Theorem cjval 10787

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 8856	. . 3 |- ( A e. CC -> E! x e. CC ( ( A + x ) e. RR /\ ( _i x. ( A - x ) ) e. RR ) )
2		riotacl 5812	. . 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 5849	. . . . . 6 |- ( y = A -> ( y + x ) = ( A + x ) )
5	4	eleq1d 2235	. . . . 5 |- ( y = A -> ( ( y + x ) e. RR <-> ( A + x ) e. RR ) )
6		oveq1 5849	. . . . . . 7 |- ( y = A -> ( y - x ) = ( A - x ) )
7	6	oveq2d 5858	. . . . . 6 |- ( y = A -> ( _i x. ( y - x ) ) = ( _i x. ( A - x ) ) )
8	7	eleq1d 2235	. . . . 5 |- ( y = A -> ( ( _i x. ( y - x ) ) e. RR <-> ( _i x. ( A - x ) ) e. RR ) )
9	5, 8	anbi12d 465	. . . 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 5800	. . 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 10784	. . 3 |- * = ( y e. CC |-> ( iota_ x e. CC ( ( y + x ) e. RR /\ ( _i x. ( y - x ) ) e. RR ) ) )
12	10, 11	fvmptg 5562	. 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 418	1 |- ( A e. CC -> ( * ` A ) = ( iota_ x e. CC ( ( A + x ) e. RR /\ ( _i x. ( A - x ) ) e. RR ) ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   E!wreu 2446   ` cfv 5188   iota_crio 5797  (class class class)co 5842   CCcc 7751   RRcr 7752   _ici 7755   + caddc 7756   x. cmul 7758   - cmin 8069   *ccj 10781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-ltxr 7938  df-sub 8071  df-neg 8072  df-reap 8473  df-cj 10784

This theorem is referenced by:  cjth  10788  remim  10802