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Theorem ecopoverg 6273
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  |-  .~  =  { <. x ,  y
>.  |  ( (
x  e.  ( S  X.  S )  /\  y  e.  ( S  X.  S ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ( z 
.+  u )  =  ( w  .+  v
) ) ) }
ecopoprg.com  |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( x  .+  y
)  =  ( y 
.+  x ) )
ecopoprg.cl  |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( x  .+  y
)  e.  S )
ecopoprg.ass  |-  ( ( x  e.  S  /\  y  e.  S  /\  z  e.  S )  ->  ( ( x  .+  y )  .+  z
)  =  ( x 
.+  ( y  .+  z ) ) )
ecopoprg.can  |-  ( ( x  e.  S  /\  y  e.  S  /\  z  e.  S )  ->  ( ( x  .+  y )  =  ( x  .+  z )  ->  y  =  z ) )
Assertion
Ref Expression
ecopoverg  |-  .~  Er  ( S  X.  S
)
Distinct variable groups:    x, y, z, w, v, u,  .+    x, S, y, z, w, v, u
Allowed substitution hints:    .~ ( x, y,
z, w, v, u)

Proof of Theorem ecopoverg
Dummy variables  f  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ecopopr.1 . . . . 5  |-  .~  =  { <. x ,  y
>.  |  ( (
x  e.  ( S  X.  S )  /\  y  e.  ( S  X.  S ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ( z 
.+  u )  =  ( w  .+  v
) ) ) }
21relopabi 4491 . . . 4  |-  Rel  .~
32a1i 9 . . 3  |-  ( T. 
->  Rel  .~  )
4 ecopoprg.com . . . . 5  |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( x  .+  y
)  =  ( y 
.+  x ) )
51, 4ecopovsymg 6271 . . . 4  |-  ( f  .~  g  ->  g  .~  f )
65adantl 271 . . 3  |-  ( ( T.  /\  f  .~  g )  ->  g  .~  f )
7 ecopoprg.cl . . . . 5  |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( x  .+  y
)  e.  S )
8 ecopoprg.ass . . . . 5  |-  ( ( x  e.  S  /\  y  e.  S  /\  z  e.  S )  ->  ( ( x  .+  y )  .+  z
)  =  ( x 
.+  ( y  .+  z ) ) )
9 ecopoprg.can . . . . 5  |-  ( ( x  e.  S  /\  y  e.  S  /\  z  e.  S )  ->  ( ( x  .+  y )  =  ( x  .+  z )  ->  y  =  z ) )
101, 4, 7, 8, 9ecopovtrng 6272 . . . 4  |-  ( ( f  .~  g  /\  g  .~  h )  -> 
f  .~  h )
1110adantl 271 . . 3  |-  ( ( T.  /\  ( f  .~  g  /\  g  .~  h ) )  -> 
f  .~  h )
124adantl 271 . . . . . . . . . . 11  |-  ( ( ( ( g  e.  S  /\  h  e.  S )  /\  (
g  e.  S  /\  h  e.  S )
)  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  =  ( y 
.+  x ) )
13 simpll 496 . . . . . . . . . . 11  |-  ( ( ( g  e.  S  /\  h  e.  S
)  /\  ( g  e.  S  /\  h  e.  S ) )  -> 
g  e.  S )
14 simplr 497 . . . . . . . . . . 11  |-  ( ( ( g  e.  S  /\  h  e.  S
)  /\  ( g  e.  S  /\  h  e.  S ) )  ->  h  e.  S )
1512, 13, 14caovcomd 5688 . . . . . . . . . 10  |-  ( ( ( g  e.  S  /\  h  e.  S
)  /\  ( g  e.  S  /\  h  e.  S ) )  -> 
( g  .+  h
)  =  ( h 
.+  g ) )
161ecopoveq 6267 . . . . . . . . . 10  |-  ( ( ( g  e.  S  /\  h  e.  S
)  /\  ( g  e.  S  /\  h  e.  S ) )  -> 
( <. g ,  h >.  .~  <. g ,  h >.  <-> 
( g  .+  h
)  =  ( h 
.+  g ) ) )
1715, 16mpbird 165 . . . . . . . . 9  |-  ( ( ( g  e.  S  /\  h  e.  S
)  /\  ( g  e.  S  /\  h  e.  S ) )  ->  <. g ,  h >.  .~ 
<. g ,  h >. )
1817anidms 389 . . . . . . . 8  |-  ( ( g  e.  S  /\  h  e.  S )  -> 
<. g ,  h >.  .~ 
<. g ,  h >. )
1918rgen2a 2418 . . . . . . 7  |-  A. g  e.  S  A. h  e.  S  <. g ,  h >.  .~  <. g ,  h >.
