Theorem cjval 11156

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 9034	. . 3 |- ( A e. CC -> E! x e. CC ( ( A + x ) e. RR /\ ( _i x. ( A - x ) ) e. RR ) )
2		riotacl 5914	. . 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 5951	. . . . . 6 |- ( y = A -> ( y + x ) = ( A + x ) )
5	4	eleq1d 2274	. . . . 5 |- ( y = A -> ( ( y + x ) e. RR <-> ( A + x ) e. RR ) )
6		oveq1 5951	. . . . . . 7 |- ( y = A -> ( y - x ) = ( A - x ) )
7	6	oveq2d 5960	. . . . . 6 |- ( y = A -> ( _i x. ( y - x ) ) = ( _i x. ( A - x ) ) )
8	7	eleq1d 2274	. . . . 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 5901	. . 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 11153	. . 3 |- * = ( y e. CC |-> ( iota_ x e. CC ( ( y + x ) e. RR /\ ( _i x. ( y - x ) ) e. RR ) ) )
12	10, 11	fvmptg 5655	. 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 1373   e. wcel 2176   E!wreu 2486   ` cfv 5271   iota_crio 5898  (class class class)co 5944   CCcc 7923   RRcr 7924   _ici 7927   + caddc 7928   x. cmul 7930   - cmin 8243   *ccj 11150

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-mulrcl 8024  ax-addcom 8025  ax-mulcom 8026  ax-addass 8027  ax-mulass 8028  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-1rid 8032  ax-0id 8033  ax-rnegex 8034  ax-precex 8035  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041  ax-pre-mulgt0 8042

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-iota 5232  df-fun 5273  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-pnf 8109  df-mnf 8110  df-ltxr 8112  df-sub 8245  df-neg 8246  df-reap 8648  df-cj 11153

This theorem is referenced by:  cjth  11157  remim  11171