P. 111 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 11001-11100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	rexuz3 11001*	Restrict the base of the upper integers set to another upper integers set. (Contributed by Mario Carneiro, 26-Dec-2013.)
|- Z = ( ZZ>= ` M )   =>    |- ( M e. ZZ -> ( E. j e. Z A. k e. ( ZZ>= ` j ) ph <-> E. j e. ZZ A. k e. ( ZZ>= ` j ) ph ) )
Theorem	rexanuz2 11002*	Combine two different upper integer properties into one. (Contributed by Mario Carneiro, 26-Dec-2013.)
|- Z = ( ZZ>= ` M )   =>    |- ( E. j e. Z A. k e. ( ZZ>= ` j ) ( ph /\ ps ) <-> ( E. j e. Z A. k e. ( ZZ>= ` j ) ph /\ E. j e. Z A. k e. ( ZZ>= ` j ) ps ) )
Theorem	r19.29uz 11003*	A version of 19.29 1620 for upper integer quantifiers. (Contributed by Mario Carneiro, 10-Feb-2014.)
|- Z = ( ZZ>= ` M )   =>    |- ( ( A. k e. Z ph /\ E. j e. Z A. k e. ( ZZ>= ` j ) ps ) -> E. j e. Z A. k e. ( ZZ>= ` j ) ( ph /\ ps ) )
Theorem	r19.2uz 11004*	A version of r19.2m 3511 for upper integer quantifiers. (Contributed by Mario Carneiro, 15-Feb-2014.)
|- Z = ( ZZ>= ` M )   =>    |- ( E. j e. Z A. k e. ( ZZ>= ` j ) ph -> E. k e. Z ph )
Theorem	recvguniqlem 11005	Lemma for recvguniq 11006. Some of the rearrangements of the expressions. (Contributed by Jim Kingdon, 8-Aug-2021.)
|- ( ph -> F : NN --> RR )   &    |- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> K e. NN )   &    |- ( ph -> A < ( ( F ` K ) + ( ( A - B ) / 2 ) ) )   &    |- ( ph -> ( F ` K ) < ( B + ( ( A - B ) / 2 ) ) )   =>    |- ( ph -> F. )
Theorem	recvguniq 11006*	Limits are unique. (Contributed by Jim Kingdon, 7-Aug-2021.)
|- ( ph -> F : NN --> RR )   &    |- ( ph -> L e. RR )   &    |- ( ph -> A. x e. RR+ E. j e. NN A. k e. ( ZZ>= ` j ) ( ( F ` k ) < ( L + x ) /\ L < ( ( F ` k ) + x ) ) )   &    |- ( ph -> M e. RR )   &    |- ( ph -> A. x e. RR+ E. j e. NN A. k e. ( ZZ>= ` j ) ( ( F ` k ) < ( M + x ) /\ M < ( ( F ` k ) + x ) ) )   =>    |- ( ph -> L = M )
4.7.4  Square root; absolute value
Syntax	csqrt 11007	Extend class notation to include square root of a complex number.
class sqr
Syntax	cabs 11008	Extend class notation to include a function for the absolute value (modulus) of a complex number.
class abs
Definition	df-rsqrt 11009*	Define a function whose value is the square root of a nonnegative real number. Defining the square root for complex numbers has one difficult part: choosing between the two roots. The usual way to define a principal square root for all complex numbers relies on excluded middle or something similar. But in the case of a nonnegative real number, we don't have the complications presented for general complex numbers, and we can choose the nonnegative root. (Contributed by Jim Kingdon, 23-Aug-2020.)
|- sqr = ( x e. RR |-> ( iota_ y e. RR ( ( y ^ 2 ) = x /\ 0 <_ y ) ) )
Definition	df-abs 11010	Define the function for the absolute value (modulus) of a complex number. (Contributed by NM, 27-Jul-1999.)
|- abs = ( x e. CC |-> ( sqr ` ( x x. ( * ` x ) ) ) )
Theorem	sqrtrval 11011*	Value of square root function. (Contributed by Jim Kingdon, 23-Aug-2020.)
