Theorem recan 11253

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 7965	. . . . 5 |- 1 e. CC
2		oveq1 5925	. . . . . . . 8 |- ( x = 1 -> ( x x. A ) = ( 1 x. A ) )
3	2	fveq2d 5558	. . . . . . 7 |- ( x = 1 -> ( Re ` ( x x. A ) ) = ( Re ` ( 1 x. A ) ) )
4		oveq1 5925	. . . . . . . 8 |- ( x = 1 -> ( x x. B ) = ( 1 x. B ) )
5	4	fveq2d 5558	. . . . . . 7 |- ( x = 1 -> ( Re ` ( x x. B ) ) = ( Re ` ( 1 x. B ) ) )
6	3, 5	eqeq12d 2208	. . . . . 6 |- ( x = 1 -> ( ( Re ` ( x x. A ) ) = ( Re ` ( x x. B ) ) <-> ( Re ` ( 1 x. A ) ) = ( Re ` ( 1 x. B ) ) ) )
7	6	rspcv 2860	. . . . 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 8220	. . . . . 6 |- -u _i e. CC
10		oveq1 5925	. . . . . . . . 9 |- ( x = -u _i -> ( x x. A ) = ( -u _i x. A ) )
11	10	fveq2d 5558	. . . . . . . 8 |- ( x = -u _i -> ( Re ` ( x x. A ) ) = ( Re ` ( -u _i x. A ) ) )
12		oveq1 5925	. . . . . . . . 9 |- ( x = -u _i -> ( x x. B ) = ( -u _i x. B ) )
13	12	fveq2d 5558	. . . . . . . 8 |- ( x = -u _i -> ( Re ` ( x x. B ) ) = ( Re ` ( -u _i x. B ) ) )
14	11, 13	eqeq12d 2208	. . . . . . 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 2860	. . . . . 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 5934	. . . 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 5936	. . 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 11003	. . . . 5 |- ( A e. CC -> A = ( ( Re ` A ) + ( _i x. ( Im ` A ) ) ) )
20		mullid 8017	. . . . . . . 8 |- ( A e. CC -> ( 1 x. A ) = A )
21	20	eqcomd 2199	. . . . . . 7 |- ( A e. CC -> A = ( 1 x. A ) )
22	21	fveq2d 5558	. . . . . 6 |- ( A e. CC -> ( Re ` A ) = ( Re ` ( 1 x. A ) ) )
23		imre 10995	. . . . . . 7 |- ( A e. CC -> ( Im ` A ) = ( Re ` ( -u _i x. A ) ) )
24	23	oveq2d 5934	. . . . . 6 |- ( A e. CC -> ( _i x. ( Im ` A ) ) = ( _i x. ( Re ` ( -u _i x. A ) ) ) )
25	22, 24	oveq12d 5936	. . . . 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 2226	. . . 4 |- ( A e. CC -> A = ( ( Re ` ( 1 x. A ) ) + ( _i x. ( Re ` ( -u _i x. A ) ) ) ) )
27		replim 11003	. . . . 5 |- ( B e. CC -> B = ( ( Re ` B ) + ( _i x. ( Im ` B ) ) ) )
28		mullid 8017	. . . . . . . 8 |- ( B e. CC -> ( 1 x. B ) = B )
29	28	eqcomd 2199	. . . . . . 7 |- ( B e. CC -> B = ( 1 x. B ) )
30	29	fveq2d 5558	. . . . . 6 |- ( B e. CC -> ( Re ` B ) = ( Re ` ( 1 x. B ) ) )
31		imre 10995	. . . . . . 7 |- ( B e. CC -> ( Im ` B ) = ( Re ` ( -u _i x. B ) ) )
32	31	oveq2d 5934	. . . . . 6 |- ( B e. CC -> ( _i x. ( Im ` B ) ) = ( _i x. ( Re ` ( -u _i x. B ) ) ) )
33	30, 32	oveq12d 5936	. . . . 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 2226	. . . 4 |- ( B e. CC -> B = ( ( Re ` ( 1 x. B ) ) + ( _i x. ( Re ` ( -u _i x. B ) ) ) ) )
35	26, 34	eqeqan12d 2209	. . 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 5926	. . . 4 |- ( A = B -> ( x x. A ) = ( x x. B ) )
38	37	fveq2d 5558	. . 3 |- ( A = B -> ( Re ` ( x x. A ) ) = ( Re ` ( x x. B ) ) )
39	38	ralrimivw 2568	. 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 2164   A.wral 2472   ` cfv 5254  (class class class)co 5918   CCcc 7870   1c1 7873   _ici 7874   + caddc 7875   x. cmul 7877   -ucneg 8191   Recre 10984   Imcim 10985

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-po 4327  df-iso 4328  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-2 9041  df-cj 10986  df-re 10987  df-im 10988

This theorem is referenced by: (None)