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Theorem List for Metamath Proof Explorer - 11601-11700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcnrecnv 11601* The inverse to the canonical bijection from  ( RR  X.  RR ) to  CC from cnref1o 10302. (Contributed by Mario Carneiro, 25-Aug-2014.)
 |-  F  =  ( x  e.  RR ,  y  e.  RR  |->  ( x  +  ( _i  x.  y
 ) ) )   =>    |-  `' F  =  ( z  e.  CC  |->  <.
 ( Re `  z
 ) ,  ( Im
 `  z ) >. )
 
Theoremsqeqd 11602 A deduction for showing two numbers whose squares are equal are themselves equal. (Contributed by Mario Carneiro, 3-Apr-2015.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  ( A ^
 2 )  =  ( B ^ 2 ) )   &    |-  ( ph  ->  0 
 <_  ( Re `  A ) )   &    |-  ( ph  ->  0 
 <_  ( Re `  B ) )   &    |-  ( ( ph  /\  ( Re `  A )  =  0  /\  ( Re `  B )  =  0 )  ->  A  =  B )   =>    |-  ( ph  ->  A  =  B )
 
Theoremrecli 11603 The real part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)
 |-  A  e.  CC   =>    |-  ( Re `  A )  e.  RR
 
Theoremimcli 11604 The imaginary part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)
 |-  A  e.  CC   =>    |-  ( Im `  A )  e.  RR
 
Theoremcjcli 11605 Closure law for complex conjugate. (Contributed by NM, 11-May-1999.)
 |-  A  e.  CC   =>    |-  ( * `  A )  e.  CC
 
Theoremreplimi 11606 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 ) ) )
 
Theoremcjcji 11607 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
 
Theoremreim0bi 11608 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 )
 
Theoremrerebi 11609 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 )
 
Theoremcjrebi 11610 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 )
 
Theoremrecji 11611 Real part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)
 |-  A  e.  CC   =>    |-  ( Re `  ( * `  A ) )  =  ( Re
 `  A )
 
Theoremimcji 11612 Imaginary part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)
 |-  A  e.  CC   =>    |-  ( Im `  ( * `  A ) )  =  -u ( Im `  A )
 
Theoremcjmulrcli 11613 A complex number times its conjugate is real. (Contributed by NM, 11-May-1999.)
 |-  A  e.  CC   =>    |-  ( A  x.  ( * `  A ) )  e.  RR
 
Theoremcjmulvali 11614 A complex number times its conjugate. (Contributed by NM, 2-Oct-1999.)
 |-  A  e.  CC   =>    |-  ( A  x.  ( * `  A ) )  =  ( ( ( Re `  A ) ^ 2 )  +  ( ( Im `  A ) ^ 2
 ) )
 
Theoremcjmulge0i 11615 A complex number times its conjugate is nonnegative. (Contributed by NM, 28-May-1999.)
 |-  A  e.  CC   =>    |-  0  <_  ( A  x.  ( * `  A ) )
 
Theoremrenegi 11616 Real part of negative. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( Re `  -u A )  =  -u ( Re `  A )
 
Theoremimnegi 11617 Imaginary part of negative. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( Im `  -u A )  =  -u ( Im `  A )
 
Theoremcjnegi 11618 Complex conjugate of negative. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( * `  -u A )  =  -u ( * `  A )
 
Theoremaddcji 11619 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 ) )
 
Theoremreaddi 11620 Real part distributes over addition. (Contributed by NM, 28-Jul-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( Re `  ( A  +  B )
 )  =  ( ( Re `  A )  +  ( Re `  B ) )
 
Theoremimaddi 11621 Imaginary part distributes over addition. (Contributed by NM, 28-Jul-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( Im `  ( A  +  B )
 )  =  ( ( Im `  A )  +  ( Im `  B ) )
 
Theoremremuli 11622 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 ) ) )
 
Theoremimmuli 11623 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 ) ) )
 
Theoremcjaddi 11624 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 ) )
 
Theoremcjmuli 11625 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 ) )
 
Theoremipcni 11626 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 ) ) )
 
Theoremcjdivi 11627 Complex conjugate distributes over division. (Contributed by NM, 29-Apr-2005.) (Revised by Mario Carneiro, 29-May-2016.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( B  =/=  0  ->  ( * `  ( A  /  B ) )  =  ( ( * `
  A )  /  ( * `  B ) ) )
 
