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Theorem List for Intuitionistic Logic Explorer - 7301-7400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsubge0d 7301 Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)       (φ → (0 ≤ (AB) ↔ BA))

Theoremsuble0d 7302 Nonpositive subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)       (φ → ((AB) ≤ 0 ↔ AB))

Theoremsubge02d 7303 Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)       (φ → (0 ≤ B ↔ (AB) ≤ A))

Theoremltadd1d 7304 Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)       (φ → (A < B ↔ (A + 𝐶) < (B + 𝐶)))

Theoremleadd1d 7305 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)       (φ → (AB ↔ (A + 𝐶) ≤ (B + 𝐶)))

Theoremleadd2d 7306 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)       (φ → (AB ↔ (𝐶 + A) ≤ (𝐶 + B)))

Theoremltsubaddd 7307 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)       (φ → ((AB) < 𝐶A < (𝐶 + B)))

Theoremlesubaddd 7308 'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)       (φ → ((AB) ≤ 𝐶A ≤ (𝐶 + B)))

Theoremltsubadd2d 7309 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)       (φ → ((AB) < 𝐶A < (B + 𝐶)))

Theoremlesubadd2d 7310 'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)       (φ → ((AB) ≤ 𝐶A ≤ (B + 𝐶)))

Theoremltaddsubd 7311 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)       (φ → ((A + B) < 𝐶A < (𝐶B)))

Theoremltaddsub2d 7312 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 29-Dec-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)       (φ → ((A + B) < 𝐶B < (𝐶A)))

Theoremleaddsub2d 7313 'Less than or equal to' relationship between and addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)       (φ → ((A + B) ≤ 𝐶B ≤ (𝐶A)))

Theoremsubled 7314 Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φ → (AB) ≤ 𝐶)       (φ → (A𝐶) ≤ B)

Theoremlesubd 7315 Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φA ≤ (B𝐶))       (φ𝐶 ≤ (BA))

Theoremltsub23d 7316 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φ → (AB) < 𝐶)       (φ → (A𝐶) < B)

Theoremltsub13d 7317 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φA < (B𝐶))       (φ𝐶 < (BA))

Theoremlesub1d 7318 Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)       (φ → (AB ↔ (A𝐶) ≤ (B𝐶)))

Theoremlesub2d 7319 Subtraction of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)       (φ → (AB ↔ (𝐶B) ≤ (𝐶A)))

Theoremltsub1d 7320 Subtraction from both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)       (φ → (A < B ↔ (A𝐶) < (B𝐶)))

Theoremltsub2d 7321 Subtraction of both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)       (φ → (A < B ↔ (𝐶B) < (𝐶A)))

Theoremltadd1dd 7322 Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by Mario Carneiro, 30-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φA < B)       (φ → (A + 𝐶) < (B + 𝐶))

Theoremltsub1dd 7323 Subtraction from both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φA < B)       (φ → (A𝐶) < (B𝐶))

Theoremltsub2dd 7324 Subtraction of both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φA < B)       (φ → (𝐶B) < (𝐶A))

Theoremleadd1dd 7325 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φAB)       (φ → (A + 𝐶) ≤ (B + 𝐶))

Theoremleadd2dd 7326 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φAB)       (φ → (𝐶 + A) ≤ (𝐶 + B))

Theoremlesub1dd 7327 Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φAB)       (φ → (A𝐶) ≤ (B𝐶))

Theoremlesub2dd 7328 Subtraction of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φAB)       (φ → (𝐶B) ≤ (𝐶A))

Theoremle2addd 7329 Adding both side of two inequalities. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φ𝐷 ℝ)    &   (φA𝐶)    &   (φB𝐷)       (φ → (A + B) ≤ (𝐶 + 𝐷))

Theoremle2subd 7330 Subtracting both sides of two 'less than or equal to' relations. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φ𝐷 ℝ)    &   (φA𝐶)    &   (φB𝐷)       (φ → (A𝐷) ≤ (𝐶B))

Theoremltleaddd 7331 Adding both sides of two orderings. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φ𝐷 ℝ)    &   (φA < 𝐶)    &   (φB𝐷)       (φ → (A + B) < (𝐶 + 𝐷))

Theoremleltaddd 7332 Adding both sides of two orderings. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φ𝐷 ℝ)    &   (φA𝐶)    &   (φB < 𝐷)       (φ → (A + B) < (𝐶 + 𝐷))

