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Theorem absval 12727
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 5691 . . . 4  |-  ( x  =  A  ->  (
* `  x )  =  ( * `  A ) )
2 oveq12 6100 . . . 4  |-  ( ( x  =  A  /\  ( * `  x
)  =  ( * `
 A ) )  ->  ( x  x.  ( * `  x
) )  =  ( A  x.  ( * `
 A ) ) )
31, 2mpdan 668 . . 3  |-  ( x  =  A  ->  (
x  x.  ( * `
 x ) )  =  ( A  x.  ( * `  A
) ) )
43fveq2d 5695 . 2  |-  ( x  =  A  ->  ( sqr `  ( x  x.  ( * `  x
) ) )  =  ( sqr `  ( A  x.  ( * `  A ) ) ) )
5 df-abs 12725 . 2  |-  abs  =  ( x  e.  CC  |->  ( sqr `  ( x  x.  ( * `  x ) ) ) )
6 fvex 5701 . 2  |-  ( sqr `  ( A  x.  (
* `  A )
) )  e.  _V
74, 5, 6fvmpt 5774 1  |-  ( A  e.  CC  ->  ( abs `  A )  =  ( sqr `  ( A  x.  ( * `  A ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   ` cfv 5418  (class class class)co 6091   CCcc 9280    x. cmul 9287   *ccj 12585   sqrcsqr 12722   abscabs 12723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-iota 5381  df-fun 5420  df-fv 5426  df-ov 6094  df-abs 12725
This theorem is referenced by:  absneg  12766  abscl  12767  abscj  12768  absvalsq  12769  absval2  12773  abs0  12774  absi  12775  absge0  12776  absrpcl  12777  absmul  12783  absid  12785  absre  12790  absf  12825  cphabscl  20704  tchcphlem2  20751  siii  24253  norm-iii-i  24541
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