P. 90 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 8901-9000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	recidapi 8901	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 8902	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 8903	A number divided by itself is one. (Contributed by NM, 9-Feb-1995.)
|- A e. CC   &    |- A # 0   =>    |- ( A / A ) = 1
Theorem	div0api 8904	Division into zero is zero. (Contributed by NM, 12-Aug-1999.)
|- A e. CC   &    |- A # 0   =>    |- ( 0 / A ) = 0
Theorem	divclapzi 8905	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 8906	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 8907	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 8908	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 8909	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 8910	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 8911	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 8912	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 8913	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 8914	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 8915	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 8916	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 8917	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 8918	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 8919	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 8920	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 8921	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 8922	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 8923	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 8924	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 8925	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 8926	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 8927	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 8928	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 8929	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 8930	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 8931	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 8932	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 8933	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 8934	Closure law for reciprocal. (Contributed by Jim Kingdon, 9-Mar-2020.)
|- A e. RR   =>    |- ( A # 0 -> ( 1 / A ) e. RR )
Theorem	rerecclapi 8935	Closure law for reciprocal. (Contributed by Jim Kingdon, 9-Mar-2020.)
|- A e. RR   &    |- A # 0   =>    |- ( 1 / A ) e. RR
Theorem	redivclapzi 8936	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 8937	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 8938	A number divided by 1 is itself. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( A / 1 ) = A )
Theorem	recclapd 8939	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 8940	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 8941	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 8942	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 8943	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 8944	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 8945	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 8946	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 8947	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 8948	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 8949	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 8950	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 8951	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 8952	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 8953	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 8954	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 8955	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 8956	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 8957	The quotient of two complex numbers is one iff they are equal. Deduction form of diveqap1 8863. Generalization of diveqap1d 8956. (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 8958	A fraction of complex numbers is zero iff its numerator is. Deduction form of diveqap0 8840. (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 8959	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 8960	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 8961	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 8962	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 8963	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 8964	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 8965	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 ) )
