Theorem recan 10530

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 7428	. . . . 5 |- 1 e. CC
2		oveq1 5651	. . . . . . . 8 |- ( x = 1 -> ( x x. A ) = ( 1 x. A ) )
3	2	fveq2d 5303	. . . . . . 7 |- ( x = 1 -> ( Re ` ( x x. A ) ) = ( Re ` ( 1 x. A ) ) )
4		oveq1 5651	. . . . . . . 8 |- ( x = 1 -> ( x x. B ) = ( 1 x. B ) )
5	4	fveq2d 5303	. . . . . . 7 |- ( x = 1 -> ( Re ` ( x x. B ) ) = ( Re ` ( 1 x. B ) ) )
6	3, 5	eqeq12d 2102	. . . . . 6 |- ( x = 1 -> ( ( Re ` ( x x. A ) ) = ( Re ` ( x x. B ) ) <-> ( Re ` ( 1 x. A ) ) = ( Re ` ( 1 x. B ) ) ) )
7	6	rspcv 2718	. . . . 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 7	. . . 4 |- ( A. x e. CC ( Re ` ( x x. A ) ) = ( Re ` ( x x. B ) ) -> ( Re ` ( 1 x. A ) ) = ( Re ` ( 1 x. B ) ) )
9		negicn 7673	. . . . . 6 |- -u _i e. CC
10		oveq1 5651	. . . . . . . . 9 |- ( x = -u _i -> ( x x. A ) = ( -u _i x. A ) )
11	10	fveq2d 5303	. . . . . . . 8 |- ( x = -u _i -> ( Re ` ( x x. A ) ) = ( Re ` ( -u _i x. A ) ) )
12		oveq1 5651	. . . . . . . . 9 |- ( x = -u _i -> ( x x. B ) = ( -u _i x. B ) )
13	12	fveq2d 5303	. . . . . . . 8 |- ( x = -u _i -> ( Re ` ( x x. B ) ) = ( Re ` ( -u _i x. B ) ) )
14	11, 13	eqeq12d 2102	. . . . . . 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 2718	. . . . . 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 7	. . . . 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 5660	. . . 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 5662	. . 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 10281	. . . . 5 |- ( A e. CC -> A = ( ( Re ` A ) + ( _i x. ( Im ` A ) ) ) )
20		mulid2 7476	. . . . . . . 8 |- ( A e. CC -> ( 1 x. A ) = A )
21	20	eqcomd 2093	. . . . . . 7 |- ( A e. CC -> A = ( 1 x. A ) )
22	21	fveq2d 5303	. . . . . 6 |- ( A e. CC -> ( Re ` A ) = ( Re ` ( 1 x. A ) ) )
23		imre 10273	. . . . . . 7 |- ( A e. CC -> ( Im ` A ) = ( Re ` ( -u _i x. A ) ) )
24	23	oveq2d 5660	. . . . . 6 |- ( A e. CC -> ( _i x. ( Im ` A ) ) = ( _i x. ( Re ` ( -u _i x. A ) ) ) )
25	22, 24	oveq12d 5662	. . . . 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 2120	. . . 4 |- ( A e. CC -> A = ( ( Re ` ( 1 x. A ) ) + ( _i x. ( Re ` ( -u _i x. A ) ) ) ) )
27		replim 10281	. . . . 5 |- ( B e. CC -> B = ( ( Re ` B ) + ( _i x. ( Im ` B ) ) ) )
28		mulid2 7476	. . . . . . . 8 |- ( B e. CC -> ( 1 x. B ) = B )
29	28	eqcomd 2093	. . . . . . 7 |- ( B e. CC -> B = ( 1 x. B ) )
30	29	fveq2d 5303	. . . . . 6 |- ( B e. CC -> ( Re ` B ) = ( Re ` ( 1 x. B ) ) )
31		imre 10273	. . . . . . 7 |- ( B e. CC -> ( Im ` B ) = ( Re ` ( -u _i x. B ) ) )
32	31	oveq2d 5660	. . . . . 6 |- ( B e. CC -> ( _i x. ( Im ` B ) ) = ( _i x. ( Re ` ( -u _i x. B ) ) ) )
33	30, 32	oveq12d 5662	. . . . 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 2120	. . . 4 |- ( B e. CC -> B = ( ( Re ` ( 1 x. B ) ) + ( _i x. ( Re ` ( -u _i x. B ) ) ) ) )
35	26, 34	eqeqan12d 2103	. . 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	syl5ibr 154	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( A. x e. CC ( Re ` ( x x. A ) ) = ( Re ` ( x x. B ) ) -> A = B ) )
37		oveq2 5652	. . . 4 |- ( A = B -> ( x x. A ) = ( x x. B ) )
38	37	fveq2d 5303	. . 3 |- ( A = B -> ( Re ` ( x x. A ) ) = ( Re ` ( x x. B ) ) )
39	38	ralrimivw 2447	. 2 |- ( A = B -> A. x e. CC ( Re ` ( x x. A ) ) = ( Re ` ( x x. B ) ) )
40	36, 39	impbid1 140	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 102   <-> wb 103   = wceq 1289   e. wcel 1438   A.wral 2359   ` cfv 5010  (class class class)co 5644   CCcc 7338   1c1 7341   _ici 7342   + caddc 7343   x. cmul 7345   -ucneg 7644   Recre 10262   Imcim 10263

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3955  ax-pow 4007  ax-pr 4034  ax-un 4258  ax-setind 4351  ax-cnex 7426  ax-resscn 7427  ax-1cn 7428  ax-1re 7429  ax-icn 7430  ax-addcl 7431  ax-addrcl 7432  ax-mulcl 7433  ax-mulrcl 7434  ax-addcom 7435  ax-mulcom 7436  ax-addass 7437  ax-mulass 7438  ax-distr 7439  ax-i2m1 7440  ax-0lt1 7441  ax-1rid 7442  ax-0id 7443  ax-rnegex 7444  ax-precex 7445  ax-cnre 7446  ax-pre-ltirr 7447  ax-pre-ltwlin 7448  ax-pre-lttrn 7449  ax-pre-apti 7450  ax-pre-ltadd 7451  ax-pre-mulgt0 7452  ax-pre-mulext 7453

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-br 3844  df-opab 3898  df-mpt 3899  df-id 4118  df-po 4121  df-iso 4122  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-dm 4446  df-rn 4447  df-res 4448  df-ima 4449  df-iota 4975  df-fun 5012  df-fn 5013  df-f 5014  df-fv 5018  df-riota 5600  df-ov 5647  df-oprab 5648  df-mpt2 5649  df-pnf 7514  df-mnf 7515  df-xr 7516  df-ltxr 7517  df-le 7518  df-sub 7645  df-neg 7646  df-reap 8042  df-ap 8049  df-div 8130  df-2 8471  df-cj 10264  df-re 10265  df-im 10266

This theorem is referenced by: (None)