Theorem absval 13978

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.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . 4 |- ( x = A -> ( * ` x ) = ( * ` A ) )
2		oveq12 6659	. . . 4 |- ( ( x = A /\ ( * ` x ) = ( * ` A ) ) -> ( x x. ( * ` x ) ) = ( A x. ( * ` A ) ) )
3	1, 2	mpdan 702	. . 3 |- ( x = A -> ( x x. ( * ` x ) ) = ( A x. ( * ` A ) ) )
4	3	fveq2d 6195	. 2 |- ( x = A -> ( sqr ` ( x x. ( * ` x ) ) ) = ( sqr ` ( A x. ( * ` A ) ) ) )
5		df-abs 13976	. 2 |- abs = ( x e. CC |-> ( sqr ` ( x x. ( * ` x ) ) ) )
6		fvex 6201	. 2 |- ( sqr ` ( A x. ( * ` A ) ) ) e. _V
7	4, 5, 6	fvmpt 6282	1 |- ( A e. CC -> ( abs ` A ) = ( sqr ` ( A x. ( * ` A ) ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   CCcc 9934   x. cmul 9941   *ccj 13836   sqrcsqrt 13973   abscabs 13974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-abs 13976

This theorem is referenced by:  absneg  14017  abscl  14018  abscj  14019  absvalsq  14020  absval2  14024  abs0  14025  absi  14026  absge0  14027  absrpcl  14028  absmul  14034  absid  14036  absre  14041  absf  14077  cphabscl  22985  cphipipcj  23000  tchcphlem2  23035  siii  27708  norm-iii-i  27996  absfico  39410