P. 92 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 9101-9200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	lt2mul2div 9101	'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 9102	'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 9103	'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 ) ) )
Theorem	lemuldiv2 9104	'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 ) ) -> ( ( C x. A ) <_ B <-> A <_ ( B / C ) ) )
Theorem	ltrec 9105	The reciprocal of both sides of 'less than'. (Contributed by NM, 26-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( A < B <-> ( 1 / B ) < ( 1 / A ) ) )
Theorem	lerec 9106	The reciprocal of both sides of 'less than or equal to'. (Contributed by NM, 3-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
|- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( A <_ B <-> ( 1 / B ) <_ ( 1 / A ) ) )
Theorem	lt2msq1 9107	Lemma for lt2msq 9108. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR /\ A < B ) -> ( A x. A ) < ( B x. B ) )
Theorem	lt2msq 9108	Two nonnegative numbers compare the same as their squares. (Contributed by Roy F. Longton, 8-Aug-2005.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( A < B <-> ( A x. A ) < ( B x. B ) ) )
Theorem	ltdiv2 9109	Division of a positive number by both sides of 'less than'. (Contributed by NM, 27-Apr-2005.)
|- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( A < B <-> ( C / B ) < ( C / A ) ) )
Theorem	ltrec1 9110	Reciprocal swap in a 'less than' relation. (Contributed by NM, 24-Feb-2005.)
|- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( ( 1 / A ) < B <-> ( 1 / B ) < A ) )
Theorem	lerec2 9111	Reciprocal swap in a 'less than or equal to' relation. (Contributed by NM, 24-Feb-2005.)
|- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( A <_ ( 1 / B ) <-> B <_ ( 1 / A ) ) )
Theorem	ledivdiv 9112	Invert ratios of positive numbers and swap their ordering. (Contributed by NM, 9-Jan-2006.)
|- ( ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) /\ ( ( C e. RR /\ 0 < C ) /\ ( D e. RR /\ 0 < D ) ) ) -> ( ( A / B ) <_ ( C / D ) <-> ( D / C ) <_ ( B / A ) ) )
Theorem	lediv2 9113	Division of a positive number by both sides of 'less than or equal to'. (Contributed by NM, 10-Jan-2006.)
|- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( A <_ B <-> ( C / B ) <_ ( C / A ) ) )
Theorem	ltdiv23 9114	Swap denominator with other side of 'less than'. (Contributed by NM, 3-Oct-1999.)
|- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( ( A / B ) < C <-> ( A / C ) < B ) )
Theorem	lediv23 9115	Swap denominator with other side of 'less than or equal to'. (Contributed by NM, 30-May-2005.)
|- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( ( A / B ) <_ C <-> ( A / C ) <_ B ) )
Theorem	lediv12a 9116	Comparison of ratio of two nonnegative numbers. (Contributed by NM, 31-Dec-2005.)
|- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ A <_ B ) ) /\ ( ( C e. RR /\ D e. RR ) /\ ( 0 < C /\ C <_ D ) ) ) -> ( A / D ) <_ ( B / C ) )
Theorem	lediv2a 9117	Division of both sides of 'less than or equal to' into a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.)
|- ( ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 <_ C ) ) /\ A <_ B ) -> ( C / B ) <_ ( C / A ) )
Theorem	reclt1 9118	The reciprocal of a positive number less than 1 is greater than 1. (Contributed by NM, 23-Feb-2005.)
|- ( ( A e. RR /\ 0 < A ) -> ( A < 1 <-> 1 < ( 1 / A ) ) )
Theorem	recgt1 9119	The reciprocal of a positive number greater than 1 is less than 1. (Contributed by NM, 28-Dec-2005.)
|- ( ( A e. RR /\ 0 < A ) -> ( 1 < A <-> ( 1 / A ) < 1 ) )
Theorem	recgt1i 9120	The reciprocal of a number greater than 1 is positive and less than 1. (Contributed by NM, 23-Feb-2005.)
|- ( ( A e. RR /\ 1 < A ) -> ( 0 < ( 1 / A ) /\ ( 1 / A ) < 1 ) )
Theorem	recp1lt1 9121	Construct a number less than 1 from any nonnegative number. (Contributed by NM, 30-Dec-2005.)
|- ( ( A e. RR /\ 0 <_ A ) -> ( A / ( 1 + A ) ) < 1 )
Theorem	recreclt 9122	Given a positive number A, construct a new positive number less than both A and 1. (Contributed by NM, 28-Dec-2005.)
