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Theorem crre 10661

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 7777	. . . 4
2		ax-icn 7739	. . . . 5
3		recn 7777	. . . . 5
4		mulcl 7771	. . . . 5 $/\$
5	2, 3, 4	sylancr 411	. . . 4
6		addcl 7769	. . . 4 $/\$
7	1, 5, 6	syl2an 287	. . 3 $/\$
8		reval 10653	. . 3
9	7, 8	syl 14	. 2 $/\$
10		cjcl 10652	. . . . . 6
11	7, 10	syl 14	. . . . 5 $/\$
12	7, 11	addcld 7809	. . . 4 $/\$
13	12	halfcld 8988	. . 3 $/\$
14	1	adantr 274	. . 3 $/\$
15		recl 10657	. . . . . . 7
16	7, 15	syl 14	. . . . . 6 $/\$
17	9, 16	eqeltrrd 2218	. . . . 5 $/\$
18		simpl 108	. . . . 5 $/\$
19	17, 18	resubcld 8167	. . . 4 $/\$
20	2	a1i 9	. . . . . . 7 $/\$
21	3	adantl 275	. . . . . . . 8 $/\$
22	2, 21, 4	sylancr 411	. . . . . . 7 $/\$
23	7, 11	subcld 8097	. . . . . . . 8 $/\$
24	23	halfcld 8988	. . . . . . 7 $/\$
25	20, 22, 24	subdid 8200	. . . . . 6 $/\$
26	14, 22, 14	pnpcand 8134	. . . . . . . . . . . . . 14 $/\$
27	22, 14, 22	pnpcan2d 8135	. . . . . . . . . . . . . 14 $/\$
28	26, 27	eqtr4d 2176	. . . . . . . . . . . . 13 $/\$
29	28	oveq1d 5797	. . . . . . . . . . . 12 $/\$
30	14, 14	addcld 7809	. . . . . . . . . . . . 13 $/\$
31	7, 11, 30	addsubd 8118	. . . . . . . . . . . 12 $/\$
32	22, 22	addcld 7809	. . . . . . . . . . . . 13 $/\$
33	32, 7, 11	subsubd 8125	. . . . . . . . . . . 12 $/\$
34	29, 31, 33	3eqtr4d 2183	. . . . . . . . . . 11 $/\$
35	14	2timesd 8986	. . . . . . . . . . . 12 $/\$
36	35	oveq2d 5798	. . . . . . . . . . 11 $/\$
37	22	2timesd 8986	. . . . . . . . . . . 12 $/\$
38	37	oveq1d 5797	. . . . . . . . . . 11 $/\$
39	34, 36, 38	3eqtr4d 2183	. . . . . . . . . 10 $/\$
40	39	oveq1d 5797	. . . . . . . . 9 $/\$
41		2cn 8815	. . . . . . . . . . 11
42		mulcl 7771	. . . . . . . . . . 11 $/\$
43	41, 14, 42	sylancr 411	. . . . . . . . . 10 $/\$
44	41	a1i 9	. . . . . . . . . 10 $/\$
45		2ap0 8837	. . . . . . . . . . 11 #
46	45	a1i 9	. . . . . . . . . 10 $/\$ #
47	12, 43, 44, 46	divsubdirapd 8614	. . . . . . . . 9 $/\$
48		mulcl 7771	. . . . . . . . . . 11 $/\$
49	41, 22, 48	sylancr 411	. . . . . . . . . 10 $/\$
50	49, 23, 44, 46	divsubdirapd 8614	. . . . . . . . 9 $/\$
51	40, 47, 50	3eqtr3d 2181	. . . . . . . 8 $/\$
52	14, 44, 46	divcanap3d 8579	. . . . . . . . 9 $/\$
53	52	oveq2d 5798	. . . . . . . 8 $/\$
54	22, 44, 46	divcanap3d 8579	. . . . . . . . 9 $/\$
55	54	oveq1d 5797	. . . . . . . 8 $/\$
56	51, 53, 55	3eqtr3d 2181	. . . . . . 7 $/\$
57	56	oveq2d 5798	. . . . . 6 $/\$
58	20, 20, 21	mulassd 7813	. . . . . . 7 $/\$
59	20, 23, 44, 46	divassapd 8610	. . . . . . 7 $/\$
60	58, 59	oveq12d 5800	. . . . . 6 $/\$
61	25, 57, 60	3eqtr4d 2183	. . . . 5 $/\$
62		ixi 8369	. . . . . . . 8
63		neg1rr 8850	. . . . . . . 8
64	62, 63	eqeltri 2213	. . . . . . 7
65		simpr 109	. . . . . . 7 $/\$
66		remulcl 7772	. . . . . . 7 $/\$
67	64, 65, 66	sylancr 411	. . . . . 6 $/\$
68		cjth 10650	. . . . . . . . 9 $/\$
69	68	simprd 113	. . . . . . . 8
70	7, 69	syl 14	. . . . . . 7 $/\$
71	70	rehalfcld 8990	. . . . . 6 $/\$
72	67, 71	resubcld 8167	. . . . 5 $/\$
73	61, 72	eqeltrd 2217	. . . 4 $/\$
74		rimul 8371	. . . 4 $/\$
75	19, 73, 74	syl2anc 409	. . 3 $/\$
76	13, 14, 75	subeq0d 8105	. 2 $/\$
77	9, 76	eqtrd 2173	1 $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wceq 1332

wcel 1481 class class class wbr 3937

cfv 5131 (class class class)co 5782

cc 7642

cr 7643

cc0 7644

c1 7645

ci 7646

caddc 7647

cmul 7649

cmin 7957

cneg 7958 # cap 8367

cdiv 8456

c2 8795

ccj 10643

cre 10644

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762

This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-po 4226 df-iso 4227 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-2 8803 df-cj 10646 df-re 10647

This theorem is referenced by: crim 10662 replim 10663 mulreap 10668 recj 10671 reneg 10672 readd 10673 remullem 10675 rei 10703 crrei 10740 crred 10780 rennim 10806 absreimsq 10871

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