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Mirrors > Home > ILE Home > Th. List > ecoviass | Unicode version |
Description: Lemma used to transfer an associative law via an equivalence relation. (Contributed by Jim Kingdon, 16-Sep-2019.) |
Ref | Expression |
---|---|
ecoviass.1 | |
ecoviass.2 | |
ecoviass.3 | |
ecoviass.4 | |
ecoviass.5 | |
ecoviass.6 | |
ecoviass.7 | |
ecoviass.8 | |
ecoviass.9 |
Ref | Expression |
---|---|
ecoviass |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ecoviass.1 | . 2 | |
2 | oveq1 5843 | . . . 4 | |
3 | 2 | oveq1d 5851 | . . 3 |
4 | oveq1 5843 | . . 3 | |
5 | 3, 4 | eqeq12d 2179 | . 2 |
6 | oveq2 5844 | . . . 4 | |
7 | 6 | oveq1d 5851 | . . 3 |
8 | oveq1 5843 | . . . 4 | |
9 | 8 | oveq2d 5852 | . . 3 |
10 | 7, 9 | eqeq12d 2179 | . 2 |
11 | oveq2 5844 | . . 3 | |
12 | oveq2 5844 | . . . 4 | |
13 | 12 | oveq2d 5852 | . . 3 |
14 | 11, 13 | eqeq12d 2179 | . 2 |
15 | ecoviass.8 | . . . 4 | |
16 | ecoviass.9 | . . . 4 | |
17 | opeq12 3754 | . . . . 5 | |
18 | 17 | eceq1d 6528 | . . . 4 |
19 | 15, 16, 18 | syl2anc 409 | . . 3 |
20 | ecoviass.2 | . . . . . . 7 | |
21 | 20 | oveq1d 5851 | . . . . . 6 |
22 | 21 | adantr 274 | . . . . 5 |
23 | ecoviass.6 | . . . . . 6 | |
24 | ecoviass.4 | . . . . . 6 | |
25 | 23, 24 | sylan 281 | . . . . 5 |
26 | 22, 25 | eqtrd 2197 | . . . 4 |
27 | 26 | 3impa 1183 | . . 3 |
28 | ecoviass.3 | . . . . . . 7 | |
29 | 28 | oveq2d 5852 | . . . . . 6 |
30 | 29 | adantl 275 | . . . . 5 |
31 | ecoviass.7 | . . . . . 6 | |
32 | ecoviass.5 | . . . . . 6 | |
33 | 31, 32 | sylan2 284 | . . . . 5 |
34 | 30, 33 | eqtrd 2197 | . . . 4 |
35 | 34 | 3impb 1188 | . . 3 |
36 | 19, 27, 35 | 3eqtr4d 2207 | . 2 |
37 | 1, 5, 10, 14, 36 | 3ecoptocl 6581 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 967 wceq 1342 wcel 2135 cop 3573 cxp 4596 (class class class)co 5836 cec 6490 cqs 6491 |
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 4094 ax-pow 4147 ax-pr 4181 |
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 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-xp 4604 df-cnv 4606 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fv 5190 df-ov 5839 df-ec 6494 df-qs 6498 |
This theorem is referenced by: addassnqg 7314 mulassnqg 7316 addasssrg 7688 mulasssrg 7690 axaddass 7804 axmulass 7805 |
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