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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 7762 | . . 3 | |
2 | recn 7753 | . . . . . . 7 | |
3 | ax-icn 7715 | . . . . . . . 8 | |
4 | recn 7753 | . . . . . . . 8 | |
5 | mulcl 7747 | . . . . . . . 8 | |
6 | 3, 4, 5 | sylancr 410 | . . . . . . 7 |
7 | subcl 7961 | . . . . . . 7 | |
8 | 2, 6, 7 | syl2an 287 | . . . . . 6 |
9 | 2 | adantr 274 | . . . . . . . 8 |
10 | 6 | adantl 275 | . . . . . . . 8 |
11 | 9, 10, 9 | ppncand 8113 | . . . . . . 7 |
12 | readdcl 7746 | . . . . . . . . 9 | |
13 | 12 | anidms 394 | . . . . . . . 8 |
14 | 13 | adantr 274 | . . . . . . 7 |
15 | 11, 14 | eqeltrd 2216 | . . . . . 6 |
16 | 9, 10, 10 | pnncand 8112 | . . . . . . . . . 10 |
17 | 3 | a1i 9 | . . . . . . . . . . 11 |
18 | 4 | adantl 275 | . . . . . . . . . . 11 |
19 | 17, 18, 18 | adddid 7790 | . . . . . . . . . 10 |
20 | 16, 19 | eqtr4d 2175 | . . . . . . . . 9 |
21 | 20 | oveq2d 5790 | . . . . . . . 8 |
22 | 18, 18 | addcld 7785 | . . . . . . . . 9 |
23 | mulass 7751 | . . . . . . . . . 10 | |
24 | 3, 3, 23 | mp3an12 1305 | . . . . . . . . 9 |
25 | 22, 24 | syl 14 | . . . . . . . 8 |
26 | 21, 25 | eqtr4d 2175 | . . . . . . 7 |
27 | ixi 8345 | . . . . . . . . 9 | |
28 | 1re 7765 | . . . . . . . . . 10 | |
29 | 28 | renegcli 8024 | . . . . . . . . 9 |
30 | 27, 29 | eqeltri 2212 | . . . . . . . 8 |
31 | simpr 109 | . . . . . . . . 9 | |
32 | 31, 31 | readdcld 7795 | . . . . . . . 8 |
33 | remulcl 7748 | . . . . . . . 8 | |
34 | 30, 32, 33 | sylancr 410 | . . . . . . 7 |
35 | 26, 34 | eqeltrd 2216 | . . . . . 6 |
36 | oveq2 5782 | . . . . . . . . 9 | |
37 | 36 | eleq1d 2208 | . . . . . . . 8 |
38 | oveq2 5782 | . . . . . . . . . 10 | |
39 | 38 | oveq2d 5790 | . . . . . . . . 9 |
40 | 39 | eleq1d 2208 | . . . . . . . 8 |
41 | 37, 40 | anbi12d 464 | . . . . . . 7 |
42 | 41 | rspcev 2789 | . . . . . 6 |
43 | 8, 15, 35, 42 | syl12anc 1214 | . . . . 5 |
44 | oveq1 5781 | . . . . . . . 8 | |
45 | 44 | eleq1d 2208 | . . . . . . 7 |
46 | oveq1 5781 | . . . . . . . . 9 | |
47 | 46 | oveq2d 5790 | . . . . . . . 8 |
48 | 47 | eleq1d 2208 | . . . . . . 7 |
49 | 45, 48 | anbi12d 464 | . . . . . 6 |
50 | 49 | rexbidv 2438 | . . . . 5 |
51 | 43, 50 | syl5ibrcom 156 | . . . 4 |
52 | 51 | rexlimivv 2555 | . . 3 |
53 | 1, 52 | syl 14 | . 2 |
54 | an4 575 | . . . 4 | |
55 | resubcl 8026 | . . . . . . 7 | |
56 | pnpcan 8001 | . . . . . . . . 9 | |
57 | 56 | 3expb 1182 | . . . . . . . 8 |
58 | 57 | eleq1d 2208 | . . . . . . 7 |
59 | 55, 58 | syl5ib 153 | . . . . . 6 |
60 | resubcl 8026 | . . . . . . . 8 | |
61 | 60 | ancoms 266 | . . . . . . 7 |
62 | 3 | a1i 9 | . . . . . . . . . 10 |
63 | subcl 7961 | . . . . . . . . . . 11 | |
64 | 63 | adantrl 469 | . . . . . . . . . 10 |
65 | subcl 7961 | . . . . . . . . . . 11 | |
66 | 65 | adantrr 470 | . . . . . . . . . 10 |
67 | 62, 64, 66 | subdid 8176 | . . . . . . . . 9 |
68 | nnncan1 7998 | . . . . . . . . . . . 12 | |
69 | 68 | 3com23 1187 | . . . . . . . . . . 11 |
70 | 69 | 3expb 1182 | . . . . . . . . . 10 |
71 | 70 | oveq2d 5790 | . . . . . . . . 9 |
72 | 67, 71 | eqtr3d 2174 | . . . . . . . 8 |
73 | 72 | eleq1d 2208 | . . . . . . 7 |
74 | 61, 73 | syl5ib 153 | . . . . . 6 |
75 | 59, 74 | anim12d 333 | . . . . 5 |
76 | rimul 8347 | . . . . . 6 | |
77 | 76 | a1i 9 | . . . . 5 |
78 | subeq0 7988 | . . . . . . 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 2514 | . 2 |
84 | oveq2 5782 | . . . . 5 | |
85 | 84 | eleq1d 2208 | . . . 4 |
86 | oveq2 5782 | . . . . . 6 | |
87 | 86 | oveq2d 5790 | . . . . 5 |
88 | 87 | eleq1d 2208 | . . . 4 |
89 | 85, 88 | anbi12d 464 | . . 3 |
90 | 89 | reu4 2878 | . 2 |
91 | 53, 83, 90 | sylanbrc 413 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 wral 2416 wrex 2417 wreu 2418 (class class class)co 5774 cc 7618 cr 7619 cc0 7620 c1 7621 ci 7622 caddc 7623 cmul 7625 cmin 7933 cneg 7934 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-ltxr 7805 df-sub 7935 df-neg 7936 df-reap 8337 |
This theorem is referenced by: cjval 10617 cjth 10618 cjf 10619 remim 10632 |
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