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Mirrors > Home > ILE Home > Th. List > ecovidi | Unicode version |
Description: Lemma used to transfer a distributive law via an equivalence relation. (Contributed by Jim Kingdon, 17-Sep-2019.) |
Ref | Expression |
---|---|
ecovidi.1 | |
ecovidi.2 | |
ecovidi.3 | |
ecovidi.4 | |
ecovidi.5 | |
ecovidi.6 | |
ecovidi.7 | |
ecovidi.8 | |
ecovidi.9 | |
ecovidi.10 | |
ecovidi.11 |
Ref | Expression |
---|---|
ecovidi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ecovidi.1 | . 2 | |
2 | oveq1 5781 | . . 3 | |
3 | oveq1 5781 | . . . 4 | |
4 | oveq1 5781 | . . . 4 | |
5 | 3, 4 | oveq12d 5792 | . . 3 |
6 | 2, 5 | eqeq12d 2154 | . 2 |
7 | oveq1 5781 | . . . 4 | |
8 | 7 | oveq2d 5790 | . . 3 |
9 | oveq2 5782 | . . . 4 | |
10 | 9 | oveq1d 5789 | . . 3 |
11 | 8, 10 | eqeq12d 2154 | . 2 |
12 | oveq2 5782 | . . . 4 | |
13 | 12 | oveq2d 5790 | . . 3 |
14 | oveq2 5782 | . . . 4 | |
15 | 14 | oveq2d 5790 | . . 3 |
16 | 13, 15 | eqeq12d 2154 | . 2 |
17 | ecovidi.10 | . . . 4 | |
18 | ecovidi.11 | . . . 4 | |
19 | opeq12 3707 | . . . . 5 | |
20 | 19 | eceq1d 6465 | . . . 4 |
21 | 17, 18, 20 | syl2anc 408 | . . 3 |
22 | ecovidi.2 | . . . . . . 7 | |
23 | 22 | oveq2d 5790 | . . . . . 6 |
24 | 23 | adantl 275 | . . . . 5 |
25 | ecovidi.7 | . . . . . 6 | |
26 | ecovidi.3 | . . . . . 6 | |
27 | 25, 26 | sylan2 284 | . . . . 5 |
28 | 24, 27 | eqtrd 2172 | . . . 4 |
29 | 28 | 3impb 1177 | . . 3 |
30 | ecovidi.4 | . . . . . 6 | |
31 | ecovidi.5 | . . . . . 6 | |
32 | 30, 31 | oveqan12d 5793 | . . . . 5 |
33 | ecovidi.8 | . . . . . 6 | |
34 | ecovidi.9 | . . . . . 6 | |
35 | ecovidi.6 | . . . . . 6 | |
36 | 33, 34, 35 | syl2an 287 | . . . . 5 |
37 | 32, 36 | eqtrd 2172 | . . . 4 |
38 | 37 | 3impdi 1271 | . . 3 |
39 | 21, 29, 38 | 3eqtr4d 2182 | . 2 |
40 | 1, 6, 11, 16, 39 | 3ecoptocl 6518 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 962 wceq 1331 wcel 1480 cop 3530 cxp 4537 (class class class)co 5774 cec 6427 cqs 6428 |
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-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-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 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 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-xp 4545 df-cnv 4547 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fv 5131 df-ov 5777 df-ec 6431 df-qs 6435 |
This theorem is referenced by: distrnqg 7195 distrsrg 7567 axdistr 7682 |
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