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Mirrors > Home > ILE Home > Th. List > ecopovsymg | Unicode version |
Description: Assuming the operation is commutative, show that the relation , specified by the first hypothesis, is symmetric. (Contributed by Jim Kingdon, 1-Sep-2019.) |
Ref | Expression |
---|---|
ecopopr.1 | |
ecopoprg.com |
Ref | Expression |
---|---|
ecopovsymg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ecopopr.1 | . . . . 5 | |
2 | opabssxp 4683 | . . . . 5 | |
3 | 1, 2 | eqsstri 3179 | . . . 4 |
4 | 3 | brel 4661 | . . 3 |
5 | eqid 2170 | . . . 4 | |
6 | breq1 3990 | . . . . 5 | |
7 | breq2 3991 | . . . . 5 | |
8 | 6, 7 | bibi12d 234 | . . . 4 |
9 | breq2 3991 | . . . . 5 | |
10 | breq1 3990 | . . . . 5 | |
11 | 9, 10 | bibi12d 234 | . . . 4 |
12 | ecopoprg.com | . . . . . . . . 9 | |
13 | 12 | adantl 275 | . . . . . . . 8 |
14 | simpll 524 | . . . . . . . 8 | |
15 | simprr 527 | . . . . . . . 8 | |
16 | 13, 14, 15 | caovcomd 6007 | . . . . . . 7 |
17 | simplr 525 | . . . . . . . 8 | |
18 | simprl 526 | . . . . . . . 8 | |
19 | 13, 17, 18 | caovcomd 6007 | . . . . . . 7 |
20 | 16, 19 | eqeq12d 2185 | . . . . . 6 |
21 | eqcom 2172 | . . . . . 6 | |
22 | 20, 21 | bitrdi 195 | . . . . 5 |
23 | 1 | ecopoveq 6606 | . . . . 5 |
24 | 1 | ecopoveq 6606 | . . . . . 6 |
25 | 24 | ancoms 266 | . . . . 5 |
26 | 22, 23, 25 | 3bitr4d 219 | . . . 4 |
27 | 5, 8, 11, 26 | 2optocl 4686 | . . 3 |
28 | 4, 27 | syl 14 | . 2 |
29 | 28 | ibi 175 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1348 wex 1485 wcel 2141 cop 3584 class class class wbr 3987 copab 4047 cxp 4607 (class class class)co 5851 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-xp 4615 df-iota 5158 df-fv 5204 df-ov 5854 |
This theorem is referenced by: ecopoverg 6612 |
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