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Theorem com2i 24582
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
Allowed substitution hints:    G( a, b)    H( a, b)

Proof of Theorem com2i
StepHypRef Expression
1 com2i.1 . . . . . 6  |-  G  =  ( 1st `  R
)
21eqcomi 2257 . . . . 5  |-  ( 1st `  R )  =  G
32eqeq2i 2263 . . . 4  |-  ( g  =  ( 1st `  R
)  <->  g  =  G )
4 rneq 4811 . . . . . 6  |-  ( g  =  G  ->  ran  g  =  ran  G )
5 com2i.3 . . . . . 6  |-  X  =  ran  G
64, 5syl6eqr 2303 . . . . 5  |-  ( g  =  G  ->  ran  g  =  X )
7 raleq 2689 . . . . . 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 2696 . . . . 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 2257 . . . . . 6  |-  ( 2nd `  R )  =  H
1312eqeq2i 2263 . . . . 5  |-  ( h  =  ( 2nd `  R
)  <->  h  =  H
)
14 oveq 5716 . . . . . 6  |-  ( h  =  H  ->  (
a h b )  =  ( a H b ) )
15 oveq 5716 . . . . . 6  |-  ( h  =  H  ->  (
b h a )  =  ( b H a ) )
1614, 15eqeq12d 2267 . . . . 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 2547 . . 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 6037 . 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 20908 . 2  |-  Com2  =  { <. g ,  h >.  |  A. a  e. 
ran  g A. b  e.  ran  g ( a h b )  =  ( b h a ) }
2119, 20eleq2s 2345 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 1619    e. wcel 1621   A.wral 2509   {copab 3973   ran crn 4581   ` cfv 4592  (class class class)co 5710   1stc1st 5972   2ndc2nd 5973   Com2ccm2 20907
This theorem is referenced by:  fldi  24593
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fv 4608  df-ov 5713  df-1st 5974  df-2nd 5975  df-com2 20908
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