Theorem cjval 11588

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 9281	. . 3 |- ( A e. CC -> E! x e. CC ( ( A + x ) e. RR /\ ( _i x. ( A - x ) ) e. RR ) )
2		riotacl 6044	. . 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 6082	. . . . . 6 |- ( y = A -> ( y + x ) = ( A + x ) )
5	4	eleq1d 2307	. . . . 5 |- ( y = A -> ( ( y + x ) e. RR <-> ( A + x ) e. RR ) )
6		oveq1 6082	. . . . . . 7 |- ( y = A -> ( y - x ) = ( A - x ) )
7	6	oveq2d 6091	. . . . . 6 |- ( y = A -> ( _i x. ( y - x ) ) = ( _i x. ( A - x ) ) )
8	7	eleq1d 2307	. . . . 5 |- ( y = A -> ( ( _i x. ( y - x ) ) e. RR <-> ( _i x. ( A - x ) ) e. RR ) )
9	5, 8	anbi12d 477	. . . 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 6030	. . 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 11585	. . 3 |- * = ( y e. CC |-> ( iota_ x e. CC ( ( y + x ) e. RR /\ ( _i x. ( y - x ) ) e. RR ) ) )
12	10, 11	fvmptg 5775	. 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 425	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 1402   e. wcel 2209   E!wreu 2530   ` cfv 5372   iota_crio 6027  (class class class)co 6075   CCcc 8167   RRcr 8168   _ici 8171   + caddc 8172   x. cmul 8174   - cmin 8487   *ccj 11582

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-ltxr 8355  df-sub 8489  df-neg 8490  df-reap 8893  df-cj 11585

This theorem is referenced by:  cjth  11589  remim  11603