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Mirrors > Home > ILE Home > Th. List > cru | Unicode version |
Description: The representation of complex numbers in terms of real and imaginary parts is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
cru |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simplrl 501 | . . . . . . 7 | |
2 | 1 | recnd 7147 | . . . . . 6 |
3 | simplll 499 | . . . . . . 7 | |
4 | 3 | recnd 7147 | . . . . . 6 |
5 | simpr 108 | . . . . . . . 8 | |
6 | ax-icn 7071 | . . . . . . . . . . 11 | |
7 | 6 | a1i 9 | . . . . . . . . . 10 |
8 | simpllr 500 | . . . . . . . . . . 11 | |
9 | 8 | recnd 7147 | . . . . . . . . . 10 |
10 | 7, 9 | mulcld 7139 | . . . . . . . . 9 |
11 | simplrr 502 | . . . . . . . . . . 11 | |
12 | 11 | recnd 7147 | . . . . . . . . . 10 |
13 | 7, 12 | mulcld 7139 | . . . . . . . . 9 |
14 | 4, 10, 2, 13 | addsubeq4d 7470 | . . . . . . . 8 |
15 | 5, 14 | mpbid 145 | . . . . . . 7 |
16 | 8, 11 | resubcld 7485 | . . . . . . . . . . 11 |
17 | 7, 9, 12 | subdid 7518 | . . . . . . . . . . . . 13 |
18 | 17, 15 | eqtr4d 2116 | . . . . . . . . . . . 12 |
19 | 1, 3 | resubcld 7485 | . . . . . . . . . . . 12 |
20 | 18, 19 | eqeltrd 2155 | . . . . . . . . . . 11 |
21 | rimul 7685 | . . . . . . . . . . 11 | |
22 | 16, 20, 21 | syl2anc 403 | . . . . . . . . . 10 |
23 | 9, 12, 22 | subeq0d 7427 | . . . . . . . . 9 |
24 | 23 | oveq2d 5548 | . . . . . . . 8 |
25 | 24 | oveq1d 5547 | . . . . . . 7 |
26 | 13 | subidd 7407 | . . . . . . 7 |
27 | 15, 25, 26 | 3eqtrd 2117 | . . . . . 6 |
28 | 2, 4, 27 | subeq0d 7427 | . . . . 5 |
29 | 28 | eqcomd 2086 | . . . 4 |
30 | 29, 23 | jca 300 | . . 3 |
31 | 30 | ex 113 | . 2 |
32 | oveq2 5540 | . . 3 | |
33 | oveq12 5541 | . . 3 | |
34 | 32, 33 | sylan2 280 | . 2 |
35 | 31, 34 | impbid1 140 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wceq 1284 wcel 1433 (class class class)co 5532 cc 6979 cr 6980 cc0 6981 ci 6983 caddc 6984 cmul 6986 cmin 7279 |
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-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: apreim 7703 apti 7722 creur 8036 creui 8037 cnref1o 8733 |
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