|- ( A e. RR -> ( sqr ` A ) = ( iota_ x e. RR ( ( x ^ 2 ) = A /\ 0 <_ x ) ) )
Theorem	absval 11012	The absolute value (modulus) of a complex number. Proposition 10-3.7(a) of [Gleason] p. 133. (Contributed by NM, 27-Jul-1999.) (Revised by Mario Carneiro, 7-Nov-2013.)
|- ( A e. CC -> ( abs ` A ) = ( sqr ` ( A x. ( * ` A ) ) ) )
Theorem	rennim 11013	A real number does not lie on the negative imaginary axis. (Contributed by Mario Carneiro, 8-Jul-2013.)
|- ( A e. RR -> ( _i x. A ) e/ RR+ )
Theorem	sqrt0rlem 11014	Lemma for sqrt0 11015. (Contributed by Jim Kingdon, 26-Aug-2020.)
|- ( ( A e. RR /\ ( ( A ^ 2 ) = 0 /\ 0 <_ A ) ) <-> A = 0 )
Theorem	sqrt0 11015	Square root of zero. (Contributed by Mario Carneiro, 9-Jul-2013.)
|- ( sqr ` 0 ) = 0
Theorem	resqrexlem1arp 11016	Lemma for resqrex 11037. 1 + A is a positive real (expressed in a way that will help apply seqf 10463 and similar theorems). (Contributed by Jim Kingdon, 28-Jul-2021.) (Revised by Jim Kingdon, 16-Oct-2022.)
|- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   =>    |- ( ( ph /\ N e. NN ) -> ( ( NN X. { ( 1 + A ) } ) ` N ) e. RR+ )
Theorem	resqrexlemp1rp 11017*	Lemma for resqrex 11037. Applying the recursion rule yields a positive real (expressed in a way that will help apply seqf 10463 and similar theorems). (Contributed by Jim Kingdon, 28-Jul-2021.) (Revised by Jim Kingdon, 16-Oct-2022.)
|- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   =>    |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> ( B ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) C ) e. RR+ )
Theorem	resqrexlemf 11018*	Lemma for resqrex 11037. The sequence is a function. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.) (Revised by Jim Kingdon, 16-Oct-2022.)
|- F = seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) )   &    |- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   =>    |- ( ph -> F : NN --> RR+ )
Theorem	resqrexlemf1 11019*	Lemma for resqrex 11037. Initial value. Although this sequence converges to the square root with any positive initial value, this choice makes various steps in the proof of convergence easier. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.) (Revised by Jim Kingdon, 16-Oct-2022.)
|- F = seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) )   &    |- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   =>    |- ( ph -> ( F ` 1 ) = ( 1 + A ) )
Theorem	resqrexlemfp1 11020*	Lemma for resqrex 11037. Recursion rule. This sequence is the ancient method for computing square roots, often known as the babylonian method, although known to many ancient cultures. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.)
|- F = seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) )   &    |- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   =>    |- ( ( ph /\ N e. NN ) -> ( F ` ( N + 1 ) ) = ( ( ( F ` N ) + ( A / ( F ` N ) ) ) / 2 ) )
Theorem	resqrexlemover 11021*	Lemma for resqrex 11037. Each element of the sequence is an overestimate. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.)
|- F = seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) )   &    |- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   =>    |- ( ( ph /\ N e. NN ) -> A < ( ( F ` N ) ^ 2 ) )
Theorem	resqrexlemdec 11022*	Lemma for resqrex 11037. The sequence is decreasing. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2021.)
|- F = seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) )   &    |- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   =>    |- ( ( ph /\ N e. NN ) -> ( F ` ( N + 1 ) ) < ( F ` N ) )
Theorem	resqrexlemdecn 11023*	Lemma for resqrex 11037. The sequence is decreasing. (Contributed by Jim Kingdon, 31-Jul-2021.)
