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	recap0d 8801	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 8802	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 8803	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 8804	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 8805	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 8806	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 8807	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 8808	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 8809	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 8810	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 8811	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 8812	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 8813	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 8814	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 8815	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 8816	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 8817	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 8818	The quotient of two complex numbers is one iff they are equal. Deduction form of diveqap1 8724. Generalization of diveqap1d 8817. (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 8819	A fraction of complex numbers is zero iff its numerator is. Deduction form of diveqap0 8701. (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 8820	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 8821	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 8822	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 8823	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 8824	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 8825	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 8826	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 8827	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 8828	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 8829	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 8830	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 8831	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 8832	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 8833	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 8834	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 8835	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 8836	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 8837	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 8838	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 8839	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 8840	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 8841	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 8842	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 8843	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 8844	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 8845	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 8846	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 8847	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 8848	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 8849	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 8850	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 8851	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 8852	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 8853	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 8854	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 8855	If two complex numbers are equal, their quotient is one. One-way deduction form of diveqap1 8724. Converse of diveqap1d 8817. (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 8856	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 8857	Swap subtrahend and minuend inside the numerator and denominator of a fraction. Deduction form of div2subap 8856. (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 8858	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 8859	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 8860	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 8861	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 8862*	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 8863	A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)
|- ( A e. RR -> A < ( A + 1 ) )
Theorem	lep1 8864	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 8865	A number minus 1 is less than itself. (Contributed by NM, 9-Apr-2006.)
|- ( A e. RR -> ( A - 1 ) < A )
Theorem	lem1 8866	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 8867	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 8868	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 8869	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 8870	Infer that a multiplicand is positive from a positive multiplier and positive product. See prodgt0 8871 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 8871	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 8872	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 8873	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 8874	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 8875	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 8876	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 ) ) )
Theorem	lemul1a 8877	Multiplication of both sides of 'less than or equal to' by a nonnegative number. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (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	lemul2a 8878	Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.)
|- ( ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 <_ C ) ) /\ A <_ B ) -> ( C x. A ) <_ ( C x. B ) )
Theorem	ltmul12a 8879	Comparison of product of two positive numbers. (Contributed by NM, 30-Dec-2005.)
|- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ A < B ) ) /\ ( ( C e. RR /\ D e. RR ) /\ ( 0 <_ C /\ C < D ) ) ) -> ( A x. C ) < ( B x. D ) )
Theorem	lemul12b 8880	Comparison of product of two nonnegative numbers. (Contributed by NM, 22-Feb-2008.)
|- ( ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR ) /\ ( C e. RR /\ ( D e. RR /\ 0 <_ D ) ) ) -> ( ( A <_ B /\ C <_ D ) -> ( A x. C ) <_ ( B x. D ) ) )
Theorem	lemul12a 8881	Comparison of product of two nonnegative numbers. (Contributed by NM, 22-Feb-2008.)
|- ( ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR ) /\ ( ( C e. RR /\ 0 <_ C ) /\ D e. RR ) ) -> ( ( A <_ B /\ C <_ D ) -> ( A x. C ) <_ ( B x. D ) ) )
Theorem	mulgt1 8882	The product of two numbers greater than 1 is greater than 1. (Contributed by NM, 13-Feb-2005.)
|- ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) -> 1 < ( A x. B ) )
Theorem	ltmulgt11 8883	Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.)
|- ( ( A e. RR /\ B e. RR /\ 0 < A ) -> ( 1 < B <-> A < ( A x. B ) ) )
Theorem	ltmulgt12 8884	Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.)
|- ( ( A e. RR /\ B e. RR /\ 0 < A ) -> ( 1 < B <-> A < ( B x. A ) ) )
Theorem	lemulge11 8885	Multiplication by a number greater than or equal to 1. (Contributed by NM, 17-Dec-2005.)
|- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 1 <_ B ) ) -> A <_ ( A x. B ) )
Theorem	lemulge12 8886	Multiplication by a number greater than or equal to 1. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 1 <_ B ) ) -> A <_ ( B x. A ) )
Theorem	ltdiv1 8887	Division of both sides of 'less than' by a positive number. (Contributed by NM, 10-Oct-2004.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A < B <-> ( A / C ) < ( B / C ) ) )
Theorem	lediv1 8888	Division of both sides of a less than or equal to relation by a positive number. (Contributed by NM, 18-Nov-2004.)
|- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A <_ B <-> ( A / C ) <_ ( B / C ) ) )
Theorem	gt0div 8889	Division of a positive number by a positive number. (Contributed by NM, 28-Sep-2005.)
|- ( ( A e. RR /\ B e. RR /\ 0 < B ) -> ( 0 < A <-> 0 < ( A / B ) ) )
Theorem	ge0div 8890	Division of a nonnegative number by a positive number. (Contributed by NM, 28-Sep-2005.)
|- ( ( A e. RR /\ B e. RR /\ 0 < B ) -> ( 0 <_ A <-> 0 <_ ( A / B ) ) )
Theorem	divgt0 8891	The ratio of two positive numbers is positive. (Contributed by NM, 12-Oct-1999.)
|- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> 0 < ( A / B ) )
Theorem	divge0 8892	The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 27-Sep-1999.)
|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 < B ) ) -> 0 <_ ( A / B ) )
Theorem	ltmuldiv 8893	'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
|- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( A x. C ) < B <-> A < ( B / C ) ) )
Theorem	ltmuldiv2 8894	'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004.)
|- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( C x. A ) < B <-> A < ( B / C ) ) )
Theorem	ltdivmul 8895	'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004.)
|- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( A / C ) < B <-> A < ( C x. B ) ) )
Theorem	ledivmul 8896	'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.)
|- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( A / C ) <_ B <-> A <_ ( C x. B ) ) )
Theorem	ltdivmul2 8897	'Less than' relationship between division and multiplication. (Contributed by NM, 24-Feb-2005.)
|- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( A / C ) < B <-> A < ( B x. C ) ) )
Theorem	lt2mul2div 8898	'Less than' relationship between division and multiplication. (Contributed by NM, 8-Jan-2006.)
|- ( ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) /\ ( C e. RR /\ ( D e. RR /\ 0 < D ) ) ) -> ( ( A x. B ) < ( C x. D ) <-> ( A / D ) < ( C / B ) ) )
Theorem	ledivmul2 8899	'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.)
|- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( A / C ) <_ B <-> A <_ ( B x. C ) ) )
Theorem	lemuldiv 8900	'Less than or equal' relationship between division and multiplication. (Contributed by NM, 10-Mar-2006.)
|- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( A x. C ) <_ B <-> A <_ ( B / C ) ) )