Theorem recan 11291

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 7989	. . . . 5 |- 1 e. CC
2		oveq1 5932	. . . . . . . 8 |- ( x = 1 -> ( x x. A ) = ( 1 x. A ) )
3	2	fveq2d 5565	. . . . . . 7 |- ( x = 1 -> ( Re ` ( x x. A ) ) = ( Re ` ( 1 x. A ) ) )
4		oveq1 5932	. . . . . . . 8 |- ( x = 1 -> ( x x. B ) = ( 1 x. B ) )
5	4	fveq2d 5565	. . . . . . 7 |- ( x = 1 -> ( Re ` ( x x. B ) ) = ( Re ` ( 1 x. B ) ) )
6	3, 5	eqeq12d 2211	. . . . . 6 |- ( x = 1 -> ( ( Re ` ( x x. A ) ) = ( Re ` ( x x. B ) ) <-> ( Re ` ( 1 x. A ) ) = ( Re ` ( 1 x. B ) ) ) )
7	6	rspcv 2864	. . . . 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 8244	. . . . . 6 |- -u _i e. CC
10		oveq1 5932	. . . . . . . . 9 |- ( x = -u _i -> ( x x. A ) = ( -u _i x. A ) )
11	10	fveq2d 5565	. . . . . . . 8 |- ( x = -u _i -> ( Re ` ( x x. A ) ) = ( Re ` ( -u _i x. A ) ) )
12		oveq1 5932	. . . . . . . . 9 |- ( x = -u _i -> ( x x. B ) = ( -u _i x. B ) )
13	12	fveq2d 5565	. . . . . . . 8 |- ( x = -u _i -> ( Re ` ( x x. B ) ) = ( Re ` ( -u _i x. B ) ) )
14	11, 13	eqeq12d 2211	. . . . . . 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 2864	. . . . . 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 5941	. . . 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 5943	. . 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 11041	. . . . 5 |- ( A e. CC -> A = ( ( Re ` A ) + ( _i x. ( Im ` A ) ) ) )
20		mullid 8041	. . . . . . . 8 |- ( A e. CC -> ( 1 x. A ) = A )
21	20	eqcomd 2202	. . . . . . 7 |- ( A e. CC -> A = ( 1 x. A ) )
22	21	fveq2d 5565	. . . . . 6 |- ( A e. CC -> ( Re ` A ) = ( Re ` ( 1 x. A ) ) )
23		imre 11033	. . . . . . 7 |- ( A e. CC -> ( Im ` A ) = ( Re ` ( -u _i x. A ) ) )
24	23	oveq2d 5941	. . . . . 6 |- ( A e. CC -> ( _i x. ( Im ` A ) ) = ( _i x. ( Re ` ( -u _i x. A ) ) ) )
25	22, 24	oveq12d 5943	. . . . 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 2229	. . . 4 |- ( A e. CC -> A = ( ( Re ` ( 1 x. A ) ) + ( _i x. ( Re ` ( -u _i x. A ) ) ) ) )
27		replim 11041	. . . . 5 |- ( B e. CC -> B = ( ( Re ` B ) + ( _i x. ( Im ` B ) ) ) )
28		mullid 8041	. . . . . . . 8 |- ( B e. CC -> ( 1 x. B ) = B )
29	28	eqcomd 2202	. . . . . . 7 |- ( B e. CC -> B = ( 1 x. B ) )
30	29	fveq2d 5565	. . . . . 6 |- ( B e. CC -> ( Re ` B ) = ( Re ` ( 1 x. B ) ) )
31		imre 11033	. . . . . . 7 |- ( B e. CC -> ( Im ` B ) = ( Re ` ( -u _i x. B ) ) )
32	31	oveq2d 5941	. . . . . 6 |- ( B e. CC -> ( _i x. ( Im ` B ) ) = ( _i x. ( Re ` ( -u _i x. B ) ) ) )
33	30, 32	oveq12d 5943	. . . . 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 2229	. . . 4 |- ( B e. CC -> B = ( ( Re ` ( 1 x. B ) ) + ( _i x. ( Re ` ( -u _i x. B ) ) ) ) )
35	26, 34	eqeqan12d 2212	. . 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 5933	. . . 4 |- ( A = B -> ( x x. A ) = ( x x. B ) )
38	37	fveq2d 5565	. . 3 |- ( A = B -> ( Re ` ( x x. A ) ) = ( Re ` ( x x. B ) ) )
39	38	ralrimivw 2571	. 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 1364   e. wcel 2167   A.wral 2475   ` cfv 5259  (class class class)co 5925   CCcc 7894   1c1 7897   _ici 7898   + caddc 7899   x. cmul 7901   -ucneg 8215   Recre 11022   Imcim 11023

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-po 4332  df-iso 4333  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-2 9066  df-cj 11024  df-re 11025  df-im 11026

This theorem is referenced by: (None)