|- ( ( A e. RR /\ 0 < A ) -> ( ( 1 / ( 1 + ( 1 / A ) ) ) < 1 /\ ( 1 / ( 1 + ( 1 / A ) ) ) < A ) )
Theorem	le2msq 9123	The square function on nonnegative reals is monotonic. (Contributed by NM, 3-Aug-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( A <_ B <-> ( A x. A ) <_ ( B x. B ) ) )
Theorem	msq11 9124	The square of a nonnegative number is a one-to-one function. (Contributed by NM, 29-Jul-1999.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( ( A x. A ) = ( B x. B ) <-> A = B ) )
Theorem	ledivp1 9125	Less-than-or-equal-to and division relation. (Lemma for computing upper bounds of products. The "+ 1" prevents division by zero.) (Contributed by NM, 28-Sep-2005.)
|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( ( A / ( B + 1 ) ) x. B ) <_ A )
Theorem	squeeze0 9126*	If a nonnegative number is less than any positive number, it is zero. (Contributed by NM, 11-Feb-2006.)
|- ( ( A e. RR /\ 0 <_ A /\ A. x e. RR ( 0 < x -> A < x ) ) -> A = 0 )
Theorem	ltp1i 9127	A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)
|- A e. RR   =>    |- A < ( A + 1 )
Theorem	recgt0i 9128	The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 15-May-1999.)
|- A e. RR   =>    |- ( 0 < A -> 0 < ( 1 / A ) )
Theorem	recgt0ii 9129	The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 15-May-1999.)
|- A e. RR   &    |- 0 < A   =>    |- 0 < ( 1 / A )
Theorem	prodgt0i 9130	Infer that a multiplicand is positive from a nonnegative multiplier and positive product. (Contributed by NM, 15-May-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( ( 0 <_ A /\ 0 < ( A x. B ) ) -> 0 < B )
Theorem	prodge0i 9131	Infer that a multiplicand is nonnegative from a positive multiplier and nonnegative product. (Contributed by NM, 2-Jul-2005.)
|- A e. RR   &    |- B e. RR   =>    |- ( ( 0 < A /\ 0 <_ ( A x. B ) ) -> 0 <_ B )
Theorem	divgt0i 9132	The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( ( 0 < A /\ 0 < B ) -> 0 < ( A / B ) )
Theorem	divge0i 9133	The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 12-Aug-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( ( 0 <_ A /\ 0 < B ) -> 0 <_ ( A / B ) )
Theorem	ltreci 9134	The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( ( 0 < A /\ 0 < B ) -> ( A < B <-> ( 1 / B ) < ( 1 / A ) ) )
Theorem	lereci 9135	The reciprocal of both sides of 'less than or equal to'. (Contributed by NM, 16-Sep-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( ( 0 < A /\ 0 < B ) -> ( A <_ B <-> ( 1 / B ) <_ ( 1 / A ) ) )
Theorem	lt2msqi 9136	The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 3-Aug-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( ( 0 <_ A /\ 0 <_ B ) -> ( A < B <-> ( A x. A ) < ( B x. B ) ) )
Theorem	le2msqi 9137	The square function on nonnegative reals is monotonic. (Contributed by NM, 2-Aug-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( ( 0 <_ A /\ 0 <_ B ) -> ( A <_ B <-> ( A x. A ) <_ ( B x. B ) ) )
Theorem	msq11i 9138	The square of a nonnegative number is a one-to-one function. (Contributed by NM, 29-Jul-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( ( 0 <_ A /\ 0 <_ B ) -> ( ( A x. A ) = ( B x. B ) <-> A = B ) )
Theorem	divgt0i2i 9139	The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.)
|- A e. RR   &    |- B e. RR   &    |- 0 < B   =>    |- ( 0 < A -> 0 < ( A / B ) )
Theorem	ltrecii 9140	The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.)
|- A e. RR   &    |- B e. RR   &    |- 0 < A   &    |- 0 < B   =>    |- ( A < B <-> ( 1 / B ) < ( 1 / A ) )
Theorem	divgt0ii 9141	The ratio of two positive numbers is positive. (Contributed by NM, 18-May-1999.)
