Theorem crre 10622

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
crre	|- ( ( A e. RR /\ B e. RR ) -> ( Re ` ( A + ( _i x. B ) ) ) = A )

Proof of Theorem crre

Step	Hyp	Ref	Expression
1		recn 7746	. . . 4 |- ( A e. RR -> A e. CC )
2		ax-icn 7708	. . . . 5 |- _i e. CC
3		recn 7746	. . . . 5 |- ( B e. RR -> B e. CC )
4		mulcl 7740	. . . . 5 |- ( ( _i e. CC /\ B e. CC ) -> ( _i x. B ) e. CC )
5	2, 3, 4	sylancr 410	. . . 4 |- ( B e. RR -> ( _i x. B ) e. CC )
6		addcl 7738	. . . 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		reval 10614	. . 3 |- ( ( A + ( _i x. B ) ) e. CC -> ( Re ` ( A + ( _i x. B ) ) ) = ( ( ( A + ( _i x. B ) ) + ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) )
9	7, 8	syl 14	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( Re ` ( A + ( _i x. B ) ) ) = ( ( ( A + ( _i x. B ) ) + ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) )
10		cjcl 10613	. . . . . 6 |- ( ( A + ( _i x. B ) ) e. CC -> ( * ` ( A + ( _i x. B ) ) ) e. CC )
11	7, 10	syl 14	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( * ` ( A + ( _i x. B ) ) ) e. CC )
12	7, 11	addcld 7778	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( ( A + ( _i x. B ) ) + ( * ` ( A + ( _i x. B ) ) ) ) e. CC )
13	12	halfcld 8957	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( A + ( _i x. B ) ) + ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) e. CC )
14	1	adantr 274	. . 3 |- ( ( A e. RR /\ B e. RR ) -> A e. CC )
15		recl 10618	. . . . . . 7 |- ( ( A + ( _i x. B ) ) e. CC -> ( Re ` ( A + ( _i x. B ) ) ) e. RR )
16	7, 15	syl 14	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( Re ` ( A + ( _i x. B ) ) ) e. RR )
17	9, 16	eqeltrrd 2215	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( A + ( _i x. B ) ) + ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) e. RR )
18		simpl 108	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> A e. RR )
19	17, 18	resubcld 8136	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( ( A + ( _i x. B ) ) + ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) - A ) e. RR )
20	2	a1i 9	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> _i e. CC )
21	3	adantl 275	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> B e. CC )
22	2, 21, 4	sylancr 410	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( _i x. B ) e. CC )
23	7, 11	subcld 8066	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( ( A + ( _i x. B ) ) - ( * ` ( A + ( _i x. B ) ) ) ) e. CC )
24	23	halfcld 8957	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( A + ( _i x. B ) ) - ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) e. CC )
25	20, 22, 24	subdid 8169	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( _i x. ( ( _i x. B ) - ( ( ( A + ( _i x. B ) ) - ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) ) ) = ( ( _i x. ( _i x. B ) ) - ( _i x. ( ( ( A + ( _i x. B ) ) - ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) ) ) )
26	14, 22, 14	pnpcand 8103	. . . . . . . . . . . . . 14 |- ( ( A e. RR /\ B e. RR ) -> ( ( A + ( _i x. B ) ) - ( A + A ) ) = ( ( _i x. B ) - A ) )
27	22, 14, 22	pnpcan2d 8104	. . . . . . . . . . . . . 14 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( _i x. B ) + ( _i x. B ) ) - ( A + ( _i x. B ) ) ) = ( ( _i x. B ) - A ) )
28	26, 27	eqtr4d 2173	. . . . . . . . . . . . 13 |- ( ( A e. RR /\ B e. RR ) -> ( ( A + ( _i x. B ) ) - ( A + A ) ) = ( ( ( _i x. B ) + ( _i x. B ) ) - ( A + ( _i x. B ) ) ) )
29	28	oveq1d 5782	. . . . . . . . . . . 12 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( A + ( _i x. B ) ) - ( A + A ) ) + ( * ` ( A + ( _i x. B ) ) ) ) = ( ( ( ( _i x. B ) + ( _i x. B ) ) - ( A + ( _i x. B ) ) ) + ( * ` ( A + ( _i x. B ) ) ) ) )
30	14, 14	addcld 7778	. . . . . . . . . . . . 13 |- ( ( A e. RR /\ B e. RR ) -> ( A + A ) e. CC )
31	7, 11, 30	addsubd 8087	. . . . . . . . . . . 12 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( A + ( _i x. B ) ) + ( * ` ( A + ( _i x. B ) ) ) ) - ( A + A ) ) = ( ( ( A + ( _i x. B ) ) - ( A + A ) ) + ( * ` ( A + ( _i x. B ) ) ) ) )
32	22, 22	addcld 7778	. . . . . . . . . . . . 13 |- ( ( A e. RR /\ B e. RR ) -> ( ( _i x. B ) + ( _i x. B ) ) e. CC )