|- F = seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) )   &    |- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   &    |- ( ph -> N e. NN )   &    |- ( ph -> M e. NN )   &    |- ( ph -> N < M )   =>    |- ( ph -> ( F ` M ) < ( F ` N ) )
Theorem	resqrexlemlo 11024*	Lemma for resqrex 11037. A (variable) lower bound for each term of the sequence. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2021.)
|- F = seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) )   &    |- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   =>    |- ( ( ph /\ N e. NN ) -> ( 1 / ( 2 ^ N ) ) < ( F ` N ) )
Theorem	resqrexlemcalc1 11025*	Lemma for resqrex 11037. Some of the calculations involved in showing that the sequence converges. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2021.)
|- F = seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) )   &    |- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   =>    |- ( ( ph /\ N e. NN ) -> ( ( ( F ` ( N + 1 ) ) ^ 2 ) - A ) = ( ( ( ( ( F ` N ) ^ 2 ) - A ) ^ 2 ) / ( 4 x. ( ( F ` N ) ^ 2 ) ) ) )
Theorem	resqrexlemcalc2 11026*	Lemma for resqrex 11037. Some of the calculations involved in showing that the sequence converges. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2021.)
|- F = seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) )   &    |- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   =>    |- ( ( ph /\ N e. NN ) -> ( ( ( F ` ( N + 1 ) ) ^ 2 ) - A ) <_ ( ( ( ( F ` N ) ^ 2 ) - A ) / 4 ) )
Theorem	resqrexlemcalc3 11027*	Lemma for resqrex 11037. Some of the calculations involved in showing that the sequence converges. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2021.)
|- F = seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) )   &    |- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   =>    |- ( ( ph /\ N e. NN ) -> ( ( ( F ` N ) ^ 2 ) - A ) <_ ( ( ( F ` 1 ) ^ 2 ) / ( 4 ^ ( N - 1 ) ) ) )
Theorem	resqrexlemnmsq 11028*	Lemma for resqrex 11037. The difference between the squares of two terms of the sequence. (Contributed by Mario Carneiro and Jim Kingdon, 30-Jul-2021.)
|- F = seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) )   &    |- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   &    |- ( ph -> N e. NN )   &    |- ( ph -> M e. NN )   &    |- ( ph -> N <_ M )   =>    |- ( ph -> ( ( ( F ` N ) ^ 2 ) - ( ( F ` M ) ^ 2 ) ) < ( ( ( F ` 1 ) ^ 2 ) / ( 4 ^ ( N - 1 ) ) ) )
Theorem	resqrexlemnm 11029*	Lemma for resqrex 11037. The difference between two terms of the sequence. (Contributed by Mario Carneiro and Jim Kingdon, 31-Jul-2021.)
|- F = seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) )   &    |- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   &    |- ( ph -> N e. NN )   &    |- ( ph -> M e. NN )   &    |- ( ph -> N <_ M )   =>    |- ( ph -> ( ( F ` N ) - ( F ` M ) ) < ( ( ( ( F ` 1 ) ^ 2 ) x. 2 ) / ( 2 ^ ( N - 1 ) ) ) )
Theorem	resqrexlemcvg 11030*	Lemma for resqrex 11037. The sequence has a limit. (Contributed by Jim Kingdon, 6-Aug-2021.)
|- F = seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) )   &    |- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   =>    |- ( ph -> E. r e. RR A. x e. RR+ E. j e. NN A. i e. ( ZZ>= ` j ) ( ( F ` i ) < ( r + x ) /\ r < ( ( F ` i ) + x ) ) )
Theorem	resqrexlemgt0 11031*	Lemma for resqrex 11037. A limit is nonnegative. (Contributed by Jim Kingdon, 7-Aug-2021.)
|- F = seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) )   &    |- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   &    |- ( ph -> L e. RR )   &    |- ( ph -> A. e e. RR+ E. j e. NN A. i e. ( ZZ>= ` j ) ( ( F ` i ) < ( L + e ) /\ L < ( ( F ` i ) + e ) ) )   =>    |- ( ph -> 0 <_ L )
Theorem	resqrexlemoverl 11032*	Lemma for resqrex 11037. Every term in the sequence is an overestimate compared with the limit L. Although this theorem is stated in terms of a particular sequence the proof could be adapted for any decreasing convergent sequence. (Contributed by Jim Kingdon, 9-Aug-2021.)
