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Theorem crre 13854

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 10026	. . . 4
2		ax-icn 9995	. . . . 5
3		recn 10026	. . . . 5
4		mulcl 10020	. . . . 5 $/\$
5	2, 3, 4	sylancr 695	. . . 4
6		addcl 10018	. . . 4 $/\$
7	1, 5, 6	syl2an 494	. . 3 $/\$
8		reval 13846	. . 3
9	7, 8	syl 17	. 2 $/\$
10		cjcl 13845	. . . . . 6
11	7, 10	syl 17	. . . . 5 $/\$
12	7, 11	addcld 10059	. . . 4 $/\$
13	12	halfcld 11277	. . 3 $/\$
14	1	adantr 481	. . 3 $/\$
15		recl 13850	. . . . . . 7
16	7, 15	syl 17	. . . . . 6 $/\$
17	9, 16	eqeltrrd 2702	. . . . 5 $/\$
18		simpl 473	. . . . 5 $/\$
19	17, 18	resubcld 10458	. . . 4 $/\$
20	2	a1i 11	. . . . . . 7 $/\$
21	3	adantl 482	. . . . . . . 8 $/\$
22	2, 21, 4	sylancr 695	. . . . . . 7 $/\$
23	7, 11	subcld 10392	. . . . . . . 8 $/\$
24	23	halfcld 11277	. . . . . . 7 $/\$
25	20, 22, 24	subdid 10486	. . . . . 6 $/\$
26	14, 22, 14	pnpcand 10429	. . . . . . . . . . . . . 14 $/\$
27	22, 14, 22	pnpcan2d 10430	. . . . . . . . . . . . . 14 $/\$
28	26, 27	eqtr4d 2659	. . . . . . . . . . . . 13 $/\$
29	28	oveq1d 6665	. . . . . . . . . . . 12 $/\$
30	14, 14	addcld 10059	. . . . . . . . . . . . 13 $/\$
31	7, 11, 30	addsubd 10413	. . . . . . . . . . . 12 $/\$
32	22, 22	addcld 10059	. . . . . . . . . . . . 13 $/\$
33	32, 7, 11	subsubd 10420	. . . . . . . . . . . 12 $/\$
34	29, 31, 33	3eqtr4d 2666	. . . . . . . . . . 11 $/\$
35	14	2timesd 11275	. . . . . . . . . . . 12 $/\$
36	35	oveq2d 6666	. . . . . . . . . . 11 $/\$
37	22	2timesd 11275	. . . . . . . . . . . 12 $/\$
38	37	oveq1d 6665	. . . . . . . . . . 11 $/\$
39	34, 36, 38	3eqtr4d 2666	. . . . . . . . . 10 $/\$
40	39	oveq1d 6665	. . . . . . . . 9 $/\$
41		2cn 11091	. . . . . . . . . . 11
42		mulcl 10020	. . . . . . . . . . 11 $/\$
43	41, 14, 42	sylancr 695	. . . . . . . . . 10 $/\$
44	41	a1i 11	. . . . . . . . . 10 $/\$
45		2ne0 11113	. . . . . . . . . . 11
46	45	a1i 11	. . . . . . . . . 10 $/\$
47	12, 43, 44, 46	divsubdird 10840	. . . . . . . . 9 $/\$
48		mulcl 10020	. . . . . . . . . . 11 $/\$
49	41, 22, 48	sylancr 695	. . . . . . . . . 10 $/\$
50	49, 23, 44, 46	divsubdird 10840	. . . . . . . . 9 $/\$
51	40, 47, 50	3eqtr3d 2664	. . . . . . . 8 $/\$
52	14, 44, 46	divcan3d 10806	. . . . . . . . 9 $/\$
53	52	oveq2d 6666	. . . . . . . 8 $/\$
54	22, 44, 46	divcan3d 10806	. . . . . . . . 9 $/\$
55	54	oveq1d 6665	. . . . . . . 8 $/\$
56	51, 53, 55	3eqtr3d 2664	. . . . . . 7 $/\$
57	56	oveq2d 6666	. . . . . 6 $/\$
58	20, 20, 21	mulassd 10063	. . . . . . 7 $/\$
59	20, 23, 44, 46	divassd 10836	. . . . . . 7 $/\$
60	58, 59	oveq12d 6668	. . . . . 6 $/\$
61	25, 57, 60	3eqtr4d 2666	. . . . 5 $/\$
62		ixi 10656	. . . . . . . 8
63		neg1rr 11125	. . . . . . . 8
64	62, 63	eqeltri 2697	. . . . . . 7
65		simpr 477	. . . . . . 7 $/\$
66		remulcl 10021	. . . . . . 7 $/\$
67	64, 65, 66	sylancr 695	. . . . . 6 $/\$
68		cjth 13843	. . . . . . . . 9 $/\$
69	68	simprd 479	. . . . . . . 8
70	7, 69	syl 17	. . . . . . 7 $/\$
71	70	rehalfcld 11279	. . . . . 6 $/\$
72	67, 71	resubcld 10458	. . . . 5 $/\$
73	61, 72	eqeltrd 2701	. . . 4 $/\$
74		rimul 11011	. . . 4 $/\$
75	19, 73, 74	syl2anc 693	. . 3 $/\$
76	13, 14, 75	subeq0d 10400	. 2 $/\$
77	9, 76	eqtrd 2656	1 $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4 $/\$ wa 384

wceq 1483

wcel 1990

wne 2794

cfv 5888 (class class class)co 6650

cc 9934

cr 9935

cc0 9936

c1 9937

ci 9938

caddc 9939

cmul 9941

cmin 10266

cneg 10267

cdiv 10684

c2 11070

ccj 13836

cre 13837

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-2 11079 df-cj 13839 df-re 13840

This theorem is referenced by: crim 13855 replim 13856 mulre 13861 recj 13864 reneg 13865 readd 13866 remullem 13868 rei 13896 crrei 13932 crred 13971 rennim 13979 absreimsq 14032 4sqlem4 15656 2sqlem2 25143 cnre2csqima 29957

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