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Theorem ecopoverg 6207
 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.)
Hypotheses
Ref Expression
ecopopr.1
ecopoprg.com
ecopoprg.cl
ecopoprg.ass
ecopoprg.can
Assertion
Ref Expression
ecopoverg
Distinct variable groups:   ,,,,,,   ,,,,,,
Allowed substitution hints:   (,,,,,)

Proof of Theorem ecopoverg
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ecopopr.1 . . . . 5
21relopabi 4463 . . . 4
32a1i 9 . . 3
4 ecopoprg.com . . . . 5
51, 4ecopovsymg 6205 . . . 4
7 ecopoprg.cl . . . . 5
8 ecopoprg.ass . . . . 5
9 ecopoprg.can . . . . 5
101, 4, 7, 8, 9ecopovtrng 6206 . . . 4
124adantl 262 . . . . . . . . . . 11
13 simpll 481 . . . . . . . . . . 11
14 simplr 482 . . . . . . . . . . 11
1512, 13, 14caovcomd 5657 . . . . . . . . . 10
161ecopoveq 6201 . . . . . . . . . 10
1715, 16mpbird 156 . . . . . . . . 9
1817anidms 377 . . . . . . . 8
1918rgen2a 2375 . . . . . . 7
20 breq12 3769 . . . . . . . . 9
2120anidms 377 . . . . . . . 8
2221ralxp 4479 . . . . . . 7
2319, 22mpbir 134 . . . . . 6
2423rspec 2373 . . . . 5
2524a1i 9 . . . 4
26 opabssxp 4414 . . . . . . 7
271, 26eqsstri 2975 . . . . . 6
2827ssbri 3806 . . . . 5
29 brxp 4375 . . . . . 6
3029simplbi 259 . . . . 5
3128, 30syl 14 . . . 4
3225, 31impbid1 130 . . 3
333, 6, 11, 32iserd 6132 . 2
3433trud 1252 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 97   wb 98   w3a 885   wceq 1243   wtru 1244  wex 1381   wcel 1393  wral 2306  cop 3378   class class class wbr 3764  copab 3817   cxp 4343   wrel 4350  (class class class)co 5512   wer 6103 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-sbc 2765  df-csb 2853  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-iun 3659  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fv 4910  df-ov 5515  df-er 6106 This theorem is referenced by:  enqer  6456  enrer  6820
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