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	divsubdirap 8801	Distribution of division over subtraction. (Contributed by NM, 4-Mar-2005.)
|- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A - B ) / C ) = ( ( A / C ) - ( B / C ) ) )
Theorem	recrecap 8802	A number is equal to the reciprocal of its reciprocal. (Contributed by Jim Kingdon, 25-Feb-2020.)
|- ( ( A e. CC /\ A # 0 ) -> ( 1 / ( 1 / A ) ) = A )
Theorem	rec11ap 8803	Reciprocal is one-to-one. (Contributed by Jim Kingdon, 25-Feb-2020.)
|- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) -> ( ( 1 / A ) = ( 1 / B ) <-> A = B ) )
Theorem	rec11rap 8804	Mutual reciprocals. (Contributed by Jim Kingdon, 25-Feb-2020.)
|- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) -> ( ( 1 / A ) = B <-> ( 1 / B ) = A ) )
Theorem	divmuldivap 8805	Multiplication of two ratios. (Contributed by Jim Kingdon, 25-Feb-2020.)
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( C e. CC /\ C # 0 ) /\ ( D e. CC /\ D # 0 ) ) ) -> ( ( A / C ) x. ( B / D ) ) = ( ( A x. B ) / ( C x. D ) ) )
Theorem	divdivdivap 8806	Division of two ratios. Theorem I.15 of [Apostol] p. 18. (Contributed by Jim Kingdon, 25-Feb-2020.)
|- ( ( ( A e. CC /\ ( B e. CC /\ B # 0 ) ) /\ ( ( C e. CC /\ C # 0 ) /\ ( D e. CC /\ D # 0 ) ) ) -> ( ( A / B ) / ( C / D ) ) = ( ( A x. D ) / ( B x. C ) ) )
Theorem	divcanap5 8807	Cancellation of common factor in a ratio. (Contributed by Jim Kingdon, 25-Feb-2020.)
|- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ ( C e. CC /\ C # 0 ) ) -> ( ( C x. A ) / ( C x. B ) ) = ( A / B ) )
Theorem	divmul13ap 8808	Swap the denominators in the product of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( C e. CC /\ C # 0 ) /\ ( D e. CC /\ D # 0 ) ) ) -> ( ( A / C ) x. ( B / D ) ) = ( ( B / C ) x. ( A / D ) ) )
Theorem	divmul24ap 8809	Swap the numerators in the product of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( C e. CC /\ C # 0 ) /\ ( D e. CC /\ D # 0 ) ) ) -> ( ( A / C ) x. ( B / D ) ) = ( ( A / D ) x. ( B / C ) ) )
Theorem	divmuleqap 8810	Cross-multiply in an equality of ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( C e. CC /\ C # 0 ) /\ ( D e. CC /\ D # 0 ) ) ) -> ( ( A / C ) = ( B / D ) <-> ( A x. D ) = ( B x. C ) ) )
Theorem	recdivap 8811	The reciprocal of a ratio. (Contributed by Jim Kingdon, 26-Feb-2020.)
|- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) -> ( 1 / ( A / B ) ) = ( B / A ) )
Theorem	divcanap6 8812	Cancellation of inverted fractions. (Contributed by Jim Kingdon, 26-Feb-2020.)
|- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) -> ( ( A / B ) x. ( B / A ) ) = 1 )
Theorem	divdiv32ap 8813	Swap denominators in a division. (Contributed by Jim Kingdon, 26-Feb-2020.)
|- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ ( C e. CC /\ C # 0 ) ) -> ( ( A / B ) / C ) = ( ( A / C ) / B ) )
Theorem	divcanap7 8814	Cancel equal divisors in a division. (Contributed by Jim Kingdon, 26-Feb-2020.)
|- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ ( C e. CC /\ C # 0 ) ) -> ( ( A / C ) / ( B / C ) ) = ( A / B ) )
Theorem	dmdcanap 8815	Cancellation law for division and multiplication. (Contributed by Jim Kingdon, 26-Feb-2020.)
|- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) /\ C e. CC ) -> ( ( A / B ) x. ( C / A ) ) = ( C / B ) )
Theorem	divdivap1 8816	Division into a fraction. (Contributed by Jim Kingdon, 26-Feb-2020.)
|- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ ( C e. CC /\ C # 0 ) ) -> ( ( A / B ) / C ) = ( A / ( B x. C ) ) )
Theorem	divdivap2 8817	Division by a fraction. (Contributed by Jim Kingdon, 26-Feb-2020.)
|- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ ( C e. CC /\ C # 0 ) ) -> ( A / ( B / C ) ) = ( ( A x. C ) / B ) )
Theorem	recdivap2 8818	Division into a reciprocal. (Contributed by Jim Kingdon, 26-Feb-2020.)
|- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) -> ( ( 1 / A ) / B ) = ( 1 / ( A x. B ) ) )
Theorem	ddcanap 8819	Cancellation in a double division. (Contributed by Jim Kingdon, 26-Feb-2020.)
|- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) -> ( A / ( A / B ) ) = B )
Theorem	divadddivap 8820	Addition of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( C e. CC /\ C # 0 ) /\ ( D e. CC /\ D # 0 ) ) ) -> ( ( A / C ) + ( B / D ) ) = ( ( ( A x. D ) + ( B x. C ) ) / ( C x. D ) ) )
Theorem	divsubdivap 8821	Subtraction of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( C e. CC /\ C # 0 ) /\ ( D e. CC /\ D # 0 ) ) ) -> ( ( A / C ) - ( B / D ) ) = ( ( ( A x. D ) - ( B x. C ) ) / ( C x. D ) ) )
Theorem	conjmulap 8822	Two numbers whose reciprocals sum to 1 are called "conjugates" and satisfy this relationship. (Contributed by Jim Kingdon, 26-Feb-2020.)
|- ( ( ( P e. CC /\ P # 0 ) /\ ( Q e. CC /\ Q # 0 ) ) -> ( ( ( 1 / P ) + ( 1 / Q ) ) = 1 <-> ( ( P - 1 ) x. ( Q - 1 ) ) = 1 ) )
Theorem	rerecclap 8823	Closure law for reciprocal. (Contributed by Jim Kingdon, 26-Feb-2020.)
|- ( ( A e. RR /\ A # 0 ) -> ( 1 / A ) e. RR )
Theorem	redivclap 8824	Closure law for division of reals. (Contributed by Jim Kingdon, 26-Feb-2020.)
|- ( ( A e. RR /\ B e. RR /\ B # 0 ) -> ( A / B ) e. RR )
Theorem	eqneg 8825	A number equal to its negative is zero. (Contributed by NM, 12-Jul-2005.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( A e. CC -> ( A = -u A <-> A = 0 ) )
Theorem	eqnegd 8826	A complex number equals its negative iff it is zero. Deduction form of eqneg 8825. (Contributed by David Moews, 28-Feb-2017.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( A = -u A <-> A = 0 ) )
Theorem	eqnegad 8827	If a complex number equals its own negative, it is zero. One-way deduction form of eqneg 8825. (Contributed by David Moews, 28-Feb-2017.)
|- ( ph -> A e. CC )   &    |- ( ph -> A = -u A )   =>    |- ( ph -> A = 0 )
Theorem	div2negap 8828	Quotient of two negatives. (Contributed by Jim Kingdon, 27-Feb-2020.)
|- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( -u A / -u B ) = ( A / B ) )
Theorem	divneg2ap 8829	Move negative sign inside of a division. (Contributed by Jim Kingdon, 27-Feb-2020.)
|- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> -u ( A / B ) = ( A / -u B ) )
Theorem	recclapzi 8830	Closure law for reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.)
