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Theorem com2i 24748
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 2260 . . . . 5  |-  ( 1st `  R )  =  G
32eqeq2i 2266 . . . 4  |-  ( g  =  ( 1st `  R
)  <->  g  =  G )
4 rneq 4857 . . . . . 6  |-  ( g  =  G  ->  ran  g  =  ran  G )
5 com2i.3 . . . . . 6  |-  X  =  ran  G
64, 5syl6eqr 2306 . . . . 5  |-  ( g  =  G  ->  ran  g  =  X )
7 raleq 2698 . . . . . 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 2705 . . . . 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 2260 . . . . . 6  |-  ( 2nd `  R )  =  H
1312eqeq2i 2266 . . . . 5  |-  ( h  =  ( 2nd `  R
)  <->  h  =  H
)
14 oveq 5763 . . . . . 6  |-  ( h  =  H  ->  (
a h b )  =  ( a H b ) )
15 oveq 5763 . . . . . 6  |-  ( h  =  H  ->  (
b h a )  =  ( b H a ) )
1614, 15eqeq12d 2270 . . . . 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 2556 . . 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 6084 . 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 21003 . 2  |-  Com2  =  { <. g ,  h >.  |  A. a  e. 
ran  g A. b  e.  ran  g ( a h b )  =  ( b h a ) }
2119, 20eleq2s 2348 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 2516   {copab 4016   ran crn 4627   ` cfv 4638  (class class class)co 5757   1stc1st 6019   2ndc2nd 6020   Com2ccm2 21002
This theorem is referenced by:  fldi  24759
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 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pr 4152  ax-un 4449
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 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-sbc 2936  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fv 4654  df-ov 5760  df-1st 6021  df-2nd 6022  df-com2 21003
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