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Mirrors > Home > ILE Home > Th. List > oviec | Unicode version |
Description: Express an operation on equivalence classes of ordered pairs in terms of equivalence class of operations on ordered pairs. See iset.mm for additional comments describing the hypotheses. (Unnecessary distinct variable restrictions were removed by David Abernethy, 4-Jun-2013.) (Contributed by NM, 6-Aug-1995.) (Revised by Mario Carneiro, 4-Jun-2013.) |
Ref | Expression |
---|---|
oviec.1 | |
oviec.2 | |
oviec.3 | |
oviec.4 | |
oviec.5 | |
oviec.7 | |
oviec.8 | |
oviec.9 | |
oviec.10 | |
oviec.11 | |
oviec.12 | |
oviec.13 | |
oviec.14 | |
oviec.15 | |
oviec.16 |
Ref | Expression |
---|---|
oviec |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oviec.4 | . . 3 | |
2 | oviec.5 | . . 3 | |
3 | oviec.16 | . . . 4 | |
4 | oviec.8 | . . . . . 6 | |
5 | oviec.7 | . . . . . 6 | |
6 | 4, 5 | opbrop 4665 | . . . . 5 |
7 | oviec.9 | . . . . . 6 | |
8 | 7, 5 | opbrop 4665 | . . . . 5 |
9 | 6, 8 | bi2anan9 596 | . . . 4 |
10 | oviec.2 | . . . . . . 7 | |
11 | oviec.11 | . . . . . . 7 | |
12 | oviec.10 | . . . . . . 7 | |
13 | 10, 11, 12 | ovi3 5957 | . . . . . 6 |
14 | oviec.3 | . . . . . . 7 | |
15 | oviec.12 | . . . . . . 7 | |
16 | 14, 15, 12 | ovi3 5957 | . . . . . 6 |
17 | 13, 16 | breqan12d 3981 | . . . . 5 |
18 | 17 | an4s 578 | . . . 4 |
19 | 3, 9, 18 | 3imtr4d 202 | . . 3 |
20 | oviec.14 | . . . 4 | |
21 | oviec.15 | . . . . . . . 8 | |
22 | 21 | eleq2i 2224 | . . . . . . 7 |
23 | 21 | eleq2i 2224 | . . . . . . 7 |
24 | 22, 23 | anbi12i 456 | . . . . . 6 |
25 | 24 | anbi1i 454 | . . . . 5 |
26 | 25 | oprabbii 5876 | . . . 4 |
27 | 20, 26 | eqtri 2178 | . . 3 |
28 | 1, 2, 19, 27 | th3q 6585 | . 2 |
29 | oviec.1 | . . . 4 | |
30 | oviec.13 | . . . 4 | |
31 | 29, 30, 12 | ovi3 5957 | . . 3 |
32 | 31 | eceq1d 6516 | . 2 |
33 | 28, 32 | eqtrd 2190 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1335 wex 1472 wcel 2128 cvv 2712 cop 3563 class class class wbr 3965 copab 4024 cxp 4584 (class class class)co 5824 coprab 5825 wer 6477 cec 6478 cqs 6479 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fv 5178 df-ov 5827 df-oprab 5828 df-er 6480 df-ec 6482 df-qs 6486 |
This theorem is referenced by: addpipqqs 7290 mulpipqqs 7293 |
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