Theorem crim 10851

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 7935	. . . 4 |- ( A e. RR -> A e. CC )
2		ax-icn 7897	. . . . 5 |- _i e. CC
3		recn 7935	. . . . 5 |- ( B e. RR -> B e. CC )
4		mulcl 7929	. . . . 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 7927	. . . 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 10843	. . 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 9131	. . . . . . 7 |- _i # 0
12		divdirap 8643	. . . . . . . 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 1203	. . . . . . 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 8635	. . . . . . . 8 |- ( ( A e. CC /\ _i e. CC /\ _i # 0 ) -> ( A / _i ) = ( ( 1 / _i ) x. A ) )
17	2, 11, 16	mp3an23 1329	. . . . . . 7 |- ( A e. CC -> ( A / _i ) = ( ( 1 / _i ) x. A ) )
18		irec 10605	. . . . . . . . 9 |- ( 1 / _i ) = -u _i
19	18	oveq1i 5879	. . . . . . . 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 8344	. . . . . . . 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 2214	. . . . . 6 |- ( A e. CC -> ( A / _i ) = ( _i x. -u A ) )
24		divcanap3 8644	. . . . . . 7 |- ( ( B e. CC /\ _i e. CC /\ _i # 0 ) -> ( ( _i x. B ) / _i ) = B )
25	2, 11, 24	mp3an23 1329	. . . . . 6 |- ( B e. CC -> ( ( _i x. B ) / _i ) = B )
26	23, 25	oveqan12d 5888	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A / _i ) + ( ( _i x. B ) / _i ) ) = ( ( _i x. -u A ) + B ) )
27		negcl 8147	. . . . . . 7 |- ( A e. CC -> -u A e. CC )
28		mulcl 7929	. . . . . . 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 8084	. . . . . 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 2215	. . . 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 5515	. 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 8208	. . 3 |- ( A e. RR -> -u A e. RR )
37		crre 10850	. . 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 2216	1 |- ( ( A e. RR /\ B e. RR ) -> ( Im ` ( A + ( _i x. B ) ) ) = B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   class class class wbr 4000   ` cfv 5212  (class class class)co 5869   CCcc 7800   RRcr 7801   0cc0 7802   1c1 7803   _ici 7804   + caddc 7805   x. cmul 7807   -ucneg 8119   # cap 8528   / cdiv 8618   Recre 10833   Imcim 10834

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-po 4293  df-iso 4294  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-2 8967  df-cj 10835  df-re 10836  df-im 10837

This theorem is referenced by:  replim  10852  reim0  10854  remullem  10864  imcj  10868  imneg  10869  imadd  10870  imi  10893  crimi  10930  crimd  10970  absreimsq  11060  4sqlem4  12373  2sqlem2  14118