Theorem cjval 11127

Description: The value of the conjugate of a complex number. (Contributed by Mario Carneiro, 6-Nov-2013.)

Assertion

Ref	Expression
cjval	|- ( A e. CC -> ( * ` A ) = ( iota_ x e. CC ( ( A + x ) e. RR /\ ( _i x. ( A - x ) ) e. RR ) ) )

Distinct variable group:   x, A

Proof of Theorem cjval

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cju 9033	. . 3 |- ( A e. CC -> E! x e. CC ( ( A + x ) e. RR /\ ( _i x. ( A - x ) ) e. RR ) )
2		riotacl 5913	. . 3 |- ( E! x e. CC ( ( A + x ) e. RR /\ ( _i x. ( A - x ) ) e. RR ) -> ( iota_ x e. CC ( ( A + x ) e. RR /\ ( _i x. ( A - x ) ) e. RR ) ) e. CC )
3	1, 2	syl 14	. 2 |- ( A e. CC -> ( iota_ x e. CC ( ( A + x ) e. RR /\ ( _i x. ( A - x ) ) e. RR ) ) e. CC )
4		oveq1 5950	. . . . . 6 |- ( y = A -> ( y + x ) = ( A + x ) )
5	4	eleq1d 2273	. . . . 5 |- ( y = A -> ( ( y + x ) e. RR <-> ( A + x ) e. RR ) )
6		oveq1 5950	. . . . . . 7 |- ( y = A -> ( y - x ) = ( A - x ) )
7	6	oveq2d 5959	. . . . . 6 |- ( y = A -> ( _i x. ( y - x ) ) = ( _i x. ( A - x ) ) )
8	7	eleq1d 2273	. . . . 5 |- ( y = A -> ( ( _i x. ( y - x ) ) e. RR <-> ( _i x. ( A - x ) ) e. RR ) )
9	5, 8	anbi12d 473	. . . 4 |- ( y = A -> ( ( ( y + x ) e. RR /\ ( _i x. ( y - x ) ) e. RR ) <-> ( ( A + x ) e. RR /\ ( _i x. ( A - x ) ) e. RR ) ) )
10	9	riotabidv 5900	. . 3 |- ( y = A -> ( iota_ x e. CC ( ( y + x ) e. RR /\ ( _i x. ( y - x ) ) e. RR ) ) = ( iota_ x e. CC ( ( A + x ) e. RR /\ ( _i x. ( A - x ) ) e. RR ) ) )
11		df-cj 11124	. . 3 |- * = ( y e. CC |-> ( iota_ x e. CC ( ( y + x ) e. RR /\ ( _i x. ( y - x ) ) e. RR ) ) )
12	10, 11	fvmptg 5654	. 2 |- ( ( A e. CC /\ ( iota_ x e. CC ( ( A + x ) e. RR /\ ( _i x. ( A - x ) ) e. RR ) ) e. CC ) -> ( * ` A ) = ( iota_ x e. CC ( ( A + x ) e. RR /\ ( _i x. ( A - x ) ) e. RR ) ) )
13	3, 12	mpdan 421	1 |- ( A e. CC -> ( * ` A ) = ( iota_ x e. CC ( ( A + x ) e. RR /\ ( _i x. ( A - x ) ) e. RR ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1372   e. wcel 2175   E!wreu 2485   ` cfv 5270   iota_crio 5897  (class class class)co 5943   CCcc 7922   RRcr 7923   _ici 7926   + caddc 7927   x. cmul 7929   - cmin 8242   *ccj 11121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-mulrcl 8023  ax-addcom 8024  ax-mulcom 8025  ax-addass 8026  ax-mulass 8027  ax-distr 8028  ax-i2m1 8029  ax-0lt1 8030  ax-1rid 8031  ax-0id 8032  ax-rnegex 8033  ax-precex 8034  ax-cnre 8035  ax-pre-ltirr 8036  ax-pre-lttrn 8038  ax-pre-apti 8039  ax-pre-ltadd 8040  ax-pre-mulgt0 8041

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-iota 5231  df-fun 5272  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-pnf 8108  df-mnf 8109  df-ltxr 8111  df-sub 8244  df-neg 8245  df-reap 8647  df-cj 11124

This theorem is referenced by:  cjth  11128  remim  11142