P. 118 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 11701-11800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cjmulrcli 11701	A complex number times its conjugate is real. (Contributed by NM, 11-May-1999.)
|- A e. CC   =>    |- ( A x. ( * ` A ) ) e. RR
Theorem	cjmulvali 11702	A complex number times its conjugate. (Contributed by NM, 2-Oct-1999.)
|- A e. CC   =>    |- ( A x. ( * ` A ) ) = ( ( ( Re ` A ) ^ 2 ) + ( ( Im ` A ) ^ 2 ) )
Theorem	cjmulge0i 11703	A complex number times its conjugate is nonnegative. (Contributed by NM, 28-May-1999.)
|- A e. CC   =>    |- 0 <_ ( A x. ( * ` A ) )
Theorem	renegi 11704	Real part of negative. (Contributed by NM, 2-Aug-1999.)
|- A e. CC   =>    |- ( Re ` -u A ) = -u ( Re ` A )
Theorem	imnegi 11705	Imaginary part of negative. (Contributed by NM, 2-Aug-1999.)
|- A e. CC   =>    |- ( Im ` -u A ) = -u ( Im ` A )
Theorem	cjnegi 11706	Complex conjugate of negative. (Contributed by NM, 2-Aug-1999.)
|- A e. CC   =>    |- ( * ` -u A ) = -u ( * ` A )
Theorem	addcji 11707	A number plus its conjugate is twice its real part. Compare Proposition 10-3.4(h) of [Gleason] p. 133. (Contributed by NM, 2-Oct-1999.)
|- A e. CC   =>    |- ( A + ( * ` A ) ) = ( 2 x. ( Re ` A ) )
Theorem	readdi 11708	Real part distributes over addition. (Contributed by NM, 28-Jul-1999.)
|- A e. CC   &    |- B e. CC   =>    |- ( Re ` ( A + B ) ) = ( ( Re ` A ) + ( Re ` B ) )
Theorem	imaddi 11709	Imaginary part distributes over addition. (Contributed by NM, 28-Jul-1999.)
|- A e. CC   &    |- B e. CC   =>    |- ( Im ` ( A + B ) ) = ( ( Im ` A ) + ( Im ` B ) )
Theorem	remuli 11710	Real part of a product. (Contributed by NM, 28-Jul-1999.)
|- A e. CC   &    |- B e. CC   =>    |- ( Re ` ( A x. B ) ) = ( ( ( Re ` A ) x. ( Re ` B ) ) - ( ( Im ` A ) x. ( Im ` B ) ) )
Theorem	immuli 11711	Imaginary part of a product. (Contributed by NM, 28-Jul-1999.)
|- A e. CC   &    |- B e. CC   =>    |- ( Im ` ( A x. B ) ) = ( ( ( Re ` A ) x. ( Im ` B ) ) + ( ( Im ` A ) x. ( Re ` B ) ) )
Theorem	cjaddi 11712	Complex conjugate distributes over addition. Proposition 10-3.4(a) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.)
|- A e. CC   &    |- B e. CC   =>    |- ( * ` ( A + B ) ) = ( ( * ` A ) + ( * ` B ) )
Theorem	cjmuli 11713	Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.)
|- A e. CC   &    |- B e. CC   =>    |- ( * ` ( A x. B ) ) = ( ( * ` A ) x. ( * ` B ) )
Theorem	ipcni 11714	Standard inner product on complex numbers. (Contributed by NM, 2-Oct-1999.)
|- A e. CC   &    |- B e. CC   =>    |- ( Re ` ( A x. ( * ` B ) ) ) = ( ( ( Re ` A ) x. ( Re ` B ) ) + ( ( Im ` A ) x. ( Im ` B ) ) )
Theorem	cjdivapi 11715	Complex conjugate distributes over division. (Contributed by Jim Kingdon, 14-Jun-2020.)
|- A e. CC   &    |- B e. CC   =>    |- ( B # 0 -> ( * ` ( A / B ) ) = ( ( * ` A ) / ( * ` B ) ) )
Theorem	crrei 11716	The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 10-May-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( Re ` ( A + ( _i x. B ) ) ) = A
Theorem	crimi 11717	The imaginary part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 10-May-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( Im ` ( A + ( _i x. B ) ) ) = B
Theorem	recld 11718	The real part of a complex number is real (closure law). (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( Re ` A ) e. RR )
Theorem	imcld 11719	The imaginary part of a complex number is real (closure law). (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( Im ` A ) e. RR )
Theorem	cjcld 11720	Closure law for complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( * ` A ) e. CC )
Theorem	replimd 11721	Construct a complex number from its real and imaginary parts. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> A = ( ( Re ` A ) + ( _i x. ( Im ` A ) ) ) )
Theorem	remimd 11722	Value of the conjugate of a complex number. The value is the real part minus _i times the imaginary part. Definition 10-3.2 of [Gleason] p. 132. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( * ` A ) = ( ( Re ` A ) - ( _i x. ( Im ` A ) ) ) )
Theorem	cjcjd 11723	The conjugate of the conjugate is the original complex number. Proposition 10-3.4(e) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( * ` ( * ` A ) ) = A )
Theorem	reim0bd 11724	A number is real iff its imaginary part is 0. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> ( Im ` A ) = 0 )   =>    |- ( ph -> A e. RR )
Theorem	rerebd 11725	A real number equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> ( Re ` A ) = A )   =>    |- ( ph -> A e. RR )
Theorem	cjrebd 11726	A number is real iff it equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> ( * ` A ) = A )   =>    |- ( ph -> A e. RR )
Theorem	cjne0d 11727	A number which is nonzero has a complex conjugate which is nonzero. Also see cjap0d 11728 which is similar but for apartness. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> A =/= 0 )   =>    |- ( ph -> ( * ` A ) =/= 0 )
Theorem	cjap0d 11728	A number which is apart from zero has a complex conjugate which is apart from zero. (Contributed by Jim Kingdon, 11-Aug-2021.)
