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	divcanap3i 8901	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 8902	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 8903	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 8904	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 8905	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 8906	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 8907	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 8908	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 8909	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 8910	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 8911	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 8912	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 8913	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 8914	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 8915	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 8916	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 8917	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 8918	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 8919	Closure law for reciprocal. (Contributed by Jim Kingdon, 9-Mar-2020.)
|- A e. RR   =>    |- ( A # 0 -> ( 1 / A ) e. RR )
Theorem	rerecclapi 8920	Closure law for reciprocal. (Contributed by Jim Kingdon, 9-Mar-2020.)
|- A e. RR   &    |- A # 0   =>    |- ( 1 / A ) e. RR
Theorem	redivclapzi 8921	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 8922	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 8923	A number divided by 1 is itself. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( A / 1 ) = A )
Theorem	recclapd 8924	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 8925	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 8926	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 8927	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 8928	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 8929	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 8930	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 8931	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 8932	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 8933	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 8934	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 8935	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 8936	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 8937	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 8938	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 8939	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 8940	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 8941	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 8942	The quotient of two complex numbers is one iff they are equal. Deduction form of diveqap1 8848. Generalization of diveqap1d 8941. (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 8943	A fraction of complex numbers is zero iff its numerator is. Deduction form of diveqap0 8825. (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 8944	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 8945	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 8946	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 8947	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 8948	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 8949	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 8950	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 8951	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 8952	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 8953	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 8954	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 8955	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 8956	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 8957	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 8958	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 8959	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 8960	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 8961	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 8962	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 8963	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 8964	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 8965	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 8966	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 8967	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 8968	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 8969	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 8970	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 8971	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 8972	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 8973	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 8974	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 8975	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 8976	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 8977	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 8978	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 8979	If two complex numbers are equal, their quotient is one. One-way deduction form of diveqap1 8848. Converse of diveqap1d 8941. (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 8980	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 8981	Swap subtrahend and minuend inside the numerator and denominator of a fraction. Deduction form of div2subap 8980. (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 8982	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 8983	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 8984	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 8985	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 ) )
Theorem	rerecapb 8986*	A real number has a multiplicative inverse if and only if it is apart from zero. Theorem 11.2.4 of [HoTT], p. (varies). (Contributed by Jim Kingdon, 18-Jan-2025.)
|- ( A e. RR -> ( A # 0 <-> E. x e. RR ( A x. x ) = 1 ) )
4.3.9  Ordering on reals (cont.)
Theorem	ltp1 8987	A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)
|- ( A e. RR -> A < ( A + 1 ) )
Theorem	lep1 8988	A number is less than or equal to itself plus 1. (Contributed by NM, 5-Jan-2006.)
|- ( A e. RR -> A <_ ( A + 1 ) )
Theorem	ltm1 8989	A number minus 1 is less than itself. (Contributed by NM, 9-Apr-2006.)
|- ( A e. RR -> ( A - 1 ) < A )
Theorem	lem1 8990	A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 2-Oct-2015.)
|- ( A e. RR -> ( A - 1 ) <_ A )
Theorem	letrp1 8991	A transitive property of 'less than or equal' and plus 1. (Contributed by NM, 5-Aug-2005.)
|- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> A <_ ( B + 1 ) )
Theorem	p1le 8992	A transitive property of plus 1 and 'less than or equal'. (Contributed by NM, 16-Aug-2005.)
|- ( ( A e. RR /\ B e. RR /\ ( A + 1 ) <_ B ) -> A <_ B )
Theorem	recgt0 8993	The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 25-Aug-1999.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( ( A e. RR /\ 0 < A ) -> 0 < ( 1 / A ) )
Theorem	prodgt0gt0 8994	Infer that a multiplicand is positive from a positive multiplier and positive product. See prodgt0 8995 for the same theorem with 0 < A replaced by the weaker condition 0 <_ A. (Contributed by Jim Kingdon, 29-Feb-2020.)
|- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < ( A x. B ) ) ) -> 0 < B )
Theorem	prodgt0 8995	Infer that a multiplicand is positive from a nonnegative multiplier and positive product. (Contributed by NM, 24-Apr-2005.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> 0 < B )
Theorem	prodgt02 8996	Infer that a multiplier is positive from a nonnegative multiplicand and positive product. (Contributed by NM, 24-Apr-2005.)
|- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ B /\ 0 < ( A x. B ) ) ) -> 0 < A )
Theorem	prodge0 8997	Infer that a multiplicand is nonnegative from a positive multiplier and nonnegative product. (Contributed by NM, 2-Jul-2005.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 <_ ( A x. B ) ) ) -> 0 <_ B )
Theorem	prodge02 8998	Infer that a multiplier is nonnegative from a positive multiplicand and nonnegative product. (Contributed by NM, 2-Jul-2005.)
|- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < B /\ 0 <_ ( A x. B ) ) ) -> 0 <_ A )
Theorem	ltmul2 8999	Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 13-Feb-2005.)
|- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A < B <-> ( C x. A ) < ( C x. B ) ) )
Theorem	lemul2 9000	Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 16-Mar-2005.)
|- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A <_ B <-> ( C x. A ) <_ ( C x. B ) ) )