cju - Metamath Proof Explorer

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Theorem cju 11016

Description: The complex conjugate of a complex number is unique. (Contributed by Mario Carneiro, 6-Nov-2013.)

**Assertion**
Ref	Expression
cju	$/\$

Distinct variable group:

Proof of Theorem cju Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnre 10036	. . 3
2		recn 10026	. . . . . . 7
3		ax-icn 9995	. . . . . . . 8
4		recn 10026	. . . . . . . 8
5		mulcl 10020	. . . . . . . 8 $/\$
6	3, 4, 5	sylancr 695	. . . . . . 7
7		subcl 10280	. . . . . . 7 $/\$
8	2, 6, 7	syl2an 494	. . . . . 6 $/\$
9	2	adantr 481	. . . . . . . 8 $/\$
10	6	adantl 482	. . . . . . . 8 $/\$
11	9, 10, 9	ppncand 10432	. . . . . . 7 $/\$
12		readdcl 10019	. . . . . . . . 9 $/\$
13	12	anidms 677	. . . . . . . 8
14	13	adantr 481	. . . . . . 7 $/\$
15	11, 14	eqeltrd 2701	. . . . . 6 $/\$
16	9, 10, 10	pnncand 10431	. . . . . . . . . 10 $/\$
17	3	a1i 11	. . . . . . . . . . 11 $/\$
18	4	adantl 482	. . . . . . . . . . 11 $/\$
19	17, 18, 18	adddid 10064	. . . . . . . . . 10 $/\$
20	16, 19	eqtr4d 2659	. . . . . . . . 9 $/\$
21	20	oveq2d 6666	. . . . . . . 8 $/\$
22	18, 18	addcld 10059	. . . . . . . . 9 $/\$
23		mulass 10024	. . . . . . . . . 10 $/\$ $/\$
24	3, 3, 23	mp3an12 1414	. . . . . . . . 9
25	22, 24	syl 17	. . . . . . . 8 $/\$
26	21, 25	eqtr4d 2659	. . . . . . 7 $/\$
27		ixi 10656	. . . . . . . . 9
28		1re 10039	. . . . . . . . . 10
29	28	renegcli 10342	. . . . . . . . 9
30	27, 29	eqeltri 2697	. . . . . . . 8
31		simpr 477	. . . . . . . . 9 $/\$
32	31, 31	readdcld 10069	. . . . . . . 8 $/\$
33		remulcl 10021	. . . . . . . 8 $/\$
34	30, 32, 33	sylancr 695	. . . . . . 7 $/\$
35	26, 34	eqeltrd 2701	. . . . . 6 $/\$
36		oveq2 6658	. . . . . . . . 9
37	36	eleq1d 2686	. . . . . . . 8
38		oveq2 6658	. . . . . . . . . 10
39	38	oveq2d 6666	. . . . . . . . 9
40	39	eleq1d 2686	. . . . . . . 8
41	37, 40	anbi12d 747	. . . . . . 7 $/\$ $/\$
42	41	rspcev 3309	. . . . . 6 $/\$ $/\$ $/\$
43	8, 15, 35, 42	syl12anc 1324	. . . . 5 $/\$ $/\$
44		oveq1 6657	. . . . . . . 8
45	44	eleq1d 2686	. . . . . . 7
46		oveq1 6657	. . . . . . . . 9
47	46	oveq2d 6666	. . . . . . . 8
48	47	eleq1d 2686	. . . . . . 7
49	45, 48	anbi12d 747	. . . . . 6 $/\$ $/\$
50	49	rexbidv 3052	. . . . 5 $/\$ $/\$
51	43, 50	syl5ibrcom 237	. . . 4 $/\$ $/\$
52	51	rexlimivv 3036	. . 3 $/\$
53	1, 52	syl 17	. 2 $/\$
54		an4 865	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
55		resubcl 10345	. . . . . . 7 $/\$
56		pnpcan 10320	. . . . . . . . 9 $/\$ $/\$
57	56	3expb 1266	. . . . . . . 8 $/\$ $/\$
58	57	eleq1d 2686	. . . . . . 7 $/\$ $/\$
59	55, 58	syl5ib 234	. . . . . 6 $/\$ $/\$ $/\$
60		resubcl 10345	. . . . . . . 8 $/\$
61	60	ancoms 469	. . . . . . 7 $/\$
62	3	a1i 11	. . . . . . . . . 10 $/\$ $/\$
63		subcl 10280	. . . . . . . . . . 11 $/\$
64	63	adantrl 752	. . . . . . . . . 10 $/\$ $/\$
65		subcl 10280	. . . . . . . . . . 11 $/\$
66	65	adantrr 753	. . . . . . . . . 10 $/\$ $/\$
67	62, 64, 66	subdid 10486	. . . . . . . . 9 $/\$ $/\$
68		nnncan1 10317	. . . . . . . . . . . 12 $/\$ $/\$
69	68	3com23 1271	. . . . . . . . . . 11 $/\$ $/\$
70	69	3expb 1266	. . . . . . . . . 10 $/\$ $/\$
71	70	oveq2d 6666	. . . . . . . . 9 $/\$ $/\$
72	67, 71	eqtr3d 2658	. . . . . . . 8 $/\$ $/\$
73	72	eleq1d 2686	. . . . . . 7 $/\$ $/\$
74	61, 73	syl5ib 234	. . . . . 6 $/\$ $/\$ $/\$
75	59, 74	anim12d 586	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
76		rimul 11011	. . . . . 6 $/\$
77	76	a1i 11	. . . . 5 $/\$ $/\$ $/\$
78		subeq0 10307	. . . . . . 7 $/\$
79	78	biimpd 219	. . . . . 6 $/\$
80	79	adantl 482	. . . . 5 $/\$ $/\$
81	75, 77, 80	3syld 60	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
82	54, 81	syl5bi 232	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
83	82	ralrimivva 2971	. 2 $/\$ $/\$ $/\$
84		oveq2 6658	. . . . 5
85	84	eleq1d 2686	. . . 4
86		oveq2 6658	. . . . . 6
87	86	oveq2d 6666	. . . . 5
88	87	eleq1d 2686	. . . 4
89	85, 88	anbi12d 747	. . 3 $/\$ $/\$
90	89	reu4 3400	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
91	53, 83, 90	sylanbrc 698	1 $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4 $/\$ wa 384

wceq 1483

wcel 1990

wral 2912

wrex 2913

wreu 2914 (class class class)co 6650

cc 9934

cr 9935

cc0 9936

c1 9937

ci 9938

caddc 9939

cmul 9941

cmin 10266

cneg 10267

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

This theorem is referenced by: cjth 13843 cjf 13844 remim 13857

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