|- A e. RR   &    |- B e. RR   &    |- 0 < A   &    |- 0 < B   =>    |- 0 < ( A / B )
Theorem	ltmul1i 9142	Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   =>    |- ( 0 < C -> ( A < B <-> ( A x. C ) < ( B x. C ) ) )
Theorem	ltdiv1i 9143	Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   =>    |- ( 0 < C -> ( A < B <-> ( A / C ) < ( B / C ) ) )
Theorem	ltmuldivi 9144	'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   =>    |- ( 0 < C -> ( ( A x. C ) < B <-> A < ( B / C ) ) )
Theorem	ltmul2i 9145	Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   =>    |- ( 0 < C -> ( A < B <-> ( C x. A ) < ( C x. B ) ) )
Theorem	lemul1i 9146	Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 2-Aug-1999.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   =>    |- ( 0 < C -> ( A <_ B <-> ( A x. C ) <_ ( B x. C ) ) )
Theorem	lemul2i 9147	Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 1-Aug-1999.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   =>    |- ( 0 < C -> ( A <_ B <-> ( C x. A ) <_ ( C x. B ) ) )
Theorem	ltdiv23i 9148	Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   =>    |- ( ( 0 < B /\ 0 < C ) -> ( ( A / B ) < C <-> ( A / C ) < B ) )
Theorem	ltdiv23ii 9149	Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   &    |- 0 < B   &    |- 0 < C   =>    |- ( ( A / B ) < C <-> ( A / C ) < B )
Theorem	ltmul1ii 9150	Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.) (Proof shortened by Paul Chapman, 25-Jan-2008.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   &    |- 0 < C   =>    |- ( A < B <-> ( A x. C ) < ( B x. C ) )
Theorem	ltdiv1ii 9151	Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   &    |- 0 < C   =>    |- ( A < B <-> ( A / C ) < ( B / C ) )
Theorem	ltp1d 9152	A number is less than itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> A < ( A + 1 ) )
Theorem	lep1d 9153	A number is less than or equal to itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> A <_ ( A + 1 ) )
Theorem	ltm1d 9154	A number minus 1 is less than itself. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> ( A - 1 ) < A )
Theorem	lem1d 9155	A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> ( A - 1 ) <_ A )
Theorem	recgt0d 9156	The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> 0 < A )   =>    |- ( ph -> 0 < ( 1 / A ) )
Theorem	divgt0d 9157	The ratio of two positive numbers is positive. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> 0 < A )   &    |- ( ph -> 0 < B )   =>    |- ( ph -> 0 < ( A / B ) )
Theorem	mulgt1d 9158	The product of two numbers greater than 1 is greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> 1 < A )   &    |- ( ph -> 1 < B )   =>    |- ( ph -> 1 < ( A x. B ) )
Theorem	lemulge11d 9159	Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> 0 <_ A )   &    |- ( ph -> 1 <_ B )   =>    |- ( ph -> A <_ ( A x. B ) )
Theorem	lemulge12d 9160	Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> 0 <_ A )   &    |- ( ph -> 1 <_ B )   =>    |- ( ph -> A <_ ( B x. A ) )
Theorem	lemul1ad 9161	Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> 0 <_ C )   &    |- ( ph -> A <_ B )   =>    |- ( ph -> ( A x. C ) <_ ( B x. C ) )
Theorem	lemul2ad 9162	Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> 0 <_ C )   &    |- ( ph -> A <_ B )   =>    |- ( ph -> ( C x. A ) <_ ( C x. B ) )
Theorem	ltmul12ad 9163	Comparison of product of two positive numbers. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> D e. RR )   &    |- ( ph -> 0 <_ A )   &    |- ( ph -> A < B )   &    |- ( ph -> 0 <_ C )   &    |- ( ph -> C < D )   =>    |- ( ph -> ( A x. C ) < ( B x. D ) )
Theorem	lemul12ad 9164	Comparison of product of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> D e. RR )   &    |- ( ph -> 0 <_ A )   &    |- ( ph -> 0 <_ C )   &    |- ( ph -> A <_ B )   &    |- ( ph -> C <_ D )   =>    |- ( ph -> ( A x. C ) <_ ( B x. D ) )
Theorem	lemul12bd 9165	Comparison of product of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> D e. RR )   &    |- ( ph -> 0 <_ A )   &    |- ( ph -> 0 <_ D )   &    |- ( ph -> A <_ B )   &    |- ( ph -> C <_ D )   =>    |- ( ph -> ( A x. C ) <_ ( B x. D ) )
Theorem	mulle0r 9166	Multiplying a nonnegative number by a nonpositive number yields a nonpositive number. (Contributed by Jim Kingdon, 28-Oct-2021.)
