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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 5539 | . . . 4 | |
3 | 2 | oveq1d 5547 | . . 3 |
4 | oveq1 5539 | . . 3 | |
5 | 3, 4 | eqeq12d 2095 | . 2 |
6 | oveq2 5540 | . . . 4 | |
7 | 6 | oveq1d 5547 | . . 3 |
8 | oveq1 5539 | . . . 4 | |
9 | 8 | oveq2d 5548 | . . 3 |
10 | 7, 9 | eqeq12d 2095 | . 2 |
11 | oveq2 5540 | . . 3 | |
12 | oveq2 5540 | . . . 4 | |
13 | 12 | oveq2d 5548 | . . 3 |
14 | 11, 13 | eqeq12d 2095 | . 2 |
15 | ecoviass.8 | . . . 4 | |
16 | ecoviass.9 | . . . 4 | |
17 | opeq12 3572 | . . . . 5 | |
18 | 17 | eceq1d 6165 | . . . 4 |
19 | 15, 16, 18 | syl2anc 403 | . . 3 |
20 | ecoviass.2 | . . . . . . 7 | |
21 | 20 | oveq1d 5547 | . . . . . 6 |
22 | 21 | adantr 270 | . . . . 5 |
23 | ecoviass.6 | . . . . . 6 | |
24 | ecoviass.4 | . . . . . 6 | |
25 | 23, 24 | sylan 277 | . . . . 5 |
26 | 22, 25 | eqtrd 2113 | . . . 4 |
27 | 26 | 3impa 1133 | . . 3 |
28 | ecoviass.3 | . . . . . . 7 | |
29 | 28 | oveq2d 5548 | . . . . . 6 |
30 | 29 | adantl 271 | . . . . 5 |
31 | ecoviass.7 | . . . . . 6 | |
32 | ecoviass.5 | . . . . . 6 | |
33 | 31, 32 | sylan2 280 | . . . . 5 |
34 | 30, 33 | eqtrd 2113 | . . . 4 |
35 | 34 | 3impb 1134 | . . 3 |
36 | 19, 27, 35 | 3eqtr4d 2123 | . 2 |
37 | 1, 5, 10, 14, 36 | 3ecoptocl 6218 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 w3a 919 wceq 1284 wcel 1433 cop 3401 cxp 4361 (class class class)co 5532 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-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-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 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 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-xp 4369 df-cnv 4371 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fv 4930 df-ov 5535 df-ec 6131 df-qs 6135 |
This theorem is referenced by: addassnqg 6572 mulassnqg 6574 addasssrg 6933 mulasssrg 6935 axaddass 7038 axmulass 7039 |
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