33	32, 7, 11	subsubd 8094	. . . . . . . . . . . 12 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( _i x. B ) + ( _i x. B ) ) - ( ( A + ( _i x. B ) ) - ( * ` ( A + ( _i x. B ) ) ) ) ) = ( ( ( ( _i x. B ) + ( _i x. B ) ) - ( A + ( _i x. B ) ) ) + ( * ` ( A + ( _i x. B ) ) ) ) )
34	29, 31, 33	3eqtr4d 2180	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( A + ( _i x. B ) ) + ( * ` ( A + ( _i x. B ) ) ) ) - ( A + A ) ) = ( ( ( _i x. B ) + ( _i x. B ) ) - ( ( A + ( _i x. B ) ) - ( * ` ( A + ( _i x. B ) ) ) ) ) )
35	14	2timesd 8955	. . . . . . . . . . . 12 |- ( ( A e. RR /\ B e. RR ) -> ( 2 x. A ) = ( A + A ) )
36	35	oveq2d 5783	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( A + ( _i x. B ) ) + ( * ` ( A + ( _i x. B ) ) ) ) - ( 2 x. A ) ) = ( ( ( A + ( _i x. B ) ) + ( * ` ( A + ( _i x. B ) ) ) ) - ( A + A ) ) )
37	22	2timesd 8955	. . . . . . . . . . . 12 |- ( ( A e. RR /\ B e. RR ) -> ( 2 x. ( _i x. B ) ) = ( ( _i x. B ) + ( _i x. B ) ) )
38	37	oveq1d 5782	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR ) -> ( ( 2 x. ( _i x. B ) ) - ( ( A + ( _i x. B ) ) - ( * ` ( A + ( _i x. B ) ) ) ) ) = ( ( ( _i x. B ) + ( _i x. B ) ) - ( ( A + ( _i x. B ) ) - ( * ` ( A + ( _i x. B ) ) ) ) ) )
39	34, 36, 38	3eqtr4d 2180	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( A + ( _i x. B ) ) + ( * ` ( A + ( _i x. B ) ) ) ) - ( 2 x. A ) ) = ( ( 2 x. ( _i x. B ) ) - ( ( A + ( _i x. B ) ) - ( * ` ( A + ( _i x. B ) ) ) ) ) )
40	39	oveq1d 5782	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( ( A + ( _i x. B ) ) + ( * ` ( A + ( _i x. B ) ) ) ) - ( 2 x. A ) ) / 2 ) = ( ( ( 2 x. ( _i x. B ) ) - ( ( A + ( _i x. B ) ) - ( * ` ( A + ( _i x. B ) ) ) ) ) / 2 ) )
41		2cn 8784	. . . . . . . . . . 11 |- 2 e. CC
42		mulcl 7740	. . . . . . . . . . 11 |- ( ( 2 e. CC /\ A e. CC ) -> ( 2 x. A ) e. CC )
43	41, 14, 42	sylancr 410	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR ) -> ( 2 x. A ) e. CC )
44	41	a1i 9	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR ) -> 2 e. CC )
45		2ap0 8806	. . . . . . . . . . 11 |- 2 # 0
46	45	a1i 9	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR ) -> 2 # 0 )
47	12, 43, 44, 46	divsubdirapd 8583	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( ( A + ( _i x. B ) ) + ( * ` ( A + ( _i x. B ) ) ) ) - ( 2 x. A ) ) / 2 ) = ( ( ( ( A + ( _i x. B ) ) + ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) - ( ( 2 x. A ) / 2 ) ) )
48		mulcl 7740	. . . . . . . . . . 11 |- ( ( 2 e. CC /\ ( _i x. B ) e. CC ) -> ( 2 x. ( _i x. B ) ) e. CC )
49	41, 22, 48	sylancr 410	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR ) -> ( 2 x. ( _i x. B ) ) e. CC )
50	49, 23, 44, 46	divsubdirapd 8583	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( 2 x. ( _i x. B ) ) - ( ( A + ( _i x. B ) ) - ( * ` ( A + ( _i x. B ) ) ) ) ) / 2 ) = ( ( ( 2 x. ( _i x. B ) ) / 2 ) - ( ( ( A + ( _i x. B ) ) - ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) ) )
51	40, 47, 50	3eqtr3d 2178	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( ( A + ( _i x. B ) ) + ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) - ( ( 2 x. A ) / 2 ) ) = ( ( ( 2 x. ( _i x. B ) ) / 2 ) - ( ( ( A + ( _i x. B ) ) - ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) ) )
52	14, 44, 46	divcanap3d 8548	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( ( 2 x. A ) / 2 ) = A )
53	52	oveq2d 5783	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( ( A + ( _i x. B ) ) + ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) - ( ( 2 x. A ) / 2 ) ) = ( ( ( ( A + ( _i x. B ) ) + ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) - A ) )
54	22, 44, 46	divcanap3d 8548	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( ( 2 x. ( _i x. B ) ) / 2 ) = ( _i x. B ) )
55	54	oveq1d 5782	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( 2 x. ( _i x. B ) ) / 2 ) - ( ( ( A + ( _i x. B ) ) - ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) ) = ( ( _i x. B ) - ( ( ( A + ( _i x. B ) ) - ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) ) )
56	51, 53, 55	3eqtr3d 2178	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( ( A + ( _i x. B ) ) + ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) - A ) = ( ( _i x. B ) - ( ( ( A + ( _i x. B ) ) - ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) ) )
