P. 99 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 9801-9900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	replimi 9801	Construct a complex number from its real and imaginary parts. (Contributed by NM, 1-Oct-1999.)
|- A e. CC   =>    |- A = ( ( Re ` A ) + ( _i x. ( Im ` A ) ) )
Theorem	cjcji 9802	The conjugate of the conjugate is the original complex number. Proposition 10-3.4(e) of [Gleason] p. 133. (Contributed by NM, 11-May-1999.)
|- A e. CC   =>    |- ( * ` ( * ` A ) ) = A
Theorem	reim0bi 9803	A number is real iff its imaginary part is 0. (Contributed by NM, 29-May-1999.)
|- A e. CC   =>    |- ( A e. RR <-> ( Im ` A ) = 0 )
Theorem	rerebi 9804	A real number equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 27-Oct-1999.)
|- A e. CC   =>    |- ( A e. RR <-> ( Re ` A ) = A )
Theorem	cjrebi 9805	A number is real iff it equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 11-Oct-1999.)
|- A e. CC   =>    |- ( A e. RR <-> ( * ` A ) = A )
Theorem	recji 9806	Real part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)
|- A e. CC   =>    |- ( Re ` ( * ` A ) ) = ( Re ` A )
Theorem	imcji 9807	Imaginary part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)
|- A e. CC   =>    |- ( Im ` ( * ` A ) ) = -u ( Im ` A )
Theorem	cjmulrcli 9808	A complex number times its conjugate is real. (Contributed by NM, 11-May-1999.)
|- A e. CC   =>    |- ( A x. ( * ` A ) ) e. RR
Theorem	cjmulvali 9809	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 9810	A complex number times its conjugate is nonnegative. (Contributed by NM, 28-May-1999.)
|- A e. CC   =>    |- 0 <_ ( A x. ( * ` A ) )
Theorem	renegi 9811	Real part of negative. (Contributed by NM, 2-Aug-1999.)
|- A e. CC   =>    |- ( Re ` -u A ) = -u ( Re ` A )
Theorem	imnegi 9812	Imaginary part of negative. (Contributed by NM, 2-Aug-1999.)
|- A e. CC   =>    |- ( Im ` -u A ) = -u ( Im ` A )
Theorem	cjnegi 9813	Complex conjugate of negative. (Contributed by NM, 2-Aug-1999.)
|- A e. CC   =>    |- ( * ` -u A ) = -u ( * ` A )
Theorem	addcji 9814	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 9815	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 9816	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 9817	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 9818	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 9819	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 9820	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 9821	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 9822	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 9823	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 9824	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 9825	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 9826	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 9827	Closure law for complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( * ` A ) e. CC )
Theorem	replimd 9828	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 9829	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 9830	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 9831	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 9832	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 9833	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 9834	A number which is nonzero has a complex conjugate which is nonzero. Also see cjap0d 9835 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 9835	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 9836	Real part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( Re ` ( * ` A ) ) = ( Re ` A ) )
Theorem	imcjd 9837	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 9838	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 9839	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 9840	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 9841	Real part of negative. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( Re ` -u A ) = -u ( Re ` A ) )
Theorem	imnegd 9842	Imaginary part of negative. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( Im ` -u A ) = -u ( Im ` A ) )
Theorem	cjnegd 9843	Complex conjugate of negative. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( * ` -u A ) = -u ( * ` A ) )
Theorem	addcjd 9844	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 9845	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 9846	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 9847	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 9848	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 9849	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 9850	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 9851	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 9852	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 9853	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 9854	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 9855	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 9856	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 9857	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 9858	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 9859	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 9860	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 9861	Real part of a division. Related to remul2 9760. (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 9862	Imaginary part of a division. Related to remul2 9760. (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 9863	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 9864	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 )
3.7.3  Sequence convergence
Theorem	caucvgrelemrec 9865*	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 9866*	Lemma for caucvgre 9867. 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 9867*	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 9868	Lemma for cvg1n 9872. 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 9869*	Lemma for cvg1n 9872. 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 9870*	Lemma for cvg1n 9872. 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 9871*	Lemma for cvg1n 9872. 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 9872*	Convergence of real sequences. This is a version of caucvgre 9867 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 9873	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 9874*	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 9875*	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 9876*	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 9877*	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 9878*	A version of 19.29 1551 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 9879*	A version of r19.2m 3329 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 9880	Lemma for recvguniq 9881. 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 9881*	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 )
3.7.4  Square root; absolute value
Syntax	csqrt 9882	Extend class notation to include square root of a complex number.
class sqr
Syntax	cabs 9883	Extend class notation to include a function for the absolute value (modulus) of a complex number.
class abs
Definition	df-rsqrt 9884*	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 9885	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 9886*	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 9887	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 9888	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 9889	Lemma for sqrt0 9890. (Contributed by Jim Kingdon, 26-Aug-2020.)
|- ( ( A e. RR /\ ( ( A ^ 2 ) = 0 /\ 0 <_ A ) ) <-> A = 0 )
Theorem	sqrt0 9890	Square root of zero. (Contributed by Mario Carneiro, 9-Jul-2013.)
|- ( sqr ` 0 ) = 0
Theorem	resqrexlem1arp 9891*	Lemma for resqrex 9912. 1 + A is a positive real (expressed in a way that will help apply iseqf 9444 and similar theorems). (Contributed by Jim Kingdon, 28-Jul-2021.)
|- F = seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) , RR+ )   &    |- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   =>    |- ( ( ph /\ N e. NN ) -> ( ( NN X. { ( 1 + A ) } ) ` N ) e. RR+ )
Theorem	resqrexlemp1rp 9892*	Lemma for resqrex 9912. Applying the recursion rule yields a positive real (expressed in a way that will help apply iseqf 9444 and similar theorems). (Contributed by Jim Kingdon, 28-Jul-2021.)
|- F = seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) , RR+ )   &    |- ( 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 9893*	Lemma for resqrex 9912. The sequence is a function. (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 ) } ) , RR+ )   &    |- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   =>    |- ( ph -> F : NN --> RR+ )
Theorem	resqrexlemf1 9894*	Lemma for resqrex 9912. 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.)
|- F = seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) , RR+ )   &    |- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   =>    |- ( ph -> ( F ` 1 ) = ( 1 + A ) )
Theorem	resqrexlemfp1 9895*	Lemma for resqrex 9912. 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 ) } ) , RR+ )   &    |- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   =>    |- ( ( ph /\ N e. NN ) -> ( F ` ( N + 1 ) ) = ( ( ( F ` N ) + ( A / ( F ` N ) ) ) / 2 ) )
Theorem	resqrexlemover 9896*	Lemma for resqrex 9912. 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 ) } ) , RR+ )   &    |- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   =>    |- ( ( ph /\ N e. NN ) -> A < ( ( F ` N ) ^ 2 ) )
Theorem	resqrexlemdec 9897*	Lemma for resqrex 9912. 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 ) } ) , RR+ )   &    |- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   =>    |- ( ( ph /\ N e. NN ) -> ( F ` ( N + 1 ) ) < ( F ` N ) )
Theorem	resqrexlemdecn 9898*	Lemma for resqrex 9912. 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 ) } ) , RR+ )   &    |- ( 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 9899*	Lemma for resqrex 9912. 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 ) } ) , RR+ )   &    |- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   =>    |- ( ( ph /\ N e. NN ) -> ( 1 / ( 2 ^ N ) ) < ( F ` N ) )
Theorem	resqrexlemcalc1 9900*	Lemma for resqrex 9912. 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 ) } ) , RR+ )   &    |- ( 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 ) ) ) )