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Theorem com2i 24769
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 4878 . . . . . 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 2708 . . . . . 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 2715 . . . . 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 5784 . . . . . 6  |-  ( h  =  H  ->  (
a h b )  =  ( a H b ) )
15 oveq 5784 . . . . . 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 2558 . . 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 6105 . 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 21024 . 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 4036   ran crn 4648   ` cfv 4659  (class class class)co 5778   1stc1st 6040   2ndc2nd 6041   Com2ccm2 21023
This theorem is referenced by:  fldi  24780
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 4101  ax-nul 4109  ax-pr 4172  ax-un 4470
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 2521  df-rex 2522  df-rab 2525  df-v 2759  df-sbc 2953  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fv 4675  df-ov 5781  df-1st 6042  df-2nd 6043  df-com2 21024
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