57	56	oveq2d 5783	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( _i x. ( ( ( ( A + ( _i x. B ) ) + ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) - A ) ) = ( _i x. ( ( _i x. B ) - ( ( ( A + ( _i x. B ) ) - ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) ) ) )
58	20, 20, 21	mulassd 7782	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( ( _i x. _i ) x. B ) = ( _i x. ( _i x. B ) ) )
59	20, 23, 44, 46	divassapd 8579	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( ( _i x. ( ( A + ( _i x. B ) ) - ( * ` ( A + ( _i x. B ) ) ) ) ) / 2 ) = ( _i x. ( ( ( A + ( _i x. B ) ) - ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) ) )
60	58, 59	oveq12d 5785	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( _i x. _i ) x. B ) - ( ( _i x. ( ( A + ( _i x. B ) ) - ( * ` ( A + ( _i x. B ) ) ) ) ) / 2 ) ) = ( ( _i x. ( _i x. B ) ) - ( _i x. ( ( ( A + ( _i x. B ) ) - ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) ) ) )
61	25, 57, 60	3eqtr4d 2180	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( _i x. ( ( ( ( A + ( _i x. B ) ) + ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) - A ) ) = ( ( ( _i x. _i ) x. B ) - ( ( _i x. ( ( A + ( _i x. B ) ) - ( * ` ( A + ( _i x. B ) ) ) ) ) / 2 ) ) )
62		ixi 8338	. . . . . . . 8 |- ( _i x. _i ) = -u 1
63		neg1rr 8819	. . . . . . . 8 |- -u 1 e. RR
64	62, 63	eqeltri 2210	. . . . . . 7 |- ( _i x. _i ) e. RR
65		simpr 109	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> B e. RR )
66		remulcl 7741	. . . . . . 7 |- ( ( ( _i x. _i ) e. RR /\ B e. RR ) -> ( ( _i x. _i ) x. B ) e. RR )
67	64, 65, 66	sylancr 410	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( ( _i x. _i ) x. B ) e. RR )
68		cjth 10611	. . . . . . . . 9 |- ( ( A + ( _i x. B ) ) e. CC -> ( ( ( A + ( _i x. B ) ) + ( * ` ( A + ( _i x. B ) ) ) ) e. RR /\ ( _i x. ( ( A + ( _i x. B ) ) - ( * ` ( A + ( _i x. B ) ) ) ) ) e. RR ) )
69	68	simprd 113	. . . . . . . 8 |- ( ( A + ( _i x. B ) ) e. CC -> ( _i x. ( ( A + ( _i x. B ) ) - ( * ` ( A + ( _i x. B ) ) ) ) ) e. RR )
70	7, 69	syl 14	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( _i x. ( ( A + ( _i x. B ) ) - ( * ` ( A + ( _i x. B ) ) ) ) ) e. RR )
71	70	rehalfcld 8959	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( ( _i x. ( ( A + ( _i x. B ) ) - ( * ` ( A + ( _i x. B ) ) ) ) ) / 2 ) e. RR )
72	67, 71	resubcld 8136	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( _i x. _i ) x. B ) - ( ( _i x. ( ( A + ( _i x. B ) ) - ( * ` ( A + ( _i x. B ) ) ) ) ) / 2 ) ) e. RR )
73	61, 72	eqeltrd 2214	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( _i x. ( ( ( ( A + ( _i x. B ) ) + ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) - A ) ) e. RR )
74		rimul 8340	. . . 4 |- ( ( ( ( ( ( A + ( _i x. B ) ) + ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) - A ) e. RR /\ ( _i x. ( ( ( ( A + ( _i x. B ) ) + ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) - A ) ) e. RR ) -> ( ( ( ( A + ( _i x. B ) ) + ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) - A ) = 0 )
75	19, 73, 74	syl2anc 408	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( ( A + ( _i x. B ) ) + ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) - A ) = 0 )
76	13, 14, 75	subeq0d 8074	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( A + ( _i x. B ) ) + ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) = A )
77	9, 76	eqtrd 2170	1 |- ( ( A e. RR /\ B e. RR ) -> ( Re ` ( A + ( _i x. B ) ) ) = A )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   class class class wbr 3924   ` cfv 5118  (class class class)co 5767   CCcc 7611   RRcr 7612   0cc0 7613   1c1 7614   _ici 7615   + caddc 7616   x. cmul 7618   - cmin 7926   -ucneg 7927   # cap 8336   / cdiv 8425   2c2 8764   *ccj 10604   Recre 10605

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730  ax-pre-mulext 7731

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-po 4213  df-iso 4214  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337  df-div 8426  df-2 8772  df-cj 10607  df-re 10608

This theorem is referenced by:  crim  10623  replim  10624  mulreap  10629  recj  10632  reneg  10633  readd  10634  remullem  10636  rei  10664  crrei  10701  crred  10741  rennim  10767  absreimsq  10832