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Mirrors > Home > MPE Home > Th. List > ecovass | Structured version Visualization version Unicode version |
Description: Lemma used to transfer an associative law via an equivalence relation. (Contributed by NM, 31-Aug-1995.) (Revised by David Abernethy, 4-Jun-2013.) |
Ref | Expression |
---|---|
ecovass.1 | |
ecovass.2 | |
ecovass.3 | |
ecovass.4 | |
ecovass.5 | |
ecovass.6 | |
ecovass.7 | |
ecovass.8 | |
ecovass.9 |
Ref | Expression |
---|---|
ecovass |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ecovass.1 | . 2 | |
2 | oveq1 6657 | . . . 4 | |
3 | 2 | oveq1d 6665 | . . 3 |
4 | oveq1 6657 | . . 3 | |
5 | 3, 4 | eqeq12d 2637 | . 2 |
6 | oveq2 6658 | . . . 4 | |
7 | 6 | oveq1d 6665 | . . 3 |
8 | oveq1 6657 | . . . 4 | |
9 | 8 | oveq2d 6666 | . . 3 |
10 | 7, 9 | eqeq12d 2637 | . 2 |
11 | oveq2 6658 | . . 3 | |
12 | oveq2 6658 | . . . 4 | |
13 | 12 | oveq2d 6666 | . . 3 |
14 | 11, 13 | eqeq12d 2637 | . 2 |
15 | ecovass.8 | . . . 4 | |
16 | ecovass.9 | . . . 4 | |
17 | opeq12 4404 | . . . . 5 | |
18 | 17 | eceq1d 7783 | . . . 4 |
19 | 15, 16, 18 | mp2an 708 | . . 3 |
20 | ecovass.2 | . . . . . . 7 | |
21 | 20 | oveq1d 6665 | . . . . . 6 |
22 | 21 | adantr 481 | . . . . 5 |
23 | ecovass.6 | . . . . . 6 | |
24 | ecovass.4 | . . . . . 6 | |
25 | 23, 24 | sylan 488 | . . . . 5 |
26 | 22, 25 | eqtrd 2656 | . . . 4 |
27 | 26 | 3impa 1259 | . . 3 |
28 | ecovass.3 | . . . . . . 7 | |
29 | 28 | oveq2d 6666 | . . . . . 6 |
30 | 29 | adantl 482 | . . . . 5 |
31 | ecovass.7 | . . . . . 6 | |
32 | ecovass.5 | . . . . . 6 | |
33 | 31, 32 | sylan2 491 | . . . . 5 |
34 | 30, 33 | eqtrd 2656 | . . . 4 |
35 | 34 | 3impb 1260 | . . 3 |
36 | 19, 27, 35 | 3eqtr4a 2682 | . 2 |
37 | 1, 5, 10, 14, 36 | 3ecoptocl 7839 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 w3a 1037 wceq 1483 wcel 1990 cop 4183 cxp 5112 (class class class)co 6650 cec 7740 cqs 7741 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-xp 5120 df-cnv 5122 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fv 5896 df-ov 6653 df-ec 7744 df-qs 7748 |
This theorem is referenced by: addasssr 9909 mulasssr 9911 axaddass 9977 axmulass 9978 |
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