Theorem crre 11622

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 8312	. . . 4 |- ( A e. RR -> A e. CC )
2		ax-icn 8274	. . . . 5 |- _i e. CC
3		recn 8312	. . . . 5 |- ( B e. RR -> B e. CC )
4		mulcl 8306	. . . . 5 |- ( ( _i e. CC /\ B e. CC ) -> ( _i x. B ) e. CC )
5	2, 3, 4	sylancr 418	. . . 4 |- ( B e. RR -> ( _i x. B ) e. CC )
6		addcl 8304	. . . 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		reval 11614	. . 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 11613	. . . . . 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 8345	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( ( A + ( _i x. B ) ) + ( * ` ( A + ( _i x. B ) ) ) ) e. CC )
13	12	halfcld 9550	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( A + ( _i x. B ) ) + ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) e. CC )
14	1	adantr 276	. . 3 |- ( ( A e. RR /\ B e. RR ) -> A e. CC )
15		recl 11618	. . . . . . 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 2316	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( A + ( _i x. B ) ) + ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) e. RR )
18		simpl 109	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> A e. RR )
19	17, 18	resubcld 8708	. . . 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 277	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> B e. CC )
22	2, 21, 4	sylancr 418	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( _i x. B ) e. CC )
23	7, 11	subcld 8637	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( ( A + ( _i x. B ) ) - ( * ` ( A + ( _i x. B ) ) ) ) e. CC )
24	23	halfcld 9550	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( A + ( _i x. B ) ) - ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) e. CC )
25	20, 22, 24	subdid 8741	. . . . . 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 8674	. . . . . . . . . . . . . 14 |- ( ( A e. RR /\ B e. RR ) -> ( ( A + ( _i x. B ) ) - ( A + A ) ) = ( ( _i x. B ) - A ) )
27	22, 14, 22	pnpcan2d 8675	. . . . . . . . . . . . . 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 2274	. . . . . . . . . . . . 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 6100	. . . . . . . . . . . 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 8345	. . . . . . . . . . . . 13 |- ( ( A e. RR /\ B e. RR ) -> ( A + A ) e. CC )
31	7, 11, 30	addsubd 8658	. . . . . . . . . . . 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 8345	. . . . . . . . . . . . 13 |- ( ( A e. RR /\ B e. RR ) -> ( ( _i x. B ) + ( _i x. B ) ) e. CC )
33	32, 7, 11	subsubd 8665	. . . . . . . . . . . 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 2281	. . . . . . . . . . 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 9548	. . . . . . . . . . . 12 |- ( ( A e. RR /\ B e. RR ) -> ( 2 x. A ) = ( A + A ) )
36	35	oveq2d 6101	. . . . . . . . . . 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 9548	. . . . . . . . . . . 12 |- ( ( A e. RR /\ B e. RR ) -> ( 2 x. ( _i x. B ) ) = ( ( _i x. B ) + ( _i x. B ) ) )
38	37	oveq1d 6100	. . . . . . . . . . 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 2281	. . . . . . . . . 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 6100	. . . . . . . . 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 9375	. . . . . . . . . . 11 |- 2 e. CC
42		mulcl 8306	. . . . . . . . . . 11 |- ( ( 2 e. CC /\ A e. CC ) -> ( 2 x. A ) e. CC )
43	41, 14, 42	sylancr 418	. . . . . . . . . 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 9397	. . . . . . . . . . 11 |- 2 # 0
46	45	a1i 9	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR ) -> 2 # 0 )
47	12, 43, 44, 46	divsubdirapd 9160	. . . . . . . . 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 8306	. . . . . . . . . . 11 |- ( ( 2 e. CC /\ ( _i x. B ) e. CC ) -> ( 2 x. ( _i x. B ) ) e. CC )
49	41, 22, 48	sylancr 418	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR ) -> ( 2 x. ( _i x. B ) ) e. CC )
50	49, 23, 44, 46	divsubdirapd 9160	. . . . . . . . 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 2279	. . . . . . . 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 9125	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( ( 2 x. A ) / 2 ) = A )
53	52	oveq2d 6101	. . . . . . . 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 9125	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( ( 2 x. ( _i x. B ) ) / 2 ) = ( _i x. B ) )
55	54	oveq1d 6100	. . . . . . . 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 2279	. . . . . . 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 6101	. . . . . 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 8349	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( ( _i x. _i ) x. B ) = ( _i x. ( _i x. B ) ) )
59	20, 23, 44, 46	divassapd 9156	. . . . . . 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 6103	. . . . . 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 2281	. . . . 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 8911	. . . . . . . 8 |- ( _i x. _i ) = -u 1
63		neg1rr 9410	. . . . . . . 8 |- -u 1 e. RR
64	62, 63	eqeltri 2311	. . . . . . 7 |- ( _i x. _i ) e. RR
65		simpr 110	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> B e. RR )
66		remulcl 8307	. . . . . . 7 |- ( ( ( _i x. _i ) e. RR /\ B e. RR ) -> ( ( _i x. _i ) x. B ) e. RR )
67	64, 65, 66	sylancr 418	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( ( _i x. _i ) x. B ) e. RR )
68		cjth 11611	. . . . . . . . 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 114	. . . . . . . 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 9552	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( ( _i x. ( ( A + ( _i x. B ) ) - ( * ` ( A + ( _i x. B ) ) ) ) ) / 2 ) e. RR )
72	67, 71	resubcld 8708	. . . . 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 2315	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( _i x. ( ( ( ( A + ( _i x. B ) ) + ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) - A ) ) e. RR )
74		rimul 8913	. . . 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 415	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( ( A + ( _i x. B ) ) + ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) - A ) = 0 )
76	13, 14, 75	subeq0d 8645	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( A + ( _i x. B ) ) + ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) = A )
77	9, 76	eqtrd 2271	1 |- ( ( A e. RR /\ B e. RR ) -> ( Re ` ( A + ( _i x. B ) ) ) = A )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   CCcc 8177   RRcr 8178   0cc0 8179   1c1 8180   _ici 8181   + caddc 8182   x. cmul 8184   - cmin 8497   -ucneg 8498   # cap 8909   / cdiv 9002   2c2 9355   *ccj 11604   Recre 11605

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-po 4441  df-iso 4442  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-2 9363  df-cj 11607  df-re 11608

This theorem is used by:  crim  11623  replim  11624  mulreap  11629  recj  11632  reneg  11633  readd  11634  remullem  11636  rei  11665  crrei  11702  crred  11742  rennim  11768  absreimsq  11833  4sqlem4  13171  2sqlem2  16234