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Theorem absval 13037
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 5866 . . . 4  |-  ( x  =  A  ->  (
* `  x )  =  ( * `  A ) )
2 oveq12 6294 . . . 4  |-  ( ( x  =  A  /\  ( * `  x
)  =  ( * `
 A ) )  ->  ( x  x.  ( * `  x
) )  =  ( A  x.  ( * `
 A ) ) )
31, 2mpdan 668 . . 3  |-  ( x  =  A  ->  (
x  x.  ( * `
 x ) )  =  ( A  x.  ( * `  A
) ) )
43fveq2d 5870 . 2  |-  ( x  =  A  ->  ( sqr `  ( x  x.  ( * `  x
) ) )  =  ( sqr `  ( A  x.  ( * `  A ) ) ) )
5 df-abs 13035 . 2  |-  abs  =  ( x  e.  CC  |->  ( sqr `  ( x  x.  ( * `  x ) ) ) )
6 fvex 5876 . 2  |-  ( sqr `  ( A  x.  (
* `  A )
) )  e.  _V
74, 5, 6fvmpt 5951 1  |-  ( A  e.  CC  ->  ( abs `  A )  =  ( sqr `  ( A  x.  ( * `  A ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   ` cfv 5588  (class class class)co 6285   CCcc 9491    x. cmul 9498   *ccj 12895   sqrcsqrt 13032   abscabs 13033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5551  df-fun 5590  df-fv 5596  df-ov 6288  df-abs 13035
This theorem is referenced by:  absneg  13076  abscl  13077  abscj  13078  absvalsq  13079  absval2  13083  abs0  13084  absi  13085  absge0  13086  absrpcl  13087  absmul  13093  absid  13095  absre  13100  absf  13136  cphabscl  21459  tchcphlem2  21506  siii  25541  norm-iii-i  25829
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