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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 5846 | . . 3 | |
3 | oveq1 5846 | . . . 4 | |
4 | oveq1 5846 | . . . 4 | |
5 | 3, 4 | oveq12d 5857 | . . 3 |
6 | 2, 5 | eqeq12d 2179 | . 2 |
7 | oveq1 5846 | . . . 4 | |
8 | 7 | oveq2d 5855 | . . 3 |
9 | oveq2 5847 | . . . 4 | |
10 | 9 | oveq1d 5854 | . . 3 |
11 | 8, 10 | eqeq12d 2179 | . 2 |
12 | oveq2 5847 | . . . 4 | |
13 | 12 | oveq2d 5855 | . . 3 |
14 | oveq2 5847 | . . . 4 | |
15 | 14 | oveq2d 5855 | . . 3 |
16 | 13, 15 | eqeq12d 2179 | . 2 |
17 | ecovidi.10 | . . . 4 | |
18 | ecovidi.11 | . . . 4 | |
19 | opeq12 3757 | . . . . 5 | |
20 | 19 | eceq1d 6531 | . . . 4 |
21 | 17, 18, 20 | syl2anc 409 | . . 3 |
22 | ecovidi.2 | . . . . . . 7 | |
23 | 22 | oveq2d 5855 | . . . . . 6 |
24 | 23 | adantl 275 | . . . . 5 |
25 | ecovidi.7 | . . . . . 6 | |
26 | ecovidi.3 | . . . . . 6 | |
27 | 25, 26 | sylan2 284 | . . . . 5 |
28 | 24, 27 | eqtrd 2197 | . . . 4 |
29 | 28 | 3impb 1188 | . . 3 |
30 | ecovidi.4 | . . . . . 6 | |
31 | ecovidi.5 | . . . . . 6 | |
32 | 30, 31 | oveqan12d 5858 | . . . . 5 |
33 | ecovidi.8 | . . . . . 6 | |
34 | ecovidi.9 | . . . . . 6 | |
35 | ecovidi.6 | . . . . . 6 | |
36 | 33, 34, 35 | syl2an 287 | . . . . 5 |
37 | 32, 36 | eqtrd 2197 | . . . 4 |
38 | 37 | 3impdi 1282 | . . 3 |
39 | 21, 29, 38 | 3eqtr4d 2207 | . 2 |
40 | 1, 6, 11, 16, 39 | 3ecoptocl 6584 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 967 wceq 1342 wcel 2135 cop 3576 cxp 4599 (class class class)co 5839 cec 6493 cqs 6494 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4097 ax-pow 4150 ax-pr 4184 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2726 df-un 3118 df-in 3120 df-ss 3127 df-pw 3558 df-sn 3579 df-pr 3580 df-op 3582 df-uni 3787 df-br 3980 df-opab 4041 df-xp 4607 df-cnv 4609 df-dm 4611 df-rn 4612 df-res 4613 df-ima 4614 df-iota 5150 df-fv 5193 df-ov 5842 df-ec 6497 df-qs 6501 |
This theorem is referenced by: distrnqg 7322 distrsrg 7694 axdistr 7809 |
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