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Theorem iscmnd 12915
Description: Properties that determine a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.)
Hypotheses
Ref Expression
iscmnd.b  |-  ( ph  ->  B  =  ( Base `  G ) )
iscmnd.p  |-  ( ph  ->  .+  =  ( +g  `  G ) )
iscmnd.g  |-  ( ph  ->  G  e.  Mnd )
iscmnd.c  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( x  .+  y )  =  ( y  .+  x ) )
Assertion
Ref Expression
iscmnd  |-  ( ph  ->  G  e. CMnd )
Distinct variable groups:    x, y, B   
x, G, y    ph, x, y
Allowed substitution hints:    .+ ( x, y)

Proof of Theorem iscmnd
StepHypRef Expression
1 iscmnd.g . . 3  |-  ( ph  ->  G  e.  Mnd )
2 iscmnd.c . . . . 5  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( x  .+  y )  =  ( y  .+  x ) )
323expib 1206 . . . 4  |-  ( ph  ->  ( ( x  e.  B  /\  y  e.  B )  ->  (
x  .+  y )  =  ( y  .+  x ) ) )
43ralrimivv 2558 . . 3  |-  ( ph  ->  A. x  e.  B  A. y  e.  B  ( x  .+  y )  =  ( y  .+  x ) )
5 iscmnd.b . . . . 5  |-  ( ph  ->  B  =  ( Base `  G ) )
6 iscmnd.p . . . . . . . 8  |-  ( ph  ->  .+  =  ( +g  `  G ) )
76oveqd 5885 . . . . . . 7  |-  ( ph  ->  ( x  .+  y
)  =  ( x ( +g  `  G
) y ) )
86oveqd 5885 . . . . . . 7  |-  ( ph  ->  ( y  .+  x
)  =  ( y ( +g  `  G
) x ) )
97, 8eqeq12d 2192 . . . . . 6  |-  ( ph  ->  ( ( x  .+  y )  =  ( y  .+  x )  <-> 
( x ( +g  `  G ) y )  =  ( y ( +g  `  G ) x ) ) )
105, 9raleqbidv 2684 . . . . 5  |-  ( ph  ->  ( A. y  e.  B  ( x  .+  y )  =  ( y  .+  x )  <->  A. y  e.  ( Base `  G ) ( x ( +g  `  G
) y )  =  ( y ( +g  `  G ) x ) ) )
115, 10raleqbidv 2684 . . . 4  |-  ( ph  ->  ( A. x  e.  B  A. y  e.  B  ( x  .+  y )  =  ( y  .+  x )  <->  A. x  e.  ( Base `  G ) A. y  e.  ( Base `  G ) ( x ( +g  `  G
) y )  =  ( y ( +g  `  G ) x ) ) )
1211anbi2d 464 . . 3  |-  ( ph  ->  ( ( G  e. 
Mnd  /\  A. x  e.  B  A. y  e.  B  ( x  .+  y )  =  ( y  .+  x ) )  <->  ( G  e. 
Mnd  /\  A. x  e.  ( Base `  G
) A. y  e.  ( Base `  G
) ( x ( +g  `  G ) y )  =  ( y ( +g  `  G
) x ) ) ) )
131, 4, 12mpbi2and 943 . 2  |-  ( ph  ->  ( G  e.  Mnd  /\ 
A. x  e.  (
Base `  G ) A. y  e.  ( Base `  G ) ( x ( +g  `  G
) y )  =  ( y ( +g  `  G ) x ) ) )
14 eqid 2177 . . 3  |-  ( Base `  G )  =  (
Base `  G )
15 eqid 2177 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
1614, 15iscmn 12910 . 2  |-  ( G  e. CMnd 
<->  ( G  e.  Mnd  /\ 
A. x  e.  (
Base `  G ) A. y  e.  ( Base `  G ) ( x ( +g  `  G
) y )  =  ( y ( +g  `  G ) x ) ) )
1713, 16sylibr 134 1  |-  ( ph  ->  G  e. CMnd )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 978    = wceq 1353    e. wcel 2148   A.wral 2455   ` cfv 5211  (class class class)co 5868   Basecbs 12432   +g cplusg 12505   Mndcmnd 12696  CMndccmn 12902
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-un 3133  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-iota 5173  df-fv 5219  df-ov 5871  df-cmn 12904
This theorem is referenced by:  isabld  12916  subcmnd  12943  iscrngd  13034
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