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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 5774 | . . . 4 | |
3 | 2 | oveq1d 5782 | . . 3 |
4 | oveq1 5774 | . . 3 | |
5 | 3, 4 | eqeq12d 2152 | . 2 |
6 | oveq2 5775 | . . . 4 | |
7 | 6 | oveq1d 5782 | . . 3 |
8 | oveq1 5774 | . . . 4 | |
9 | 8 | oveq2d 5783 | . . 3 |
10 | 7, 9 | eqeq12d 2152 | . 2 |
11 | oveq2 5775 | . . 3 | |
12 | oveq2 5775 | . . . 4 | |
13 | 12 | oveq2d 5783 | . . 3 |
14 | 11, 13 | eqeq12d 2152 | . 2 |
15 | ecoviass.8 | . . . 4 | |
16 | ecoviass.9 | . . . 4 | |
17 | opeq12 3702 | . . . . 5 | |
18 | 17 | eceq1d 6458 | . . . 4 |
19 | 15, 16, 18 | syl2anc 408 | . . 3 |
20 | ecoviass.2 | . . . . . . 7 | |
21 | 20 | oveq1d 5782 | . . . . . 6 |
22 | 21 | adantr 274 | . . . . 5 |
23 | ecoviass.6 | . . . . . 6 | |
24 | ecoviass.4 | . . . . . 6 | |
25 | 23, 24 | sylan 281 | . . . . 5 |
26 | 22, 25 | eqtrd 2170 | . . . 4 |
27 | 26 | 3impa 1176 | . . 3 |
28 | ecoviass.3 | . . . . . . 7 | |
29 | 28 | oveq2d 5783 | . . . . . 6 |
30 | 29 | adantl 275 | . . . . 5 |
31 | ecoviass.7 | . . . . . 6 | |
32 | ecoviass.5 | . . . . . 6 | |
33 | 31, 32 | sylan2 284 | . . . . 5 |
34 | 30, 33 | eqtrd 2170 | . . . 4 |
35 | 34 | 3impb 1177 | . . 3 |
36 | 19, 27, 35 | 3eqtr4d 2180 | . 2 |
37 | 1, 5, 10, 14, 36 | 3ecoptocl 6511 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 962 wceq 1331 wcel 1480 cop 3525 cxp 4532 (class class class)co 5767 cec 6420 cqs 6421 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-xp 4540 df-cnv 4542 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fv 5126 df-ov 5770 df-ec 6424 df-qs 6428 |
This theorem is referenced by: addassnqg 7183 mulassnqg 7185 addasssrg 7557 mulasssrg 7559 axaddass 7673 axmulass 7674 |
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