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Theorem com2i 25519
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
Dummy variables  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 com2i.1 . . . . . 6  |-  G  =  ( 1st `  R
)
21eqcomi 2300 . . . . 5  |-  ( 1st `  R )  =  G
32eqeq2i 2306 . . . 4  |-  ( g  =  ( 1st `  R
)  <->  g  =  G )
4 rneq 4920 . . . . . 6  |-  ( g  =  G  ->  ran  g  =  ran  G )
5 com2i.3 . . . . . 6  |-  X  =  ran  G
64, 5syl6eqr 2346 . . . . 5  |-  ( g  =  G  ->  ran  g  =  X )
7 raleq 2749 . . . . . 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 2757 . . . . 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 15 . . . 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 187 . . 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 2300 . . . . . 6  |-  ( 2nd `  R )  =  H
1312eqeq2i 2306 . . . . 5  |-  ( h  =  ( 2nd `  R
)  <->  h  =  H
)
14 oveq 5880 . . . . . 6  |-  ( h  =  H  ->  (
a h b )  =  ( a H b ) )
15 oveq 5880 . . . . . 6  |-  ( h  =  H  ->  (
b h a )  =  ( b H a ) )
1614, 15eqeq12d 2310 . . . . 5  |-  ( h  =  H  ->  (
( a h b )  =  ( b h a )  <->  ( a H b )  =  ( b H a ) ) )
1713, 16sylbi 187 . . . 4  |-  ( h  =  ( 2nd `  R
)  ->  ( (
a h b )  =  ( b h a )  <->  ( a H b )  =  ( b H a ) ) )
18172ralbidv 2598 . . 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 6201 . 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 21094 . 2  |-  Com2  =  { <. g ,  h >.  |  A. a  e. 
ran  g A. b  e.  ran  g ( a h b )  =  ( b h a ) }
2119, 20eleq2s 2388 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 4    <-> wb 176    = wceq 1632    e. wcel 1696   A.wral 2556   {copab 4092   ran crn 4706   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   Com2ccm2 21093
This theorem is referenced by:  fldi  25530
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-1st 6138  df-2nd 6139  df-com2 21094
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