Theoremlt2addd 7333 Adding both side of two inequalities. Theorem I.25 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φ𝐷 ℝ)    &   (φA < 𝐶)    &   (φB < 𝐷)       (φ → (A + B) < (𝐶 + 𝐷))

Theoremlt2subd 7334 Subtracting both sides of two 'less than' relations. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φ𝐷 ℝ)    &   (φA < 𝐶)    &   (φB < 𝐷)       (φ → (A𝐷) < (𝐶B))

Theoremltaddsublt 7335 Addition and subtraction on one side of 'less than'. (Contributed by AV, 24-Nov-2018.)
((A B 𝐶 ℝ) → (B < 𝐶 ↔ ((A + B) − 𝐶) < A))

Theorem1le1 7336 1 ≤ 1. Common special case. (Contributed by David A. Wheeler, 16-Jul-2016.)
1 ≤ 1

Theoremgt0add 7337 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 B 0 < (A + B)) → (0 < A 0 < B))

3.3.5  Real Apartness

Syntaxcreap 7338 Class of real apartness relation.
class #

Definitiondf-reap 7339* Define real apartness. Definition in Section 11.2.1 of [HoTT], p. (varies). Although # is an apartness relation on the reals (see df-ap 7346 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, # and # agree (apreap 7351). (Contributed by Jim Kingdon, 26-Jan-2020.)
# = {⟨x, y⟩ ∣ ((x y ℝ) (x < y y < x))}

Theoremreapval 7340 Real apartness in terms of classes. Beyond the development of # itself, proofs should use reaplt 7352 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 29-Jan-2020.)
((A B ℝ) → (A # B ↔ (A < B B < A)))

Theoremreapirr 7341 Real apartness is irreflexive. Part of Definition 11.2.7(v) of [HoTT], p. (varies). Beyond the development of # itself, proofs should use apirr 7369 instead. (Contributed by Jim Kingdon, 26-Jan-2020.)
(A ℝ → ¬ A # A)

Theoremrecexre 7342* Existence of reciprocal of real number. (Contributed by Jim Kingdon, 29-Jan-2020.)
((A A # 0) → x ℝ (A · x) = 1)

Theoremreapti 7343 Real apartness is tight. Beyond the development of apartness itself, proofs should use apti 7386. (Contributed by Jim Kingdon, 30-Jan-2020.) (New usage is discouraged.)
((A B ℝ) → (A = B ↔ ¬ A # B))

Theoremrecexgt0 7344* Existence of reciprocal of positive real number. (Contributed by Jim Kingdon, 6-Feb-2020.)
((A 0 < A) → x ℝ (0 < x (A · x) = 1))

3.3.6  Complex Apartness

Syntaxcap 7345 Class of complex apartness relation.
class #

Definitiondf-ap 7346* 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 7420 which says that a number apart from zero has a reciprocal).

The defining characteristics of an apartness are irreflexivity (apirr 7369), symmetry (apsym 7370), and cotransitivity (apcotr 7371). Apartness implies negated equality, as seen at apne 7387, 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 7386).

(Contributed by Jim Kingdon, 26-Jan-2020.)

# = {⟨x, y⟩ ∣ 𝑟 𝑠 𝑡 u ℝ ((x = (𝑟 + (i · 𝑠)) y = (𝑡 + (i · u))) (𝑟 # 𝑡 𝑠 # u))}

Theoremixi 7347 i times itself is minus 1. (Contributed by NM, 6-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
(i · i) = -1

Theoreminelr 7348 The imaginary unit i is not a real number. (Contributed by NM, 6-May-1999.)
¬ i

Theoremrimul 7349 A real number times the imaginary unit is real only if the number is 0. (Contributed by NM, 28-May-1999.) (Revised by Mario Carneiro, 27-May-2016.)
((A (i · A) ℝ) → A = 0)

Theoremrereim 7350 Decomposition of a real number into real part (itself) and imaginary part (zero). (Contributed by Jim Kingdon, 30-Jan-2020.)
(((A B ℝ) (𝐶 A = (B + (i · 𝐶)))) → (B = A 𝐶 = 0))

Theoremapreap 7351 Complex apartness and real apartness agree on the real numbers. (Contributed by Jim Kingdon, 31-Jan-2020.)
((A B ℝ) → (A # BA # B))

Theoremreaplt 7352 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 B ℝ) → (A # B ↔ (A < B B < A)))

