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Theorem List for Intuitionistic Logic Explorer - 7301-7400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremltsub2d 7301 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 7302 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 7303 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 7304 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 7305 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 7306 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 7307 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 7308 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 7309 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 7310 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 7311 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 7312 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 7313 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 7314 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 7315 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 7316 1 ≤ 1. Common special case. (Contributed by David A. Wheeler, 16-Jul-2016.)
1 ≤ 1
 
Theoremgt0add 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 B 0 < (A + B)) → (0 < A 0 < B))
 
3.3.5  Real Apartness
 
Syntaxcreap 7318 Class of real apartness relation.
class #
 
Definitiondf-reap 7319* Define real apartness. Definition in Section 11.2.1 of [HoTT], p. (varies). Although # 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, # and # agree (apreap 7331). (Contributed by Jim Kingdon, 26-Jan-2020.)
# = {⟨x, y⟩ ∣ ((x y ℝ) (x < y y < x))}
 
Theoremreapval 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 B ℝ) → (A # B ↔ (A < B B < A)))
 
Theoremreapirr 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 # A)
 
Theoremrecexre 7322* Existence of reciprocal of real number. (Contributed by Jim Kingdon, 29-Jan-2020.)
((A A # 0) → x ℝ (A · x) = 1)
 
Theoremreapti 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 B ℝ) → (A = B ↔ ¬ A # B))
 
Theoremrecexgt0 7324* 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 7325 Class of complex apartness relation.
class #
 
Definitiondf-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 𝑠 𝑡 u ℝ ((x = (𝑟 + (i · 𝑠)) y = (𝑡 + (i · u))) (𝑟 # 𝑡 𝑠 # u))}
 
Theoremixi 7327 i times itself is minus 1. (Contributed by NM, 6-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
(i · i) = -1
 
Theoreminelr 7328 The imaginary unit i is not a real number. (Contributed by NM, 6-May-1999.)
¬ i
 
Theoremrimul 7329 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 7330 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 7331 Complex apartness and real apartness agree on the real numbers. (Contributed by Jim Kingdon, 31-Jan-2020.)
((A B ℝ) → (A # BA # B))
 
Theoremreaplt 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 B ℝ) → (A # B ↔ (A < B B < A)))
 
Theoremreapltxor 7333 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 7334 One is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.)
1 # 0
 
Theoremltmul1a 7335 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 7336 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 7337 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 7338 Lemma for reapmul1 7339. (Contributed by Jim Kingdon, 8-Feb-2020.)
((A B (𝐶 0 < 𝐶)) → (A # B ↔ (A · 𝐶) # (B · 𝐶)))
 
Theoremreapmul1 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 B (𝐶 𝐶 # 0)) → (A # B ↔ (A · 𝐶) # (B · 𝐶)))
 
Theoremreapadd1 7340 Real addition respects apartness. (Contributed by Jim Kingdon, 13-Feb-2020.)
((A B 𝐶 ℝ) → (A # B ↔ (A + 𝐶) # (B + 𝐶)))
 
Theoremreapneg 7341 Real negation respects apartness. (Contributed by Jim Kingdon, 13-Feb-2020.)
((A B ℝ) → (A # B ↔ -A # -B))
 
Theoremreapcotr 7342 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 7343 Left extensionality for multiplication. (Contributed by Jim Kingdon, 19-Feb-2020.)
((A B 𝐶 ℝ) → ((A · 𝐶) # (B · 𝐶) → A # B))
 
Theoremremulext2 7344 Right extensionality for real multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
((A B 𝐶 ℝ) → ((𝐶 · A) # (𝐶 · B) → A # B))
 
Theoremapsqgt0 7345 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 7346 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 7347 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 7348 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 7349 Apartness is irreflexive. (Contributed by Jim Kingdon, 16-Feb-2020.)
(A ℂ → ¬ A # A)
 
Theoremapsym 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 B ℂ) → (A # BB # A))
 
