Theorem crim 11419

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 8165	. . . 4 |- ( A e. RR -> A e. CC )
2		ax-icn 8127	. . . . 5 |- _i e. CC
3		recn 8165	. . . . 5 |- ( B e. RR -> B e. CC )
4		mulcl 8159	. . . . 5 |- ( ( _i e. CC /\ B e. CC ) -> ( _i x. B ) e. CC )
5	2, 3, 4	sylancr 414	. . . 4 |- ( B e. RR -> ( _i x. B ) e. CC )
6		addcl 8157	. . . 4 |- ( ( A e. CC /\ ( _i x. B ) e. CC ) -> ( A + ( _i x. B ) ) e. CC )
7	1, 5, 6	syl2an 289	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( A + ( _i x. B ) ) e. CC )
8		imval 11411	. . 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 424	. . . . . 6 |- ( B e. CC -> ( _i x. B ) e. CC )
11		iap0 9367	. . . . . . 7 |- _i # 0
12		divdirap 8877	. . . . . . . 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 1229	. . . . . . 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 439	. . . . . 6 |- ( ( A e. CC /\ ( _i x. B ) e. CC ) -> ( ( A + ( _i x. B ) ) / _i ) = ( ( A / _i ) + ( ( _i x. B ) / _i ) ) )
15	10, 14	sylan2 286	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + ( _i x. B ) ) / _i ) = ( ( A / _i ) + ( ( _i x. B ) / _i ) ) )
16		divrecap2 8869	. . . . . . . 8 |- ( ( A e. CC /\ _i e. CC /\ _i # 0 ) -> ( A / _i ) = ( ( 1 / _i ) x. A ) )
17	2, 11, 16	mp3an23 1365	. . . . . . 7 |- ( A e. CC -> ( A / _i ) = ( ( 1 / _i ) x. A ) )
18		irec 10901	. . . . . . . . 9 |- ( 1 / _i ) = -u _i
19	18	oveq1i 6028	. . . . . . . 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 8576	. . . . . . . 8 |- ( ( _i e. CC /\ A e. CC ) -> ( -u _i x. A ) = ( _i x. -u A ) )
22	2, 21	mpan 424	. . . . . . 7 |- ( A e. CC -> ( -u _i x. A ) = ( _i x. -u A ) )
23	17, 20, 22	3eqtrd 2268	. . . . . 6 |- ( A e. CC -> ( A / _i ) = ( _i x. -u A ) )
24		divcanap3 8878	. . . . . . 7 |- ( ( B e. CC /\ _i e. CC /\ _i # 0 ) -> ( ( _i x. B ) / _i ) = B )
25	2, 11, 24	mp3an23 1365	. . . . . 6 |- ( B e. CC -> ( ( _i x. B ) / _i ) = B )
26	23, 25	oveqan12d 6037	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A / _i ) + ( ( _i x. B ) / _i ) ) = ( ( _i x. -u A ) + B ) )
27		negcl 8379	. . . . . . 7 |- ( A e. CC -> -u A e. CC )
28		mulcl 8159	. . . . . . 7 |- ( ( _i e. CC /\ -u A e. CC ) -> ( _i x. -u A ) e. CC )
29	2, 27, 28	sylancr 414	. . . . . 6 |- ( A e. CC -> ( _i x. -u A ) e. CC )
30		addcom 8316	. . . . . 6 |- ( ( ( _i x. -u A ) e. CC /\ B e. CC ) -> ( ( _i x. -u A ) + B ) = ( B + ( _i x. -u A ) ) )
31	29, 30	sylan 283	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( _i x. -u A ) + B ) = ( B + ( _i x. -u A ) ) )
32	15, 26, 31	3eqtrrd 2269	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( B + ( _i x. -u A ) ) = ( ( A + ( _i x. B ) ) / _i ) )
33	1, 3, 32	syl2an 289	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( B + ( _i x. -u A ) ) = ( ( A + ( _i x. B ) ) / _i ) )
34	33	fveq2d 5643	. 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 8440	. . 3 |- ( A e. RR -> -u A e. RR )
37		crre 11418	. . 3 |- ( ( B e. RR /\ -u A e. RR ) -> ( Re ` ( B + ( _i x. -u A ) ) ) = B )
38	35, 36, 37	syl2anr 290	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( Re ` ( B + ( _i x. -u A ) ) ) = B )
39	9, 34, 38	3eqtr2d 2270	1 |- ( ( A e. RR /\ B e. RR ) -> ( Im ` ( A + ( _i x. B ) ) ) = B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   class class class wbr 4088   ` cfv 5326  (class class class)co 6018   CCcc 8030   RRcr 8031   0cc0 8032   1c1 8033   _ici 8034   + caddc 8035   x. cmul 8037   -ucneg 8351   # cap 8761   / cdiv 8852   Recre 11401   Imcim 11402

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-po 4393  df-iso 4394  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-2 9202  df-cj 11403  df-re 11404  df-im 11405

This theorem is referenced by:  replim  11420  reim0  11422  remullem  11432  imcj  11436  imneg  11437  imadd  11438  imi  11461  crimi  11498  crimd  11538  absreimsq  11628  4sqlem4  12966  2sqlem2  15846