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	imaddd 11001	Imaginary part distributes over addition. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( Im ` ( A + B ) ) = ( ( Im ` A ) + ( Im ` B ) ) )
Theorem	resubd 11002	Real part distributes over subtraction. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( Re ` ( A - B ) ) = ( ( Re ` A ) - ( Re ` B ) ) )
Theorem	imsubd 11003	Imaginary part distributes over subtraction. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( Im ` ( A - B ) ) = ( ( Im ` A ) - ( Im ` B ) ) )
Theorem	remuld 11004	Real part of a product. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( Re ` ( A x. B ) ) = ( ( ( Re ` A ) x. ( Re ` B ) ) - ( ( Im ` A ) x. ( Im ` B ) ) ) )
Theorem	immuld 11005	Imaginary part of a product. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( Im ` ( A x. B ) ) = ( ( ( Re ` A ) x. ( Im ` B ) ) + ( ( Im ` A ) x. ( Re ` B ) ) ) )
Theorem	cjaddd 11006	Complex conjugate distributes over addition. Proposition 10-3.4(a) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( * ` ( A + B ) ) = ( ( * ` A ) + ( * ` B ) ) )
Theorem	cjmuld 11007	Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( * ` ( A x. B ) ) = ( ( * ` A ) x. ( * ` B ) ) )
Theorem	ipcnd 11008	Standard inner product on complex numbers. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( Re ` ( A x. ( * ` B ) ) ) = ( ( ( Re ` A ) x. ( Re ` B ) ) + ( ( Im ` A ) x. ( Im ` B ) ) ) )
Theorem	cjdivapd 11009	Complex conjugate distributes over division. (Contributed by Jim Kingdon, 15-Jun-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> B # 0 )   =>    |- ( ph -> ( * ` ( A / B ) ) = ( ( * ` A ) / ( * ` B ) ) )
Theorem	rered 11010	A real number equals its real part. One direction of Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> ( Re ` A ) = A )
Theorem	reim0d 11011	The imaginary part of a real number is 0. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> ( Im ` A ) = 0 )
Theorem	cjred 11012	A real number equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> ( * ` A ) = A )
Theorem	remul2d 11013	Real part of a product. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( Re ` ( A x. B ) ) = ( A x. ( Re ` B ) ) )
Theorem	immul2d 11014	Imaginary part of a product. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( Im ` ( A x. B ) ) = ( A x. ( Im ` B ) ) )
Theorem	redivapd 11015	Real part of a division. Related to remul2 10914. (Contributed by Jim Kingdon, 15-Jun-2020.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. CC )   &    |- ( ph -> A # 0 )   =>    |- ( ph -> ( Re ` ( B / A ) ) = ( ( Re ` B ) / A ) )
Theorem	imdivapd 11016	Imaginary part of a division. Related to remul2 10914. (Contributed by Jim Kingdon, 15-Jun-2020.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. CC )   &    |- ( ph -> A # 0 )   =>    |- ( ph -> ( Im ` ( B / A ) ) = ( ( Im ` B ) / A ) )
Theorem	crred 11017	The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( Re ` ( A + ( _i x. B ) ) ) = A )
Theorem	crimd 11018	The imaginary part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( Im ` ( A + ( _i x. B ) ) ) = B )
Theorem	cnreim 11019	Complex apartness in terms of real and imaginary parts. See also apreim 8590 which is similar but with different notation. (Contributed by Jim Kingdon, 16-Dec-2023.)
|- ( ( A e. CC /\ B e. CC ) -> ( A # B <-> ( ( Re ` A ) # ( Re ` B ) \/ ( Im ` A ) # ( Im ` B ) ) ) )
4.7.3  Sequence convergence
Theorem	caucvgrelemrec 11020*	Two ways to express a reciprocal. (Contributed by Jim Kingdon, 20-Jul-2021.)
