Theorem cjth 10746

Description: The defining property of the complex conjugate. (Contributed by Mario Carneiro, 6-Nov-2013.)

Assertion

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

Proof of Theorem cjth

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cju 8832	. . . 4 |- ( A e. CC -> E! x e. CC ( ( A + x ) e. RR /\ ( _i x. ( A - x ) ) e. RR ) )
2		riotasbc 5795	. . . 4 |- ( 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 ) ) / x ]. ( ( A + x ) e. RR /\ ( _i x. ( A - x ) ) e. RR ) )
3	1, 2	syl 14	. . 3 |- ( A e. CC -> [. ( iota_ x e. CC ( ( A + x ) e. RR /\ ( _i x. ( A - x ) ) e. RR ) ) / x ]. ( ( A + x ) e. RR /\ ( _i x. ( A - x ) ) e. RR ) )
4		cjval 10745	. . . 4 |- ( A e. CC -> ( * ` A ) = ( iota_ x e. CC ( ( A + x ) e. RR /\ ( _i x. ( A - x ) ) e. RR ) ) )
5	4	sbceq1d 2942	. . 3 |- ( A e. CC -> ( [. ( * ` A ) / x ]. ( ( A + x ) e. RR /\ ( _i x. ( A - x ) ) e. RR ) <-> [. ( iota_ x e. CC ( ( A + x ) e. RR /\ ( _i x. ( A - x ) ) e. RR ) ) / x ]. ( ( A + x ) e. RR /\ ( _i x. ( A - x ) ) e. RR ) ) )
6	3, 5	mpbird 166	. 2 |- ( A e. CC -> [. ( * ` A ) / x ]. ( ( A + x ) e. RR /\ ( _i x. ( A - x ) ) e. RR ) )
7		riotacl 5794	. . . . 5 |- ( 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 )
8	1, 7	syl 14	. . . 4 |- ( A e. CC -> ( iota_ x e. CC ( ( A + x ) e. RR /\ ( _i x. ( A - x ) ) e. RR ) ) e. CC )
9	4, 8	eqeltrd 2234	. . 3 |- ( A e. CC -> ( * ` A ) e. CC )
10		oveq2 5832	. . . . . 6 |- ( x = ( * ` A ) -> ( A + x ) = ( A + ( * ` A ) ) )
11	10	eleq1d 2226	. . . . 5 |- ( x = ( * ` A ) -> ( ( A + x ) e. RR <-> ( A + ( * ` A ) ) e. RR ) )
12		oveq2 5832	. . . . . . 7 |- ( x = ( * ` A ) -> ( A - x ) = ( A - ( * ` A ) ) )
13	12	oveq2d 5840	. . . . . 6 |- ( x = ( * ` A ) -> ( _i x. ( A - x ) ) = ( _i x. ( A - ( * ` A ) ) ) )
14	13	eleq1d 2226	. . . . 5 |- ( x = ( * ` A ) -> ( ( _i x. ( A - x ) ) e. RR <-> ( _i x. ( A - ( * ` A ) ) ) e. RR ) )
15	11, 14	anbi12d 465	. . . 4 |- ( x = ( * ` A ) -> ( ( ( A + x ) e. RR /\ ( _i x. ( A - x ) ) e. RR ) <-> ( ( A + ( * ` A ) ) e. RR /\ ( _i x. ( A - ( * ` A ) ) ) e. RR ) ) )
16	15	sbcieg 2969	. . 3 |- ( ( * ` A ) e. CC -> ( [. ( * ` A ) / x ]. ( ( A + x ) e. RR /\ ( _i x. ( A - x ) ) e. RR ) <-> ( ( A + ( * ` A ) ) e. RR /\ ( _i x. ( A - ( * ` A ) ) ) e. RR ) ) )
17	9, 16	syl 14	. 2 |- ( A e. CC -> ( [. ( * ` A ) / x ]. ( ( A + x ) e. RR /\ ( _i x. ( A - x ) ) e. RR ) <-> ( ( A + ( * ` A ) ) e. RR /\ ( _i x. ( A - ( * ` A ) ) ) e. RR ) ) )
18	6, 17	mpbid 146	1 |- ( A e. CC -> ( ( A + ( * ` A ) ) e. RR /\ ( _i x. ( A - ( * ` A ) ) ) e. RR ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1335   e. wcel 2128   E!wreu 2437   [.wsbc 2937   ` cfv 5170   iota_crio 5779  (class class class)co 5824   CCcc 7730   RRcr 7731   _ici 7734   + caddc 7735   x. cmul 7737   - cmin 8046   *ccj 10739

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-mulrcl 7831  ax-addcom 7832  ax-mulcom 7833  ax-addass 7834  ax-mulass 7835  ax-distr 7836  ax-i2m1 7837  ax-0lt1 7838  ax-1rid 7839  ax-0id 7840  ax-rnegex 7841  ax-precex 7842  ax-cnre 7843  ax-pre-ltirr 7844  ax-pre-lttrn 7846  ax-pre-apti 7847  ax-pre-ltadd 7848  ax-pre-mulgt0 7849

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-iota 5135  df-fun 5172  df-fv 5178  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-pnf 7914  df-mnf 7915  df-ltxr 7917  df-sub 8048  df-neg 8049  df-reap 8450  df-cj 10742

This theorem is referenced by:  recl  10753  crre  10757