|- A e. CC   =>    |- ( A # 0 -> ( 1 / A ) e. CC )
Theorem	recap0apzi 8831	The reciprocal of a number apart from zero is apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
|- A e. CC   =>    |- ( A # 0 -> ( 1 / A ) # 0 )
Theorem	recidapzi 8832	Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.)
|- A e. CC   =>    |- ( A # 0 -> ( A x. ( 1 / A ) ) = 1 )
Theorem	div1i 8833	A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.)
|- A e. CC   =>    |- ( A / 1 ) = A
Theorem	eqnegi 8834	A number equal to its negative is zero. (Contributed by NM, 29-May-1999.)
|- A e. CC   =>    |- ( A = -u A <-> A = 0 )
Theorem	recclapi 8835	Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
|- A e. CC   &    |- A # 0   =>    |- ( 1 / A ) e. CC
Theorem	recidapi 8836	Multiplication of a number and its reciprocal. (Contributed by NM, 9-Feb-1995.)
|- A e. CC   &    |- A # 0   =>    |- ( A x. ( 1 / A ) ) = 1
Theorem	recrecapi 8837	A number is equal to the reciprocal of its reciprocal. Theorem I.10 of [Apostol] p. 18. (Contributed by NM, 9-Feb-1995.)
|- A e. CC   &    |- A # 0   =>    |- ( 1 / ( 1 / A ) ) = A
Theorem	dividapi 8838	A number divided by itself is one. (Contributed by NM, 9-Feb-1995.)
|- A e. CC   &    |- A # 0   =>    |- ( A / A ) = 1
Theorem	div0api 8839	Division into zero is zero. (Contributed by NM, 12-Aug-1999.)
|- A e. CC   &    |- A # 0   =>    |- ( 0 / A ) = 0
Theorem	divclapzi 8840	Closure law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
|- A e. CC   &    |- B e. CC   =>    |- ( B # 0 -> ( A / B ) e. CC )
Theorem	divcanap1zi 8841	A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
|- A e. CC   &    |- B e. CC   =>    |- ( B # 0 -> ( ( A / B ) x. B ) = A )
Theorem	divcanap2zi 8842	A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
|- A e. CC   &    |- B e. CC   =>    |- ( B # 0 -> ( B x. ( A / B ) ) = A )
Theorem	divrecapzi 8843	Relationship between division and reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.)
|- A e. CC   &    |- B e. CC   =>    |- ( B # 0 -> ( A / B ) = ( A x. ( 1 / B ) ) )
Theorem	divcanap3zi 8844	A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
|- A e. CC   &    |- B e. CC   =>    |- ( B # 0 -> ( ( B x. A ) / B ) = A )
Theorem	divcanap4zi 8845	A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
|- A e. CC   &    |- B e. CC   =>    |- ( B # 0 -> ( ( A x. B ) / B ) = A )
Theorem	rec11api 8846	Reciprocal is one-to-one. (Contributed by Jim Kingdon, 28-Feb-2020.)
|- A e. CC   &    |- B e. CC   =>    |- ( ( A # 0 /\ B # 0 ) -> ( ( 1 / A ) = ( 1 / B ) <-> A = B ) )
Theorem	divclapi 8847	Closure law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
|- A e. CC   &    |- B e. CC   &    |- B # 0   =>    |- ( A / B ) e. CC
Theorem	divcanap2i 8848	A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
|- A e. CC   &    |- B e. CC   &    |- B # 0   =>    |- ( B x. ( A / B ) ) = A
Theorem	divcanap1i 8849	A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
|- A e. CC   &    |- B e. CC   &    |- B # 0   =>    |- ( ( A / B ) x. B ) = A
Theorem	divrecapi 8850	Relationship between division and reciprocal. (Contributed by Jim Kingdon, 28-Feb-2020.)
|- A e. CC   &    |- B e. CC   &    |- B # 0   =>    |- ( A / B ) = ( A x. ( 1 / B ) )
Theorem	divcanap3i 8851	A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
|- A e. CC   &    |- B e. CC   &    |- B # 0   =>    |- ( ( B x. A ) / B ) = A
Theorem	divcanap4i 8852	A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
|- A e. CC   &    |- B e. CC   &    |- B # 0   =>    |- ( ( A x. B ) / B ) = A
Theorem	divap0i 8853	The ratio of numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 28-Feb-2020.)
