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Theorem iscmnd 13368
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 1208 . . . 4  |-  ( ph  ->  ( ( x  e.  B  /\  y  e.  B )  ->  (
x  .+  y )  =  ( y  .+  x ) ) )
43ralrimivv 2575 . . 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 5935 . . . . . . 7  |-  ( ph  ->  ( x  .+  y
)  =  ( x ( +g  `  G
) y ) )
86oveqd 5935 . . . . . . 7  |-  ( ph  ->  ( y  .+  x
)  =  ( y ( +g  `  G
) x ) )
97, 8eqeq12d 2208 . . . . . 6  |-  ( ph  ->  ( ( x  .+  y )  =  ( y  .+  x )  <-> 
( x ( +g  `  G ) y )  =  ( y ( +g  `  G ) x ) ) )
105, 9raleqbidv 2706 . . . . 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 2706 . . . 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 945 . 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 2193 . . 3  |-  ( Base `  G )  =  (
Base `  G )
15 eqid 2193 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
1614, 15iscmn 13363 . 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 980    = wceq 1364    e. wcel 2164   A.wral 2472   ` cfv 5254  (class class class)co 5918   Basecbs 12618   +g cplusg 12695   Mndcmnd 12997  CMndccmn 13354
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-un 3157  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-iota 5215  df-fv 5262  df-ov 5921  df-cmn 13356
This theorem is referenced by:  isabld  13369  subcmnd  13403  iscrngd  13538
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