P. 89 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 8801-8900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	reapti 8801	Real apartness is tight. Beyond the development of apartness itself, proofs should use apti 8844. (Contributed by Jim Kingdon, 30-Jan-2020.) (New usage is discouraged.)
|- ( ( A e. RR /\ B e. RR ) -> ( A = B <-> -. A #ℝ B ) )
Theorem	recexgt0 8802*	Existence of reciprocal of positive real number. (Contributed by Jim Kingdon, 6-Feb-2020.)
|- ( ( A e. RR /\ 0 < A ) -> E. x e. RR ( 0 < x /\ ( A x. x ) = 1 ) )
4.3.6  Complex Apartness
Syntax	cap 8803	Class of complex apartness relation.
class #
Definition	df-ap 8804*	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 8901 which says that a number apart from zero has a reciprocal). The defining characteristics of an apartness are irreflexivity (apirr 8827), symmetry (apsym 8828), and cotransitivity (apcotr 8829). Apartness implies negated equality, as seen at apne 8845, 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 8844). (Contributed by Jim Kingdon, 26-Jan-2020.)
|- # = { <. x , y >. | E. r e. RR E. s e. RR E. t e. RR E. u e. RR ( ( x = ( r + ( _i x. s ) ) /\ y = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) }
Theorem	ixi 8805	_i times itself is minus 1. (Contributed by NM, 6-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
|- ( _i x. _i ) = -u 1
Theorem	inelr 8806	The imaginary unit _i is not a real number. (Contributed by NM, 6-May-1999.)
|- -. _i e. RR
Theorem	rimul 8807	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 e. RR /\ ( _i x. A ) e. RR ) -> A = 0 )
Theorem	rereim 8808	Decomposition of a real number into real part (itself) and imaginary part (zero). (Contributed by Jim Kingdon, 30-Jan-2020.)
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ A = ( B + ( _i x. C ) ) ) ) -> ( B = A /\ C = 0 ) )
Theorem	apreap 8809	Complex apartness and real apartness agree on the real numbers. (Contributed by Jim Kingdon, 31-Jan-2020.)
|- ( ( A e. RR /\ B e. RR ) -> ( A # B <-> A #ℝ B ) )
Theorem	reaplt 8810	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 e. RR /\ B e. RR ) -> ( A # B <-> ( A < B \/ B < A ) ) )
Theorem	reapltxor 8811	Real apartness in terms of less than (exclusive-or version). (Contributed by Jim Kingdon, 23-Mar-2020.)
|- ( ( A e. RR /\ B e. RR ) -> ( A # B <-> ( A < B \/_ B < A ) ) )
Theorem	1ap0 8812	One is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.)
|- 1 # 0
Theorem	ltmul1a 8813	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 e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) /\ A < B ) -> ( A x. C ) < ( B x. C ) )
Theorem	ltmul1 8814	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 e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A < B <-> ( A x. C ) < ( B x. C ) ) )
Theorem	lemul1 8815	Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 21-Feb-2005.)
|- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A <_ B <-> ( A x. C ) <_ ( B x. C ) ) )
Theorem	reapmul1lem 8816	Lemma for reapmul1 8817. (Contributed by Jim Kingdon, 8-Feb-2020.)
|- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A # B <-> ( A x. C ) # ( B x. C ) ) )
Theorem	reapmul1 8817	Multiplication of both sides of real apartness by a real number apart from zero. Special case of apmul1 9010. (Contributed by Jim Kingdon, 8-Feb-2020.)
|- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C # 0 ) ) -> ( A # B <-> ( A x. C ) # ( B x. C ) ) )
Theorem	reapadd1 8818	Real addition respects apartness. (Contributed by Jim Kingdon, 13-Feb-2020.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A # B <-> ( A + C ) # ( B + C ) ) )
Theorem	reapneg 8819	Real negation respects apartness. (Contributed by Jim Kingdon, 13-Feb-2020.)
|- ( ( A e. RR /\ B e. RR ) -> ( A # B <-> -u A # -u B ) )
Theorem	reapcotr 8820	Real apartness is cotransitive. Part of Definition 11.2.7(v) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Feb-2020.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A # B -> ( A # C \/ B # C ) ) )
Theorem	remulext1 8821	Left extensionality for multiplication. (Contributed by Jim Kingdon, 19-Feb-2020.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A x. C ) # ( B x. C ) -> A # B ) )
Theorem	remulext2 8822	Right extensionality for real multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( C x. A ) # ( C x. B ) -> A # B ) )
Theorem	apsqgt0 8823	The square of a real number apart from zero is positive. (Contributed by Jim Kingdon, 7-Feb-2020.)
|- ( ( A e. RR /\ A # 0 ) -> 0 < ( A x. A ) )
Theorem	cru 8824	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 e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) <-> ( A = C /\ B = D ) ) )
Theorem	apreim 8825	Complex apartness in terms of real and imaginary parts. (Contributed by Jim Kingdon, 12-Feb-2020.)
