Theorem crim 11202

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 8060	. . . 4 |- ( A e. RR -> A e. CC )
2		ax-icn 8022	. . . . 5 |- _i e. CC
3		recn 8060	. . . . 5 |- ( B e. RR -> B e. CC )
4		mulcl 8054	. . . . 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 8052	. . . 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 11194	. . 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 9262	. . . . . . 7 |- _i # 0
12		divdirap 8772	. . . . . . . 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 1206	. . . . . . 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 8764	. . . . . . . 8 |- ( ( A e. CC /\ _i e. CC /\ _i # 0 ) -> ( A / _i ) = ( ( 1 / _i ) x. A ) )
17	2, 11, 16	mp3an23 1342	. . . . . . 7 |- ( A e. CC -> ( A / _i ) = ( ( 1 / _i ) x. A ) )
18		irec 10786	. . . . . . . . 9 |- ( 1 / _i ) = -u _i
19	18	oveq1i 5956	. . . . . . . 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 8471	. . . . . . . 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 2242	. . . . . 6 |- ( A e. CC -> ( A / _i ) = ( _i x. -u A ) )
24		divcanap3 8773	. . . . . . 7 |- ( ( B e. CC /\ _i e. CC /\ _i # 0 ) -> ( ( _i x. B ) / _i ) = B )
25	2, 11, 24	mp3an23 1342	. . . . . 6 |- ( B e. CC -> ( ( _i x. B ) / _i ) = B )
26	23, 25	oveqan12d 5965	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A / _i ) + ( ( _i x. B ) / _i ) ) = ( ( _i x. -u A ) + B ) )
27		negcl 8274	. . . . . . 7 |- ( A e. CC -> -u A e. CC )
28		mulcl 8054	. . . . . . 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 8211	. . . . . 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 2243	. . . 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 5582	. 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 8335	. . 3 |- ( A e. RR -> -u A e. RR )
37		crre 11201	. . 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 2244	1 |- ( ( A e. RR /\ B e. RR ) -> ( Im ` ( A + ( _i x. B ) ) ) = B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   class class class wbr 4045   ` cfv 5272  (class class class)co 5946   CCcc 7925   RRcr 7926   0cc0 7927   1c1 7928   _ici 7929   + caddc 7930   x. cmul 7932   -ucneg 8246   # cap 8656   / cdiv 8747   Recre 11184   Imcim 11185

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-mulrcl 8026  ax-addcom 8027  ax-mulcom 8028  ax-addass 8029  ax-mulass 8030  ax-distr 8031  ax-i2m1 8032  ax-0lt1 8033  ax-1rid 8034  ax-0id 8035  ax-rnegex 8036  ax-precex 8037  ax-cnre 8038  ax-pre-ltirr 8039  ax-pre-ltwlin 8040  ax-pre-lttrn 8041  ax-pre-apti 8042  ax-pre-ltadd 8043  ax-pre-mulgt0 8044  ax-pre-mulext 8045

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4046  df-opab 4107  df-mpt 4108  df-id 4341  df-po 4344  df-iso 4345  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-fv 5280  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-pnf 8111  df-mnf 8112  df-xr 8113  df-ltxr 8114  df-le 8115  df-sub 8247  df-neg 8248  df-reap 8650  df-ap 8657  df-div 8748  df-2 9097  df-cj 11186  df-re 11187  df-im 11188

This theorem is referenced by:  replim  11203  reim0  11205  remullem  11215  imcj  11219  imneg  11220  imadd  11221  imi  11244  crimi  11281  crimd  11321  absreimsq  11411  4sqlem4  12748  2sqlem2  15625