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Theorem crre 11368

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	$/\$

**Proof of Theorem crre**
Step	Hyp	Ref	Expression
1		recn 8132	. . . 4
2		ax-icn 8094	. . . . 5
3		recn 8132	. . . . 5
4		mulcl 8126	. . . . 5 $/\$
5	2, 3, 4	sylancr 414	. . . 4
6		addcl 8124	. . . 4 $/\$
7	1, 5, 6	syl2an 289	. . 3 $/\$
8		reval 11360	. . 3
9	7, 8	syl 14	. 2 $/\$
10		cjcl 11359	. . . . . 6
11	7, 10	syl 14	. . . . 5 $/\$
12	7, 11	addcld 8166	. . . 4 $/\$
13	12	halfcld 9356	. . 3 $/\$
14	1	adantr 276	. . 3 $/\$
15		recl 11364	. . . . . . 7
16	7, 15	syl 14	. . . . . 6 $/\$
17	9, 16	eqeltrrd 2307	. . . . 5 $/\$
18		simpl 109	. . . . 5 $/\$
19	17, 18	resubcld 8527	. . . 4 $/\$
20	2	a1i 9	. . . . . . 7 $/\$
21	3	adantl 277	. . . . . . . 8 $/\$
22	2, 21, 4	sylancr 414	. . . . . . 7 $/\$
23	7, 11	subcld 8457	. . . . . . . 8 $/\$
24	23	halfcld 9356	. . . . . . 7 $/\$
25	20, 22, 24	subdid 8560	. . . . . 6 $/\$
26	14, 22, 14	pnpcand 8494	. . . . . . . . . . . . . 14 $/\$
27	22, 14, 22	pnpcan2d 8495	. . . . . . . . . . . . . 14 $/\$
28	26, 27	eqtr4d 2265	. . . . . . . . . . . . 13 $/\$
29	28	oveq1d 6016	. . . . . . . . . . . 12 $/\$
30	14, 14	addcld 8166	. . . . . . . . . . . . 13 $/\$
31	7, 11, 30	addsubd 8478	. . . . . . . . . . . 12 $/\$
32	22, 22	addcld 8166	. . . . . . . . . . . . 13 $/\$
33	32, 7, 11	subsubd 8485	. . . . . . . . . . . 12 $/\$
34	29, 31, 33	3eqtr4d 2272	. . . . . . . . . . 11 $/\$
35	14	2timesd 9354	. . . . . . . . . . . 12 $/\$
36	35	oveq2d 6017	. . . . . . . . . . 11 $/\$
37	22	2timesd 9354	. . . . . . . . . . . 12 $/\$
38	37	oveq1d 6016	. . . . . . . . . . 11 $/\$
39	34, 36, 38	3eqtr4d 2272	. . . . . . . . . 10 $/\$
40	39	oveq1d 6016	. . . . . . . . 9 $/\$
41		2cn 9181	. . . . . . . . . . 11
42		mulcl 8126	. . . . . . . . . . 11 $/\$
43	41, 14, 42	sylancr 414	. . . . . . . . . 10 $/\$
44	41	a1i 9	. . . . . . . . . 10 $/\$
45		2ap0 9203	. . . . . . . . . . 11 #
46	45	a1i 9	. . . . . . . . . 10 $/\$ #
47	12, 43, 44, 46	divsubdirapd 8977	. . . . . . . . 9 $/\$
48		mulcl 8126	. . . . . . . . . . 11 $/\$
49	41, 22, 48	sylancr 414	. . . . . . . . . 10 $/\$
50	49, 23, 44, 46	divsubdirapd 8977	. . . . . . . . 9 $/\$
51	40, 47, 50	3eqtr3d 2270	. . . . . . . 8 $/\$
52	14, 44, 46	divcanap3d 8942	. . . . . . . . 9 $/\$
53	52	oveq2d 6017	. . . . . . . 8 $/\$
54	22, 44, 46	divcanap3d 8942	. . . . . . . . 9 $/\$
55	54	oveq1d 6016	. . . . . . . 8 $/\$
56	51, 53, 55	3eqtr3d 2270	. . . . . . 7 $/\$
57	56	oveq2d 6017	. . . . . 6 $/\$
58	20, 20, 21	mulassd 8170	. . . . . . 7 $/\$
59	20, 23, 44, 46	divassapd 8973	. . . . . . 7 $/\$
60	58, 59	oveq12d 6019	. . . . . 6 $/\$
61	25, 57, 60	3eqtr4d 2272	. . . . 5 $/\$
62		ixi 8730	. . . . . . . 8
63		neg1rr 9216	. . . . . . . 8
64	62, 63	eqeltri 2302	. . . . . . 7
65		simpr 110	. . . . . . 7 $/\$
66		remulcl 8127	. . . . . . 7 $/\$
67	64, 65, 66	sylancr 414	. . . . . 6 $/\$
68		cjth 11357	. . . . . . . . 9 $/\$
69	68	simprd 114	. . . . . . . 8
70	7, 69	syl 14	. . . . . . 7 $/\$
71	70	rehalfcld 9358	. . . . . 6 $/\$
72	67, 71	resubcld 8527	. . . . 5 $/\$
73	61, 72	eqeltrd 2306	. . . 4 $/\$
74		rimul 8732	. . . 4 $/\$
75	19, 73, 74	syl2anc 411	. . 3 $/\$
76	13, 14, 75	subeq0d 8465	. 2 $/\$
77	9, 76	eqtrd 2262	1 $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wceq 1395

wcel 2200 class class class wbr 4083

cfv 5318 (class class class)co 6001

cc 7997

cr 7998

cc0 7999

c1 8000

ci 8001

caddc 8002

cmul 8004

cmin 8317

cneg 8318 # cap 8728

cdiv 8819

c2 9161

ccj 11350

cre 11351

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-sep 4202 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-cnex 8090 ax-resscn 8091 ax-1cn 8092 ax-1re 8093 ax-icn 8094 ax-addcl 8095 ax-addrcl 8096 ax-mulcl 8097 ax-mulrcl 8098 ax-addcom 8099 ax-mulcom 8100 ax-addass 8101 ax-mulass 8102 ax-distr 8103 ax-i2m1 8104 ax-0lt1 8105 ax-1rid 8106 ax-0id 8107 ax-rnegex 8108 ax-precex 8109 ax-cnre 8110 ax-pre-ltirr 8111 ax-pre-ltwlin 8112 ax-pre-lttrn 8113 ax-pre-apti 8114 ax-pre-ltadd 8115 ax-pre-mulgt0 8116 ax-pre-mulext 8117

This theorem depends on definitions: df-bi 117 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-br 4084 df-opab 4146 df-mpt 4147 df-id 4384 df-po 4387 df-iso 4388 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-fv 5326 df-riota 5954 df-ov 6004 df-oprab 6005 df-mpo 6006 df-pnf 8183 df-mnf 8184 df-xr 8185 df-ltxr 8186 df-le 8187 df-sub 8319 df-neg 8320 df-reap 8722 df-ap 8729 df-div 8820 df-2 9169 df-cj 11353 df-re 11354

This theorem is referenced by: crim 11369 replim 11370 mulreap 11375 recj 11378 reneg 11379 readd 11380 remullem 11382 rei 11410 crrei 11447 crred 11487 rennim 11513 absreimsq 11578 4sqlem4 12915 2sqlem2 15794

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