|- ( ( ( A e. RR /\ B e. RR ) /\ ( A <_ 0 /\ 0 <_ B ) ) -> ( A x. B ) <_ 0 )
4.3.10  Suprema
Theorem	lbreu 9167*	If a set of reals contains a lower bound, it contains a unique lower bound. (Contributed by NM, 9-Oct-2005.)
|- ( ( S C_ RR /\ E. x e. S A. y e. S x <_ y ) -> E! x e. S A. y e. S x <_ y )
Theorem	lbcl 9168*	If a set of reals contains a lower bound, it contains a unique lower bound that belongs to the set. (Contributed by NM, 9-Oct-2005.) (Revised by Mario Carneiro, 24-Dec-2016.)
|- ( ( S C_ RR /\ E. x e. S A. y e. S x <_ y ) -> ( iota_ x e. S A. y e. S x <_ y ) e. S )
Theorem	lble 9169*	If a set of reals contains a lower bound, the lower bound is less than or equal to all members of the set. (Contributed by NM, 9-Oct-2005.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
|- ( ( S C_ RR /\ E. x e. S A. y e. S x <_ y /\ A e. S ) -> ( iota_ x e. S A. y e. S x <_ y ) <_ A )
Theorem	lbinf 9170*	If a set of reals contains a lower bound, the lower bound is its infimum. (Contributed by NM, 9-Oct-2005.) (Revised by AV, 4-Sep-2020.)
|- ( ( S C_ RR /\ E. x e. S A. y e. S x <_ y ) -> inf ( S , RR , < ) = ( iota_ x e. S A. y e. S x <_ y ) )
Theorem	lbinfcl 9171*	If a set of reals contains a lower bound, it contains its infimum. (Contributed by NM, 11-Oct-2005.) (Revised by AV, 4-Sep-2020.)
|- ( ( S C_ RR /\ E. x e. S A. y e. S x <_ y ) -> inf ( S , RR , < ) e. S )
Theorem	lbinfle 9172*	If a set of reals contains a lower bound, its infimum is less than or equal to all members of the set. (Contributed by NM, 11-Oct-2005.) (Revised by AV, 4-Sep-2020.)
|- ( ( S C_ RR /\ E. x e. S A. y e. S x <_ y /\ A e. S ) -> inf ( S , RR , < ) <_ A )
Theorem	suprubex 9173*	A member of a nonempty bounded set of reals is less than or equal to the set's upper bound. (Contributed by Jim Kingdon, 18-Jan-2022.)
|- ( ph -> E. x e. RR ( A. y e. A -. x < y /\ A. y e. RR ( y < x -> E. z e. A y < z ) ) )   &    |- ( ph -> A C_ RR )   &    |- ( ph -> B e. A )   =>    |- ( ph -> B <_ sup ( A , RR , < ) )
Theorem	suprlubex 9174*	The supremum of a nonempty bounded set of reals is the least upper bound. (Contributed by Jim Kingdon, 19-Jan-2022.)
|- ( ph -> E. x e. RR ( A. y e. A -. x < y /\ A. y e. RR ( y < x -> E. z e. A y < z ) ) )   &    |- ( ph -> A C_ RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( B < sup ( A , RR , < ) <-> E. z e. A B < z ) )
Theorem	suprnubex 9175*	An upper bound is not less than the supremum of a nonempty bounded set of reals. (Contributed by Jim Kingdon, 19-Jan-2022.)
|- ( ph -> E. x e. RR ( A. y e. A -. x < y /\ A. y e. RR ( y < x -> E. z e. A y < z ) ) )   &    |- ( ph -> A C_ RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( -. B < sup ( A , RR , < ) <-> A. z e. A -. B < z ) )
Theorem	suprleubex 9176*	The supremum of a nonempty bounded set of reals is less than or equal to an upper bound. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 6-Sep-2014.)
|- ( ph -> E. x e. RR ( A. y e. A -. x < y /\ A. y e. RR ( y < x -> E. z e. A y < z ) ) )   &    |- ( ph -> A C_ RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( sup ( A , RR , < ) <_ B <-> A. z e. A z <_ B ) )
Theorem	negiso 9177	Negation is an order anti-isomorphism of the real numbers, which is its own inverse. (Contributed by Mario Carneiro, 24-Dec-2016.)
|- F = ( x e. RR |-> -u x )   =>    |- ( F Isom < , `' < ( RR , RR ) /\ `' F = F )
Theorem	dfinfre 9178*	The infimum of a set of reals A. (Contributed by NM, 9-Oct-2005.) (Revised by AV, 4-Sep-2020.)