|- A e. CC   &    |- B e. CC   &    |- A # 0   &    |- B # 0   =>    |- ( A / B ) # 0
Theorem	rec11apii 8854	Reciprocal is one-to-one. (Contributed by Jim Kingdon, 28-Feb-2020.)
|- A e. CC   &    |- B e. CC   &    |- A # 0   &    |- B # 0   =>    |- ( ( 1 / A ) = ( 1 / B ) <-> A = B )
Theorem	divassapzi 8855	An associative law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
|- A e. CC   &    |- B e. CC   &    |- C e. CC   =>    |- ( C # 0 -> ( ( A x. B ) / C ) = ( A x. ( B / C ) ) )
Theorem	divmulapzi 8856	Relationship between division and multiplication. (Contributed by Jim Kingdon, 28-Feb-2020.)
|- A e. CC   &    |- B e. CC   &    |- C e. CC   =>    |- ( B # 0 -> ( ( A / B ) = C <-> ( B x. C ) = A ) )
Theorem	divdirapzi 8857	Distribution of division over addition. (Contributed by Jim Kingdon, 28-Feb-2020.)
|- A e. CC   &    |- B e. CC   &    |- C e. CC   =>    |- ( C # 0 -> ( ( A + B ) / C ) = ( ( A / C ) + ( B / C ) ) )
Theorem	divdiv23apzi 8858	Swap denominators in a division. (Contributed by Jim Kingdon, 28-Feb-2020.)
|- A e. CC   &    |- B e. CC   &    |- C e. CC   =>    |- ( ( B # 0 /\ C # 0 ) -> ( ( A / B ) / C ) = ( ( A / C ) / B ) )
Theorem	divmulapi 8859	Relationship between division and multiplication. (Contributed by Jim Kingdon, 29-Feb-2020.)
|- A e. CC   &    |- B e. CC   &    |- C e. CC   &    |- B # 0   =>    |- ( ( A / B ) = C <-> ( B x. C ) = A )
Theorem	divdiv32api 8860	Swap denominators in a division. (Contributed by Jim Kingdon, 29-Feb-2020.)
|- A e. CC   &    |- B e. CC   &    |- C e. CC   &    |- B # 0   &    |- C # 0   =>    |- ( ( A / B ) / C ) = ( ( A / C ) / B )
Theorem	divassapi 8861	An associative law for division. (Contributed by Jim Kingdon, 9-Mar-2020.)
|- A e. CC   &    |- B e. CC   &    |- C e. CC   &    |- C # 0   =>    |- ( ( A x. B ) / C ) = ( A x. ( B / C ) )
Theorem	divdirapi 8862	Distribution of division over addition. (Contributed by Jim Kingdon, 9-Mar-2020.)
|- A e. CC   &    |- B e. CC   &    |- C e. CC   &    |- C # 0   =>    |- ( ( A + B ) / C ) = ( ( A / C ) + ( B / C ) )
Theorem	div23api 8863	A commutative/associative law for division. (Contributed by Jim Kingdon, 9-Mar-2020.)
|- A e. CC   &    |- B e. CC   &    |- C e. CC   &    |- C # 0   =>    |- ( ( A x. B ) / C ) = ( ( A / C ) x. B )
Theorem	div11api 8864	One-to-one relationship for division. (Contributed by Jim Kingdon, 9-Mar-2020.)
