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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 4437 | . . . . 5 |
7 | oviec.9 | . . . . . 6 | |
8 | 7, 5 | opbrop 4437 | . . . . 5 |
9 | 6, 8 | bi2anan9 570 | . . . 4 |
10 | oviec.2 | . . . . . . 7 | |
11 | oviec.11 | . . . . . . 7 | |
12 | oviec.10 | . . . . . . 7 | |
13 | 10, 11, 12 | ovi3 5657 | . . . . . 6 |
14 | oviec.3 | . . . . . . 7 | |
15 | oviec.12 | . . . . . . 7 | |
16 | 14, 15, 12 | ovi3 5657 | . . . . . 6 |
17 | 13, 16 | breqan12d 3800 | . . . . 5 |
18 | 17 | an4s 552 | . . . 4 |
19 | 3, 9, 18 | 3imtr4d 201 | . . 3 |
20 | oviec.14 | . . . 4 | |
21 | oviec.15 | . . . . . . . 8 | |
22 | 21 | eleq2i 2145 | . . . . . . 7 |
23 | 21 | eleq2i 2145 | . . . . . . 7 |
24 | 22, 23 | anbi12i 447 | . . . . . 6 |
25 | 24 | anbi1i 445 | . . . . 5 |
26 | 25 | oprabbii 5580 | . . . 4 |
27 | 20, 26 | eqtri 2101 | . . 3 |
28 | 1, 2, 19, 27 | th3q 6234 | . 2 |
29 | oviec.1 | . . . 4 | |
30 | oviec.13 | . . . 4 | |
31 | 29, 30, 12 | ovi3 5657 | . . 3 |
32 | 31 | eceq1d 6165 | . 2 |
33 | 28, 32 | eqtrd 2113 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wceq 1284 wex 1421 wcel 1433 cvv 2601 cop 3401 class class class wbr 3785 copab 3838 cxp 4361 (class class class)co 5532 coprab 5533 wer 6126 cec 6127 cqs 6128 |
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 |
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-ral 2353 df-rex 2354 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-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fv 4930 df-ov 5535 df-oprab 5536 df-er 6129 df-ec 6131 df-qs 6135 |
This theorem is referenced by: addpipqqs 6560 mulpipqqs 6563 |
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