|- ( A C_ RR -> inf ( A , RR , < ) = U. { x e. RR | ( A. y e. A x <_ y /\ A. y e. RR ( x < y -> E. z e. A z < y ) ) } )
Theorem	sup3exmid 9179*	If any inhabited set of real numbers bounded from above has a supremum, excluded middle follows. (Contributed by Jim Kingdon, 2-Apr-2023.)
|- ( ( u C_ RR /\ E. w w e. u /\ E. x e. RR A. y e. u y <_ x ) -> E. x e. RR ( A. y e. u -. x < y /\ A. y e. RR ( y < x -> E. z e. u y < z ) ) )   =>    |- DECID ph
4.3.11  Imaginary and complex number properties
Theorem	crap0 9180	The real representation of complex numbers is apart from zero iff one of its terms is apart from zero. (Contributed by Jim Kingdon, 5-Mar-2020.)
|- ( ( A e. RR /\ B e. RR ) -> ( ( A # 0 \/ B # 0 ) <-> ( A + ( _i x. B ) ) # 0 ) )
Theorem	creur 9181*	The real part of a complex number is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
|- ( A e. CC -> E! x e. RR E. y e. RR A = ( x + ( _i x. y ) ) )
Theorem	creui 9182*	The imaginary part of a complex number is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
|- ( A e. CC -> E! y e. RR E. x e. RR A = ( x + ( _i x. y ) ) )
Theorem	cju 9183*	The complex conjugate of a complex number is unique. (Contributed by Mario Carneiro, 6-Nov-2013.)
|- ( A e. CC -> E! x e. CC ( ( A + x ) e. RR /\ ( _i x. ( A - x ) ) e. RR ) )
4.3.12  Function operation analogue theorems
Theorem	ofnegsub 9184	Function analogue of negsub 8469. (Contributed by Mario Carneiro, 24-Jul-2014.)
|- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> ( F oF + ( ( A X. { -u 1 } ) oF x. G ) ) = ( F oF - G ) )
4.4  Integer sets
4.4.1  Positive integers (as a subset of complex numbers)
Syntax	cn 9185	Extend class notation to include the class of positive integers.
class NN
Definition	df-inn 9186*	Definition of the set of positive integers. For naming consistency with the Metamath Proof Explorer usages should refer to dfnn2 9187 instead. (Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro, 3-May-2014.) (New usage is discouraged.)
|- NN = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) }
Theorem	dfnn2 9187*	Definition of the set of positive integers. Another name for df-inn 9186. (Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro, 3-May-2014.)
|- NN = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) }
Theorem	peano5nni 9188*	Peano's inductive postulate. Theorem I.36 (principle of mathematical induction) of [Apostol] p. 34. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.)
|- ( ( 1 e. A /\ A. x e. A ( x + 1 ) e. A ) -> NN C_ A )
Theorem	nnssre 9189	The positive integers are a subset of the reals. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.)
|- NN C_ RR
Theorem	nnsscn 9190	The positive integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
|- NN C_ CC
Theorem	nnex 9191	The set of positive integers exists. (Contributed by NM, 3-Oct-1999.) (Revised by Mario Carneiro, 17-Nov-2014.)
|- NN e. _V
Theorem	nnre 9192	A positive integer is a real number. (Contributed by NM, 18-Aug-1999.)
|- ( A e. NN -> A e. RR )
Theorem	nncn 9193	A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.)
|- ( A e. NN -> A e. CC )
Theorem	nnrei 9194	A positive integer is a real number. (Contributed by NM, 18-Aug-1999.)
|- A e. NN   =>    |- A e. RR
Theorem	nncni 9195	A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.)
|- A e. NN   =>    |- A e. CC
Theorem	1nn 9196	Peano postulate: 1 is a positive integer. (Contributed by NM, 11-Jan-1997.)
|- 1 e. NN
Theorem	peano2nn 9197	Peano postulate: a successor of a positive integer is a positive integer. (Contributed by NM, 11-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.)
|- ( A e. NN -> ( A + 1 ) e. NN )
Theorem	nnred 9198	A positive integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN )   =>    |- ( ph -> A e. RR )
Theorem	nncnd 9199	A positive integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN )   =>    |- ( ph -> A e. CC )
Theorem	peano2nnd 9200	Peano postulate: a successor of a positive integer is a positive integer. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN )   =>    |- ( ph -> ( A + 1 ) e. NN )