Theoremcrrei 11628 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
 
Theoremcrimi 11629 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
 
Theoremrecld 11630 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 )
 
Theoremimcld 11631 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 )
 
Theoremcjcld 11632 Closure law for complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( * `  A )  e. 
 CC )
 
Theoremreplimd 11633 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 ) ) ) )
 
Theoremremimd 11634 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 ) ) ) )
 
Theoremcjcjd 11635 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 )
 
Theoremreim0bd 11636 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 )
 
Theoremrerebd 11637 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 )
 
Theoremcjrebd 11638 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 )
 
Theoremcjne0d 11639 A number is nonzero iff its complex conjugate is nonzero. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   =>    |-  ( ph  ->  ( * `  A )  =/=  0 )
 
Theoremrecjd 11640 Real part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( Re `  ( * `  A ) )  =  ( Re `  A ) )
 
Theoremimcjd 11641 Imaginary part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( Im `  ( * `  A ) )  =  -u ( Im `  A ) )
 
Theoremcjmulrcld 11642 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 )
 
Theoremcjmulvald 11643 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 ) ) )
 
Theoremcjmulge0d 11644 A complex number times its conjugate is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  0  <_  ( A  x.  ( * `  A ) ) )
 
Theoremrenegd 11645 Real part of negative. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( Re `  -u A )  =  -u ( Re `  A ) )
 
Theoremimnegd 11646 Imaginary part of negative. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( Im `  -u A )  =  -u ( Im `  A ) )
 
Theoremcjnegd 11647 Complex conjugate of negative. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( * `  -u A )  =  -u ( * `  A ) )
 
Theoremaddcjd 11648 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 ) ) )
 
Theoremcjexpd 11649 Complex conjugate of natural number exponentiation. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  ( * `  ( A ^ N ) )  =  ( ( * `  A ) ^ N ) )
 
Theoremreaddd 11650 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 ) ) )
 
Theoremimaddd 11651 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 ) ) )
 
Theoremresubd 11652 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 ) ) )
 
Theoremimsubd 11653 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 ) ) )
 
Theoremremuld 11654 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 ) ) ) )
 
Theoremimmuld 11655 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 ) ) ) )
 
Theoremcjaddd 11656 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 ) ) )
 
Theoremcjmuld 11657 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 ) ) )
 
Theoremipcnd 11658 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 ) ) ) )
 
Theoremcjdivd 11659 Complex conjugate distributes over division. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B  =/=  0
 )   =>    |-  ( ph  ->  ( * `  ( A  /  B ) )  =  ( ( * `  A )  /  ( * `  B ) ) )
 
Theoremrered 11660 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 )
 
Theoremreim0d 11661 The imaginary part of a real number is 0. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( Im `  A )  =  0 )
 
Theoremcjred 11662 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 )
 
Theoremremul2d 11663 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 ) ) )
 
Theoremimmul2d 11664 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 ) ) )
 
Theoremredivd 11665 Real part of a division. Related to remul2 11566. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  A  =/=  0
 )   =>    |-  ( ph  ->  ( Re `  ( B  /  A ) )  =  ( ( Re `  B )  /  A ) )
 
Theoremimdivd 11666 Imaginary part of a division. Related to remul2 11566. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  A  =/=  0
 )   =>    |-  ( ph  ->  ( Im `  ( B  /  A ) )  =  ( ( Im `  B )  /  A ) )
 
Theoremcrred 11667 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 )
 
Theoremcrimd 11668 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 )
 
5.7.3  Square root; absolute value
 
Syntaxcsqr 11669 Extend class notation to include square root of a complex number.
 class  sqr
 
Syntaxcabs 11670 Extend class notation to include a function for the absolute value (modulus) of a complex number.
 class  abs
 
Definitiondf-sqr 11671* Define a function whose value is the square root of a complex number. Since  ( y ^
2 )  =  x iff  ( -u y ^
2 )  =  x, we ensure uniqueness by restricting the range to numbers with positive real part, or numbers with 0 real part and nonnegative imaginary part. A description can be found under "Principal square root of a complex number" at http://en.wikipedia.org/wiki/Square_root. The square root symbol was introduced in 1525 by Christoff Rudolff.