|- ( ( A e. RR /\ A # 0 ) -> ( iota_ r e. RR ( A x. r ) = 1 ) = ( 1 / A ) )
Theorem	caucvgrelemcau 11021*	Lemma for caucvgre 11022. Converting the Cauchy condition. (Contributed by Jim Kingdon, 20-Jul-2021.)
|- ( ph -> F : NN --> RR )   &    |- ( ph -> A. n e. NN A. k e. ( ZZ>= ` n ) ( ( F ` n ) < ( ( F ` k ) + ( 1 / n ) ) /\ ( F ` k ) < ( ( F ` n ) + ( 1 / n ) ) ) )   =>    |- ( ph -> A. n e. NN A. k e. NN ( n <RR k -> ( ( F ` n ) <RR ( ( F ` k ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) /\ ( F ` k ) <RR ( ( F ` n ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) ) ) )
Theorem	caucvgre 11022*	Convergence of real sequences. A Cauchy sequence (as defined here, which has a rate of convergence built in) of real numbers converges to a real number. Specifically on rate of convergence, all terms after the nth term must be within 1 / n of the nth term. (Contributed by Jim Kingdon, 19-Jul-2021.)
|- ( ph -> F : NN --> RR )   &    |- ( ph -> A. n e. NN A. k e. ( ZZ>= ` n ) ( ( F ` n ) < ( ( F ` k ) + ( 1 / n ) ) /\ ( F ` k ) < ( ( F ` n ) + ( 1 / n ) ) ) )   =>    |- ( ph -> E. y e. RR A. x e. RR+ E. j e. NN A. i e. ( ZZ>= ` j ) ( ( F ` i ) < ( y + x ) /\ y < ( ( F ` i ) + x ) ) )
Theorem	cvg1nlemcxze 11023	Lemma for cvg1n 11027. Rearranging an expression related to the rate of convergence. (Contributed by Jim Kingdon, 6-Aug-2021.)
|- ( ph -> C e. RR+ )   &    |- ( ph -> X e. RR+ )   &    |- ( ph -> Z e. NN )   &    |- ( ph -> E e. NN )   &    |- ( ph -> A e. NN )   &    |- ( ph -> ( ( ( ( C x. 2 ) / X ) / Z ) + A ) < E )   =>    |- ( ph -> ( C / ( E x. Z ) ) < ( X / 2 ) )
Theorem	cvg1nlemf 11024*	Lemma for cvg1n 11027. The modified sequence G is a sequence. (Contributed by Jim Kingdon, 1-Aug-2021.)
|- ( ph -> F : NN --> RR )   &    |- ( ph -> C e. RR+ )   &    |- ( ph -> A. n e. NN A. k e. ( ZZ>= ` n ) ( ( F ` n ) < ( ( F ` k ) + ( C / n ) ) /\ ( F ` k ) < ( ( F ` n ) + ( C / n ) ) ) )   &    |- G = ( j e. NN |-> ( F ` ( j x. Z ) ) )   &    |- ( ph -> Z e. NN )   &    |- ( ph -> C < Z )   =>    |- ( ph -> G : NN --> RR )
Theorem	cvg1nlemcau 11025*	Lemma for cvg1n 11027. By selecting spaced out terms for the modified sequence G, the terms are within 1 / n (without the constant C). (Contributed by Jim Kingdon, 1-Aug-2021.)
|- ( ph -> F : NN --> RR )   &    |- ( ph -> C e. RR+ )   &    |- ( ph -> A. n e. NN A. k e. ( ZZ>= ` n ) ( ( F ` n ) < ( ( F ` k ) + ( C / n ) ) /\ ( F ` k ) < ( ( F ` n ) + ( C / n ) ) ) )   &    |- G = ( j e. NN |-> ( F ` ( j x. Z ) ) )   &    |- ( ph -> Z e. NN )   &    |- ( ph -> C < Z )   =>    |- ( ph -> A. n e. NN A. k e. ( ZZ>= ` n ) ( ( G ` n ) < ( ( G ` k ) + ( 1 / n ) ) /\ ( G ` k ) < ( ( G ` n ) + ( 1 / n ) ) ) )
Theorem	cvg1nlemres 11026*	Lemma for cvg1n 11027. The original sequence F has a limit (turns out it is the same as the limit of the modified sequence G). (Contributed by Jim Kingdon, 1-Aug-2021.)