Theorem	recdivap2d 8966	Division into a reciprocal. (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 ) = ( 1 / ( A x. B ) ) )
Theorem	divcanap6d 8967	Cancellation of inverted fractions. (Contributed by Jim Kingdon, 3-Mar-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> A # 0 )   &    |- ( ph -> B # 0 )   =>    |- ( ph -> ( ( A / B ) x. ( B / A ) ) = 1 )
Theorem	ddcanapd 8968	Cancellation in a double division. (Contributed by Jim Kingdon, 3-Mar-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> A # 0 )   &    |- ( ph -> B # 0 )   =>    |- ( ph -> ( A / ( A / B ) ) = B )
Theorem	rec11apd 8969	Reciprocal is one-to-one. (Contributed by Jim Kingdon, 3-Mar-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> A # 0 )   &    |- ( ph -> B # 0 )   &    |- ( ph -> ( 1 / A ) = ( 1 / B ) )   =>    |- ( ph -> A = B )
Theorem	divmulapd 8970	Relationship between division and multiplication. (Contributed by Jim Kingdon, 8-Mar-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> B # 0 )   =>    |- ( ph -> ( ( A / B ) = C <-> ( B x. C ) = A ) )
Theorem	apdivmuld 8971	Relationship between division and multiplication. (Contributed by Jim Kingdon, 26-Dec-2022.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> B # 0 )   =>    |- ( ph -> ( ( A / B ) # C <-> ( B x. C ) # A ) )
Theorem	div32apd 8972	A commutative/associative law for division. (Contributed by Jim Kingdon, 8-Mar-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> B # 0 )   =>    |- ( ph -> ( ( A / B ) x. C ) = ( A x. ( C / B ) ) )
Theorem	div13apd 8973	A commutative/associative law for division. (Contributed by Jim Kingdon, 8-Mar-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> B # 0 )   =>    |- ( ph -> ( ( A / B ) x. C ) = ( ( C / B ) x. A ) )
Theorem	divdiv32apd 8974	Swap denominators in a division. (Contributed by Jim Kingdon, 8-Mar-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> B # 0 )   &    |- ( ph -> C # 0 )   =>    |- ( ph -> ( ( A / B ) / C ) = ( ( A / C ) / B ) )
Theorem	divcanap5d 8975	Cancellation of common factor in a ratio. (Contributed by Jim Kingdon, 8-Mar-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> B # 0 )   &    |- ( ph -> C # 0 )   =>    |- ( ph -> ( ( C x. A ) / ( C x. B ) ) = ( A / B ) )
Theorem	divcanap5rd 8976	Cancellation of common factor in a ratio. (Contributed by Jim Kingdon, 8-Mar-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> B # 0 )   &    |- ( ph -> C # 0 )   =>    |- ( ph -> ( ( A x. C ) / ( B x. C ) ) = ( A / B ) )
Theorem	divcanap7d 8977	Cancel equal divisors in a division. (Contributed by Jim Kingdon, 8-Mar-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> B # 0 )   &    |- ( ph -> C # 0 )   =>    |- ( ph -> ( ( A / C ) / ( B / C ) ) = ( A / B ) )
Theorem	dmdcanapd 8978	Cancellation law for division and multiplication. (Contributed by Jim Kingdon, 8-Mar-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> B # 0 )   &    |- ( ph -> C # 0 )   =>    |- ( ph -> ( ( B / C ) x. ( A / B ) ) = ( A / C ) )
Theorem	dmdcanap2d 8979	Cancellation law for division and multiplication. (Contributed by Jim Kingdon, 8-Mar-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> B # 0 )   &    |- ( ph -> C # 0 )   =>    |- ( ph -> ( ( A / B ) x. ( B / C ) ) = ( A / C ) )
Theorem	divdivap1d 8980	Division into a fraction. (Contributed by Jim Kingdon, 8-Mar-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> B # 0 )   &    |- ( ph -> C # 0 )   =>    |- ( ph -> ( ( A / B ) / C ) = ( A / ( B x. C ) ) )
Theorem	divdivap2d 8981	Division by a fraction. (Contributed by Jim Kingdon, 8-Mar-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> B # 0 )   &    |- ( ph -> C # 0 )   =>    |- ( ph -> ( A / ( B / C ) ) = ( ( A x. C ) / B ) )
Theorem	divmulap2d 8982	Relationship between division and multiplication. (Contributed by Jim Kingdon, 2-Mar-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> C # 0 )   =>    |- ( ph -> ( ( A / C ) = B <-> A = ( C x. B ) ) )
