Theorem crim 10822

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 7907	. . . 4 |- ( A e. RR -> A e. CC )
2		ax-icn 7869	. . . . 5 |- _i e. CC
3		recn 7907	. . . . 5 |- ( B e. RR -> B e. CC )
4		mulcl 7901	. . . . 5 |- ( ( _i e. CC /\ B e. CC ) -> ( _i x. B ) e. CC )
5	2, 3, 4	sylancr 412	. . . 4 |- ( B e. RR -> ( _i x. B ) e. CC )
6		addcl 7899	. . . 4 |- ( ( A e. CC /\ ( _i x. B ) e. CC ) -> ( A + ( _i x. B ) ) e. CC )
7	1, 5, 6	syl2an 287	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( A + ( _i x. B ) ) e. CC )
8		imval 10814	. . 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 422	. . . . . 6 |- ( B e. CC -> ( _i x. B ) e. CC )
11		iap0 9101	. . . . . . 7 |- _i # 0
12		divdirap 8614	. . . . . . . 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 1198	. . . . . . 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 437	. . . . . 6 |- ( ( A e. CC /\ ( _i x. B ) e. CC ) -> ( ( A + ( _i x. B ) ) / _i ) = ( ( A / _i ) + ( ( _i x. B ) / _i ) ) )
15	10, 14	sylan2 284	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + ( _i x. B ) ) / _i ) = ( ( A / _i ) + ( ( _i x. B ) / _i ) ) )
16		divrecap2 8606	. . . . . . . 8 |- ( ( A e. CC /\ _i e. CC /\ _i # 0 ) -> ( A / _i ) = ( ( 1 / _i ) x. A ) )
17	2, 11, 16	mp3an23 1324	. . . . . . 7 |- ( A e. CC -> ( A / _i ) = ( ( 1 / _i ) x. A ) )
18		irec 10575	. . . . . . . . 9 |- ( 1 / _i ) = -u _i
19	18	oveq1i 5863	. . . . . . . 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 8316	. . . . . . . 8 |- ( ( _i e. CC /\ A e. CC ) -> ( -u _i x. A ) = ( _i x. -u A ) )
22	2, 21	mpan 422	. . . . . . 7 |- ( A e. CC -> ( -u _i x. A ) = ( _i x. -u A ) )
23	17, 20, 22	3eqtrd 2207	. . . . . 6 |- ( A e. CC -> ( A / _i ) = ( _i x. -u A ) )
24		divcanap3 8615	. . . . . . 7 |- ( ( B e. CC /\ _i e. CC /\ _i # 0 ) -> ( ( _i x. B ) / _i ) = B )
25	2, 11, 24	mp3an23 1324	. . . . . 6 |- ( B e. CC -> ( ( _i x. B ) / _i ) = B )
26	23, 25	oveqan12d 5872	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A / _i ) + ( ( _i x. B ) / _i ) ) = ( ( _i x. -u A ) + B ) )
27		negcl 8119	. . . . . . 7 |- ( A e. CC -> -u A e. CC )
28		mulcl 7901	. . . . . . 7 |- ( ( _i e. CC /\ -u A e. CC ) -> ( _i x. -u A ) e. CC )
29	2, 27, 28	sylancr 412	. . . . . 6 |- ( A e. CC -> ( _i x. -u A ) e. CC )
30		addcom 8056	. . . . . 6 |- ( ( ( _i x. -u A ) e. CC /\ B e. CC ) -> ( ( _i x. -u A ) + B ) = ( B + ( _i x. -u A ) ) )
31	29, 30	sylan 281	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( _i x. -u A ) + B ) = ( B + ( _i x. -u A ) ) )
32	15, 26, 31	3eqtrrd 2208	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( B + ( _i x. -u A ) ) = ( ( A + ( _i x. B ) ) / _i ) )
33	1, 3, 32	syl2an 287	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( B + ( _i x. -u A ) ) = ( ( A + ( _i x. B ) ) / _i ) )
34	33	fveq2d 5500	. 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 8180	. . 3 |- ( A e. RR -> -u A e. RR )
37		crre 10821	. . 3 |- ( ( B e. RR /\ -u A e. RR ) -> ( Re ` ( B + ( _i x. -u A ) ) ) = B )
38	35, 36, 37	syl2anr 288	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( Re ` ( B + ( _i x. -u A ) ) ) = B )
39	9, 34, 38	3eqtr2d 2209	1 |- ( ( A e. RR /\ B e. RR ) -> ( Im ` ( A + ( _i x. B ) ) ) = B )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   CCcc 7772   RRcr 7773   0cc0 7774   1c1 7775   _ici 7776   + caddc 7777   x. cmul 7779   -ucneg 8091   # cap 8500   / cdiv 8589   Recre 10804   Imcim 10805

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-po 4281  df-iso 4282  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-2 8937  df-cj 10806  df-re 10807  df-im 10808

This theorem is referenced by:  replim  10823  reim0  10825  remullem  10835  imcj  10839  imneg  10840  imadd  10841  imi  10864  crimi  10901  crimd  10941  absreimsq  11031  4sqlem4  12344  2sqlem2  13745