See sqrcl 11796 for its closure, sqrval 11673 for its value, sqrth 11799 and sqsqri 11810 for its relationship to squares, and sqr11i 11819 for uniqueness. (Contributed by NM, 27-Jul-1999.) (Revised by Mario Carneiro, 8-Jul-2013.)

 |- 
 sqr  =  ( x  e.  CC  |->  ( iota_ y  e. 
 CC ( ( y ^ 2 )  =  x  /\  0  <_  ( Re `  y ) 
 /\  ( _i  x.  y )  e/  RR+ )
 ) )
 
Definitiondf-abs 11672 Define the function for the absolute value (modulus) of a complex number. See abscli 11829 for its closure and absval 11674 or absval2i 11831 for its value. (Contributed by NM, 27-Jul-1999.)
 |- 
 abs  =  ( x  e.  CC  |->  ( sqr `  ( x  x.  ( * `  x ) ) ) )
 
Theoremsqrval 11673* Value of square root function. (Contributed by Mario Carneiro, 8-Jul-2013.)
 |-  ( A  e.  CC  ->  ( sqr `  A )  =  ( iota_ x  e. 
 CC ( ( x ^ 2 )  =  A  /\  0  <_  ( Re `  x ) 
 /\  ( _i  x.  x )  e/  RR+ )
 ) )
 
Theoremabsval 11674 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 ) ) ) )
 
Theoremrennim 11675 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+ )
 
Theoremcnpart 11676 The specification of restriction to the right half-plane partitions the complex plane without 0 into two disjoint pieces, which are related by a reflection about the origin (under the map  x 
|->  -u x). (Contributed by Mario Carneiro, 8-Jul-2013.)
 |-  ( ( A  e.  CC  /\  A  =/=  0
 )  ->  ( (
 0  <_  ( Re `  A )  /\  ( _i  x.  A )  e/  RR+ )  <->  -.  ( 0  <_  ( Re `  -u A )  /\  ( _i  x.  -u A )  e/  RR+ )
 ) )
 
Theoremsqr0lem 11677 Square root of zero. (Contributed by Mario Carneiro, 9-Jul-2013.)
 |-  ( ( A  e.  CC  /\  ( ( A ^ 2 )  =  0  /\  0  <_  ( Re `  A ) 
 /\  ( _i  x.  A )  e/  RR+ )
 ) 
 <->  A  =  0 )
 
Theoremsqr0 11678 Square root of zero. (Contributed by Mario Carneiro, 9-Jul-2013.)
 |-  ( sqr `  0
 )  =  0
 
Theoremsqrlem1 11679* Lemma for 01sqrex 11686. (Contributed by Mario Carneiro, 10-Jul-2013.)
 |-  S  =  { x  e.  RR+  |  ( x ^ 2 )  <_  A }   &    |-  B  =  sup ( S ,  RR ,  <  )   =>    |-  ( ( A  e.  RR+  /\  A  <_  1 )  ->  A. y  e.  S  y  <_  1 )
 
Theoremsqrlem2 11680* Lemma for 01sqrex 11686. (Contributed by Mario Carneiro, 10-Jul-2013.)
 |-  S  =  { x  e.  RR+  |  ( x ^ 2 )  <_  A }   &    |-  B  =  sup ( S ,  RR ,  <  )   =>    |-  ( ( A  e.  RR+  /\  A  <_  1 )  ->  A  e.  S )
 
Theoremsqrlem3 11681* Lemma for 01sqrex 11686. (Contributed by Mario Carneiro, 10-Jul-2013.)
 |-  S  =  { x  e.  RR+  |  ( x ^ 2 )  <_  A }   &    |-  B  =  sup ( S ,  RR ,  <  )   =>    |-  ( ( A  e.  RR+  /\  A  <_  1 )  ->  ( S  C_  RR  /\  S  =/=  (/)  /\  E. z  e.  RR  A. y  e.  S  y  <_  z
 ) )
 
Theoremsqrlem4 11682* Lemma for 01sqrex 11686. (Contributed by Mario Carneiro, 10-Jul-2013.)
 |-  S  =  { x  e.  RR+  |  ( x ^ 2 )  <_  A }   &    |-  B  =  sup ( S ,  RR ,  <  )   =>    |-  ( ( A  e.  RR+  /\  A  <_  1 )  ->  ( B  e.  RR+  /\  B  <_  1 )
 )
 
