Theorem crre 9678

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 7071	. . . 4 |- ( A e. RR -> A e. CC )
2		ax-icn 7036	. . . . 5 |- _i e. CC
3		recn 7071	. . . . 5 |- ( B e. RR -> B e. CC )
4		mulcl 7065	. . . . 5 |- ( ( _i e. CC /\ B e. CC ) -> ( _i x. B ) e. CC )
5	2, 3, 4	sylancr 399	. . . 4 |- ( B e. RR -> ( _i x. B ) e. CC )
6		addcl 7063	. . . 4 |- ( ( A e. CC /\ ( _i x. B ) e. CC ) -> ( A + ( _i x. B ) ) e. CC )
7	1, 5, 6	syl2an 277	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( A + ( _i x. B ) ) e. CC )
8		reval 9670	. . 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 9669	. . . . . 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 7103	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( ( A + ( _i x. B ) ) + ( * ` ( A + ( _i x. B ) ) ) ) e. CC )
13	12	halfcld 8225	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( A + ( _i x. B ) ) + ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) e. CC )
14	1	adantr 265	. . 3 |- ( ( A e. RR /\ B e. RR ) -> A e. CC )
15		recl 9674	. . . . . . 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 2131	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( A + ( _i x. B ) ) + ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) e. RR )
18		simpl 106	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> A e. RR )
19	17, 18	resubcld 7450	. . . 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 266	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> B e. CC )
22	2, 21, 4	sylancr 399	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( _i x. B ) e. CC )
23	7, 11	subcld 7384	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( ( A + ( _i x. B ) ) - ( * ` ( A + ( _i x. B ) ) ) ) e. CC )
24	23	halfcld 8225	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( A + ( _i x. B ) ) - ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) e. CC )
25	20, 22, 24	subdid 7482	. . . . . 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 7421	. . . . . . . . . . . . . 14 |- ( ( A e. RR /\ B e. RR ) -> ( ( A + ( _i x. B ) ) - ( A + A ) ) = ( ( _i x. B ) - A ) )
27	22, 14, 22	pnpcan2d 7422	. . . . . . . . . . . . . 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 2091	. . . . . . . . . . . . 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 5554	. . . . . . . . . . . 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 7103	. . . . . . . . . . . . 13 |- ( ( A e. RR /\ B e. RR ) -> ( A + A ) e. CC )
31	7, 11, 30	addsubd 7405	. . . . . . . . . . . 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 7103	. . . . . . . . . . . . 13 |- ( ( A e. RR /\ B e. RR ) -> ( ( _i x. B ) + ( _i x. B ) ) e. CC )
33	32, 7, 11	subsubd 7412	. . . . . . . . . . . 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 2098	. . . . . . . . . . 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 8223	. . . . . . . . . . . 12 |- ( ( A e. RR /\ B e. RR ) -> ( 2 x. A ) = ( A + A ) )
36	35	oveq2d 5555	. . . . . . . . . . 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 8223	. . . . . . . . . . . 12 |- ( ( A e. RR /\ B e. RR ) -> ( 2 x. ( _i x. B ) ) = ( ( _i x. B ) + ( _i x. B ) ) )
38	37	oveq1d 5554	. . . . . . . . . . 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 2098	. . . . . . . . . 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 5554	. . . . . . . . 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 8060	. . . . . . . . . . 11 |- 2 e. CC
42		mulcl 7065	. . . . . . . . . . 11 |- ( ( 2 e. CC /\ A e. CC ) -> ( 2 x. A ) e. CC )
43	41, 14, 42	sylancr 399	. . . . . . . . . 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 8082	. . . . . . . . . . 11 |- 2 # 0
46	45	a1i 9	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR ) -> 2 # 0 )
47	12, 43, 44, 46	divsubdirapd 7878	. . . . . . . . 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 7065	. . . . . . . . . . 11 |- ( ( 2 e. CC /\ ( _i x. B ) e. CC ) -> ( 2 x. ( _i x. B ) ) e. CC )
49	41, 22, 48	sylancr 399	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR ) -> ( 2 x. ( _i x. B ) ) e. CC )
50	49, 23, 44, 46	divsubdirapd 7878	. . . . . . . . 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 2096	. . . . . . . 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 7844	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( ( 2 x. A ) / 2 ) = A )
53	52	oveq2d 5555	. . . . . . . 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 7844	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( ( 2 x. ( _i x. B ) ) / 2 ) = ( _i x. B ) )
55	54	oveq1d 5554	. . . . . . . 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 2096	. . . . . . 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 5555	. . . . . 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 7107	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( ( _i x. _i ) x. B ) = ( _i x. ( _i x. B ) ) )
59	20, 23, 44, 46	divassapd 7874	. . . . . . 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 5557	. . . . . 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 2098	. . . . 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 7647	. . . . . . . 8 |- ( _i x. _i ) = -u 1
63		neg1rr 8095	. . . . . . . 8 |- -u 1 e. RR
64	62, 63	eqeltri 2126	. . . . . . 7 |- ( _i x. _i ) e. RR
65		simpr 107	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> B e. RR )
66		remulcl 7066	. . . . . . 7 |- ( ( ( _i x. _i ) e. RR /\ B e. RR ) -> ( ( _i x. _i ) x. B ) e. RR )
67	64, 65, 66	sylancr 399	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( ( _i x. _i ) x. B ) e. RR )
68		cjth 9667	. . . . . . . . 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 111	. . . . . . . 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 8227	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( ( _i x. ( ( A + ( _i x. B ) ) - ( * ` ( A + ( _i x. B ) ) ) ) ) / 2 ) e. RR )
72	67, 71	resubcld 7450	. . . . 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 2130	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( _i x. ( ( ( ( A + ( _i x. B ) ) + ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) - A ) ) e. RR )
74		rimul 7649	. . . 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 397	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( ( A + ( _i x. B ) ) + ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) - A ) = 0 )
76	13, 14, 75	subeq0d 7392	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( A + ( _i x. B ) ) + ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) = A )
77	9, 76	eqtrd 2088	1 |- ( ( A e. RR /\ B e. RR ) -> ( Re ` ( A + ( _i x. B ) ) ) = A )

