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Mirrors > Home > ILE Home > Th. List > cju | Unicode version |
Description: The complex conjugate of a complex number is unique. (Contributed by Mario Carneiro, 6-Nov-2013.) |
Ref | Expression |
---|---|
cju |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnre 7115 | . . 3 | |
2 | recn 7106 | . . . . . . 7 | |
3 | ax-icn 7071 | . . . . . . . 8 | |
4 | recn 7106 | . . . . . . . 8 | |
5 | mulcl 7100 | . . . . . . . 8 | |
6 | 3, 4, 5 | sylancr 405 | . . . . . . 7 |
7 | subcl 7307 | . . . . . . 7 | |
8 | 2, 6, 7 | syl2an 283 | . . . . . 6 |
9 | 2 | adantr 270 | . . . . . . . 8 |
10 | 6 | adantl 271 | . . . . . . . 8 |
11 | 9, 10, 9 | ppncand 7459 | . . . . . . 7 |
12 | readdcl 7099 | . . . . . . . . 9 | |
13 | 12 | anidms 389 | . . . . . . . 8 |
14 | 13 | adantr 270 | . . . . . . 7 |
15 | 11, 14 | eqeltrd 2155 | . . . . . 6 |
16 | 9, 10, 10 | pnncand 7458 | . . . . . . . . . 10 |
17 | 3 | a1i 9 | . . . . . . . . . . 11 |
18 | 4 | adantl 271 | . . . . . . . . . . 11 |
19 | 17, 18, 18 | adddid 7143 | . . . . . . . . . 10 |
20 | 16, 19 | eqtr4d 2116 | . . . . . . . . 9 |
21 | 20 | oveq2d 5548 | . . . . . . . 8 |
22 | 18, 18 | addcld 7138 | . . . . . . . . 9 |
23 | mulass 7104 | . . . . . . . . . 10 | |
24 | 3, 3, 23 | mp3an12 1258 | . . . . . . . . 9 |
25 | 22, 24 | syl 14 | . . . . . . . 8 |
26 | 21, 25 | eqtr4d 2116 | . . . . . . 7 |
27 | ixi 7683 | . . . . . . . . 9 | |
28 | 1re 7118 | . . . . . . . . . 10 | |
29 | 28 | renegcli 7370 | . . . . . . . . 9 |
30 | 27, 29 | eqeltri 2151 | . . . . . . . 8 |
31 | simpr 108 | . . . . . . . . 9 | |
32 | 31, 31 | readdcld 7148 | . . . . . . . 8 |
33 | remulcl 7101 | . . . . . . . 8 | |
34 | 30, 32, 33 | sylancr 405 | . . . . . . 7 |
35 | 26, 34 | eqeltrd 2155 | . . . . . 6 |
36 | oveq2 5540 | . . . . . . . . 9 | |
37 | 36 | eleq1d 2147 | . . . . . . . 8 |
38 | oveq2 5540 | . . . . . . . . . 10 | |
39 | 38 | oveq2d 5548 | . . . . . . . . 9 |
40 | 39 | eleq1d 2147 | . . . . . . . 8 |
41 | 37, 40 | anbi12d 456 | . . . . . . 7 |
42 | 41 | rspcev 2701 | . . . . . 6 |
43 | 8, 15, 35, 42 | syl12anc 1167 | . . . . 5 |
44 | oveq1 5539 | . . . . . . . 8 | |
45 | 44 | eleq1d 2147 | . . . . . . 7 |
46 | oveq1 5539 | . . . . . . . . 9 | |
47 | 46 | oveq2d 5548 | . . . . . . . 8 |
48 | 47 | eleq1d 2147 | . . . . . . 7 |
49 | 45, 48 | anbi12d 456 | . . . . . 6 |
50 | 49 | rexbidv 2369 | . . . . 5 |
51 | 43, 50 | syl5ibrcom 155 | . . . 4 |
52 | 51 | rexlimivv 2482 | . . 3 |
53 | 1, 52 | syl 14 | . 2 |
54 | an4 550 | . . . 4 | |
55 | resubcl 7372 | . . . . . . 7 | |
56 | pnpcan 7347 | . . . . . . . . 9 | |
57 | 56 | 3expb 1139 | . . . . . . . 8 |
58 | 57 | eleq1d 2147 | . . . . . . 7 |
59 | 55, 58 | syl5ib 152 | . . . . . 6 |
60 | resubcl 7372 | . . . . . . . 8 | |
61 | 60 | ancoms 264 | . . . . . . 7 |
62 | 3 | a1i 9 | . . . . . . . . . 10 |
63 | subcl 7307 | . . . . . . . . . . 11 | |
64 | 63 | adantrl 461 | . . . . . . . . . 10 |
65 | subcl 7307 | . . . . . . . . . . 11 | |
66 | 65 | adantrr 462 | . . . . . . . . . 10 |
67 | 62, 64, 66 | subdid 7518 | . . . . . . . . 9 |
68 | nnncan1 7344 | . . . . . . . . . . . 12 | |
69 | 68 | 3com23 1144 | . . . . . . . . . . 11 |
70 | 69 | 3expb 1139 | . . . . . . . . . 10 |
71 | 70 | oveq2d 5548 | . . . . . . . . 9 |
72 | 67, 71 | eqtr3d 2115 | . . . . . . . 8 |
73 | 72 | eleq1d 2147 | . . . . . . 7 |
74 | 61, 73 | syl5ib 152 | . . . . . 6 |
75 | 59, 74 | anim12d 328 | . . . . 5 |
76 | rimul 7685 | . . . . . 6 | |
77 | 76 | a1i 9 | . . . . 5 |
78 | subeq0 7334 | . . . . . . 7 | |
79 | 78 | biimpd 142 | . . . . . 6 |
80 | 79 | adantl 271 | . . . . 5 |
81 | 75, 77, 80 | 3syld 56 | . . . 4 |
82 | 54, 81 | syl5bi 150 | . . 3 |
83 | 82 | ralrimivva 2443 | . 2 |
84 | oveq2 5540 | . . . . 5 | |
85 | 84 | eleq1d 2147 | . . . 4 |
86 | oveq2 5540 | . . . . . 6 | |
87 | 86 | oveq2d 5548 | . . . . 5 |
88 | 87 | eleq1d 2147 | . . . 4 |
89 | 85, 88 | anbi12d 456 | . . 3 |
90 | 89 | reu4 2786 | . 2 |
91 | 53, 83, 90 | sylanbrc 408 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wceq 1284 wcel 1433 wral 2348 wrex 2349 wreu 2350 (class class class)co 5532 cc 6979 cr 6980 cc0 6981 c1 6982 ci 6983 caddc 6984 cmul 6986 cmin 7279 cneg 7280 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-pnf 7155 df-mnf 7156 df-ltxr 7158 df-sub 7281 df-neg 7282 df-reap 7675 |
This theorem is referenced by: cjval 9732 cjth 9733 cjf 9734 remim 9747 |
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