Theorem cjth 11611

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 9291	. . . 4 |- ( A e. CC -> E! x e. CC ( ( A + x ) e. RR /\ ( _i x. ( A - x ) ) e. RR ) )
2		riotasbc 6055	. . . 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 11610	. . . 4 |- ( A e. CC -> ( * ` A ) = ( iota_ x e. CC ( ( A + x ) e. RR /\ ( _i x. ( A - x ) ) e. RR ) ) )
5	4	sbceq1d 3056	. . 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 167	. 2 |- ( A e. CC -> [. ( * ` A ) / x ]. ( ( A + x ) e. RR /\ ( _i x. ( A - x ) ) e. RR ) )
7		riotacl 6054	. . . . 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 2315	. . 3 |- ( A e. CC -> ( * ` A ) e. CC )
10		oveq2 6093	. . . . . 6 |- ( x = ( * ` A ) -> ( A + x ) = ( A + ( * ` A ) ) )
11	10	eleq1d 2307	. . . . 5 |- ( x = ( * ` A ) -> ( ( A + x ) e. RR <-> ( A + ( * ` A ) ) e. RR ) )
12		oveq2 6093	. . . . . . 7 |- ( x = ( * ` A ) -> ( A - x ) = ( A - ( * ` A ) ) )
13	12	oveq2d 6101	. . . . . 6 |- ( x = ( * ` A ) -> ( _i x. ( A - x ) ) = ( _i x. ( A - ( * ` A ) ) ) )
14	13	eleq1d 2307	. . . . 5 |- ( x = ( * ` A ) -> ( ( _i x. ( A - x ) ) e. RR <-> ( _i x. ( A - ( * ` A ) ) ) e. RR ) )
15	11, 14	anbi12d 477	. . . 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 3084	. . 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 147	1 |- ( A e. CC -> ( ( A + ( * ` A ) ) e. RR /\ ( _i x. ( A - ( * ` A ) ) ) e. RR ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   E!wreu 2530   [.wsbc 3051   ` cfv 5377   iota_crio 6037  (class class class)co 6085   CCcc 8177   RRcr 8178   _ici 8181   + caddc 8182   x. cmul 8184   - cmin 8497   *ccj 11604

This proof depends on 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 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296

This proof 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 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-sub 8499  df-neg 8500  df-reap 8903  df-cj 11607

This theorem is used by:  recl  11618  crre  11622