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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 4608 | . . . . 5 | |
3 | 1, 2 | eqsstri 3124 | . . . 4 |
4 | 3 | brel 4586 | . . 3 |
5 | eqid 2137 | . . . 4 | |
6 | breq1 3927 | . . . . 5 | |
7 | breq2 3928 | . . . . 5 | |
8 | 6, 7 | bibi12d 234 | . . . 4 |
9 | breq2 3928 | . . . . 5 | |
10 | breq1 3927 | . . . . 5 | |
11 | 9, 10 | bibi12d 234 | . . . 4 |
12 | ecopoprg.com | . . . . . . . . 9 | |
13 | 12 | adantl 275 | . . . . . . . 8 |
14 | simpll 518 | . . . . . . . 8 | |
15 | simprr 521 | . . . . . . . 8 | |
16 | 13, 14, 15 | caovcomd 5920 | . . . . . . 7 |
17 | simplr 519 | . . . . . . . 8 | |
18 | simprl 520 | . . . . . . . 8 | |
19 | 13, 17, 18 | caovcomd 5920 | . . . . . . 7 |
20 | 16, 19 | eqeq12d 2152 | . . . . . 6 |
21 | eqcom 2139 | . . . . . 6 | |
22 | 20, 21 | syl6bb 195 | . . . . 5 |
23 | 1 | ecopoveq 6517 | . . . . 5 |
24 | 1 | ecopoveq 6517 | . . . . . 6 |
25 | 24 | ancoms 266 | . . . . 5 |
26 | 22, 23, 25 | 3bitr4d 219 | . . . 4 |
27 | 5, 8, 11, 26 | 2optocl 4611 | . . 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 1331 wex 1468 wcel 1480 cop 3525 class class class wbr 3924 copab 3983 cxp 4532 (class class class)co 5767 |
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-eu 2000 df-mo 2001 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-iota 5083 df-fv 5126 df-ov 5770 |
This theorem is referenced by: ecopoverg 6523 |
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