|- ( ph -> F : NN --> RR )   &    |- ( ph -> C e. RR+ )   &    |- ( ph -> A. n e. NN A. k e. ( ZZ>= ` n ) ( ( F ` n ) < ( ( F ` k ) + ( C / n ) ) /\ ( F ` k ) < ( ( F ` n ) + ( C / n ) ) ) )   &    |- G = ( j e. NN |-> ( F ` ( j x. Z ) ) )   &    |- ( ph -> Z e. NN )   &    |- ( ph -> C < Z )   =>    |- ( ph -> E. y e. RR A. x e. RR+ E. j e. NN A. i e. ( ZZ>= ` j ) ( ( F ` i ) < ( y + x ) /\ y < ( ( F ` i ) + x ) ) )
Theorem	cvg1n 11027*	Convergence of real sequences. This is a version of caucvgre 11022 with a constant multiplier C on the rate of convergence. That is, all terms after the nth term must be within C / n of the nth term. (Contributed by Jim Kingdon, 1-Aug-2021.)
|- ( ph -> F : NN --> RR )   &    |- ( ph -> C e. RR+ )   &    |- ( ph -> A. n e. NN A. k e. ( ZZ>= ` n ) ( ( F ` n ) < ( ( F ` k ) + ( C / n ) ) /\ ( F ` k ) < ( ( F ` n ) + ( C / n ) ) ) )   =>    |- ( ph -> E. y e. RR A. x e. RR+ E. j e. NN A. i e. ( ZZ>= ` j ) ( ( F ` i ) < ( y + x ) /\ y < ( ( F ` i ) + x ) ) )
Theorem	uzin2 11028	The upper integers are closed under intersection. (Contributed by Mario Carneiro, 24-Dec-2013.)
|- ( ( A e. ran ZZ>= /\ B e. ran ZZ>= ) -> ( A i^i B ) e. ran ZZ>= )
Theorem	rexanuz 11029*	Combine two different upper integer properties into one. (Contributed by Mario Carneiro, 25-Dec-2013.)
|- ( E. j e. ZZ A. k e. ( ZZ>= ` j ) ( ph /\ ps ) <-> ( E. j e. ZZ A. k e. ( ZZ>= ` j ) ph /\ E. j e. ZZ A. k e. ( ZZ>= ` j ) ps ) )
Theorem	rexfiuz 11030*	Combine finitely many different upper integer properties into one. (Contributed by Mario Carneiro, 6-Jun-2014.)
|- ( A e. Fin -> ( E. j e. ZZ A. k e. ( ZZ>= ` j ) A. n e. A ph <-> A. n e. A E. j e. ZZ A. k e. ( ZZ>= ` j ) ph ) )
Theorem	rexuz3 11031*	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 11032*	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 11033*	A version of 19.29 1631 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 11034*	A version of r19.2m 3524 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 11035	Lemma for recvguniq 11036. 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 11036*	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 11037	Extend class notation to include square root of a complex number.
class sqr
Syntax	cabs 11038	Extend class notation to include a function for the absolute value (modulus) of a complex number.
class abs
Definition	df-rsqrt 11039*	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 11040	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 11041*	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 11042	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 11043	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 11044	Lemma for sqrt0 11045. (Contributed by Jim Kingdon, 26-Aug-2020.)
|- ( ( A e. RR /\ ( ( A ^ 2 ) = 0 /\ 0 <_ A ) ) <-> A = 0 )
Theorem	sqrt0 11045	Square root of zero. (Contributed by Mario Carneiro, 9-Jul-2013.)
