Theorem recan 11120

Description: Cancellation law involving the real part of a complex number. (Contributed by NM, 12-May-2005.)

Assertion

Ref	Expression
recan	|- ( ( A e. CC /\ B e. CC ) -> ( A. x e. CC ( Re ` ( x x. A ) ) = ( Re ` ( x x. B ) ) <-> A = B ) )

Distinct variable groups:   x, A   x, B

Proof of Theorem recan

Step	Hyp	Ref	Expression
1		ax-1cn 7906	. . . . 5 |- 1 e. CC
2		oveq1 5884	. . . . . . . 8 |- ( x = 1 -> ( x x. A ) = ( 1 x. A ) )
3	2	fveq2d 5521	. . . . . . 7 |- ( x = 1 -> ( Re ` ( x x. A ) ) = ( Re ` ( 1 x. A ) ) )
4		oveq1 5884	. . . . . . . 8 |- ( x = 1 -> ( x x. B ) = ( 1 x. B ) )
5	4	fveq2d 5521	. . . . . . 7 |- ( x = 1 -> ( Re ` ( x x. B ) ) = ( Re ` ( 1 x. B ) ) )
6	3, 5	eqeq12d 2192	. . . . . 6 |- ( x = 1 -> ( ( Re ` ( x x. A ) ) = ( Re ` ( x x. B ) ) <-> ( Re ` ( 1 x. A ) ) = ( Re ` ( 1 x. B ) ) ) )
7	6	rspcv 2839	. . . . 5 |- ( 1 e. CC -> ( A. x e. CC ( Re ` ( x x. A ) ) = ( Re ` ( x x. B ) ) -> ( Re ` ( 1 x. A ) ) = ( Re ` ( 1 x. B ) ) ) )
8	1, 7	ax-mp 5	. . . 4 |- ( A. x e. CC ( Re ` ( x x. A ) ) = ( Re ` ( x x. B ) ) -> ( Re ` ( 1 x. A ) ) = ( Re ` ( 1 x. B ) ) )
9		negicn 8160	. . . . . 6 |- -u _i e. CC
10		oveq1 5884	. . . . . . . . 9 |- ( x = -u _i -> ( x x. A ) = ( -u _i x. A ) )
11	10	fveq2d 5521	. . . . . . . 8 |- ( x = -u _i -> ( Re ` ( x x. A ) ) = ( Re ` ( -u _i x. A ) ) )
12		oveq1 5884	. . . . . . . . 9 |- ( x = -u _i -> ( x x. B ) = ( -u _i x. B ) )
13	12	fveq2d 5521	. . . . . . . 8 |- ( x = -u _i -> ( Re ` ( x x. B ) ) = ( Re ` ( -u _i x. B ) ) )
14	11, 13	eqeq12d 2192	. . . . . . 7 |- ( x = -u _i -> ( ( Re ` ( x x. A ) ) = ( Re ` ( x x. B ) ) <-> ( Re ` ( -u _i x. A ) ) = ( Re ` ( -u _i x. B ) ) ) )
15	14	rspcv 2839	. . . . . 6 |- ( -u _i e. CC -> ( A. x e. CC ( Re ` ( x x. A ) ) = ( Re ` ( x x. B ) ) -> ( Re ` ( -u _i x. A ) ) = ( Re ` ( -u _i x. B ) ) ) )
16	9, 15	ax-mp 5	. . . . 5 |- ( A. x e. CC ( Re ` ( x x. A ) ) = ( Re ` ( x x. B ) ) -> ( Re ` ( -u _i x. A ) ) = ( Re ` ( -u _i x. B ) ) )
17	16	oveq2d 5893	. . . 4 |- ( A. x e. CC ( Re ` ( x x. A ) ) = ( Re ` ( x x. B ) ) -> ( _i x. ( Re ` ( -u _i x. A ) ) ) = ( _i x. ( Re ` ( -u _i x. B ) ) ) )
18	8, 17	oveq12d 5895	. . 3 |- ( A. x e. CC ( Re ` ( x x. A ) ) = ( Re ` ( x x. B ) ) -> ( ( Re ` ( 1 x. A ) ) + ( _i x. ( Re ` ( -u _i x. A ) ) ) ) = ( ( Re ` ( 1 x. B ) ) + ( _i x. ( Re ` ( -u _i x. B ) ) ) ) )
19		replim 10870	. . . . 5 |- ( A e. CC -> A = ( ( Re ` A ) + ( _i x. ( Im ` A ) ) ) )
20		mullid 7957	. . . . . . . 8 |- ( A e. CC -> ( 1 x. A ) = A )
21	20	eqcomd 2183	. . . . . . 7 |- ( A e. CC -> A = ( 1 x. A ) )