Theoremreapltxor 7353 Real apartness in terms of less than (exclusive-or version). (Contributed by Jim Kingdon, 23-Mar-2020.)
((A B ℝ) → (A # B ↔ (A < BB < A)))

Theorem1ap0 7354 One is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.)
1 # 0

Theoremltmul1a 7355 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 15-May-1999.) (Revised by Mario Carneiro, 27-May-2016.)
(((A B (𝐶 0 < 𝐶)) A < B) → (A · 𝐶) < (B · 𝐶))

Theoremltmul1 7356 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 13-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.)
((A B (𝐶 0 < 𝐶)) → (A < B ↔ (A · 𝐶) < (B · 𝐶)))

Theoremlemul1 7357 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 21-Feb-2005.)
((A B (𝐶 0 < 𝐶)) → (AB ↔ (A · 𝐶) ≤ (B · 𝐶)))

Theoremreapmul1lem 7358 Lemma for reapmul1 7359. (Contributed by Jim Kingdon, 8-Feb-2020.)
((A B (𝐶 0 < 𝐶)) → (A # B ↔ (A · 𝐶) # (B · 𝐶)))

Theoremreapmul1 7359 Multiplication of both sides of real apartness by a real number apart from zero. Special case of apmul1 7526. (Contributed by Jim Kingdon, 8-Feb-2020.)
((A B (𝐶 𝐶 # 0)) → (A # B ↔ (A · 𝐶) # (B · 𝐶)))

((A B 𝐶 ℝ) → (A # B ↔ (A + 𝐶) # (B + 𝐶)))

Theoremreapneg 7361 Real negation respects apartness. (Contributed by Jim Kingdon, 13-Feb-2020.)
((A B ℝ) → (A # B ↔ -A # -B))

Theoremreapcotr 7362 Real apartness is cotransitive. Part of Definition 11.2.7(v) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Feb-2020.)
((A B 𝐶 ℝ) → (A # B → (A # 𝐶 B # 𝐶)))

Theoremremulext1 7363 Left extensionality for multiplication. (Contributed by Jim Kingdon, 19-Feb-2020.)
((A B 𝐶 ℝ) → ((A · 𝐶) # (B · 𝐶) → A # B))

Theoremremulext2 7364 Right extensionality for real multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
((A B 𝐶 ℝ) → ((𝐶 · A) # (𝐶 · B) → A # B))

Theoremapsqgt0 7365 The square of a real number apart from zero is positive. (Contributed by Jim Kingdon, 7-Feb-2020.)
((A A # 0) → 0 < (A · A))

Theoremcru 7366 The representation of complex numbers in terms of real and imaginary parts is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
(((A B ℝ) (𝐶 𝐷 ℝ)) → ((A + (i · B)) = (𝐶 + (i · 𝐷)) ↔ (A = 𝐶 B = 𝐷)))

Theoremapreim 7367 Complex apartness in terms of real and imaginary parts. (Contributed by Jim Kingdon, 12-Feb-2020.)
(((A B ℝ) (𝐶 𝐷 ℝ)) → ((A + (i · B)) # (𝐶 + (i · 𝐷)) ↔ (A # 𝐶 B # 𝐷)))

Theoremmulreim 7368 Complex multiplication in terms of real and imaginary parts. (Contributed by Jim Kingdon, 23-Feb-2020.)
(((A B ℝ) (𝐶 𝐷 ℝ)) → ((A + (i · B)) · (𝐶 + (i · 𝐷))) = (((A · 𝐶) + -(B · 𝐷)) + (i · ((𝐶 · B) + (𝐷 · A)))))

Theoremapirr 7369 Apartness is irreflexive. (Contributed by Jim Kingdon, 16-Feb-2020.)
(A ℂ → ¬ A # A)

Theoremapsym 7370 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 B ℂ) → (A # BB # A))

Theoremapcotr 7371 Apartness is cotransitive. (Contributed by Jim Kingdon, 16-Feb-2020.)
((A B 𝐶 ℂ) → (A # B → (A # 𝐶 B # 𝐶)))

((A B 𝐶 ℂ) → (A # B ↔ (A + 𝐶) # (B + 𝐶)))

((A B 𝐶 ℂ) → (A # B ↔ (𝐶 + A) # (𝐶 + B)))

Theoremaddext 7374 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 B ℂ) (𝐶 𝐷 ℂ)) → ((A + B) # (𝐶 + 𝐷) → (A # 𝐶 B # 𝐷)))

