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Mirrors > Home > ILE Home > Th. List > ecovdi | Unicode version |
Description: Lemma used to transfer a distributive law via an equivalence relation. Most likely ecovidi 6218 will be more helpful. (Contributed by NM, 2-Sep-1995.) (Revised by David Abernethy, 4-Jun-2013.) |
Ref | Expression |
---|---|
ecovdi.1 | |
ecovdi.2 | |
ecovdi.3 | |
ecovdi.4 | |
ecovdi.5 | |
ecovdi.6 | |
ecovdi.7 | |
ecovdi.8 | |
ecovdi.9 | |
ecovdi.10 | |
ecovdi.11 |
Ref | Expression |
---|---|
ecovdi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ecovdi.1 | . 2 | |
2 | oveq1 5519 | . . 3 | |
3 | oveq1 5519 | . . . 4 | |
4 | oveq1 5519 | . . . 4 | |
5 | 3, 4 | oveq12d 5530 | . . 3 |
6 | 2, 5 | eqeq12d 2054 | . 2 |
7 | oveq1 5519 | . . . 4 | |
8 | 7 | oveq2d 5528 | . . 3 |
9 | oveq2 5520 | . . . 4 | |
10 | 9 | oveq1d 5527 | . . 3 |
11 | 8, 10 | eqeq12d 2054 | . 2 |
12 | oveq2 5520 | . . . 4 | |
13 | 12 | oveq2d 5528 | . . 3 |
14 | oveq2 5520 | . . . 4 | |
15 | 14 | oveq2d 5528 | . . 3 |
16 | 13, 15 | eqeq12d 2054 | . 2 |
17 | ecovdi.10 | . . . 4 | |
18 | ecovdi.11 | . . . 4 | |
19 | opeq12 3551 | . . . . 5 | |
20 | 19 | eceq1d 6142 | . . . 4 |
21 | 17, 18, 20 | mp2an 402 | . . 3 |
22 | ecovdi.2 | . . . . . . 7 | |
23 | 22 | oveq2d 5528 | . . . . . 6 |
24 | 23 | adantl 262 | . . . . 5 |
25 | ecovdi.7 | . . . . . 6 | |
26 | ecovdi.3 | . . . . . 6 | |
27 | 25, 26 | sylan2 270 | . . . . 5 |
28 | 24, 27 | eqtrd 2072 | . . . 4 |
29 | 28 | 3impb 1100 | . . 3 |
30 | ecovdi.4 | . . . . . 6 | |
31 | ecovdi.5 | . . . . . 6 | |
32 | 30, 31 | oveqan12d 5531 | . . . . 5 |
33 | ecovdi.8 | . . . . . 6 | |
34 | ecovdi.9 | . . . . . 6 | |
35 | ecovdi.6 | . . . . . 6 | |
36 | 33, 34, 35 | syl2an 273 | . . . . 5 |
37 | 32, 36 | eqtrd 2072 | . . . 4 |
38 | 37 | 3impdi 1190 | . . 3 |
39 | 21, 29, 38 | 3eqtr4a 2098 | . 2 |
40 | 1, 6, 11, 16, 39 | 3ecoptocl 6195 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 w3a 885 wceq 1243 wcel 1393 cop 3378 cxp 4343 (class class class)co 5512 cec 6104 cqs 6105 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-xp 4351 df-cnv 4353 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fv 4910 df-ov 5515 df-ec 6108 df-qs 6112 |
This theorem is referenced by: (None) |
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