|- F = seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) )   &    |- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   &    |- ( ph -> L e. RR )   &    |- ( ph -> A. e e. RR+ E. j e. NN A. i e. ( ZZ>= ` j ) ( ( F ` i ) < ( L + e ) /\ L < ( ( F ` i ) + e ) ) )   &    |- ( ph -> K e. NN )   =>    |- ( ph -> L <_ ( F ` K ) )
Theorem	resqrexlemglsq 11033*	Lemma for resqrex 11037. The sequence formed by squaring each term of F converges to ( L ^ 2 ). (Contributed by Mario Carneiro and Jim Kingdon, 8-Aug-2021.)
|- F = seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) )   &    |- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   &    |- ( ph -> L e. RR )   &    |- ( ph -> A. e e. RR+ E. j e. NN A. i e. ( ZZ>= ` j ) ( ( F ` i ) < ( L + e ) /\ L < ( ( F ` i ) + e ) ) )   &    |- G = ( x e. NN |-> ( ( F ` x ) ^ 2 ) )   =>    |- ( ph -> A. e e. RR+ E. j e. NN A. k e. ( ZZ>= ` j ) ( ( G ` k ) < ( ( L ^ 2 ) + e ) /\ ( L ^ 2 ) < ( ( G ` k ) + e ) ) )
Theorem	resqrexlemga 11034*	Lemma for resqrex 11037. The sequence formed by squaring each term of F converges to A. (Contributed by Mario Carneiro and Jim Kingdon, 8-Aug-2021.)
|- F = seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) )   &    |- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   &    |- ( ph -> L e. RR )   &    |- ( ph -> A. e e. RR+ E. j e. NN A. i e. ( ZZ>= ` j ) ( ( F ` i ) < ( L + e ) /\ L < ( ( F ` i ) + e ) ) )   &    |- G = ( x e. NN |-> ( ( F ` x ) ^ 2 ) )   =>    |- ( ph -> A. e e. RR+ E. j e. NN A. k e. ( ZZ>= ` j ) ( ( G ` k ) < ( A + e ) /\ A < ( ( G ` k ) + e ) ) )
Theorem	resqrexlemsqa 11035*	Lemma for resqrex 11037. The square of a limit is A. (Contributed by Jim Kingdon, 7-Aug-2021.)
|- F = seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) )   &    |- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   &    |- ( ph -> L e. RR )   &    |- ( ph -> A. e e. RR+ E. j e. NN A. i e. ( ZZ>= ` j ) ( ( F ` i ) < ( L + e ) /\ L < ( ( F ` i ) + e ) ) )   =>    |- ( ph -> ( L ^ 2 ) = A )
Theorem	resqrexlemex 11036*	Lemma for resqrex 11037. Existence of square root given a sequence which converges to the square root. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.)
|- F = seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) )   &    |- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   =>    |- ( ph -> E. x e. RR ( 0 <_ x /\ ( x ^ 2 ) = A ) )
Theorem	resqrex 11037*	Existence of a square root for positive reals. (Contributed by Mario Carneiro, 9-Jul-2013.)
|- ( ( A e. RR /\ 0 <_ A ) -> E. x e. RR ( 0 <_ x /\ ( x ^ 2 ) = A ) )
Theorem	rsqrmo 11038*	Uniqueness for the square root function. (Contributed by Jim Kingdon, 10-Aug-2021.)
|- ( ( A e. RR /\ 0 <_ A ) -> E* x e. RR ( ( x ^ 2 ) = A /\ 0 <_ x ) )
Theorem	rersqreu 11039*	Existence and uniqueness for the real square root function. (Contributed by Jim Kingdon, 10-Aug-2021.)
|- ( ( A e. RR /\ 0 <_ A ) -> E! x e. RR ( ( x ^ 2 ) = A /\ 0 <_ x ) )
Theorem	resqrtcl 11040	Closure of the square root function. (Contributed by Mario Carneiro, 9-Jul-2013.)
