Theorem cjth 11357

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 9108	. . . 4 |- ( A e. CC -> E! x e. CC ( ( A + x ) e. RR /\ ( _i x. ( A - x ) ) e. RR ) )
2		riotasbc 5971	. . . 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 11356	. . . 4 |- ( A e. CC -> ( * ` A ) = ( iota_ x e. CC ( ( A + x ) e. RR /\ ( _i x. ( A - x ) ) e. RR ) ) )
5	4	sbceq1d 3033	. . 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 5970	. . . . 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 2306	. . 3 |- ( A e. CC -> ( * ` A ) e. CC )
10		oveq2 6009	. . . . . 6 |- ( x = ( * ` A ) -> ( A + x ) = ( A + ( * ` A ) ) )
11	10	eleq1d 2298	. . . . 5 |- ( x = ( * ` A ) -> ( ( A + x ) e. RR <-> ( A + ( * ` A ) ) e. RR ) )
12		oveq2 6009	. . . . . . 7 |- ( x = ( * ` A ) -> ( A - x ) = ( A - ( * ` A ) ) )
13	12	oveq2d 6017	. . . . . 6 |- ( x = ( * ` A ) -> ( _i x. ( A - x ) ) = ( _i x. ( A - ( * ` A ) ) ) )
14	13	eleq1d 2298	. . . . 5 |- ( x = ( * ` A ) -> ( ( _i x. ( A - x ) ) e. RR <-> ( _i x. ( A - ( * ` A ) ) ) e. RR ) )
15	11, 14	anbi12d 473	. . . 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 3061	. . 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 ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   E!wreu 2510   [.wsbc 3028   ` cfv 5318   iota_crio 5953  (class class class)co 6001   CCcc 7997   RRcr 7998   _ici 8001   + caddc 8002   x. cmul 8004   - cmin 8317   *ccj 11350

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-pnf 8183  df-mnf 8184  df-ltxr 8186  df-sub 8319  df-neg 8320  df-reap 8722  df-cj 11353

This theorem is referenced by:  recl  11364  crre  11368