Theorem recan 11875

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 8272	. . . . 5 |- 1 e. CC
2		oveq1 6092	. . . . . . . 8 |- ( x = 1 -> ( x x. A ) = ( 1 x. A ) )
3	2	fveq2d 5699	. . . . . . 7 |- ( x = 1 -> ( Re ` ( x x. A ) ) = ( Re ` ( 1 x. A ) ) )
4		oveq1 6092	. . . . . . . 8 |- ( x = 1 -> ( x x. B ) = ( 1 x. B ) )
5	4	fveq2d 5699	. . . . . . 7 |- ( x = 1 -> ( Re ` ( x x. B ) ) = ( Re ` ( 1 x. B ) ) )
6	3, 5	eqeq12d 2253	. . . . . 6 |- ( x = 1 -> ( ( Re ` ( x x. A ) ) = ( Re ` ( x x. B ) ) <-> ( Re ` ( 1 x. A ) ) = ( Re ` ( 1 x. B ) ) ) )
7	6	rspcv 2925	. . . . 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 8527	. . . . . 6 |- -u _i e. CC
10		oveq1 6092	. . . . . . . . 9 |- ( x = -u _i -> ( x x. A ) = ( -u _i x. A ) )
11	10	fveq2d 5699	. . . . . . . 8 |- ( x = -u _i -> ( Re ` ( x x. A ) ) = ( Re ` ( -u _i x. A ) ) )
12		oveq1 6092	. . . . . . . . 9 |- ( x = -u _i -> ( x x. B ) = ( -u _i x. B ) )
13	12	fveq2d 5699	. . . . . . . 8 |- ( x = -u _i -> ( Re ` ( x x. B ) ) = ( Re ` ( -u _i x. B ) ) )
14	11, 13	eqeq12d 2253	. . . . . . 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 2925	. . . . . 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 6101	. . . 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 6103	. . 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 11624	. . . . 5 |- ( A e. CC -> A = ( ( Re ` A ) + ( _i x. ( Im ` A ) ) ) )
20		mullid 8324	. . . . . . . 8 |- ( A e. CC -> ( 1 x. A ) = A )
21	20	eqcomd 2244	. . . . . . 7 |- ( A e. CC -> A = ( 1 x. A ) )
22	21	fveq2d 5699	. . . . . 6 |- ( A e. CC -> ( Re ` A ) = ( Re ` ( 1 x. A ) ) )
23		imre 11616	. . . . . . 7 |- ( A e. CC -> ( Im ` A ) = ( Re ` ( -u _i x. A ) ) )
24	23	oveq2d 6101	. . . . . 6 |- ( A e. CC -> ( _i x. ( Im ` A ) ) = ( _i x. ( Re ` ( -u _i x. A ) ) ) )
25	22, 24	oveq12d 6103	. . . . 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 2271	. . . 4 |- ( A e. CC -> A = ( ( Re ` ( 1 x. A ) ) + ( _i x. ( Re ` ( -u _i x. A ) ) ) ) )
27		replim 11624	. . . . 5 |- ( B e. CC -> B = ( ( Re ` B ) + ( _i x. ( Im ` B ) ) ) )
28		mullid 8324	. . . . . . . 8 |- ( B e. CC -> ( 1 x. B ) = B )
29	28	eqcomd 2244	. . . . . . 7 |- ( B e. CC -> B = ( 1 x. B ) )
30	29	fveq2d 5699	. . . . . 6 |- ( B e. CC -> ( Re ` B ) = ( Re ` ( 1 x. B ) ) )
31		imre 11616	. . . . . . 7 |- ( B e. CC -> ( Im ` B ) = ( Re ` ( -u _i x. B ) ) )
32	31	oveq2d 6101	. . . . . 6 |- ( B e. CC -> ( _i x. ( Im ` B ) ) = ( _i x. ( Re ` ( -u _i x. B ) ) ) )
33	30, 32	oveq12d 6103	. . . . 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 2271	. . . 4 |- ( B e. CC -> B = ( ( Re ` ( 1 x. B ) ) + ( _i x. ( Re ` ( -u _i x. B ) ) ) ) )
35	26, 34	eqeqan12d 2254	. . 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 6093	. . . 4 |- ( A = B -> ( x x. A ) = ( x x. B ) )
38	37	fveq2d 5699	. . 3 |- ( A = B -> ( Re ` ( x x. A ) ) = ( Re ` ( x x. B ) ) )
39	38	ralrimivw 2624	. 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 ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   ` cfv 5377  (class class class)co 6085   CCcc 8177   1c1 8180   _ici 8181   + caddc 8182   x. cmul 8184   -ucneg 8498   Recre 11605   Imcim 11606

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-po 4441  df-iso 4442  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-2 9363  df-cj 11607  df-re 11608  df-im 11609

This theorem is used by: (None)