distribution of division over addition. (Contributed by Jim Kingdon, 2-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐶 # 0)       (𝜑 → ((𝐴 + 𝐵) / 𝐶) = ((𝐴 / 𝐶) + (𝐵 / 𝐶)))

Theoremdivsubdirapd 7780 Distribution of division over subtraction. (Contributed by Jim Kingdon, 2-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐶 # 0)       (𝜑 → ((𝐴𝐵) / 𝐶) = ((𝐴 / 𝐶) − (𝐵 / 𝐶)))

Theoremdiv11apd 7781 One-to-one relationship for division. (Contributed by Jim Kingdon, 2-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐶 # 0)    &   (𝜑 → (𝐴 / 𝐶) = (𝐵 / 𝐶))       (𝜑𝐴 = 𝐵)

Theoremdivmuldivapd 7782 Multiplication of two ratios. (Contributed by Jim Kingdon, 30-Jul-2021.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)    &   (𝜑𝐵 # 0)    &   (𝜑𝐷 # 0)       (𝜑 → ((𝐴 / 𝐵) · (𝐶 / 𝐷)) = ((𝐴 · 𝐶) / (𝐵 · 𝐷)))

Theoremrerecclapd 7783 Closure law for reciprocal. (Contributed by Jim Kingdon, 29-Feb-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐴 # 0)       (𝜑 → (1 / 𝐴) ∈ ℝ)

Theoremredivclapd 7784 Closure law for division of reals. (Contributed by Jim Kingdon, 29-Feb-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐵 # 0)       (𝜑 → (𝐴 / 𝐵) ∈ ℝ)

Theoremmvllmulapd 7785 Move LHS left multiplication to RHS. (Contributed by Jim Kingdon, 10-Jun-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐴 # 0)    &   (𝜑 → (𝐴 · 𝐵) = 𝐶)       (𝜑𝐵 = (𝐶 / 𝐴))

3.3.9  Ordering on reals (cont.)

Theoremltp1 7786 A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)
(𝐴 ∈ ℝ → 𝐴 < (𝐴 + 1))

Theoremlep1 7787 A number is less than or equal to itself plus 1. (Contributed by NM, 5-Jan-2006.)
(𝐴 ∈ ℝ → 𝐴 ≤ (𝐴 + 1))

Theoremltm1 7788 A number minus 1 is less than itself. (Contributed by NM, 9-Apr-2006.)
(𝐴 ∈ ℝ → (𝐴 − 1) < 𝐴)

Theoremlem1 7789 A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 2-Oct-2015.)
(𝐴 ∈ ℝ → (𝐴 − 1) ≤ 𝐴)

Theoremletrp1 7790 A transitive property of 'less than or equal' and plus 1. (Contributed by NM, 5-Aug-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → 𝐴 ≤ (𝐵 + 1))

Theoremp1le 7791 A transitive property of plus 1 and 'less than or equal'. (Contributed by NM, 16-Aug-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐴 + 1) ≤ 𝐵) → 𝐴𝐵)

Theoremrecgt0 7792 The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 25-Aug-1999.) (Revised by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 0 < (1 / 𝐴))

Theoremprodgt0gt0 7793 Infer that a multiplicand is positive from a positive multiplier and positive product. See prodgt0 7794 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)

Theoremprodgt0 7794 Infer that a multiplicand is positive from a nonnegative multiplier and positive product. (Contributed by NM, 24-Apr-2005.) (Revised by Mario Carneiro, 27-May-2016.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 < (𝐴 · 𝐵))) → 0 < 𝐵)

Theoremprodgt02 7795 Infer that a multiplier is positive from a nonnegative multiplicand and positive product. (Contributed by NM, 24-Apr-2005.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐵 ∧ 0 < (𝐴 · 𝐵))) → 0 < 𝐴)

Theoremprodge0 7796 Infer that a multiplicand is nonnegative from a positive multiplier and nonnegative product. (Contributed by NM, 2-Jul-2005.) (Revised by Mario Carneiro, 27-May-2016.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 < 𝐴 ∧ 0 ≤ (𝐴 · 𝐵))) → 0 ≤ 𝐵)

Theoremprodge02 7797 Infer that a multiplier is nonnegative from a positive multiplicand and nonnegative product. (Contributed by NM, 2-Jul-2005.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 < 𝐵 ∧ 0 ≤ (𝐴 · 𝐵))) → 0 ≤ 𝐴)

Theoremltmul2 7798 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 13-Feb-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴 < 𝐵 ↔ (𝐶 · 𝐴) < (𝐶 · 𝐵)))

Theoremlemul2 7799 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 16-Mar-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴𝐵 ↔ (𝐶 · 𝐴) ≤ (𝐶 · 𝐵)))

Theoremlemul1a 7800 Multiplication of both sides of 'less than or equal to' by a nonnegative number. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 21-Feb-2005.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) ∧ 𝐴𝐵) → (𝐴 · 𝐶) ≤ (𝐵 · 𝐶))

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