Theoremapcotr 7351 Apartness is cotransitive. (Contributed by Jim Kingdon, 16-Feb-2020.)
((A B 𝐶 ℂ) → (A # B → (A # 𝐶 B # 𝐶)))
 
Theoremapadd1 7352 Addition respects apartness. Analogue of addcan 6948 for apartness. (Contributed by Jim Kingdon, 13-Feb-2020.)
((A B 𝐶 ℂ) → (A # B ↔ (A + 𝐶) # (B + 𝐶)))
 
Theoremapadd2 7353 Addition respects apartness. (Contributed by Jim Kingdon, 16-Feb-2020.)
((A B 𝐶 ℂ) → (A # B ↔ (𝐶 + A) # (𝐶 + B)))
 
Theoremaddext 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 B ℂ) (𝐶 𝐷 ℂ)) → ((A + B) # (𝐶 + 𝐷) → (A # 𝐶 B # 𝐷)))
 
Theoremapneg 7355 Negation respects apartness. (Contributed by Jim Kingdon, 14-Feb-2020.)
((A B ℂ) → (A # B ↔ -A # -B))
 
Theoremmulext1 7356 Left extensionality for complex multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
((A B 𝐶 ℂ) → ((A · 𝐶) # (B · 𝐶) → A # B))
 
Theoremmulext2 7357 Right extensionality for complex multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
((A B 𝐶 ℂ) → ((𝐶 · A) # (𝐶 · B) → A # B))
 
Theoremmulext 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 B ℂ) (𝐶 𝐷 ℂ)) → ((A · B) # (𝐶 · 𝐷) → (A # 𝐶 B # 𝐷)))
 
Theoremmulap0r 7359 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 7360 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 7361 A square is nonnegative. (Contributed by NM, 14-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
A        0 ≤ (A · A)
 
Theoremmsqge0d 7362 A square is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)       (φ → 0 ≤ (A · A))
 
Theoremmulge0 7363 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 7364 The product of two nonnegative numbers is nonnegative. (Contributed by NM, 30-Jul-1999.)
A     &   B        ((0 ≤ A 0 ≤ B) → 0 ≤ (A · B))
 
Theoremmulge0d 7365 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 7366 Complex apartness is tight. (Contributed by Jim Kingdon, 21-Feb-2020.)
((A B ℂ) → (A = B ↔ ¬ A # B))
 
Theoremapne 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 B ℂ) → (A # BAB))
 
Theoremgt0ap0 7368 Positive implies apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
((A 0 < A) → A # 0)
 
Theoremgt0ap0i 7369 Positive means apart from zero (useful for ordering theorems involving division). (Contributed by Jim Kingdon, 27-Feb-2020.)
A        (0 < AA # 0)
 
Theoremgt0ap0ii 7370 Positive implies apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
A     &   0 < A       A # 0
 
Theoremgt0ap0d 7371 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 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))
 
Theoremltleap 7373 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 7374 Lemma for recexap 7376. (Contributed by Eric Schmidt, 23-May-2007.)
((A B ℂ) → ((A + (i · B)) · (A − (i · B))) = ((A · A) + (B · B)))
 
Theoremrecexaplem2 7375 Lemma for recexap 7376. (Contributed by Jim Kingdon, 20-Feb-2020.)
((A B (A + (i · B)) # 0) → ((A · A) + (B · B)) # 0)
 
Theoremrecexap 7376* Existence of reciprocal of nonzero complex number. (Contributed by Jim Kingdon, 20-Feb-2020.)
((A A # 0) → x ℂ (A · x) = 1)
 
Theoremmulap0 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 A # 0) (B B # 0)) → (A · B) # 0)
 
Theoremmulap0b 7378 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 7379 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 7380 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))
 
Theoremmulap0d 7381 The product of two numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 23-Feb-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φA # 0)    &   (φB # 0)       (φ → (A · B) # 0)
 
