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Theorem iscmnd 15426
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 1157 . . . 4  |-  ( ph  ->  ( ( x  e.  B  /\  y  e.  B )  ->  (
x  .+  y )  =  ( y  .+  x ) ) )
43ralrimivv 2799 . . 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 6100 . . . . . . 7  |-  ( ph  ->  ( x  .+  y
)  =  ( x ( +g  `  G
) y ) )
86oveqd 6100 . . . . . . 7  |-  ( ph  ->  ( y  .+  x
)  =  ( y ( +g  `  G
) x ) )
97, 8eqeq12d 2452 . . . . . 6  |-  ( ph  ->  ( ( x  .+  y )  =  ( y  .+  x )  <-> 
( x ( +g  `  G ) y )  =  ( y ( +g  `  G ) x ) ) )
105, 9raleqbidv 2918 . . . . 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 2918 . . . 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 686 . . 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 889 . 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 2438 . . 3  |-  ( Base `  G )  =  (
Base `  G )
15 eqid 2438 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
1614, 15iscmn 15421 . 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 205 1  |-  ( ph  ->  G  e. CMnd )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707   ` cfv 5456  (class class class)co 6083   Basecbs 13471   +g cplusg 13531   Mndcmnd 14686  CMndccmn 15414
This theorem is referenced by:  isabld  15427  subcmn  15458  prdscmnd  15478  iscrngd  15701  psrcrng  16478  xrsmcmn  16726
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-iota 5420  df-fv 5464  df-ov 6086  df-cmn 15416
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