|- ( ( A e. RR /\ 0 <_ A ) -> ( sqr ` A ) e. RR )
Theorem	rersqrtthlem 11041	Lemma for resqrtth 11042. (Contributed by Jim Kingdon, 10-Aug-2021.)
|- ( ( A e. RR /\ 0 <_ A ) -> ( ( ( sqr ` A ) ^ 2 ) = A /\ 0 <_ ( sqr ` A ) ) )
Theorem	resqrtth 11042	Square root theorem over the reals. Theorem I.35 of [Apostol] p. 29. (Contributed by Mario Carneiro, 9-Jul-2013.)
|- ( ( A e. RR /\ 0 <_ A ) -> ( ( sqr ` A ) ^ 2 ) = A )
Theorem	remsqsqrt 11043	Square of square root. (Contributed by Mario Carneiro, 10-Jul-2013.)
|- ( ( A e. RR /\ 0 <_ A ) -> ( ( sqr ` A ) x. ( sqr ` A ) ) = A )
Theorem	sqrtge0 11044	The square root function is nonnegative for nonnegative input. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 9-Jul-2013.)
|- ( ( A e. RR /\ 0 <_ A ) -> 0 <_ ( sqr ` A ) )
Theorem	sqrtgt0 11045	The square root function is positive for positive input. (Contributed by Mario Carneiro, 10-Jul-2013.) (Revised by Mario Carneiro, 6-Sep-2013.)
|- ( ( A e. RR /\ 0 < A ) -> 0 < ( sqr ` A ) )
Theorem	sqrtmul 11046	Square root distributes over multiplication. (Contributed by NM, 30-Jul-1999.) (Revised by Mario Carneiro, 29-May-2016.)
|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( sqr ` ( A x. B ) ) = ( ( sqr ` A ) x. ( sqr ` B ) ) )
Theorem	sqrtle 11047	Square root is monotonic. (Contributed by NM, 17-Mar-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)
|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( A <_ B <-> ( sqr ` A ) <_ ( sqr ` B ) ) )
Theorem	sqrtlt 11048	Square root is strictly monotonic. Closed form of sqrtlti 11148. (Contributed by Scott Fenton, 17-Apr-2014.) (Proof shortened by Mario Carneiro, 29-May-2016.)
|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( A < B <-> ( sqr ` A ) < ( sqr ` B ) ) )
Theorem	sqrt11ap 11049	Analogue to sqrt11 11050 but for apartness. (Contributed by Jim Kingdon, 11-Aug-2021.)
|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( ( sqr ` A ) # ( sqr ` B ) <-> A # B ) )
Theorem	sqrt11 11050	The square root function is one-to-one. Also see sqrt11ap 11049 which would follow easily from this given excluded middle, but which is proved another way without it. (Contributed by Scott Fenton, 11-Jun-2013.)
|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( ( sqr ` A ) = ( sqr ` B ) <-> A = B ) )
Theorem	sqrt00 11051	A square root is zero iff its argument is 0. (Contributed by NM, 27-Jul-1999.) (Proof shortened by Mario Carneiro, 29-May-2016.)
|- ( ( A e. RR /\ 0 <_ A ) -> ( ( sqr ` A ) = 0 <-> A = 0 ) )
Theorem	rpsqrtcl 11052	The square root of a positive real is a positive real. (Contributed by NM, 22-Feb-2008.)
|- ( A e. RR+ -> ( sqr ` A ) e. RR+ )
Theorem	sqrtdiv 11053	Square root distributes over division. (Contributed by Mario Carneiro, 5-May-2016.)
|- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> ( sqr ` ( A / B ) ) = ( ( sqr ` A ) / ( sqr ` B ) ) )
Theorem	sqrtsq2 11054	Relationship between square root and squares. (Contributed by NM, 31-Jul-1999.) (Revised by Mario Carneiro, 29-May-2016.)
