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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 4414 | . . . . 5 | |
3 | 1, 2 | eqsstri 2975 | . . . 4 |
4 | 3 | brel 4392 | . . 3 |
5 | eqid 2040 | . . . 4 | |
6 | breq1 3767 | . . . . 5 | |
7 | breq2 3768 | . . . . 5 | |
8 | 6, 7 | bibi12d 224 | . . . 4 |
9 | breq2 3768 | . . . . 5 | |
10 | breq1 3767 | . . . . 5 | |
11 | 9, 10 | bibi12d 224 | . . . 4 |
12 | ecopoprg.com | . . . . . . . . 9 | |
13 | 12 | adantl 262 | . . . . . . . 8 |
14 | simpll 481 | . . . . . . . 8 | |
15 | simprr 484 | . . . . . . . 8 | |
16 | 13, 14, 15 | caovcomd 5657 | . . . . . . 7 |
17 | simplr 482 | . . . . . . . 8 | |
18 | simprl 483 | . . . . . . . 8 | |
19 | 13, 17, 18 | caovcomd 5657 | . . . . . . 7 |
20 | 16, 19 | eqeq12d 2054 | . . . . . 6 |
21 | eqcom 2042 | . . . . . 6 | |
22 | 20, 21 | syl6bb 185 | . . . . 5 |
23 | 1 | ecopoveq 6201 | . . . . 5 |
24 | 1 | ecopoveq 6201 | . . . . . 6 |
25 | 24 | ancoms 255 | . . . . 5 |
26 | 22, 23, 25 | 3bitr4d 209 | . . . 4 |
27 | 5, 8, 11, 26 | 2optocl 4417 | . . 3 |
28 | 4, 27 | syl 14 | . 2 |
29 | 28 | ibi 165 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wex 1381 wcel 1393 cop 3378 class class class wbr 3764 copab 3817 cxp 4343 (class class class)co 5512 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-xp 4351 df-iota 4867 df-fv 4910 df-ov 5515 |
This theorem is referenced by: ecopoverg 6207 |
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