|- A e. CC   &    |- B e. CC   &    |- C e. CC   &    |- C # 0   =>    |- ( ( A / C ) = ( B / C ) <-> A = B )
Theorem	divmuldivapi 8865	Multiplication of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
|- A e. CC   &    |- B e. CC   &    |- C e. CC   &    |- D e. CC   &    |- B # 0   &    |- D # 0   =>    |- ( ( A / B ) x. ( C / D ) ) = ( ( A x. C ) / ( B x. D ) )
Theorem	divmul13api 8866	Swap denominators of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
|- A e. CC   &    |- B e. CC   &    |- C e. CC   &    |- D e. CC   &    |- B # 0   &    |- D # 0   =>    |- ( ( A / B ) x. ( C / D ) ) = ( ( C / B ) x. ( A / D ) )
Theorem	divadddivapi 8867	Addition of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
|- A e. CC   &    |- B e. CC   &    |- C e. CC   &    |- D e. CC   &    |- B # 0   &    |- D # 0   =>    |- ( ( A / B ) + ( C / D ) ) = ( ( ( A x. D ) + ( C x. B ) ) / ( B x. D ) )
Theorem	divdivdivapi 8868	Division of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
|- A e. CC   &    |- B e. CC   &    |- C e. CC   &    |- D e. CC   &    |- B # 0   &    |- D # 0   &    |- C # 0   =>    |- ( ( A / B ) / ( C / D ) ) = ( ( A x. D ) / ( B x. C ) )
Theorem	rerecclapzi 8869	Closure law for reciprocal. (Contributed by Jim Kingdon, 9-Mar-2020.)
|- A e. RR   =>    |- ( A # 0 -> ( 1 / A ) e. RR )
Theorem	rerecclapi 8870	Closure law for reciprocal. (Contributed by Jim Kingdon, 9-Mar-2020.)
|- A e. RR   &    |- A # 0   =>    |- ( 1 / A ) e. RR
Theorem	redivclapzi 8871	Closure law for division of reals. (Contributed by Jim Kingdon, 9-Mar-2020.)
|- A e. RR   &    |- B e. RR   =>    |- ( B # 0 -> ( A / B ) e. RR )
Theorem	redivclapi 8872	Closure law for division of reals. (Contributed by Jim Kingdon, 9-Mar-2020.)
|- A e. RR   &    |- B e. RR   &    |- B # 0   =>    |- ( A / B ) e. RR
Theorem	div1d 8873	A number divided by 1 is itself. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( A / 1 ) = A )
Theorem	recclapd 8874	Closure law for reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> A # 0 )   =>    |- ( ph -> ( 1 / A ) e. CC )
Theorem	recap0d 8875	The reciprocal of a number apart from zero is apart from zero. (Contributed by Jim Kingdon, 3-Mar-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> A # 0 )   =>    |- ( ph -> ( 1 / A ) # 0 )
Theorem	recidapd 8876	Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> A # 0 )   =>    |- ( ph -> ( A x. ( 1 / A ) ) = 1 )
Theorem	recidap2d 8877	Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> A # 0 )   =>    |- ( ph -> ( ( 1 / A ) x. A ) = 1 )
Theorem	recrecapd 8878	A number is equal to the reciprocal of its reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> A # 0 )   =>    |- ( ph -> ( 1 / ( 1 / A ) ) = A )
Theorem	dividapd 8879	A number divided by itself is one. (Contributed by Jim Kingdon, 3-Mar-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> A # 0 )   =>    |- ( ph -> ( A / A ) = 1 )
Theorem	div0apd 8880	Division into zero is zero. (Contributed by Jim Kingdon, 3-Mar-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> A # 0 )   =>    |- ( ph -> ( 0 / A ) = 0 )
Theorem	apmul1 8881	Multiplication of both sides of complex apartness by a complex number apart from zero. (Contributed by Jim Kingdon, 20-Mar-2020.)
|- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( A # B <-> ( A x. C ) # ( B x. C ) ) )
Theorem	apmul2 8882	Multiplication of both sides of complex apartness by a complex number apart from zero. (Contributed by Jim Kingdon, 6-Jan-2023.)
|- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( A # B <-> ( C x. A ) # ( C x. B ) ) )
Theorem	divclapd 8883	Closure law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> B # 0 )   =>    |- ( ph -> ( A / B ) e. CC )
Theorem	divcanap1d 8884	A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> B # 0 )   =>    |- ( ph -> ( ( A / B ) x. B ) = A )
Theorem	divcanap2d 8885	A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> B # 0 )   =>    |- ( ph -> ( B x. ( A / B ) ) = A )
Theorem	divrecapd 8886	Relationship between division and reciprocal. Theorem I.9 of [Apostol] p. 18. (Contributed by Jim Kingdon, 29-Feb-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> B # 0 )   =>    |- ( ph -> ( A / B ) = ( A x. ( 1 / B ) ) )
Theorem	divrecap2d 8887	Relationship between division and reciprocal. (Contributed by Jim Kingdon, 29-Feb-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> B # 0 )   =>    |- ( ph -> ( A / B ) = ( ( 1 / B ) x. A ) )
Theorem	divcanap3d 8888	A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> B # 0 )   =>    |- ( ph -> ( ( B x. A ) / B ) = A )
Theorem	divcanap4d 8889	A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> B # 0 )   =>    |- ( ph -> ( ( A x. B ) / B ) = A )
Theorem	diveqap0d 8890	If a ratio is zero, the numerator is zero. (Contributed by Jim Kingdon, 19-Mar-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> B # 0 )   &    |- ( ph -> ( A / B ) = 0 )   =>    |- ( ph -> A = 0 )
Theorem	diveqap1d 8891	Equality in terms of unit ratio. (Contributed by Jim Kingdon, 19-Mar-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> B # 0 )   &    |- ( ph -> ( A / B ) = 1 )   =>    |- ( ph -> A = B )
Theorem	diveqap1ad 8892	The quotient of two complex numbers is one iff they are equal. Deduction form of diveqap1 8798. Generalization of diveqap1d 8891. (Contributed by Jim Kingdon, 19-Mar-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> B # 0 )   =>    |- ( ph -> ( ( A / B ) = 1 <-> A = B ) )
Theorem	diveqap0ad 8893	A fraction of complex numbers is zero iff its numerator is. Deduction form of diveqap0 8775. (Contributed by Jim Kingdon, 19-Mar-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> B # 0 )   =>    |- ( ph -> ( ( A / B ) = 0 <-> A = 0 ) )
Theorem	divap1d 8894	If two complex numbers are apart, their quotient is apart from one. (Contributed by Jim Kingdon, 20-Mar-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> B # 0 )   &    |- ( ph -> A # B )   =>    |- ( ph -> ( A / B ) # 1 )
Theorem	divap0bd 8895	A ratio is zero iff the numerator is zero. (Contributed by Jim Kingdon, 19-Mar-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> B # 0 )   =>    |- ( ph -> ( A # 0 <-> ( A / B ) # 0 ) )
Theorem	divnegapd 8896	Move negative sign inside of a division. (Contributed by Jim Kingdon, 19-Mar-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> B # 0 )   =>    |- ( ph -> -u ( A / B ) = ( -u A / B ) )
Theorem	divneg2apd 8897	Move negative sign inside of a division. (Contributed by Jim Kingdon, 19-Mar-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> B # 0 )   =>    |- ( ph -> -u ( A / B ) = ( A / -u B ) )
Theorem	div2negapd 8898	Quotient of two negatives. (Contributed by Jim Kingdon, 19-Mar-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> B # 0 )   =>    |- ( ph -> ( -u A / -u B ) = ( A / B ) )
Theorem	divap0d 8899	The ratio of numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 3-Mar-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> A # 0 )   &    |- ( ph -> B # 0 )   =>    |- ( ph -> ( A / B ) # 0 )
Theorem	recdivapd 8900	The reciprocal of a ratio. (Contributed by Jim Kingdon, 3-Mar-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> A # 0 )   &    |- ( ph -> B # 0 )   =>    |- ( ph -> ( 1 / ( A / B ) ) = ( B / A ) )