22	21	fveq2d 5521	. . . . . 6 |- ( A e. CC -> ( Re ` A ) = ( Re ` ( 1 x. A ) ) )
23		imre 10862	. . . . . . 7 |- ( A e. CC -> ( Im ` A ) = ( Re ` ( -u _i x. A ) ) )
24	23	oveq2d 5893	. . . . . 6 |- ( A e. CC -> ( _i x. ( Im ` A ) ) = ( _i x. ( Re ` ( -u _i x. A ) ) ) )
25	22, 24	oveq12d 5895	. . . . 5 |- ( A e. CC -> ( ( Re ` A ) + ( _i x. ( Im ` A ) ) ) = ( ( Re ` ( 1 x. A ) ) + ( _i x. ( Re ` ( -u _i x. A ) ) ) ) )
26	19, 25	eqtrd 2210	. . . 4 |- ( A e. CC -> A = ( ( Re ` ( 1 x. A ) ) + ( _i x. ( Re ` ( -u _i x. A ) ) ) ) )
27		replim 10870	. . . . 5 |- ( B e. CC -> B = ( ( Re ` B ) + ( _i x. ( Im ` B ) ) ) )
28		mullid 7957	. . . . . . . 8 |- ( B e. CC -> ( 1 x. B ) = B )
29	28	eqcomd 2183	. . . . . . 7 |- ( B e. CC -> B = ( 1 x. B ) )
30	29	fveq2d 5521	. . . . . 6 |- ( B e. CC -> ( Re ` B ) = ( Re ` ( 1 x. B ) ) )
31		imre 10862	. . . . . . 7 |- ( B e. CC -> ( Im ` B ) = ( Re ` ( -u _i x. B ) ) )
32	31	oveq2d 5893	. . . . . 6 |- ( B e. CC -> ( _i x. ( Im ` B ) ) = ( _i x. ( Re ` ( -u _i x. B ) ) ) )
33	30, 32	oveq12d 5895	. . . . 5 |- ( B e. CC -> ( ( Re ` B ) + ( _i x. ( Im ` B ) ) ) = ( ( Re ` ( 1 x. B ) ) + ( _i x. ( Re ` ( -u _i x. B ) ) ) ) )
34	27, 33	eqtrd 2210	. . . 4 |- ( B e. CC -> B = ( ( Re ` ( 1 x. B ) ) + ( _i x. ( Re ` ( -u _i x. B ) ) ) ) )
35	26, 34	eqeqan12d 2193	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( A = B <-> ( ( Re ` ( 1 x. A ) ) + ( _i x. ( Re ` ( -u _i x. A ) ) ) ) = ( ( Re ` ( 1 x. B ) ) + ( _i x. ( Re ` ( -u _i x. B ) ) ) ) ) )
36	18, 35	imbitrrid 156	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( A. x e. CC ( Re ` ( x x. A ) ) = ( Re ` ( x x. B ) ) -> A = B ) )
37		oveq2 5885	. . . 4 |- ( A = B -> ( x x. A ) = ( x x. B ) )
38	37	fveq2d 5521	. . 3 |- ( A = B -> ( Re ` ( x x. A ) ) = ( Re ` ( x x. B ) ) )
39	38	ralrimivw 2551	. 2 |- ( A = B -> A. x e. CC ( Re ` ( x x. A ) ) = ( Re ` ( x x. B ) ) )
40	36, 39	impbid1 142	1 |- ( ( A e. CC /\ B e. CC ) -> ( A. x e. CC ( Re ` ( x x. A ) ) = ( Re ` ( x x. B ) ) <-> A = B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   A.wral 2455   ` cfv 5218  (class class class)co 5877   CCcc 7811   1c1 7814   _ici 7815   + caddc 7816   x. cmul 7818   -ucneg 8131   Recre 10851   Imcim 10852

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-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931

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-po 4298  df-iso 4299  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-2 8980  df-cj 10853  df-re 10854  df-im 10855

This theorem is referenced by: (None)