|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( ( sqr ` A ) = B <-> A = ( B ^ 2 ) ) )
Theorem	sqrtsq 11055	Square root of square. (Contributed by NM, 14-Jan-2006.) (Revised by Mario Carneiro, 29-May-2016.)
|- ( ( A e. RR /\ 0 <_ A ) -> ( sqr ` ( A ^ 2 ) ) = A )
Theorem	sqrtmsq 11056	Square root of square. (Contributed by NM, 2-Aug-1999.) (Revised by Mario Carneiro, 29-May-2016.)
|- ( ( A e. RR /\ 0 <_ A ) -> ( sqr ` ( A x. A ) ) = A )
Theorem	sqrt1 11057	The square root of 1 is 1. (Contributed by NM, 31-Jul-1999.)
|- ( sqr ` 1 ) = 1
Theorem	sqrt4 11058	The square root of 4 is 2. (Contributed by NM, 3-Aug-1999.)
|- ( sqr ` 4 ) = 2
Theorem	sqrt9 11059	The square root of 9 is 3. (Contributed by NM, 11-May-2004.)
|- ( sqr ` 9 ) = 3
Theorem	sqrt2gt1lt2 11060	The square root of 2 is bounded by 1 and 2. (Contributed by Roy F. Longton, 8-Aug-2005.) (Revised by Mario Carneiro, 6-Sep-2013.)
|- ( 1 < ( sqr ` 2 ) /\ ( sqr ` 2 ) < 2 )
Theorem	absneg 11061	Absolute value of negative. (Contributed by NM, 27-Feb-2005.)
|- ( A e. CC -> ( abs ` -u A ) = ( abs ` A ) )
Theorem	abscl 11062	Real closure of absolute value. (Contributed by NM, 3-Oct-1999.)
|- ( A e. CC -> ( abs ` A ) e. RR )
Theorem	abscj 11063	The absolute value of a number and its conjugate are the same. Proposition 10-3.7(b) of [Gleason] p. 133. (Contributed by NM, 28-Apr-2005.)
|- ( A e. CC -> ( abs ` ( * ` A ) ) = ( abs ` A ) )
Theorem	absvalsq 11064	Square of value of absolute value function. (Contributed by NM, 16-Jan-2006.)
|- ( A e. CC -> ( ( abs ` A ) ^ 2 ) = ( A x. ( * ` A ) ) )
Theorem	absvalsq2 11065	Square of value of absolute value function. (Contributed by NM, 1-Feb-2007.)
|- ( A e. CC -> ( ( abs ` A ) ^ 2 ) = ( ( ( Re ` A ) ^ 2 ) + ( ( Im ` A ) ^ 2 ) ) )
Theorem	sqabsadd 11066	Square of absolute value of sum. Proposition 10-3.7(g) of [Gleason] p. 133. (Contributed by NM, 21-Jan-2007.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( abs ` ( A + B ) ) ^ 2 ) = ( ( ( ( abs ` A ) ^ 2 ) + ( ( abs ` B ) ^ 2 ) ) + ( 2 x. ( Re ` ( A x. ( * ` B ) ) ) ) ) )
Theorem	sqabssub 11067	Square of absolute value of difference. (Contributed by NM, 21-Jan-2007.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( abs ` ( A - B ) ) ^ 2 ) = ( ( ( ( abs ` A ) ^ 2 ) + ( ( abs ` B ) ^ 2 ) ) - ( 2 x. ( Re ` ( A x. ( * ` B ) ) ) ) ) )
Theorem	absval2 11068	Value of absolute value function. Definition 10.36 of [Gleason] p. 133. (Contributed by NM, 17-Mar-2005.)
|- ( A e. CC -> ( abs ` A ) = ( sqr ` ( ( ( Re ` A ) ^ 2 ) + ( ( Im ` A ) ^ 2 ) ) ) )
Theorem	abs0 11069	The absolute value of 0. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 29-May-2016.)
|- ( abs ` 0 ) = 0
Theorem	absi 11070	The absolute value of the imaginary unit. (Contributed by NM, 26-Mar-2005.)