Syntax hints:   -> wi 4   /\ wa 101   = wceq 1259   e. wcel 1409   class class class wbr 3791   ` cfv 4929  (class class class)co 5539   CCcc 6944   RRcr 6945   0cc0 6946   1c1 6947   _ici 6948   + caddc 6949   x. cmul 6951   - cmin 7244   -ucneg 7245   # cap 7645   / cdiv 7724   2c2 8039   *ccj 9660   Recre 9661

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338  ax-cnex 7032  ax-resscn 7033  ax-1cn 7034  ax-1re 7035  ax-icn 7036  ax-addcl 7037  ax-addrcl 7038  ax-mulcl 7039  ax-mulrcl 7040  ax-addcom 7041  ax-mulcom 7042  ax-addass 7043  ax-mulass 7044  ax-distr 7045  ax-i2m1 7046  ax-1rid 7048  ax-0id 7049  ax-rnegex 7050  ax-precex 7051  ax-cnre 7052  ax-pre-ltirr 7053  ax-pre-ltwlin 7054  ax-pre-lttrn 7055  ax-pre-apti 7056  ax-pre-ltadd 7057  ax-pre-mulgt0 7058  ax-pre-mulext 7059

This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-reu 2330  df-rmo 2331  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-eprel 4053  df-id 4057  df-po 4060  df-iso 4061  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-riota 5495  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-1o 6031  df-2o 6032  df-oadd 6035  df-omul 6036  df-er 6136  df-ec 6138  df-qs 6142  df-ni 6459  df-pli 6460  df-mi 6461  df-lti 6462  df-plpq 6499  df-mpq 6500  df-enq 6502  df-nqqs 6503  df-plqqs 6504  df-mqqs 6505  df-1nqqs 6506  df-rq 6507  df-ltnqqs 6508  df-enq0 6579  df-nq0 6580  df-0nq0 6581  df-plq0 6582  df-mq0 6583  df-inp 6621  df-i1p 6622  df-iplp 6623  df-iltp 6625  df-enr 6868  df-nr 6869  df-ltr 6872  df-0r 6873  df-1r 6874  df-0 6953  df-1 6954  df-r 6956  df-lt 6959  df-pnf 7120  df-mnf 7121  df-xr 7122  df-ltxr 7123  df-le 7124  df-sub 7246  df-neg 7247  df-reap 7639  df-ap 7646  df-div 7725  df-2 8048  df-cj 9663  df-re 9664

This theorem is referenced by:  crim  9679  replim  9680  mulreap  9685  recj  9688  reneg  9689  readd  9690  remullem  9692  rei  9720  crrei  9757  crred  9797  rennim  9821  absreimsq  9886