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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 7730 | . . 3 | |
2 | recn 7721 | . . . . . . 7 | |
3 | ax-icn 7683 | . . . . . . . 8 | |
4 | recn 7721 | . . . . . . . 8 | |
5 | mulcl 7715 | . . . . . . . 8 | |
6 | 3, 4, 5 | sylancr 410 | . . . . . . 7 |
7 | subcl 7929 | . . . . . . 7 | |
8 | 2, 6, 7 | syl2an 287 | . . . . . 6 |
9 | 2 | adantr 274 | . . . . . . . 8 |
10 | 6 | adantl 275 | . . . . . . . 8 |
11 | 9, 10, 9 | ppncand 8081 | . . . . . . 7 |
12 | readdcl 7714 | . . . . . . . . 9 | |
13 | 12 | anidms 394 | . . . . . . . 8 |
14 | 13 | adantr 274 | . . . . . . 7 |
15 | 11, 14 | eqeltrd 2194 | . . . . . 6 |
16 | 9, 10, 10 | pnncand 8080 | . . . . . . . . . 10 |
17 | 3 | a1i 9 | . . . . . . . . . . 11 |
18 | 4 | adantl 275 | . . . . . . . . . . 11 |
19 | 17, 18, 18 | adddid 7758 | . . . . . . . . . 10 |
20 | 16, 19 | eqtr4d 2153 | . . . . . . . . 9 |
21 | 20 | oveq2d 5758 | . . . . . . . 8 |
22 | 18, 18 | addcld 7753 | . . . . . . . . 9 |
23 | mulass 7719 | . . . . . . . . . 10 | |
24 | 3, 3, 23 | mp3an12 1290 | . . . . . . . . 9 |
25 | 22, 24 | syl 14 | . . . . . . . 8 |
26 | 21, 25 | eqtr4d 2153 | . . . . . . 7 |
27 | ixi 8312 | . . . . . . . . 9 | |
28 | 1re 7733 | . . . . . . . . . 10 | |
29 | 28 | renegcli 7992 | . . . . . . . . 9 |
30 | 27, 29 | eqeltri 2190 | . . . . . . . 8 |
31 | simpr 109 | . . . . . . . . 9 | |
32 | 31, 31 | readdcld 7763 | . . . . . . . 8 |
33 | remulcl 7716 | . . . . . . . 8 | |
34 | 30, 32, 33 | sylancr 410 | . . . . . . 7 |
35 | 26, 34 | eqeltrd 2194 | . . . . . 6 |
36 | oveq2 5750 | . . . . . . . . 9 | |
37 | 36 | eleq1d 2186 | . . . . . . . 8 |
38 | oveq2 5750 | . . . . . . . . . 10 | |
39 | 38 | oveq2d 5758 | . . . . . . . . 9 |
40 | 39 | eleq1d 2186 | . . . . . . . 8 |
41 | 37, 40 | anbi12d 464 | . . . . . . 7 |
42 | 41 | rspcev 2763 | . . . . . 6 |
43 | 8, 15, 35, 42 | syl12anc 1199 | . . . . 5 |
44 | oveq1 5749 | . . . . . . . 8 | |
45 | 44 | eleq1d 2186 | . . . . . . 7 |
46 | oveq1 5749 | . . . . . . . . 9 | |
47 | 46 | oveq2d 5758 | . . . . . . . 8 |
48 | 47 | eleq1d 2186 | . . . . . . 7 |
49 | 45, 48 | anbi12d 464 | . . . . . 6 |
50 | 49 | rexbidv 2415 | . . . . 5 |
51 | 43, 50 | syl5ibrcom 156 | . . . 4 |
52 | 51 | rexlimivv 2532 | . . 3 |
53 | 1, 52 | syl 14 | . 2 |
54 | an4 560 | . . . 4 | |
55 | resubcl 7994 | . . . . . . 7 | |
56 | pnpcan 7969 | . . . . . . . . 9 | |
57 | 56 | 3expb 1167 | . . . . . . . 8 |
58 | 57 | eleq1d 2186 | . . . . . . 7 |
59 | 55, 58 | syl5ib 153 | . . . . . 6 |
60 | resubcl 7994 | . . . . . . . 8 | |
61 | 60 | ancoms 266 | . . . . . . 7 |
62 | 3 | a1i 9 | . . . . . . . . . 10 |
63 | subcl 7929 | . . . . . . . . . . 11 | |
64 | 63 | adantrl 469 | . . . . . . . . . 10 |
65 | subcl 7929 | . . . . . . . . . . 11 | |
66 | 65 | adantrr 470 | . . . . . . . . . 10 |
67 | 62, 64, 66 | subdid 8144 | . . . . . . . . 9 |
68 | nnncan1 7966 | . . . . . . . . . . . 12 | |
69 | 68 | 3com23 1172 | . . . . . . . . . . 11 |
70 | 69 | 3expb 1167 | . . . . . . . . . 10 |
71 | 70 | oveq2d 5758 | . . . . . . . . 9 |
72 | 67, 71 | eqtr3d 2152 | . . . . . . . 8 |
73 | 72 | eleq1d 2186 | . . . . . . 7 |
74 | 61, 73 | syl5ib 153 | . . . . . 6 |
75 | 59, 74 | anim12d 333 | . . . . 5 |
76 | rimul 8314 | . . . . . 6 | |
77 | 76 | a1i 9 | . . . . 5 |
78 | subeq0 7956 | . . . . . . 7 | |
79 | 78 | biimpd 143 | . . . . . 6 |
80 | 79 | adantl 275 | . . . . 5 |
81 | 75, 77, 80 | 3syld 57 | . . . 4 |
82 | 54, 81 | syl5bi 151 | . . 3 |
83 | 82 | ralrimivva 2491 | . 2 |
84 | oveq2 5750 | . . . . 5 | |
85 | 84 | eleq1d 2186 | . . . 4 |
86 | oveq2 5750 | . . . . . 6 | |
87 | 86 | oveq2d 5758 | . . . . 5 |
88 | 87 | eleq1d 2186 | . . . 4 |
89 | 85, 88 | anbi12d 464 | . . 3 |
90 | 89 | reu4 2851 | . 2 |
91 | 53, 83, 90 | sylanbrc 413 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1316 wcel 1465 wral 2393 wrex 2394 wreu 2395 (class class class)co 5742 cc 7586 cr 7587 cc0 7588 c1 7589 ci 7590 caddc 7591 cmul 7593 cmin 7901 cneg 7902 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rmo 2401 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-pnf 7770 df-mnf 7771 df-ltxr 7773 df-sub 7903 df-neg 7904 df-reap 8304 |
This theorem is referenced by: cjval 10572 cjth 10573 cjf 10574 remim 10587 |
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