|- ( abs ` _i ) = 1
Theorem	absge0 11071	Absolute value is nonnegative. (Contributed by NM, 20-Nov-2004.) (Revised by Mario Carneiro, 29-May-2016.)
|- ( A e. CC -> 0 <_ ( abs ` A ) )
Theorem	absrpclap 11072	The absolute value of a number apart from zero is a positive real. (Contributed by Jim Kingdon, 11-Aug-2021.)
|- ( ( A e. CC /\ A # 0 ) -> ( abs ` A ) e. RR+ )
Theorem	abs00ap 11073	The absolute value of a number is apart from zero iff the number is apart from zero. (Contributed by Jim Kingdon, 11-Aug-2021.)
|- ( A e. CC -> ( ( abs ` A ) # 0 <-> A # 0 ) )
Theorem	absext 11074	Strong extensionality for absolute value. (Contributed by Jim Kingdon, 12-Aug-2021.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( abs ` A ) # ( abs ` B ) -> A # B ) )
Theorem	abs00 11075	The absolute value of a number is zero iff the number is zero. Also see abs00ap 11073 which is similar but for apartness. Proposition 10-3.7(c) of [Gleason] p. 133. (Contributed by NM, 26-Sep-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)
|- ( A e. CC -> ( ( abs ` A ) = 0 <-> A = 0 ) )
Theorem	abs00ad 11076	A complex number is zero iff its absolute value is zero. Deduction form of abs00 11075. (Contributed by David Moews, 28-Feb-2017.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( ( abs ` A ) = 0 <-> A = 0 ) )
Theorem	abs00bd 11077	If a complex number is zero, its absolute value is zero. (Contributed by David Moews, 28-Feb-2017.)
|- ( ph -> A = 0 )   =>    |- ( ph -> ( abs ` A ) = 0 )
Theorem	absreimsq 11078	Square of the absolute value of a number that has been decomposed into real and imaginary parts. (Contributed by NM, 1-Feb-2007.)
|- ( ( A e. RR /\ B e. RR ) -> ( ( abs ` ( A + ( _i x. B ) ) ) ^ 2 ) = ( ( A ^ 2 ) + ( B ^ 2 ) ) )
Theorem	absreim 11079	Absolute value of a number that has been decomposed into real and imaginary parts. (Contributed by NM, 14-Jan-2006.)
|- ( ( A e. RR /\ B e. RR ) -> ( abs ` ( A + ( _i x. B ) ) ) = ( sqr ` ( ( A ^ 2 ) + ( B ^ 2 ) ) ) )
Theorem	absmul 11080	Absolute value distributes over multiplication. Proposition 10-3.7(f) of [Gleason] p. 133. (Contributed by NM, 11-Oct-1999.) (Revised by Mario Carneiro, 29-May-2016.)
|- ( ( A e. CC /\ B e. CC ) -> ( abs ` ( A x. B ) ) = ( ( abs ` A ) x. ( abs ` B ) ) )
Theorem	absdivap 11081	Absolute value distributes over division. (Contributed by Jim Kingdon, 11-Aug-2021.)
|- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( abs ` ( A / B ) ) = ( ( abs ` A ) / ( abs ` B ) ) )
Theorem	absid 11082	A nonnegative number is its own absolute value. (Contributed by NM, 11-Oct-1999.) (Revised by Mario Carneiro, 29-May-2016.)
|- ( ( A e. RR /\ 0 <_ A ) -> ( abs ` A ) = A )
Theorem	abs1 11083	The absolute value of 1. Common special case. (Contributed by David A. Wheeler, 16-Jul-2016.)
|- ( abs ` 1 ) = 1
Theorem	absnid 11084	A negative number is the negative of its own absolute value. (Contributed by NM, 27-Feb-2005.)
|- ( ( A e. RR /\ A <_ 0 ) -> ( abs ` A ) = -u A )
Theorem	leabs 11085	A real number is less than or equal to its absolute value. (Contributed by NM, 27-Feb-2005.)
|- ( A e. RR -> A <_ ( abs ` A ) )
Theorem	qabsor 11086	The absolute value of a rational number is either that number or its negative. (Contributed by Jim Kingdon, 8-Nov-2021.)