|- ( sqr ` 0 ) = 0
Theorem	resqrexlem1arp 11046	Lemma for resqrex 11067. 1 + A is a positive real (expressed in a way that will help apply seqf 10492 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 11047*	Lemma for resqrex 11067. Applying the recursion rule yields a positive real (expressed in a way that will help apply seqf 10492 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 11048*	Lemma for resqrex 11067. 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 11049*	Lemma for resqrex 11067. 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 11050*	Lemma for resqrex 11067. 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 11051*	Lemma for resqrex 11067. 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 11052*	Lemma for resqrex 11067. 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 11053*	Lemma for resqrex 11067. 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 11054*	Lemma for resqrex 11067. 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 11055*	Lemma for resqrex 11067. 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 11056*	Lemma for resqrex 11067. 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 11057*	Lemma for resqrex 11067. 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 11058*	Lemma for resqrex 11067. 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 11059*	Lemma for resqrex 11067. 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 11060*	Lemma for resqrex 11067. 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 11061*	Lemma for resqrex 11067. 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 11062*	Lemma for resqrex 11067. 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 11063*	Lemma for resqrex 11067. 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 11064*	Lemma for resqrex 11067. 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 11065*	Lemma for resqrex 11067. 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 11066*	Lemma for resqrex 11067. 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 11067*	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 11068*	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 11069*	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 11070	Closure of the square root function. (Contributed by Mario Carneiro, 9-Jul-2013.)
|- ( ( A e. RR /\ 0 <_ A ) -> ( sqr ` A ) e. RR )
Theorem	rersqrtthlem 11071	Lemma for resqrtth 11072. (Contributed by Jim Kingdon, 10-Aug-2021.)
|- ( ( A e. RR /\ 0 <_ A ) -> ( ( ( sqr ` A ) ^ 2 ) = A /\ 0 <_ ( sqr ` A ) ) )
Theorem	resqrtth 11072	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 11073	Square of square root. (Contributed by Mario Carneiro, 10-Jul-2013.)
|- ( ( A e. RR /\ 0 <_ A ) -> ( ( sqr ` A ) x. ( sqr ` A ) ) = A )
Theorem	sqrtge0 11074	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 11075	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 11076	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 11077	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 11078	Square root is strictly monotonic. Closed form of sqrtlti 11178. (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 11079	Analogue to sqrt11 11080 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 11080	The square root function is one-to-one. Also see sqrt11ap 11079 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 11081	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 11082	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 11083	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 11084	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 11085	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 11086	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 11087	The square root of 1 is 1. (Contributed by NM, 31-Jul-1999.)
|- ( sqr ` 1 ) = 1
Theorem	sqrt4 11088	The square root of 4 is 2. (Contributed by NM, 3-Aug-1999.)
|- ( sqr ` 4 ) = 2
Theorem	sqrt9 11089	The square root of 9 is 3. (Contributed by NM, 11-May-2004.)
|- ( sqr ` 9 ) = 3
Theorem	sqrt2gt1lt2 11090	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 11091	Absolute value of negative. (Contributed by NM, 27-Feb-2005.)
|- ( A e. CC -> ( abs ` -u A ) = ( abs ` A ) )
Theorem	abscl 11092	Real closure of absolute value. (Contributed by NM, 3-Oct-1999.)
|- ( A e. CC -> ( abs ` A ) e. RR )
Theorem	abscj 11093	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 11094	Square of value of absolute value function. (Contributed by NM, 16-Jan-2006.)
|- ( A e. CC -> ( ( abs ` A ) ^ 2 ) = ( A x. ( * ` A ) ) )
Theorem	absvalsq2 11095	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 11096	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 11097	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 11098	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 11099	The absolute value of 0. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 29-May-2016.)
|- ( abs ` 0 ) = 0
Theorem	absi 11100	The absolute value of the imaginary unit. (Contributed by NM, 26-Mar-2005.)
|- ( abs ` _i ) = 1