Theorem crre 11567

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 8276	. . . 4 |- ( A e. RR -> A e. CC )
2		ax-icn 8238	. . . . 5 |- _i e. CC
3		recn 8276	. . . . 5 |- ( B e. RR -> B e. CC )
4		mulcl 8270	. . . . 5 |- ( ( _i e. CC /\ B e. CC ) -> ( _i x. B ) e. CC )
5	2, 3, 4	sylancr 414	. . . 4 |- ( B e. RR -> ( _i x. B ) e. CC )
6		addcl 8268	. . . 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 11559	. . 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 11558	. . . . . 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 8309	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( ( A + ( _i x. B ) ) + ( * ` ( A + ( _i x. B ) ) ) ) e. CC )
13	12	halfcld 9500	. . 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 11563	. . . . . . 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 2312	. . . . 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 8671	. . . 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 414	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( _i x. B ) e. CC )
23	7, 11	subcld 8600	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( ( A + ( _i x. B ) ) - ( * ` ( A + ( _i x. B ) ) ) ) e. CC )
24	23	halfcld 9500	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( A + ( _i x. B ) ) - ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) e. CC )
25	20, 22, 24	subdid 8704	. . . . . 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 8637	. . . . . . . . . . . . . 14 |- ( ( A e. RR /\ B e. RR ) -> ( ( A + ( _i x. B ) ) - ( A + A ) ) = ( ( _i x. B ) - A ) )
27	22, 14, 22	pnpcan2d 8638	. . . . . . . . . . . . . 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 2270	. . . . . . . . . . . . 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 6073	. . . . . . . . . . . 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 8309	. . . . . . . . . . . . 13 |- ( ( A e. RR /\ B e. RR ) -> ( A + A ) e. CC )
31	7, 11, 30	addsubd 8621	. . . . . . . . . . . 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 8309	. . . . . . . . . . . . 13 |- ( ( A e. RR /\ B e. RR ) -> ( ( _i x. B ) + ( _i x. B ) ) e. CC )
33	32, 7, 11	subsubd 8628	. . . . . . . . . . . 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 2277	. . . . . . . . . . 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 9498	. . . . . . . . . . . 12 |- ( ( A e. RR /\ B e. RR ) -> ( 2 x. A ) = ( A + A ) )
36	35	oveq2d 6074	. . . . . . . . . . 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 9498	. . . . . . . . . . . 12 |- ( ( A e. RR /\ B e. RR ) -> ( 2 x. ( _i x. B ) ) = ( ( _i x. B ) + ( _i x. B ) ) )
38	37	oveq1d 6073	. . . . . . . . . . 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 2277	. . . . . . . . . 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 6073	. . . . . . . . 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 9325	. . . . . . . . . . 11 |- 2 e. CC
42		mulcl 8270	. . . . . . . . . . 11 |- ( ( 2 e. CC /\ A e. CC ) -> ( 2 x. A ) e. CC )
43	41, 14, 42	sylancr 414	. . . . . . . . . 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 9347	. . . . . . . . . . 11 |- 2 # 0
46	45	a1i 9	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR ) -> 2 # 0 )
47	12, 43, 44, 46	divsubdirapd 9121	. . . . . . . . 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 8270	. . . . . . . . . . 11 |- ( ( 2 e. CC /\ ( _i x. B ) e. CC ) -> ( 2 x. ( _i x. B ) ) e. CC )
49	41, 22, 48	sylancr 414	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR ) -> ( 2 x. ( _i x. B ) ) e. CC )
50	49, 23, 44, 46	divsubdirapd 9121	. . . . . . . . 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 2275	. . . . . . . 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 9086	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( ( 2 x. A ) / 2 ) = A )
53	52	oveq2d 6074	. . . . . . . 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 9086	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( ( 2 x. ( _i x. B ) ) / 2 ) = ( _i x. B ) )
55	54	oveq1d 6073	. . . . . . . 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 2275	. . . . . . 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 6074	. . . . . 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 8313	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( ( _i x. _i ) x. B ) = ( _i x. ( _i x. B ) ) )
59	20, 23, 44, 46	divassapd 9117	. . . . . . 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 6076	. . . . . 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 2277	. . . . 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 8874	. . . . . . . 8 |- ( _i x. _i ) = -u 1
63		neg1rr 9360	. . . . . . . 8 |- -u 1 e. RR
64	62, 63	eqeltri 2307	. . . . . . 7 |- ( _i x. _i ) e. RR
65		simpr 110	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> B e. RR )
66		remulcl 8271	. . . . . . 7 |- ( ( ( _i x. _i ) e. RR /\ B e. RR ) -> ( ( _i x. _i ) x. B ) e. RR )
67	64, 65, 66	sylancr 414	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( ( _i x. _i ) x. B ) e. RR )
68		cjth 11556	. . . . . . . . 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 9502	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( ( _i x. ( ( A + ( _i x. B ) ) - ( * ` ( A + ( _i x. B ) ) ) ) ) / 2 ) e. RR )
72	67, 71	resubcld 8671	. . . . 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 2311	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( _i x. ( ( ( ( A + ( _i x. B ) ) + ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) - A ) ) e. RR )
74		rimul 8876	. . . 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 411	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( ( A + ( _i x. B ) ) + ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) - A ) = 0 )
76	13, 14, 75	subeq0d 8608	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( A + ( _i x. B ) ) + ( * ` ( A + ( _i x. B ) ) ) ) / 2 ) = A )
77	9, 76	eqtrd 2267	1 |- ( ( A e. RR /\ B e. RR ) -> ( Re ` ( A + ( _i x. B ) ) ) = A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   CCcc 8141   RRcr 8142   0cc0 8143   1c1 8144   _ici 8145   + caddc 8146   x. cmul 8148   - cmin 8460   -ucneg 8461   # cap 8872   / cdiv 8963   2c2 9305   *ccj 11549   Recre 11550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-po 4422  df-iso 4423  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-2 9313  df-cj 11552  df-re 11553

This theorem is referenced by:  crim  11568  replim  11569  mulreap  11574  recj  11577  reneg  11578  readd  11579  remullem  11581  rei  11609  crrei  11646  crred  11686  rennim  11712  absreimsq  11777  4sqlem4  13115  2sqlem2  16114