Theorem cjth 10857

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 8920	. . . 4 |- ( A e. CC -> E! x e. CC ( ( A + x ) e. RR /\ ( _i x. ( A - x ) ) e. RR ) )
2		riotasbc 5848	. . . 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 10856	. . . 4 |- ( A e. CC -> ( * ` A ) = ( iota_ x e. CC ( ( A + x ) e. RR /\ ( _i x. ( A - x ) ) e. RR ) ) )
5	4	sbceq1d 2969	. . 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 5847	. . . . 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 2254	. . 3 |- ( A e. CC -> ( * ` A ) e. CC )
10		oveq2 5885	. . . . . 6 |- ( x = ( * ` A ) -> ( A + x ) = ( A + ( * ` A ) ) )
11	10	eleq1d 2246	. . . . 5 |- ( x = ( * ` A ) -> ( ( A + x ) e. RR <-> ( A + ( * ` A ) ) e. RR ) )
12		oveq2 5885	. . . . . . 7 |- ( x = ( * ` A ) -> ( A - x ) = ( A - ( * ` A ) ) )
13	12	oveq2d 5893	. . . . . 6 |- ( x = ( * ` A ) -> ( _i x. ( A - x ) ) = ( _i x. ( A - ( * ` A ) ) ) )
14	13	eleq1d 2246	. . . . 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 2997	. . 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 1353   e. wcel 2148   E!wreu 2457   [.wsbc 2964   ` cfv 5218   iota_crio 5832  (class class class)co 5877   CCcc 7811   RRcr 7812   _ici 7815   + caddc 7816   x. cmul 7818   - cmin 8130   *ccj 10850

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-ltxr 7999  df-sub 8132  df-neg 8133  df-reap 8534  df-cj 10853

This theorem is referenced by:  recl  10864  crre  10868