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Theorem absval 11723
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 5525 . . . 4  |-  ( x  =  A  ->  (
* `  x )  =  ( * `  A ) )
2 oveq12 5867 . . . 4  |-  ( ( x  =  A  /\  ( * `  x
)  =  ( * `
 A ) )  ->  ( x  x.  ( * `  x
) )  =  ( A  x.  ( * `
 A ) ) )
31, 2mpdan 649 . . 3  |-  ( x  =  A  ->  (
x  x.  ( * `
 x ) )  =  ( A  x.  ( * `  A
) ) )
43fveq2d 5529 . 2  |-  ( x  =  A  ->  ( sqr `  ( x  x.  ( * `  x
) ) )  =  ( sqr `  ( A  x.  ( * `  A ) ) ) )
5 df-abs 11721 . 2  |-  abs  =  ( x  e.  CC  |->  ( sqr `  ( x  x.  ( * `  x ) ) ) )
6 fvex 5539 . 2  |-  ( sqr `  ( A  x.  (
* `  A )
) )  e.  _V
74, 5, 6fvmpt 5602 1  |-  ( A  e.  CC  ->  ( abs `  A )  =  ( sqr `  ( A  x.  ( * `  A ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   CCcc 8735    x. cmul 8742   *ccj 11581   sqrcsqr 11718   abscabs 11719
This theorem is referenced by:  absneg  11762  abscl  11763  abscj  11764  absvalsq  11765  absval2  11769  abs0  11770  absi  11771  absge0  11772  absrpcl  11773  absmul  11779  absid  11781  absre  11786  absf  11821  cphabscl  18621  tchcphlem2  18666  siii  21431  norm-iii-i  21718
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-abs 11721
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