|- ( A e. QQ -> ( ( abs ` A ) = A \/ ( abs ` A ) = -u A ) )
Theorem	qabsord 11087	The absolute value of a rational number is either that number or its negative. (Contributed by Jim Kingdon, 8-Nov-2021.)
|- ( ph -> A e. QQ )   =>    |- ( ph -> ( ( abs ` A ) = A \/ ( abs ` A ) = -u A ) )
Theorem	absre 11088	Absolute value of a real number. (Contributed by NM, 17-Mar-2005.)
|- ( A e. RR -> ( abs ` A ) = ( sqr ` ( A ^ 2 ) ) )
Theorem	absresq 11089	Square of the absolute value of a real number. (Contributed by NM, 16-Jan-2006.)
|- ( A e. RR -> ( ( abs ` A ) ^ 2 ) = ( A ^ 2 ) )
Theorem	absexp 11090	Absolute value of positive integer exponentiation. (Contributed by NM, 5-Jan-2006.)
|- ( ( A e. CC /\ N e. NN0 ) -> ( abs ` ( A ^ N ) ) = ( ( abs ` A ) ^ N ) )
Theorem	absexpzap 11091	Absolute value of integer exponentiation. (Contributed by Jim Kingdon, 11-Aug-2021.)
|- ( ( A e. CC /\ A # 0 /\ N e. ZZ ) -> ( abs ` ( A ^ N ) ) = ( ( abs ` A ) ^ N ) )
Theorem	abssq 11092	Square can be moved in and out of absolute value. (Contributed by Scott Fenton, 18-Apr-2014.) (Proof shortened by Mario Carneiro, 29-May-2016.)
|- ( A e. CC -> ( ( abs ` A ) ^ 2 ) = ( abs ` ( A ^ 2 ) ) )
Theorem	sqabs 11093	The squares of two reals are equal iff their absolute values are equal. (Contributed by NM, 6-Mar-2009.)
|- ( ( A e. RR /\ B e. RR ) -> ( ( A ^ 2 ) = ( B ^ 2 ) <-> ( abs ` A ) = ( abs ` B ) ) )
Theorem	absrele 11094	The absolute value of a complex number is greater than or equal to the absolute value of its real part. (Contributed by NM, 1-Apr-2005.)
|- ( A e. CC -> ( abs ` ( Re ` A ) ) <_ ( abs ` A ) )
Theorem	absimle 11095	The absolute value of a complex number is greater than or equal to the absolute value of its imaginary part. (Contributed by NM, 17-Mar-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)
|- ( A e. CC -> ( abs ` ( Im ` A ) ) <_ ( abs ` A ) )
Theorem	nn0abscl 11096	The absolute value of an integer is a nonnegative integer. (Contributed by NM, 27-Feb-2005.)
|- ( A e. ZZ -> ( abs ` A ) e. NN0 )
Theorem	zabscl 11097	The absolute value of an integer is an integer. (Contributed by Stefan O'Rear, 24-Sep-2014.)
|- ( A e. ZZ -> ( abs ` A ) e. ZZ )
Theorem	ltabs 11098	A number which is less than its absolute value is negative. (Contributed by Jim Kingdon, 12-Aug-2021.)
|- ( ( A e. RR /\ A < ( abs ` A ) ) -> A < 0 )
Theorem	abslt 11099	Absolute value and 'less than' relation. (Contributed by NM, 6-Apr-2005.) (Revised by Mario Carneiro, 29-May-2016.)
|- ( ( A e. RR /\ B e. RR ) -> ( ( abs ` A ) < B <-> ( -u B < A /\ A < B ) ) )
Theorem	absle 11100	Absolute value and 'less than or equal to' relation. (Contributed by NM, 6-Apr-2005.) (Revised by Mario Carneiro, 29-May-2016.)
|- ( ( A e. RR /\ B e. RR ) -> ( ( abs ` A ) <_ B <-> ( -u B <_ A /\ A <_ B ) ) )