20 breq12 3798 . . . . . . . . 9  |-  ( ( f  =  <. g ,  h >.  /\  f  =  <. g ,  h >. )  ->  ( f  .~  f  <->  <. g ,  h >.  .~  <. g ,  h >. ) )
2120anidms 389 . . . . . . . 8  |-  ( f  =  <. g ,  h >.  ->  ( f  .~  f 
<-> 
<. g ,  h >.  .~ 
<. g ,  h >. ) )
2221ralxp 4507 . . . . . . 7  |-  ( A. f  e.  ( S  X.  S ) f  .~  f 
<-> 
A. g  e.  S  A. h  e.  S  <. g ,  h >.  .~ 
<. g ,  h >. )
2319, 22mpbir 144 . . . . . 6  |-  A. f  e.  ( S  X.  S
) f  .~  f
2423rspec 2416 . . . . 5  |-  ( f  e.  ( S  X.  S )  ->  f  .~  f )
2524a1i 9 . . . 4  |-  ( T. 
->  ( f  e.  ( S  X.  S )  ->  f  .~  f
) )
26 opabssxp 4440 . . . . . . 7  |-  { <. x ,  y >.  |  ( ( x  e.  ( S  X.  S )  /\  y  e.  ( S  X.  S ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ( z  .+  u
)  =  ( w 
.+  v ) ) ) }  C_  (
( S  X.  S
)  X.  ( S  X.  S ) )
271, 26eqsstri 3030 . . . . . 6  |-  .~  C_  (
( S  X.  S
)  X.  ( S  X.  S ) )
2827ssbri 3835 . . . . 5  |-  ( f  .~  f  ->  f
( ( S  X.  S )  X.  ( S  X.  S ) ) f )
29 brxp 4401 . . . . . 6  |-  ( f ( ( S  X.  S )  X.  ( S  X.  S ) ) f  <->  ( f  e.  ( S  X.  S
)  /\  f  e.  ( S  X.  S
) ) )
3029simplbi 268 . . . . 5  |-  ( f ( ( S  X.  S )  X.  ( S  X.  S ) ) f  ->  f  e.  ( S  X.  S
) )
3128, 30syl 14 . . . 4  |-  ( f  .~  f  ->  f  e.  ( S  X.  S
) )
3225, 31impbid1 140 . . 3  |-  ( T. 
->  ( f  e.  ( S  X.  S )  <-> 
f  .~  f )
)
333, 6, 11, 32iserd 6198 . 2  |-  ( T. 
->  .~  Er  ( S  X.  S ) )
3433trud 1294 1  |-  .~  Er  ( S  X.  S
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 920    = wceq 1285   T. wtru 1286   E.wex 1422    e. wcel 1434   A.wral 2349   <.cop 3409   class class class wbr 3793   {copab 3846    X. cxp 4369   Rel wrel 4376  (class class class)co 5543    Er wer 6169
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-csb 2910  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-iun 3688  df-br 3794  df-opab 3848  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-iota 4897  df-fv 4940  df-ov 5546  df-er 6172
This theorem is referenced by:  enqer  6610  enrer  6974
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