Theorem crim 9745

Description: The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 12-May-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)

Assertion

Ref	Expression
crim	|- ( ( A e. RR /\ B e. RR ) -> ( Im ` ( A + ( _i x. B ) ) ) = B )

Proof of Theorem crim

Step	Hyp	Ref	Expression
1		recn 7106	. . . 4 |- ( A e. RR -> A e. CC )
2		ax-icn 7071	. . . . 5 |- _i e. CC
3		recn 7106	. . . . 5 |- ( B e. RR -> B e. CC )
4		mulcl 7100	. . . . 5 |- ( ( _i e. CC /\ B e. CC ) -> ( _i x. B ) e. CC )
5	2, 3, 4	sylancr 405	. . . 4 |- ( B e. RR -> ( _i x. B ) e. CC )
6		addcl 7098	. . . 4 |- ( ( A e. CC /\ ( _i x. B ) e. CC ) -> ( A + ( _i x. B ) ) e. CC )
7	1, 5, 6	syl2an 283	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( A + ( _i x. B ) ) e. CC )
8		imval 9737	. . 3 |- ( ( A + ( _i x. B ) ) e. CC -> ( Im ` ( A + ( _i x. B ) ) ) = ( Re ` ( ( A + ( _i x. B ) ) / _i ) ) )
9	7, 8	syl 14	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( Im ` ( A + ( _i x. B ) ) ) = ( Re ` ( ( A + ( _i x. B ) ) / _i ) ) )
10	2, 4	mpan 414	. . . . . 6 |- ( B e. CC -> ( _i x. B ) e. CC )
11		iap0 8254	. . . . . . 7 |- _i # 0
12		divdirap 7785	. . . . . . . 8 |- ( ( A e. CC /\ ( _i x. B ) e. CC /\ ( _i e. CC /\ _i # 0 ) ) -> ( ( A + ( _i x. B ) ) / _i ) = ( ( A / _i ) + ( ( _i x. B ) / _i ) ) )
13	12	3expa 1138	. . . . . . 7 |- ( ( ( A e. CC /\ ( _i x. B ) e. CC ) /\ ( _i e. CC /\ _i # 0 ) ) -> ( ( A + ( _i x. B ) ) / _i ) = ( ( A / _i ) + ( ( _i x. B ) / _i ) ) )
14	2, 11, 13	mpanr12 429	. . . . . 6 |- ( ( A e. CC /\ ( _i x. B ) e. CC ) -> ( ( A + ( _i x. B ) ) / _i ) = ( ( A / _i ) + ( ( _i x. B ) / _i ) ) )
15	10, 14	sylan2 280	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + ( _i x. B ) ) / _i ) = ( ( A / _i ) + ( ( _i x. B ) / _i ) ) )
16		divrecap2 7777	. . . . . . . 8 |- ( ( A e. CC /\ _i e. CC /\ _i # 0 ) -> ( A / _i ) = ( ( 1 / _i ) x. A ) )
17	2, 11, 16	mp3an23 1260	. . . . . . 7 |- ( A e. CC -> ( A / _i ) = ( ( 1 / _i ) x. A ) )
18		irec 9574	. . . . . . . . 9 |- ( 1 / _i ) = -u _i
19	18	oveq1i 5542	. . . . . . . 8 |- ( ( 1 / _i ) x. A ) = ( -u _i x. A )
20	19	a1i 9	. . . . . . 7 |- ( A e. CC -> ( ( 1 / _i ) x. A ) = ( -u _i x. A ) )
21		mulneg12 7501	. . . . . . . 8 |- ( ( _i e. CC /\ A e. CC ) -> ( -u _i x. A ) = ( _i x. -u A ) )
22	2, 21	mpan 414	. . . . . . 7 |- ( A e. CC -> ( -u _i x. A ) = ( _i x. -u A ) )
23	17, 20, 22	3eqtrd 2117	. . . . . 6 |- ( A e. CC -> ( A / _i ) = ( _i x. -u A ) )
24		divcanap3 7786	. . . . . . 7 |- ( ( B e. CC /\ _i e. CC /\ _i # 0 ) -> ( ( _i x. B ) / _i ) = B )
25	2, 11, 24	mp3an23 1260	. . . . . 6 |- ( B e. CC -> ( ( _i x. B ) / _i ) = B )
26	23, 25	oveqan12d 5551	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A / _i ) + ( ( _i x. B ) / _i ) ) = ( ( _i x. -u A ) + B ) )
27		negcl 7308	. . . . . . 7 |- ( A e. CC -> -u A e. CC )
28		mulcl 7100	. . . . . . 7 |- ( ( _i e. CC /\ -u A e. CC ) -> ( _i x. -u A ) e. CC )
29	2, 27, 28	sylancr 405	. . . . . 6 |- ( A e. CC -> ( _i x. -u A ) e. CC )
30		addcom 7245	. . . . . 6 |- ( ( ( _i x. -u A ) e. CC /\ B e. CC ) -> ( ( _i x. -u A ) + B ) = ( B + ( _i x. -u A ) ) )
31	29, 30	sylan 277	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( _i x. -u A ) + B ) = ( B + ( _i x. -u A ) ) )
32	15, 26, 31	3eqtrrd 2118	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( B + ( _i x. -u A ) ) = ( ( A + ( _i x. B ) ) / _i ) )
33	1, 3, 32	syl2an 283	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( B + ( _i x. -u A ) ) = ( ( A + ( _i x. B ) ) / _i ) )
34	33	fveq2d 5202	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( Re ` ( B + ( _i x. -u A ) ) ) = ( Re ` ( ( A + ( _i x. B ) ) / _i ) ) )
35		id 19	. . 3 |- ( B e. RR -> B e. RR )
36		renegcl 7369	. . 3 |- ( A e. RR -> -u A e. RR )
37		crre 9744	. . 3 |- ( ( B e. RR /\ -u A e. RR ) -> ( Re ` ( B + ( _i x. -u A ) ) ) = B )
38	35, 36, 37	syl2anr 284	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( Re ` ( B + ( _i x. -u A ) ) ) = B )
39	9, 34, 38	3eqtr2d 2119	1 |- ( ( A e. RR /\ B e. RR ) -> ( Im ` ( A + ( _i x. B ) ) ) = B )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   class class class wbr 3785   ` cfv 4922  (class class class)co 5532   CCcc 6979   RRcr 6980   0cc0 6981   1c1 6982   _ici 6983   + caddc 6984   x. cmul 6986   -ucneg 7280   # cap 7681   / cdiv 7760   Recre 9727   Imcim 9728

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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-mulrcl 7075  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-precex 7086  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092  ax-pre-mulgt0 7093  ax-pre-mulext 7094

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-po 4051  df-iso 4052  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-reap 7675  df-ap 7682  df-div 7761  df-2 8098  df-cj 9729  df-re 9730  df-im 9731

This theorem is referenced by:  replim  9746  reim0  9748  remullem  9758  imcj  9762  imneg  9763  imadd  9764  imi  9787  crimi  9824  crimd  9864  absreimsq  9953