Theorem crre 9945

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 7220	. . . 4 |- ( A e. RR -> A e. CC )
2		ax-icn 7185	. . . . 5 |- _i e. CC
3		recn 7220	. . . . 5 |- ( B e. RR -> B e. CC )
4		mulcl 7214	. . . . 5 |- ( ( _i e. CC /\ B e. CC ) -> ( _i x. B ) e. CC )
5	2, 3, 4	sylancr 405	. . . 4 |- ( B e. RR -> ( _i x. B ) e. CC )
6		addcl 7212	. . . 4 |- ( ( A e. CC /\ ( _i x. B ) e. CC ) -> ( A + ( _i x. B ) ) e. CC )
7	1, 5, 6	syl2an 283	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( A + ( _i x. B ) ) e. CC )
8		reval 9937	. . 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 9936	. . . . . 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 7252	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( ( A + ( _i x. B ) ) + ( * ` ( A + ( _i x. B ) ) ) ) e. CC )
13	12	halfcld 8394	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( A + ( _i x. B ) ) + ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) e. CC )
14	1	adantr 270	. . 3 |- ( ( A e. RR /\ B e. RR ) -> A e. CC )
15		recl 9941	. . . . . . 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 2160	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( A + ( _i x. B ) ) + ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) e. RR )
18		simpl 107	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> A e. RR )
19	17, 18	resubcld 7604	. . . 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 271	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> B e. CC )
22	2, 21, 4	sylancr 405	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( _i x. B ) e. CC )
23	7, 11	subcld 7538	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( ( A + ( _i x. B ) ) - ( * ` ( A + ( _i x. B ) ) ) ) e. CC )
24	23	halfcld 8394	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( A + ( _i x. B ) ) - ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) e. CC )
25	20, 22, 24	subdid 7637	. . . . . 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 7575	. . . . . . . . . . . . . 14 |- ( ( A e. RR /\ B e. RR ) -> ( ( A + ( _i x. B ) ) - ( A + A ) ) = ( ( _i x. B ) - A ) )
27	22, 14, 22	pnpcan2d 7576	. . . . . . . . . . . . . 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 2118	. . . . . . . . . . . . 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 5578	. . . . . . . . . . . 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 7252	. . . . . . . . . . . . 13 |- ( ( A e. RR /\ B e. RR ) -> ( A + A ) e. CC )
31	7, 11, 30	addsubd 7559	. . . . . . . . . . . 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 7252	. . . . . . . . . . . . 13 |- ( ( A e. RR /\ B e. RR ) -> ( ( _i x. B ) + ( _i x. B ) ) e. CC )
33	32, 7, 11	subsubd 7566	. . . . . . . . . . . 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 2125	. . . . . . . . . . 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 8392	. . . . . . . . . . . 12 |- ( ( A e. RR /\ B e. RR ) -> ( 2 x. A ) = ( A + A ) )
36	35	oveq2d 5579	. . . . . . . . . . 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 8392	. . . . . . . . . . . 12 |- ( ( A e. RR /\ B e. RR ) -> ( 2 x. ( _i x. B ) ) = ( ( _i x. B ) + ( _i x. B ) ) )
38	37	oveq1d 5578	. . . . . . . . . . 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 2125	. . . . . . . . . 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 5578	. . . . . . . . 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 8229	. . . . . . . . . . 11 |- 2 e. CC
42		mulcl 7214	. . . . . . . . . . 11 |- ( ( 2 e. CC /\ A e. CC ) -> ( 2 x. A ) e. CC )
43	41, 14, 42	sylancr 405	. . . . . . . . . 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 8251	. . . . . . . . . . 11 |- 2 # 0
46	45	a1i 9	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR ) -> 2 # 0 )
47	12, 43, 44, 46	divsubdirapd 8035	. . . . . . . . 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 7214	. . . . . . . . . . 11 |- ( ( 2 e. CC /\ ( _i x. B ) e. CC ) -> ( 2 x. ( _i x. B ) ) e. CC )
49	41, 22, 48	sylancr 405	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR ) -> ( 2 x. ( _i x. B ) ) e. CC )
50	49, 23, 44, 46	divsubdirapd 8035	. . . . . . . . 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 2123	. . . . . . . 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 8001	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( ( 2 x. A ) / 2 ) = A )
53	52	oveq2d 5579	. . . . . . . 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 8001	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( ( 2 x. ( _i x. B ) ) / 2 ) = ( _i x. B ) )
55	54	oveq1d 5578	. . . . . . . 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 2123	. . . . . . 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 5579	. . . . . 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 7256	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( ( _i x. _i ) x. B ) = ( _i x. ( _i x. B ) ) )
59	20, 23, 44, 46	divassapd 8031	. . . . . . 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 5581	. . . . . 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 2125	. . . . 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 7802	. . . . . . . 8 |- ( _i x. _i ) = -u 1
63		neg1rr 8264	. . . . . . . 8 |- -u 1 e. RR
64	62, 63	eqeltri 2155	. . . . . . 7 |- ( _i x. _i ) e. RR
65		simpr 108	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> B e. RR )
66		remulcl 7215	. . . . . . 7 |- ( ( ( _i x. _i ) e. RR /\ B e. RR ) -> ( ( _i x. _i ) x. B ) e. RR )
67	64, 65, 66	sylancr 405	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( ( _i x. _i ) x. B ) e. RR )
68		cjth 9934	. . . . . . . . 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 112	. . . . . . . 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 8396	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( ( _i x. ( ( A + ( _i x. B ) ) - ( * ` ( A + ( _i x. B ) ) ) ) ) / 2 ) e. RR )
72	67, 71	resubcld 7604	. . . . 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 2159	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( _i x. ( ( ( ( A + ( _i x. B ) ) + ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) - A ) ) e. RR )
74		rimul 7804	. . . 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 403	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( ( A + ( _i x. B ) ) + ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) - A ) = 0 )
76	13, 14, 75	subeq0d 7546	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( A + ( _i x. B ) ) + ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) = A )
77	9, 76	eqtrd 2115	1 |- ( ( A e. RR /\ B e. RR ) -> ( Re ` ( A + ( _i x. B ) ) ) = A )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1285   e. wcel 1434   class class class wbr 3805   ` cfv 4952  (class class class)co 5563   CCcc 7093   RRcr 7094   0cc0 7095   1c1 7096   _ici 7097   + caddc 7098   x. cmul 7100   - cmin 7398   -ucneg 7399   # cap 7800   / cdiv 7879   2c2 8208   *ccj 9927   Recre 9928

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-mulrcl 7189  ax-addcom 7190  ax-mulcom 7191  ax-addass 7192  ax-mulass 7193  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-1rid 7197  ax-0id 7198  ax-rnegex 7199  ax-precex 7200  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-apti 7205  ax-pre-ltadd 7206  ax-pre-mulgt0 7207  ax-pre-mulext 7208

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2825  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-mpt 3861  df-id 4076  df-po 4079  df-iso 4080  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-reap 7794  df-ap 7801  df-div 7880  df-2 8217  df-cj 9930  df-re 9931

This theorem is referenced by:  crim  9946  replim  9947  mulreap  9952  recj  9955  reneg  9956  readd  9957  remullem  9959  rei  9987  crrei  10024  crred  10064  rennim  10089  absreimsq  10154