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Theorem com2i 24815
Description: Property of a commutative structure with two operation. (Contributed by FL, 14-Feb-2010.)
Hypotheses
Ref Expression
com2i.1  |-  G  =  ( 1st `  R
)
com2i.2  |-  H  =  ( 2nd `  R
)
com2i.3  |-  X  =  ran  G
Assertion
Ref Expression
com2i  |-  ( R  e.  Com2  ->  A. a  e.  X  A. b  e.  X  ( a H b )  =  ( b H a ) )
Distinct variable groups:    R, a,
b    X, a, b
Dummy variables  g  h are mutually distinct and distinct from all other variables.
Allowed substitution hints:    G( a, b)    H( a, b)

Proof of Theorem com2i
StepHypRef Expression
1 com2i.1 . . . . . 6  |-  G  =  ( 1st `  R
)
21eqcomi 2288 . . . . 5  |-  ( 1st `  R )  =  G
32eqeq2i 2294 . . . 4  |-  ( g  =  ( 1st `  R
)  <->  g  =  G )
4 rneq 4903 . . . . . 6  |-  ( g  =  G  ->  ran  g  =  ran  G )
5 com2i.3 . . . . . 6  |-  X  =  ran  G
64, 5syl6eqr 2334 . . . . 5  |-  ( g  =  G  ->  ran  g  =  X )
7 raleq 2737 . . . . . 6  |-  ( ran  g  =  X  -> 
( A. b  e. 
ran  g ( a h b )  =  ( b h a )  <->  A. b  e.  X  ( a h b )  =  ( b h a ) ) )
87raleqbi1dv 2745 . . . . 5  |-  ( ran  g  =  X  -> 
( A. a  e. 
ran  g A. b  e.  ran  g ( a h b )  =  ( b h a )  <->  A. a  e.  X  A. b  e.  X  ( a h b )  =  ( b h a ) ) )
96, 8syl 17 . . . 4  |-  ( g  =  G  ->  ( A. a  e.  ran  g A. b  e.  ran  g ( a h b )  =  ( b h a )  <->  A. a  e.  X  A. b  e.  X  ( a h b )  =  ( b h a ) ) )
103, 9sylbi 189 . . 3  |-  ( g  =  ( 1st `  R
)  ->  ( A. a  e.  ran  g A. b  e.  ran  g ( a h b )  =  ( b h a )  <->  A. a  e.  X  A. b  e.  X  ( a
h b )  =  ( b h a ) ) )
11 com2i.2 . . . . . . 7  |-  H  =  ( 2nd `  R
)
1211eqcomi 2288 . . . . . 6  |-  ( 2nd `  R )  =  H
1312eqeq2i 2294 . . . . 5  |-  ( h  =  ( 2nd `  R
)  <->  h  =  H
)
14 oveq 5825 . . . . . 6  |-  ( h  =  H  ->  (
a h b )  =  ( a H b ) )
15 oveq 5825 . . . . . 6  |-  ( h  =  H  ->  (
b h a )  =  ( b H a ) )
1614, 15eqeq12d 2298 . . . . 5  |-  ( h  =  H  ->  (
( a h b )  =  ( b h a )  <->  ( a H b )  =  ( b H a ) ) )
1713, 16sylbi 189 . . . 4  |-  ( h  =  ( 2nd `  R
)  ->  ( (
a h b )  =  ( b h a )  <->  ( a H b )  =  ( b H a ) ) )
18172ralbidv 2586 . . 3  |-  ( h  =  ( 2nd `  R
)  ->  ( A. a  e.  X  A. b  e.  X  (
a h b )  =  ( b h a )  <->  A. a  e.  X  A. b  e.  X  ( a H b )  =  ( b H a ) ) )
1910, 18elopabi 6146 . 2  |-  ( R  e.  { <. g ,  h >.  |  A. a  e.  ran  g A. b  e.  ran  g ( a h b )  =  ( b h a ) }  ->  A. a  e.  X  A. b  e.  X  (
a H b )  =  ( b H a ) )
20 df-com2 21070 . 2  |-  Com2  =  { <. g ,  h >.  |  A. a  e. 
ran  g A. b  e.  ran  g ( a h b )  =  ( b h a ) }
2119, 20eleq2s 2376 1  |-  ( R  e.  Com2  ->  A. a  e.  X  A. b  e.  X  ( a H b )  =  ( b H a ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    = wceq 1624    e. wcel 1685   A.wral 2544   {copab 4077   ran crn 4689   ` cfv 5221  (class class class)co 5819   1stc1st 6081   2ndc2nd 6082   Com2ccm2 21069
This theorem is referenced by:  fldi  24826
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fv 5229  df-ov 5822  df-1st 6083  df-2nd 6084  df-com2 21070
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