P. 87 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 8601-8700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	divap0 8601	The ratio of numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 22-Feb-2020.)
|- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) -> ( A / B ) # 0 )
Theorem	recap0 8602	The reciprocal of a number apart from zero is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.)
|- ( ( A e. CC /\ A # 0 ) -> ( 1 / A ) # 0 )
Theorem	recidap 8603	Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 24-Feb-2020.)
|- ( ( A e. CC /\ A # 0 ) -> ( A x. ( 1 / A ) ) = 1 )
Theorem	recidap2 8604	Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 24-Feb-2020.)
|- ( ( A e. CC /\ A # 0 ) -> ( ( 1 / A ) x. A ) = 1 )
Theorem	divrecap 8605	Relationship between division and reciprocal. (Contributed by Jim Kingdon, 24-Feb-2020.)
|- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( A / B ) = ( A x. ( 1 / B ) ) )
Theorem	divrecap2 8606	Relationship between division and reciprocal. (Contributed by Jim Kingdon, 25-Feb-2020.)
|- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( A / B ) = ( ( 1 / B ) x. A ) )
Theorem	divassap 8607	An associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
|- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A x. B ) / C ) = ( A x. ( B / C ) ) )
Theorem	div23ap 8608	A commutative/associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
|- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A x. B ) / C ) = ( ( A / C ) x. B ) )
Theorem	div32ap 8609	A commutative/associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
|- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ C e. CC ) -> ( ( A / B ) x. C ) = ( A x. ( C / B ) ) )
Theorem	div13ap 8610	A commutative/associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
|- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ C e. CC ) -> ( ( A / B ) x. C ) = ( ( C / B ) x. A ) )
Theorem	div12ap 8611	A commutative/associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
|- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( A x. ( B / C ) ) = ( B x. ( A / C ) ) )
Theorem	divmulassap 8612	An associative law for division and multiplication. (Contributed by Jim Kingdon, 24-Jan-2022.)
|- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( D e. CC /\ D # 0 ) ) -> ( ( A x. ( B / D ) ) x. C ) = ( ( A x. B ) x. ( C / D ) ) )
Theorem	divmulasscomap 8613	An associative/commutative law for division and multiplication. (Contributed by Jim Kingdon, 24-Jan-2022.)
|- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( D e. CC /\ D # 0 ) ) -> ( ( A x. ( B / D ) ) x. C ) = ( B x. ( ( A x. C ) / D ) ) )
Theorem	divdirap 8614	Distribution of division over addition. (Contributed by Jim Kingdon, 25-Feb-2020.)
|- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A + B ) / C ) = ( ( A / C ) + ( B / C ) ) )
Theorem	divcanap3 8615	A cancellation law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
|- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( ( B x. A ) / B ) = A )
Theorem	divcanap4 8616	A cancellation law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
|- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( ( A x. B ) / B ) = A )
Theorem	div11ap 8617	One-to-one relationship for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
|- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A / C ) = ( B / C ) <-> A = B ) )
Theorem	dividap 8618	A number divided by itself is one. (Contributed by Jim Kingdon, 25-Feb-2020.)
|- ( ( A e. CC /\ A # 0 ) -> ( A / A ) = 1 )
Theorem	div0ap 8619	Division into zero is zero. (Contributed by Jim Kingdon, 25-Feb-2020.)
|- ( ( A e. CC /\ A # 0 ) -> ( 0 / A ) = 0 )
Theorem	div1 8620	A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.) (Proof shortened by Mario Carneiro, 27-May-2016.)
|- ( A e. CC -> ( A / 1 ) = A )
Theorem	1div1e1 8621	1 divided by 1 is 1 (common case). (Contributed by David A. Wheeler, 7-Dec-2018.)
|- ( 1 / 1 ) = 1
Theorem	diveqap1 8622	Equality in terms of unit ratio. (Contributed by Jim Kingdon, 25-Feb-2020.)
|- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( ( A / B ) = 1 <-> A = B ) )
Theorem	divnegap 8623	Move negative sign inside of a division. (Contributed by Jim Kingdon, 25-Feb-2020.)
|- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> -u ( A / B ) = ( -u A / B ) )
Theorem	muldivdirap 8624	Distribution of division over addition with a multiplication. (Contributed by Jim Kingdon, 11-Nov-2021.)
|- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( ( C x. A ) + B ) / C ) = ( A + ( B / C ) ) )
Theorem	divsubdirap 8625	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 8626	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 8627	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 8628	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 8629	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 8630	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 8631	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 8632	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 8633	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 8634	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 8635	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 8636	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 8637	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 8638	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 8639	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 8640	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 8641	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 8642	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 8643	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 8644	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 8645	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 8646	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 8647	Closure law for reciprocal. (Contributed by Jim Kingdon, 26-Feb-2020.)
|- ( ( A e. RR /\ A # 0 ) -> ( 1 / A ) e. RR )
Theorem	redivclap 8648	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 8649	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 8650	A complex number equals its negative iff it is zero. Deduction form of eqneg 8649. (Contributed by David Moews, 28-Feb-2017.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( A = -u A <-> A = 0 ) )
Theorem	eqnegad 8651	If a complex number equals its own negative, it is zero. One-way deduction form of eqneg 8649. (Contributed by David Moews, 28-Feb-2017.)
|- ( ph -> A e. CC )   &    |- ( ph -> A = -u A )   =>    |- ( ph -> A = 0 )
Theorem	div2negap 8652	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 8653	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 8654	Closure law for reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.)
|- A e. CC   =>    |- ( A # 0 -> ( 1 / A ) e. CC )
Theorem	recap0apzi 8655	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 8656	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 8657	A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.)
|- A e. CC   =>    |- ( A / 1 ) = A
Theorem	eqnegi 8658	A number equal to its negative is zero. (Contributed by NM, 29-May-1999.)
|- A e. CC   =>    |- ( A = -u A <-> A = 0 )
Theorem	recclapi 8659	Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
|- A e. CC   &    |- A # 0   =>    |- ( 1 / A ) e. CC
Theorem	recidapi 8660	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 8661	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 8662	A number divided by itself is one. (Contributed by NM, 9-Feb-1995.)
|- A e. CC   &    |- A # 0   =>    |- ( A / A ) = 1
Theorem	div0api 8663	Division into zero is zero. (Contributed by NM, 12-Aug-1999.)
|- A e. CC   &    |- A # 0   =>    |- ( 0 / A ) = 0
Theorem	divclapzi 8664	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 8665	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 8666	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 8667	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 8668	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 8669	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 8670	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 8671	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 8672	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 8673	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 8674	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 8675	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 8676	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 8677	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 8678	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 8679	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 8680	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 8681	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 8682	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 8683	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 8684	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 8685	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 8686	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 8687	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 8688	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 8689	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 8690	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 8691	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 8692	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 8693	Closure law for reciprocal. (Contributed by Jim Kingdon, 9-Mar-2020.)
|- A e. RR   =>    |- ( A # 0 -> ( 1 / A ) e. RR )
Theorem	rerecclapi 8694	Closure law for reciprocal. (Contributed by Jim Kingdon, 9-Mar-2020.)
|- A e. RR   &    |- A # 0   =>    |- ( 1 / A ) e. RR
Theorem	redivclapzi 8695	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 8696	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 8697	A number divided by 1 is itself. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( A / 1 ) = A )
Theorem	recclapd 8698	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 8699	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 8700	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 )