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Mirrors > Home > ILE Home > Th. List > ecopoverg | Unicode version |
Description: Assuming that operation is commutative (second hypothesis), closed (third hypothesis), associative (fourth hypothesis), and has the cancellation property (fifth hypothesis), show that the relation , specified by the first hypothesis, is an equivalence relation. (Contributed by Jim Kingdon, 1-Sep-2019.) |
Ref | Expression |
---|---|
ecopopr.1 | |
ecopoprg.com | |
ecopoprg.cl | |
ecopoprg.ass | |
ecopoprg.can |
Ref | Expression |
---|---|
ecopoverg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ecopopr.1 | . . . . 5 | |
2 | 1 | relopabi 4660 | . . . 4 |
3 | 2 | a1i 9 | . . 3 |
4 | ecopoprg.com | . . . . 5 | |
5 | 1, 4 | ecopovsymg 6521 | . . . 4 |
6 | 5 | adantl 275 | . . 3 |
7 | ecopoprg.cl | . . . . 5 | |
8 | ecopoprg.ass | . . . . 5 | |
9 | ecopoprg.can | . . . . 5 | |
10 | 1, 4, 7, 8, 9 | ecopovtrng 6522 | . . . 4 |
11 | 10 | adantl 275 | . . 3 |
12 | 4 | adantl 275 | . . . . . . . . . . 11 |
13 | simpll 518 | . . . . . . . . . . 11 | |
14 | simplr 519 | . . . . . . . . . . 11 | |
15 | 12, 13, 14 | caovcomd 5920 | . . . . . . . . . 10 |
16 | 1 | ecopoveq 6517 | . . . . . . . . . 10 |
17 | 15, 16 | mpbird 166 | . . . . . . . . 9 |
18 | 17 | anidms 394 | . . . . . . . 8 |
19 | 18 | rgen2a 2484 | . . . . . . 7 |
20 | breq12 3929 | . . . . . . . . 9 | |
21 | 20 | anidms 394 | . . . . . . . 8 |
22 | 21 | ralxp 4677 | . . . . . . 7 |
23 | 19, 22 | mpbir 145 | . . . . . 6 |
24 | 23 | rspec 2482 | . . . . 5 |
25 | 24 | a1i 9 | . . . 4 |
26 | opabssxp 4608 | . . . . . . 7 | |
27 | 1, 26 | eqsstri 3124 | . . . . . 6 |
28 | 27 | ssbri 3967 | . . . . 5 |
29 | brxp 4565 | . . . . . 6 | |
30 | 29 | simplbi 272 | . . . . 5 |
31 | 28, 30 | syl 14 | . . . 4 |
32 | 25, 31 | impbid1 141 | . . 3 |
33 | 3, 6, 11, 32 | iserd 6448 | . 2 |
34 | 33 | mptru 1340 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wceq 1331 wtru 1332 wex 1468 wcel 1480 wral 2414 cop 3525 class class class wbr 3924 copab 3983 cxp 4532 wrel 4539 (class class class)co 5767 wer 6419 |
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-sbc 2905 df-csb 2999 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-iun 3810 df-br 3925 df-opab 3985 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fv 5126 df-ov 5770 df-er 6422 |
This theorem is referenced by: enqer 7159 enrer 7536 |
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