Theoremapneg 7375 Negation respects apartness. (Contributed by Jim Kingdon, 14-Feb-2020.)
((A B ℂ) → (A # B ↔ -A # -B))

Theoremmulext1 7376 Left extensionality for complex multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
((A B 𝐶 ℂ) → ((A · 𝐶) # (B · 𝐶) → A # B))

Theoremmulext2 7377 Right extensionality for complex multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
((A B 𝐶 ℂ) → ((𝐶 · A) # (𝐶 · B) → A # B))

Theoremmulext 7378 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 B ℂ) (𝐶 𝐷 ℂ)) → ((A · B) # (𝐶 · 𝐷) → (A # 𝐶 B # 𝐷)))

Theoremmulap0r 7379 A product apart from zero. Lemma 2.13 of [Geuvers], p. 6. (Contributed by Jim Kingdon, 24-Feb-2020.)
((A B (A · B) # 0) → (A # 0 B # 0))

Theoremmsqge0 7380 A square is nonnegative. Lemma 2.35 of [Geuvers], p. 9. (Contributed by NM, 23-May-2007.) (Revised by Mario Carneiro, 27-May-2016.)
(A ℝ → 0 ≤ (A · A))

Theoremmsqge0i 7381 A square is nonnegative. (Contributed by NM, 14-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
A        0 ≤ (A · A)

Theoremmsqge0d 7382 A square is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)       (φ → 0 ≤ (A · A))

Theoremmulge0 7383 The product of two nonnegative numbers is nonnegative. (Contributed by NM, 8-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.)
(((A 0 ≤ A) (B 0 ≤ B)) → 0 ≤ (A · B))

Theoremmulge0i 7384 The product of two nonnegative numbers is nonnegative. (Contributed by NM, 30-Jul-1999.)
A     &   B        ((0 ≤ A 0 ≤ B) → 0 ≤ (A · B))

Theoremmulge0d 7385 The product of two nonnegative numbers is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ → 0 ≤ A)    &   (φ → 0 ≤ B)       (φ → 0 ≤ (A · B))

Theoremapti 7386 Complex apartness is tight. (Contributed by Jim Kingdon, 21-Feb-2020.)
((A B ℂ) → (A = B ↔ ¬ A # B))

Theoremapne 7387 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 B ℂ) → (A # BAB))

Theoremgt0ap0 7388 Positive implies apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
((A 0 < A) → A # 0)

Theoremgt0ap0i 7389 Positive means apart from zero (useful for ordering theorems involving division). (Contributed by Jim Kingdon, 27-Feb-2020.)
A        (0 < AA # 0)

Theoremgt0ap0ii 7390 Positive implies apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
A     &   0 < A       A # 0

Theoremgt0ap0d 7391 Positive implies apart from zero. Because of the way we define #, A must be an element of , not just *. (Contributed by Jim Kingdon, 27-Feb-2020.)
(φA ℝ)    &   (φ → 0 < A)       (φA # 0)

Theoremnegap0 7392 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))

Theoremltleap 7393 Less than in terms of non-strict order and apartness. (Contributed by Jim Kingdon, 28-Feb-2020.)
((A B ℝ) → (A < B ↔ (AB A # B)))

3.3.7  Reciprocals

Theoremrecextlem1 7394 Lemma for recexap 7396. (Contributed by Eric Schmidt, 23-May-2007.)
((A B ℂ) → ((A + (i · B)) · (A − (i · B))) = ((A · A) + (B · B)))

Theoremrecexaplem2 7395 Lemma for recexap 7396. (Contributed by Jim Kingdon, 20-Feb-2020.)
((A B (A + (i · B)) # 0) → ((A · A) + (B · B)) # 0)

Theoremrecexap 7396* Existence of reciprocal of nonzero complex number. (Contributed by Jim Kingdon, 20-Feb-2020.)
((A A # 0) → x ℂ (A · x) = 1)

Theoremmulap0 7397 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 A # 0) (B B # 0)) → (A · B) # 0)

Theoremmulap0b 7398 The product of two numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.)
((A B ℂ) → ((A # 0 B # 0) ↔ (A · B) # 0))

Theoremmulap0i 7399 The product of two numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 23-Feb-2020.)
A     &   B     &   A # 0    &   B # 0       (A · B) # 0

Theoremmulap0bd 7400 The product of two numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.)
(φA ℂ)    &   (φB ℂ)       (φ → ((A # 0 B # 0) ↔ (A · B) # 0))

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