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A + ( _i x. B ) ) # ( C + ( _i x. D ) ) <-> ( A # C \/ B # D ) ) )
Theorem	mulreim 8826	Complex multiplication in terms of real and imaginary parts. (Contributed by Jim Kingdon, 23-Feb-2020.)
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A + ( _i x. B ) ) x. ( C + ( _i x. D ) ) ) = ( ( ( A x. C ) + -u ( B x. D ) ) + ( _i x. ( ( C x. B ) + ( D x. A ) ) ) ) )
Theorem	apirr 8827	Apartness is irreflexive. (Contributed by Jim Kingdon, 16-Feb-2020.)
|- ( A e. CC -> -. A # A )
Theorem	apsym 8828	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 e. CC /\ B e. CC ) -> ( A # B <-> B # A ) )
Theorem	apcotr 8829	Apartness is cotransitive. (Contributed by Jim Kingdon, 16-Feb-2020.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A # B -> ( A # C \/ B # C ) ) )
Theorem	apadd1 8830	Addition respects apartness. Analogue of addcan 8401 for apartness. (Contributed by Jim Kingdon, 13-Feb-2020.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A # B <-> ( A + C ) # ( B + C ) ) )
Theorem	apadd2 8831	Addition respects apartness. (Contributed by Jim Kingdon, 16-Feb-2020.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A # B <-> ( C + A ) # ( C + B ) ) )
Theorem	addext 8832	Strong extensionality for addition. Given excluded middle, apartness would be equivalent to negated equality and this would follow readily (for all operations) from oveq12 6037. For us, it is proved a different way. (Contributed by Jim Kingdon, 15-Feb-2020.)
|- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A + B ) # ( C + D ) -> ( A # C \/ B # D ) ) )
Theorem	apneg 8833	Negation respects apartness. (Contributed by Jim Kingdon, 14-Feb-2020.)
|- ( ( A e. CC /\ B e. CC ) -> ( A # B <-> -u A # -u B ) )
Theorem	mulext1 8834	Left extensionality for complex multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. C ) # ( B x. C ) -> A # B ) )
Theorem	mulext2 8835	Right extensionality for complex multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( C x. A ) # ( C x. B ) -> A # B ) )
Theorem	mulext 8836	Strong extensionality for multiplication. Given excluded middle, apartness would be equivalent to negated equality and this would follow readily (for all operations) from oveq12 6037. For us, it is proved a different way. (Contributed by Jim Kingdon, 23-Feb-2020.)
|- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A x. B ) # ( C x. D ) -> ( A # C \/ B # D ) ) )
Theorem	mulap0r 8837	A product apart from zero. Lemma 2.13 of [Geuvers], p. 6. (Contributed by Jim Kingdon, 24-Feb-2020.)
|- ( ( A e. CC /\ B e. CC /\ ( A x. B ) # 0 ) -> ( A # 0 /\ B # 0 ) )
Theorem	msqge0 8838	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 e. RR -> 0 <_ ( A x. A ) )
Theorem	msqge0i 8839	A square is nonnegative. (Contributed by NM, 14-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
|- A e. RR   =>    |- 0 <_ ( A x. A )
Theorem	msqge0d 8840	A square is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> 0 <_ ( A x. A ) )
Theorem	mulge0 8841	The product of two nonnegative numbers is nonnegative. (Contributed by NM, 8-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> 0 <_ ( A x. B ) )
Theorem	mulge0i 8842	The product of two nonnegative numbers is nonnegative. (Contributed by NM, 30-Jul-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( ( 0 <_ A /\ 0 <_ B ) -> 0 <_ ( A x. B ) )
Theorem	mulge0d 8843	The product of two nonnegative numbers is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> 0 <_ A )   &    |- ( ph -> 0 <_ B )   =>    |- ( ph -> 0 <_ ( A x. B ) )
Theorem	apti 8844	Complex apartness is tight. (Contributed by Jim Kingdon, 21-Feb-2020.)
