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Theorem gsumvallem2 14774
Description: Lemma for properties of the set of identities of  G. The set of identities of a monoid is exactly the unique identity element. (Contributed by Mario Carneiro, 7-Dec-2014.)
Hypotheses
Ref Expression
gsumvallem1.b  |-  B  =  ( Base `  G
)
gsumvallem1.z  |-  .0.  =  ( 0g `  G )
gsumvallem1.p  |-  .+  =  ( +g  `  G )
gsumvallem1.o  |-  O  =  { x  e.  B  |  A. y  e.  B  ( ( x  .+  y )  =  y  /\  ( y  .+  x )  =  y ) }
Assertion
Ref Expression
gsumvallem2  |-  ( G  e.  Mnd  ->  O  =  {  .0.  } )
Distinct variable groups:    x, y, B    x, G, y    x,  .+ , y    x,  .0. , y
Allowed substitution hints:    O( x, y)

Proof of Theorem gsumvallem2
StepHypRef Expression
1 gsumvallem1.b . . 3  |-  B  =  ( Base `  G
)
2 gsumvallem1.z . . 3  |-  .0.  =  ( 0g `  G )
3 gsumvallem1.p . . 3  |-  .+  =  ( +g  `  G )
4 gsumvallem1.o . . 3  |-  O  =  { x  e.  B  |  A. y  e.  B  ( ( x  .+  y )  =  y  /\  ( y  .+  x )  =  y ) }
51, 2, 3, 4gsumvallem1 14773 . 2  |-  ( G  e.  Mnd  ->  O  C_ 
{  .0.  } )
61, 2mndidcl 14716 . . . 4  |-  ( G  e.  Mnd  ->  .0.  e.  B )
71, 3, 2mndlrid 14717 . . . . 5  |-  ( ( G  e.  Mnd  /\  y  e.  B )  ->  ( (  .0.  .+  y )  =  y  /\  ( y  .+  .0.  )  =  y
) )
87ralrimiva 2791 . . . 4  |-  ( G  e.  Mnd  ->  A. y  e.  B  ( (  .0.  .+  y )  =  y  /\  ( y 
.+  .0.  )  =  y ) )
9 oveq1 6090 . . . . . . . 8  |-  ( x  =  .0.  ->  (
x  .+  y )  =  (  .0.  .+  y
) )
109eqeq1d 2446 . . . . . . 7  |-  ( x  =  .0.  ->  (
( x  .+  y
)  =  y  <->  (  .0.  .+  y )  =  y ) )
11 oveq2 6091 . . . . . . . 8  |-  ( x  =  .0.  ->  (
y  .+  x )  =  ( y  .+  .0.  ) )
1211eqeq1d 2446 . . . . . . 7  |-  ( x  =  .0.  ->  (
( y  .+  x
)  =  y  <->  ( y  .+  .0.  )  =  y ) )
1310, 12anbi12d 693 . . . . . 6  |-  ( x  =  .0.  ->  (
( ( x  .+  y )  =  y  /\  ( y  .+  x )  =  y )  <->  ( (  .0.  .+  y )  =  y  /\  ( y  .+  .0.  )  =  y
) ) )
1413ralbidv 2727 . . . . 5  |-  ( x  =  .0.  ->  ( A. y  e.  B  ( ( x  .+  y )  =  y  /\  ( y  .+  x )  =  y )  <->  A. y  e.  B  ( (  .0.  .+  y )  =  y  /\  ( y  .+  .0.  )  =  y
) ) )
1514, 4elrab2 3096 . . . 4  |-  (  .0. 
e.  O  <->  (  .0.  e.  B  /\  A. y  e.  B  ( (  .0.  .+  y )  =  y  /\  ( y 
.+  .0.  )  =  y ) ) )
166, 8, 15sylanbrc 647 . . 3  |-  ( G  e.  Mnd  ->  .0.  e.  O )
1716snssd 3945 . 2  |-  ( G  e.  Mnd  ->  {  .0.  } 
C_  O )
185, 17eqssd 3367 1  |-  ( G  e.  Mnd  ->  O  =  {  .0.  } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   {crab 2711   {csn 3816   ` cfv 5456  (class class class)co 6083   Basecbs 13471   +g cplusg 13531   0gc0g 13725   Mndcmnd 14686
This theorem is referenced by:  gsumz  14783  gsumval3a  15514
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-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
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-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  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-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-ov 6086  df-riota 6551  df-0g 13729  df-mnd 14692
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