Theorem	divmulap3d 8983	Relationship between division and multiplication. (Contributed by Jim Kingdon, 2-Mar-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> C # 0 )   =>    |- ( ph -> ( ( A / C ) = B <-> A = ( B x. C ) ) )
Theorem	divassapd 8984	An associative law for division. (Contributed by Jim Kingdon, 2-Mar-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> C # 0 )   =>    |- ( ph -> ( ( A x. B ) / C ) = ( A x. ( B / C ) ) )
Theorem	div12apd 8985	A commutative/associative law for division. (Contributed by Jim Kingdon, 2-Mar-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> C # 0 )   =>    |- ( ph -> ( A x. ( B / C ) ) = ( B x. ( A / C ) ) )
Theorem	div23apd 8986	A commutative/associative law for division. (Contributed by Jim Kingdon, 2-Mar-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> C # 0 )   =>    |- ( ph -> ( ( A x. B ) / C ) = ( ( A / C ) x. B ) )
Theorem	divdirapd 8987	Distribution of division over addition. (Contributed by Jim Kingdon, 2-Mar-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> C # 0 )   =>    |- ( ph -> ( ( A + B ) / C ) = ( ( A / C ) + ( B / C ) ) )
Theorem	divsubdirapd 8988	Distribution of division over subtraction. (Contributed by Jim Kingdon, 2-Mar-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> C # 0 )   =>    |- ( ph -> ( ( A - B ) / C ) = ( ( A / C ) - ( B / C ) ) )
Theorem	div11apd 8989	One-to-one relationship for division. (Contributed by Jim Kingdon, 2-Mar-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> C # 0 )   &    |- ( ph -> ( A / C ) = ( B / C ) )   =>    |- ( ph -> A = B )
Theorem	divmuldivapd 8990	Multiplication of two ratios. (Contributed by Jim Kingdon, 30-Jul-2021.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> D e. CC )   &    |- ( ph -> B # 0 )   &    |- ( ph -> D # 0 )   =>    |- ( ph -> ( ( A / B ) x. ( C / D ) ) = ( ( A x. C ) / ( B x. D ) ) )
Theorem	divmuleqapd 8991	Cross-multiply in an equality of ratios. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> D e. CC )   &    |- ( ph -> B # 0 )   &    |- ( ph -> D # 0 )   =>    |- ( ph -> ( ( A / B ) = ( C / D ) <-> ( A x. D ) = ( C x. B ) ) )
Theorem	rerecclapd 8992	Closure law for reciprocal. (Contributed by Jim Kingdon, 29-Feb-2020.)
|- ( ph -> A e. RR )   &    |- ( ph -> A # 0 )   =>    |- ( ph -> ( 1 / A ) e. RR )
Theorem	redivclapd 8993	Closure law for division of reals. (Contributed by Jim Kingdon, 29-Feb-2020.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> B # 0 )   =>    |- ( ph -> ( A / B ) e. RR )
Theorem	diveqap1bd 8994	If two complex numbers are equal, their quotient is one. One-way deduction form of diveqap1 8863. Converse of diveqap1d 8956. (Contributed by David Moews, 28-Feb-2017.) (Revised by Jim Kingdon, 2-Aug-2023.)
|- ( ph -> B e. CC )   &    |- ( ph -> B # 0 )   &    |- ( ph -> A = B )   =>    |- ( ph -> ( A / B ) = 1 )
Theorem	div2subap 8995	Swap the order of subtraction in a division. (Contributed by Scott Fenton, 24-Jun-2013.)
|- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC /\ C # D ) ) -> ( ( A - B ) / ( C - D ) ) = ( ( B - A ) / ( D - C ) ) )
Theorem	div2subapd 8996	Swap subtrahend and minuend inside the numerator and denominator of a fraction. Deduction form of div2subap 8995. (Contributed by David Moews, 28-Feb-2017.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> D e. CC )   &    |- ( ph -> C # D )   =>    |- ( ph -> ( ( A - B ) / ( C - D ) ) = ( ( B - A ) / ( D - C ) ) )
Theorem	subrecap 8997	Subtraction of reciprocals. (Contributed by Scott Fenton, 9-Jul-2015.)
|- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) -> ( ( 1 / A ) - ( 1 / B ) ) = ( ( B - A ) / ( A x. B ) ) )
Theorem	subrecapi 8998	Subtraction of reciprocals. (Contributed by Scott Fenton, 9-Jan-2017.)
|- A e. CC   &    |- B e. CC   &    |- A # 0   &    |- B # 0   =>    |- ( ( 1 / A ) - ( 1 / B ) ) = ( ( B - A ) / ( A x. B ) )
Theorem	subrecapd 8999	Subtraction of reciprocals. (Contributed by Scott Fenton, 9-Jan-2017.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> A # 0 )   &    |- ( ph -> B # 0 )   =>    |- ( ph -> ( ( 1 / A ) - ( 1 / B ) ) = ( ( B - A ) / ( A x. B ) ) )
Theorem	mvllmulapd 9000	Move LHS left multiplication to RHS. (Contributed by Jim Kingdon, 10-Jun-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> A # 0 )   &    |- ( ph -> ( A x. B ) = C )   =>    |- ( ph -> B = ( C / A ) )