|- ( ( A e. CC /\ B e. CC ) -> ( A = B <-> -. A # B ) )
Theorem	apne 8845	Apartness implies negated equality. We cannot in general prove the converse (as shown at neapmkv 16784), which is the whole point of having separate notations for apartness and negated equality. (Contributed by Jim Kingdon, 21-Feb-2020.)
|- ( ( A e. CC /\ B e. CC ) -> ( A # B -> A =/= B ) )
Theorem	apcon4bid 8846	Contrapositive law deduction for apartness. (Contributed by Jim Kingdon, 31-Jul-2023.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> D e. CC )   &    |- ( ph -> ( A # B <-> C # D ) )   =>    |- ( ph -> ( A = B <-> C = D ) )
Theorem	leltap 8847	<_ implies 'less than' is 'apart'. (Contributed by Jim Kingdon, 13-Aug-2021.)
|- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( A < B <-> B # A ) )
Theorem	gt0ap0 8848	Positive implies apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
|- ( ( A e. RR /\ 0 < A ) -> A # 0 )
Theorem	gt0ap0i 8849	Positive means apart from zero (useful for ordering theorems involving division). (Contributed by Jim Kingdon, 27-Feb-2020.)
|- A e. RR   =>    |- ( 0 < A -> A # 0 )
Theorem	gt0ap0ii 8850	Positive implies apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
|- A e. RR   &    |- 0 < A   =>    |- A # 0
Theorem	gt0ap0d 8851	Positive implies apart from zero. Because of the way we define #, A must be an element of RR, not just RR*. (Contributed by Jim Kingdon, 27-Feb-2020.)
|- ( ph -> A e. RR )   &    |- ( ph -> 0 < A )   =>    |- ( ph -> A # 0 )
Theorem	negap0 8852	A number is apart from zero iff its negative is apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
|- ( A e. CC -> ( A # 0 <-> -u A # 0 ) )
Theorem	negap0d 8853	The negative of a number apart from zero is apart from zero. (Contributed by Jim Kingdon, 25-Feb-2024.)
|- ( ph -> A e. CC )   &    |- ( ph -> A # 0 )   =>    |- ( ph -> -u A # 0 )
Theorem	ltleap 8854	Less than in terms of non-strict order and apartness. (Contributed by Jim Kingdon, 28-Feb-2020.)
|- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> ( A <_ B /\ A # B ) ) )
Theorem	ltap 8855	'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
|- ( ( A e. RR /\ B e. RR /\ A < B ) -> B # A )
Theorem	gtapii 8856	'Greater than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
|- A e. RR   &    |- B e. RR   &    |- A < B   =>    |- B # A
Theorem	ltapii 8857	'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
|- A e. RR   &    |- B e. RR   &    |- A < B   =>    |- A # B
Theorem	ltapi 8858	'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
|- A e. RR   &    |- B e. RR   =>    |- ( A < B -> B # A )
Theorem	gtapd 8859	'Greater than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> A < B )   =>    |- ( ph -> B # A )
Theorem	ltapd 8860	'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> A < B )   =>    |- ( ph -> A # B )
Theorem	leltapd 8861	<_ implies 'less than' is 'apart'. (Contributed by Jim Kingdon, 13-Aug-2021.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> A <_ B )   =>    |- ( ph -> ( A < B <-> B # A ) )
Theorem	ap0gt0 8862	A nonnegative number is apart from zero if and only if it is positive. (Contributed by Jim Kingdon, 11-Aug-2021.)
|- ( ( A e. RR /\ 0 <_ A ) -> ( A # 0 <-> 0 < A ) )
Theorem	ap0gt0d 8863	A nonzero nonnegative number is positive. (Contributed by Jim Kingdon, 11-Aug-2021.)
|- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   &    |- ( ph -> A # 0 )   =>    |- ( ph -> 0 < A )
Theorem	apsub1 8864	Subtraction respects apartness. Analogue of subcan2 8446 for apartness. (Contributed by Jim Kingdon, 6-Jan-2022.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A # B <-> ( A - C ) # ( B - C ) ) )
Theorem	subap0 8865	Two numbers being apart is equivalent to their difference being apart from zero. (Contributed by Jim Kingdon, 25-Dec-2022.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( A - B ) # 0 <-> A # B ) )
Theorem	subap0d 8866	Two numbers apart from each other have difference apart from zero. (Contributed by Jim Kingdon, 12-Aug-2021.) (Proof shortened by BJ, 15-Aug-2024.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> A # B )   =>    |- ( ph -> ( A - B ) # 0 )
Theorem	cnstab 8867	Equality of complex numbers is stable. Stability here means -. -. A = B -> A = B as defined at df-stab 839. This theorem for real numbers is Proposition 5.2 of [BauerHanson], p. 27. (Contributed by Jim Kingdon, 1-Aug-2023.) (Proof shortened by BJ, 15-Aug-2024.)
|- ( ( A e. CC /\ B e. CC ) -> STAB A = B )
Theorem	aprcl 8868	Reverse closure for apartness. (Contributed by Jim Kingdon, 19-Dec-2023.)
|- ( A # B -> ( A e. CC /\ B e. CC ) )
Theorem	apsscn 8869*	The points apart from a given point are complex numbers. (Contributed by Jim Kingdon, 19-Dec-2023.)
|- { x e. A | x # B } C_ CC
Theorem	lt0ap0 8870	A number which is less than zero is apart from zero. (Contributed by Jim Kingdon, 25-Feb-2024.)
|- ( ( A e. RR /\ A < 0 ) -> A # 0 )
Theorem	lt0ap0d 8871	A real number less than zero is apart from zero. Deduction form. (Contributed by Jim Kingdon, 24-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> A < 0 )   =>    |- ( ph -> A # 0 )
Theorem	aptap 8872	Complex apartness (as defined at df-ap 8804) is a tight apartness (as defined at df-tap 7512). (Contributed by Jim Kingdon, 16-Feb-2025.)
|- # TAp CC
4.3.7  Reciprocals
Theorem	recextlem1 8873	Lemma for recexap 8875. (Contributed by Eric Schmidt, 23-May-2007.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( A + ( _i x. B ) ) x. ( A - ( _i x. B ) ) ) = ( ( A x. A ) + ( B x. B ) ) )
Theorem	recexaplem2 8874	Lemma for recexap 8875. (Contributed by Jim Kingdon, 20-Feb-2020.)
|- ( ( A e. RR /\ B e. RR /\ ( A + ( _i x. B ) ) # 0 ) -> ( ( A x. A ) + ( B x. B ) ) # 0 )
Theorem	recexap 8875*	Existence of reciprocal of nonzero complex number. (Contributed by Jim Kingdon, 20-Feb-2020.)
|- ( ( A e. CC /\ A # 0 ) -> E. x e. CC ( A x. x ) = 1 )
Theorem	mulap0 8876	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 e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) -> ( A x. B ) # 0 )
Theorem	mulap0b 8877	The product of two numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( A # 0 /\ B # 0 ) <-> ( A x. B ) # 0 ) )
Theorem	mulap0i 8878	The product of two numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 23-Feb-2020.)
|- A e. CC   &    |- B e. CC   &    |- A # 0   &    |- B # 0   =>    |- ( A x. B ) # 0
Theorem	mulap0bd 8879	The product of two numbers apart from zero is apart from zero. Exercise 11.11 of [HoTT], p. (varies). (Contributed by Jim Kingdon, 24-Feb-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( ( A # 0 /\ B # 0 ) <-> ( A x. B ) # 0 ) )
Theorem	mulap0d 8880	The product of two numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 23-Feb-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> A # 0 )   &    |- ( ph -> B # 0 )   =>    |- ( ph -> ( A x. B ) # 0 )
Theorem	mulap0bad 8881	A factor of a complex number apart from zero is apart from zero. Partial converse of mulap0d 8880 and consequence of mulap0bd 8879. (Contributed by Jim Kingdon, 24-Feb-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> ( A x. B ) # 0 )   =>    |- ( ph -> A # 0 )
Theorem	mulap0bbd 8882	A factor of a complex number apart from zero is apart from zero. Partial converse of mulap0d 8880 and consequence of mulap0bd 8879. (Contributed by Jim Kingdon, 24-Feb-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> ( A x. B ) # 0 )   =>    |- ( ph -> B # 0 )
Theorem	mulcanapd 8883	Cancellation law for multiplication. (Contributed by Jim Kingdon, 21-Feb-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> C # 0 )   =>    |- ( ph -> ( ( C x. A ) = ( C x. B ) <-> A = B ) )
Theorem	mulcanap2d 8884	Cancellation law for multiplication. (Contributed by Jim Kingdon, 21-Feb-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> C # 0 )   =>    |- ( ph -> ( ( A x. C ) = ( B x. C ) <-> A = B ) )
Theorem	mulcanapad 8885	Cancellation of a nonzero factor on the left in an equation. One-way deduction form of mulcanapd 8883. (Contributed by Jim Kingdon, 21-Feb-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> C # 0 )   &    |- ( ph -> ( C x. A ) = ( C x. B ) )   =>    |- ( ph -> A = B )
Theorem	mulcanap2ad 8886	Cancellation of a nonzero factor on the right in an equation. One-way deduction form of mulcanap2d 8884. (Contributed by Jim Kingdon, 21-Feb-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> C # 0 )   &    |- ( ph -> ( A x. C ) = ( B x. C ) )   =>    |- ( ph -> A = B )
Theorem	mulcanap 8887	Cancellation law for multiplication (full theorem form). (Contributed by Jim Kingdon, 21-Feb-2020.)
|- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( C x. A ) = ( C x. B ) <-> A = B ) )
Theorem	mulcanap2 8888	Cancellation law for multiplication. (Contributed by Jim Kingdon, 21-Feb-2020.)
|- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A x. C ) = ( B x. C ) <-> A = B ) )
Theorem	mulcanapi 8889	Cancellation law for multiplication. (Contributed by Jim Kingdon, 21-Feb-2020.)
|- A e. CC   &    |- B e. CC   &    |- C e. CC   &    |- C # 0   =>    |- ( ( C x. A ) = ( C x. B ) <-> A = B )
Theorem	muleqadd 8890	Property of numbers whose product equals their sum. Equation 5 of [Kreyszig] p. 12. (Contributed by NM, 13-Nov-2006.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) = ( A + B ) <-> ( ( A - 1 ) x. ( B - 1 ) ) = 1 ) )
Theorem	receuap 8891*	Existential uniqueness of reciprocals. (Contributed by Jim Kingdon, 21-Feb-2020.)
|- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> E! x e. CC ( B x. x ) = A )
Theorem	mul0eqap 8892	If two numbers are apart from each other and their product is zero, one of them must be zero. (Contributed by Jim Kingdon, 31-Jul-2023.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> A # B )   &    |- ( ph -> ( A x. B ) = 0 )   =>    |- ( ph -> ( A = 0 \/ B = 0 ) )
Theorem	recapb 8893*	A complex number has a multiplicative inverse if and only if it is apart from zero. Theorem 11.2.4 of [HoTT], p. (varies), generalized from real to complex numbers. (Contributed by Jim Kingdon, 18-Jan-2025.)
|- ( A e. CC -> ( A # 0 <-> E. x e. CC ( A x. x ) = 1 ) )
4.3.8  Division
Syntax	cdiv 8894	Extend class notation to include division.
class /
Definition	df-div 8895*	Define division. Theorem divmulap 8897 relates it to multiplication, and divclap 8900 and redivclap 8953 prove its closure laws. (Contributed by NM, 2-Feb-1995.) Use divvalap 8896 instead. (Revised by Mario Carneiro, 1-Apr-2014.) (New usage is discouraged.)
|- / = ( x e. CC , y e. ( CC \ { 0 } ) |-> ( iota_ z e. CC ( y x. z ) = x ) )
Theorem	divvalap 8896*	Value of division: the (unique) element x such that ( B x. x ) = A. This is meaningful only when B is apart from zero. (Contributed by Jim Kingdon, 21-Feb-2020.)
|- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( A / B ) = ( iota_ x e. CC ( B x. x ) = A ) )
Theorem	divmulap 8897	Relationship between division and multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
|- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A / C ) = B <-> ( C x. B ) = A ) )
Theorem	divmulap2 8898	Relationship between division and multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
|- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A / C ) = B <-> A = ( C x. B ) ) )
Theorem	divmulap3 8899	Relationship between division and multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
|- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A / C ) = B <-> A = ( B x. C ) ) )
Theorem	divclap 8900	Closure law for division. (Contributed by Jim Kingdon, 22-Feb-2020.)
|- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( A / B ) e. CC )