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Theorem absval 12043
Description: 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.)
Assertion
Ref Expression
absval  |-  ( A  e.  CC  ->  ( abs `  A )  =  ( sqr `  ( A  x.  ( * `  A ) ) ) )

Proof of Theorem absval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fveq2 5728 . . . 4  |-  ( x  =  A  ->  (
* `  x )  =  ( * `  A ) )
2 oveq12 6090 . . . 4  |-  ( ( x  =  A  /\  ( * `  x
)  =  ( * `
 A ) )  ->  ( x  x.  ( * `  x
) )  =  ( A  x.  ( * `
 A ) ) )
31, 2mpdan 650 . . 3  |-  ( x  =  A  ->  (
x  x.  ( * `
 x ) )  =  ( A  x.  ( * `  A
) ) )
43fveq2d 5732 . 2  |-  ( x  =  A  ->  ( sqr `  ( x  x.  ( * `  x
) ) )  =  ( sqr `  ( A  x.  ( * `  A ) ) ) )
5 df-abs 12041 . 2  |-  abs  =  ( x  e.  CC  |->  ( sqr `  ( x  x.  ( * `  x ) ) ) )
6 fvex 5742 . 2  |-  ( sqr `  ( A  x.  (
* `  A )
) )  e.  _V
74, 5, 6fvmpt 5806 1  |-  ( A  e.  CC  ->  ( abs `  A )  =  ( sqr `  ( A  x.  ( * `  A ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   ` cfv 5454  (class class class)co 6081   CCcc 8988    x. cmul 8995   *ccj 11901   sqrcsqr 12038   abscabs 12039
This theorem is referenced by:  absneg  12082  abscl  12083  abscj  12084  absvalsq  12085  absval2  12089  abs0  12090  absi  12091  absge0  12092  absrpcl  12093  absmul  12099  absid  12101  absre  12106  absf  12141  cphabscl  19148  tchcphlem2  19193  siii  22354  norm-iii-i  22641
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-abs 12041
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