|- ( ph -> A e. CC )   &    |- ( ph -> A # 0 )   =>    |- ( ph -> ( * ` A ) # 0 )
Theorem	recjd 11729	Real part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( Re ` ( * ` A ) ) = ( Re ` A ) )
Theorem	imcjd 11730	Imaginary part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( Im ` ( * ` A ) ) = -u ( Im ` A ) )
Theorem	cjmulrcld 11731	A complex number times its conjugate is real. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( A x. ( * ` A ) ) e. RR )
Theorem	cjmulvald 11732	A complex number times its conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( A x. ( * ` A ) ) = ( ( ( Re ` A ) ^ 2 ) + ( ( Im ` A ) ^ 2 ) ) )
Theorem	cjmulge0d 11733	A complex number times its conjugate is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> 0 <_ ( A x. ( * ` A ) ) )
Theorem	renegd 11734	Real part of negative. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( Re ` -u A ) = -u ( Re ` A ) )
Theorem	imnegd 11735	Imaginary part of negative. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( Im ` -u A ) = -u ( Im ` A ) )
Theorem	cjnegd 11736	Complex conjugate of negative. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( * ` -u A ) = -u ( * ` A ) )
Theorem	addcjd 11737	A number plus its conjugate is twice its real part. Compare Proposition 10-3.4(h) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( A + ( * ` A ) ) = ( 2 x. ( Re ` A ) ) )
Theorem	cjexpd 11738	Complex conjugate of positive integer exponentiation. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> N e. NN0 )   =>    |- ( ph -> ( * ` ( A ^ N ) ) = ( ( * ` A ) ^ N ) )
Theorem	readdd 11739	Real part distributes over addition. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( Re ` ( A + B ) ) = ( ( Re ` A ) + ( Re ` B ) ) )
Theorem	imaddd 11740	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 11741	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 11742	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 11743	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 11744	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 11745	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 11746	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 11747	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 11748	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 11749	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 11750	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 11751	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 11752	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 11753	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 11754	Real part of a division. Related to remul2 11652. (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 11755	Imaginary part of a division. Related to remul2 11652. (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 11756	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 11757	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 11758	Complex apartness in terms of real and imaginary parts. See also apreim 8933 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 ) ) ) )
4.8.3  Sequence convergence
Theorem	caucvgrelemrec 11759*	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 11760*	Lemma for caucvgre 11761. 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 11761*	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 11762	Lemma for cvg1n 11766. 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 11763*	Lemma for cvg1n 11766. 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 11764*	Lemma for cvg1n 11766. 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 11765*	Lemma for cvg1n 11766. 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 11766*	Convergence of real sequences. This is a version of caucvgre 11761 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 11767	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 11768*	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 11769*	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 11770*	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 11771*	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 11772*	A version of 19.29 1673 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 11773*	A version of r19.2m 3614 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 11774	Lemma for recvguniq 11775. 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 11775*	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 )
4.8.4  Square root; absolute value
Syntax	csqrt 11776	Extend class notation to include square root of a complex number.
class sqr
Syntax	cabs 11777	Extend class notation to include a function for the absolute value (modulus) of a complex number.
class abs
Definition	df-rsqrt 11778*	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 11779	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 11780*	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 11781	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 11782	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 11783	Lemma for sqrt0 11784. (Contributed by Jim Kingdon, 26-Aug-2020.)
|- ( ( A e. RR /\ ( ( A ^ 2 ) = 0 /\ 0 <_ A ) ) <-> A = 0 )
Theorem	sqrt0 11784	Square root of zero. (Contributed by Mario Carneiro, 9-Jul-2013.)
|- ( sqr ` 0 ) = 0
Theorem	resqrexlem1arp 11785	Lemma for resqrex 11806. 1 + A is a positive real (expressed in a way that will help apply seqf 10914 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 11786*	Lemma for resqrex 11806. Applying the recursion rule yields a positive real (expressed in a way that will help apply seqf 10914 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 11787*	Lemma for resqrex 11806. 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 11788*	Lemma for resqrex 11806. 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 11789*	Lemma for resqrex 11806. 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 11790*	Lemma for resqrex 11806. 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 11791*	Lemma for resqrex 11806. 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 11792*	Lemma for resqrex 11806. 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 11793*	Lemma for resqrex 11806. 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 11794*	Lemma for resqrex 11806. 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 11795*	Lemma for resqrex 11806. 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 11796*	Lemma for resqrex 11806. 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 11797*	Lemma for resqrex 11806. 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 11798*	Lemma for resqrex 11806. 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 11799*	Lemma for resqrex 11806. 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 11800*	Lemma for resqrex 11806. 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 )