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Theorem ecopoverg 6143
Description: Assuming that operation  F 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  .~  { <. , 
>.  |  S  X.  S  S  X.  S 
<. ,  >.  <. ,  >.  .+  .+  }
ecopoprg.com  S  S  .+  .+
ecopoprg.cl  S  S  .+  S
ecopoprg.ass  S  S  S  .+  .+  .+  .+
ecopoprg.can  S  S  S  .+  .+
Assertion
Ref Expression
ecopoverg  .~  Er  S  X.  S
Distinct variable groups:   ,,,,,,  .+   , S,,,,,
Allowed substitution hints:    .~ (,,,,,)

Proof of Theorem ecopoverg
Dummy variables  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ecopopr.1 . . . . 5  .~  { <. , 
>.  |  S  X.  S  S  X.  S 
<. ,  >.  <. ,  >.  .+  .+  }
21relopabi 4406 . . . 4  Rel  .~
32a1i 9 . . 3  Rel  .~
4 ecopoprg.com . . . . 5  S  S  .+  .+
51, 4ecopovsymg 6141 . . . 4 
.~  .~
65adantl 262 . . 3  .~  .~
7 ecopoprg.cl . . . . 5  S  S  .+  S
8 ecopoprg.ass . . . . 5  S  S  S  .+  .+  .+  .+
9 ecopoprg.can . . . . 5  S  S  S  .+  .+
101, 4, 7, 8, 9ecopovtrng 6142 . . . 4  .~  .~  h  .~  h
1110adantl 262 . . 3 
.~  .~  h  .~  h
124adantl 262 . . . . . . . . . . 11  S  h  S  S  h  S  S  S  .+  .+
13 simpll 481 . . . . . . . . . . 11  S  h  S  S  h  S  S
14 simplr 482 . . . . . . . . . . 11  S  h  S  S  h  S 
h  S
1512, 13, 14caovcomd 5599 . . . . . . . . . 10  S  h  S  S  h  S  .+  h  h  .+
161ecopoveq 6137 . . . . . . . . . 10  S  h  S  S  h  S  <. ,  h >.  .~  <. ,  h >.  .+  h  h  .+
1715, 16mpbird 156 . . . . . . . . 9  S  h  S  S  h  S  <. ,  h >.  .~ 
<. ,  h >.
1817anidms 377 . . . . . . . 8  S  h  S  <. ,  h >.  .~  <. ,  h >.
1918rgen2a 2369 . . . . . . 7  S  h  S  <. ,  h >.  .~  <. ,  h >.
20 breq12 3760 . . . . . . . . 9  <. ,  h >.  <. ,  h >. 
.~  <. ,  h >.  .~  <. ,  h >.
2120anidms 377 . . . . . . . 8  <. ,  h >.  .~ 
<. ,  h >.  .~ 
<. ,  h >.
2221ralxp 4422 . . . . . . 7  S  X.  S  .~  S  h  S  <. ,  h >.  .~ 
<. ,  h >.
2319, 22mpbir 134 . . . . . 6  S  X.  S  .~
2423rspec 2367 . . . . 5  S  X.  S  .~
2524a1i 9 . . . 4  S  X.  S  .~
26 opabssxp 4357 . . . . . . 7  { <. ,  >.  |  S  X.  S  S  X.  S  <. ,  >.  <. ,  >. 
.+  .+  }  C_  S  X.  S  X.  S  X.  S
271, 26eqsstri 2969 . . . . . 6  .~  C_  S  X.  S  X.  S  X.  S
2827ssbri 3797 . . . . 5 
.~  S  X.  S  X.  S  X.  S
29 brxp 4318 . . . . . 6  S  X.  S  X.  S  X.  S  S  X.  S  S  X.  S
3029simplbi 259 . . . . 5  S  X.  S  X.  S  X.  S  S  X.  S
3128, 30syl 14 . . . 4 
.~  S  X.  S
3225, 31impbid1 130 . . 3  S  X.  S  .~
333, 6, 11, 32iserd 6068 . 2  .~  Er  S  X.  S
3433trud 1251 1  .~  Er  S  X.  S
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   w3a 884   wceq 1242   wtru 1243  wex 1378   wcel 1390  wral 2300   <.cop 3370   class class class wbr 3755   {copab 3808    X. cxp 4286   Rel wrel 4293  (class class class)co 5455    Er wer 6039
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fv 4853  df-ov 5458  df-er 6042
This theorem is referenced by:  enqer  6342  enrer  6623
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