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Theorem iscmnd 15101
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 1154 . . . 4  |-  ( ph  ->  ( ( x  e.  B  /\  y  e.  B )  ->  (
x  .+  y )  =  ( y  .+  x ) ) )
43ralrimivv 2634 . . 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 5875 . . . . . . 7  |-  ( ph  ->  ( x  .+  y
)  =  ( x ( +g  `  G
) y ) )
86oveqd 5875 . . . . . . 7  |-  ( ph  ->  ( y  .+  x
)  =  ( y ( +g  `  G
) x ) )
97, 8eqeq12d 2297 . . . . . 6  |-  ( ph  ->  ( ( x  .+  y )  =  ( y  .+  x )  <-> 
( x ( +g  `  G ) y )  =  ( y ( +g  `  G ) x ) ) )
105, 9raleqbidv 2748 . . . . 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 2748 . . . 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 684 . . 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 887 . 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 2283 . . 3  |-  ( Base `  G )  =  (
Base `  G )
15 eqid 2283 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
1614, 15iscmn 15096 . 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 203 1  |-  ( ph  ->  G  e. CMnd )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   Mndcmnd 14361  CMndccmn 15089
This theorem is referenced by:  isabld  15102  subcmn  15133  prdscmnd  15153  iscrngd  15376  psrcrng  16157  xrsmcmn  16397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-cmn 15091
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