Theoremsqrlem5 11683* Lemma for 01sqrex 11686. (Contributed by Mario Carneiro, 10-Jul-2013.)
 |-  S  =  { x  e.  RR+  |  ( x ^ 2 )  <_  A }   &    |-  B  =  sup ( S ,  RR ,  <  )   &    |-  T  =  {
 y  |  E. a  e.  S  E. b  e.  S  y  =  ( a  x.  b ) }   =>    |-  ( ( A  e.  RR+  /\  A  <_  1 )  ->  ( ( T  C_  RR  /\  T  =/=  (/)  /\  E. v  e.  RR  A. u  e.  T  u  <_  v
 )  /\  ( B ^ 2 )  = 
 sup ( T ,  RR ,  <  ) ) )
 
Theoremsqrlem6 11684* Lemma for 01sqrex 11686. (Contributed by Mario Carneiro, 10-Jul-2013.)
 |-  S  =  { x  e.  RR+  |  ( x ^ 2 )  <_  A }   &    |-  B  =  sup ( S ,  RR ,  <  )   &    |-  T  =  {
 y  |  E. a  e.  S  E. b  e.  S  y  =  ( a  x.  b ) }   =>    |-  ( ( A  e.  RR+  /\  A  <_  1 )  ->  ( B ^ 2
 )  <_  A )
 
Theoremsqrlem7 11685* Lemma for 01sqrex 11686. (Contributed by Mario Carneiro, 10-Jul-2013.)
 |-  S  =  { x  e.  RR+  |  ( x ^ 2 )  <_  A }   &    |-  B  =  sup ( S ,  RR ,  <  )   &    |-  T  =  {
 y  |  E. a  e.  S  E. b  e.  S  y  =  ( a  x.  b ) }   =>    |-  ( ( A  e.  RR+  /\  A  <_  1 )  ->  ( B ^ 2
 )  =  A )
 
Theorem01sqrex 11686* Existence of a square root for reals in the interval  ( 0 ,  1 ]. (Contributed by Mario Carneiro, 10-Jul-2013.)
 |-  ( ( A  e.  RR+  /\  A  <_  1 )  ->  E. x  e.  RR+  ( x  <_  1  /\  ( x ^ 2 )  =  A ) )
 
Theoremresqrex 11687* 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 ) )
 
Theoremsqrmo 11688* Uniqueness for the square root function. (Contributed by Mario Carneiro, 9-Jul-2013.) (Revised by NM, 17-Jun-2017.)
 |-  ( A  e.  CC  ->  E* x  e.  CC ( ( x ^
 2 )  =  A  /\  0  <_  ( Re
 `  x )  /\  ( _i  x.  x )  e/  RR+ ) )
 
Theoremresqreu 11689* Existence and uniqueness for the real square root function. (Contributed by Mario Carneiro, 9-Jul-2013.)
 |-  ( ( A  e.  RR  /\  0  <_  A )  ->  E! x  e. 
 CC  ( ( x ^ 2 )  =  A  /\  0  <_  ( Re `  x ) 
 /\  ( _i  x.  x )  e/  RR+ )
 )
 
Theoremresqrcl 11690 Closure of the square root function. (Contributed by Mario Carneiro, 9-Jul-2013.)
 |-  ( ( A  e.  RR  /\  0  <_  A )  ->  ( sqr `  A )  e.  RR )
 
Theoremresqrthlem 11691 Lemma for resqrth 11692. (Contributed by Mario Carneiro, 9-Jul-2013.)
 |-  ( ( A  e.  RR  /\  0  <_  A )  ->  ( ( ( sqr `  A ) ^ 2 )  =  A  /\  0  <_  ( Re `  ( sqr `  A ) )  /\  ( _i  x.  ( sqr `  A ) ) 
 e/  RR+ ) )
 
Theoremresqrth 11692 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 )
 
Theoremremsqsqr 11693 Square of square root. (Contributed by Mario Carneiro, 10-Jul-2013.)
 |-  ( ( A  e.  RR  /\  0  <_  A )  ->  ( ( sqr `  A )  x.  ( sqr `  A ) )  =  A )
 
Theoremsqrge0 11694 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 ) )
 
Theoremsqrgt0 11695 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 ) )
 
Theoremsqrmul 11696 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 ) ) )
 
Theoremsqrle 11697 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 ) ) )
 
Theoremsqrlt 11698 Square root is strictly monotonic. Closed form of sqrlti 11824. (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 ) ) )
 
Theoremsqr11 11699 The square root function is one to one. (Contributed by Scott Fenton, 11-Jun-2013.)
 |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  ->  ( ( sqr `  A )  =  ( sqr `  B )  <->  A  =  B ) )
 
Theoremsqr00 11700 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 )
 )
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