Theoremmulap0bad 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.)
(φA ℂ)    &   (φB ℂ)    &   (φ → (A · B) # 0)       (φA # 0)
 
Theoremmulap0bbd 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.)
(φA ℂ)    &   (φB ℂ)    &   (φ → (A · B) # 0)       (φB # 0)
 
Theoremmulcanapd 7384 Cancellation law for multiplication. (Contributed by Jim Kingdon, 21-Feb-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φ𝐶 # 0)       (φ → ((𝐶 · A) = (𝐶 · B) ↔ A = B))
 
Theoremmulcanap2d 7385 Cancellation law for multiplication. (Contributed by Jim Kingdon, 21-Feb-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φ𝐶 # 0)       (φ → ((A · 𝐶) = (B · 𝐶) ↔ A = B))
 
Theoremmulcanapad 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.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φ𝐶 # 0)    &   (φ → (𝐶 · A) = (𝐶 · B))       (φA = B)
 
Theoremmulcanap2ad 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.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φ𝐶 # 0)    &   (φ → (A · 𝐶) = (B · 𝐶))       (φA = B)
 
Theoremmulcanap 7388 Cancellation law for multiplication (full theorem form). (Contributed by Jim Kingdon, 21-Feb-2020.)
((A B (𝐶 𝐶 # 0)) → ((𝐶 · A) = (𝐶 · B) ↔ A = B))
 
Theoremmulcanap2 7389 Cancellation law for multiplication. (Contributed by Jim Kingdon, 21-Feb-2020.)
((A B (𝐶 𝐶 # 0)) → ((A · 𝐶) = (B · 𝐶) ↔ A = B))
 
Theoremmulcanapi 7390 Cancellation law for multiplication. (Contributed by Jim Kingdon, 21-Feb-2020.)
A     &   B     &   𝐶     &   𝐶 # 0       ((𝐶 · A) = (𝐶 · B) ↔ A = B)
 
Theoremmuleqadd 7391 Property of numbers whose product equals their sum. Equation 5 of [Kreyszig] p. 12. (Contributed by NM, 13-Nov-2006.)
((A B ℂ) → ((A · B) = (A + B) ↔ ((A − 1) · (B − 1)) = 1))
 
Theoremreceuap 7392* Existential uniqueness of reciprocals. (Contributed by Jim Kingdon, 21-Feb-2020.)
((A B B # 0) → ∃!x ℂ (B · x) = A)
 
3.3.8  Division
 
Syntaxcdiv 7393 Extend class notation to include division.
class /
 
Definitiondf-div 7394* Define division. Theorem divmulap 7396 relates it to multiplication, and divclap 7399 and redivclap 7449 prove its closure laws. (Contributed by NM, 2-Feb-1995.) (Revised by Mario Carneiro, 1-Apr-2014.) (New usage is discouraged.)
/ = (x ℂ, y (ℂ ∖ {0}) ↦ (z ℂ (y · z) = x))
 
Theoremdivvalap 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 B B # 0) → (A / B) = (x ℂ (B · x) = A))
 
Theoremdivmulap 7396 Relationship between division and multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
((A B (𝐶 𝐶 # 0)) → ((A / 𝐶) = B ↔ (𝐶 · B) = A))
 
Theoremdivmulap2 7397 Relationship between division and multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
((A B (𝐶 𝐶 # 0)) → ((A / 𝐶) = BA = (𝐶 · B)))
 
Theoremdivmulap3 7398 Relationship between division and multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
((A B (𝐶 𝐶 # 0)) → ((A / 𝐶) = BA = (B · 𝐶)))
 
Theoremdivclap 7399 Closure law for division. (Contributed by Jim Kingdon, 22-Feb-2020.)
((A B B # 0) → (A / B) ℂ)
 
Theoremrecclap 7400 Closure law for reciprocal. (Contributed by Jim Kingdon, 22-Feb-2020.)
((A A # 0) → (1 / A) ℂ)
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