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Theorem gsumvallem2 13055
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
gsumvallem2.b  |-  B  =  ( Base `  G
)
gsumvallem2.z  |-  .0.  =  ( 0g `  G )
gsumvallem2.p  |-  .+  =  ( +g  `  G )
gsumvallem2.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 gsumvallem2.b . . 3  |-  B  =  ( Base `  G
)
2 gsumvallem2.z . . 3  |-  .0.  =  ( 0g `  G )
3 gsumvallem2.p . . 3  |-  .+  =  ( +g  `  G )
4 gsumvallem2.o . . 3  |-  O  =  { x  e.  B  |  A. y  e.  B  ( ( x  .+  y )  =  y  /\  ( y  .+  x )  =  y ) }
51, 2, 3, 4mgmidsssn0 12957 . 2  |-  ( G  e.  Mnd  ->  O  C_ 
{  .0.  } )
61, 2mndidcl 13001 . . . 4  |-  ( G  e.  Mnd  ->  .0.  e.  B )
71, 3, 2mndlrid 13005 . . . . 5  |-  ( ( G  e.  Mnd  /\  y  e.  B )  ->  ( (  .0.  .+  y )  =  y  /\  ( y  .+  .0.  )  =  y
) )
87ralrimiva 2567 . . . 4  |-  ( G  e.  Mnd  ->  A. y  e.  B  ( (  .0.  .+  y )  =  y  /\  ( y 
.+  .0.  )  =  y ) )
9 oveq1 5917 . . . . . . 7  |-  ( x  =  .0.  ->  (
x  .+  y )  =  (  .0.  .+  y
) )
109eqeq1d 2202 . . . . . 6  |-  ( x  =  .0.  ->  (
( x  .+  y
)  =  y  <->  (  .0.  .+  y )  =  y ) )
1110ovanraleqv 5934 . . . . 5  |-  ( x  =  .0.  ->  ( A. y  e.  B  ( ( x  .+  y )  =  y  /\  ( y  .+  x )  =  y )  <->  A. y  e.  B  ( (  .0.  .+  y )  =  y  /\  ( y  .+  .0.  )  =  y
) ) )
1211, 4elrab2 2919 . . . 4  |-  (  .0. 
e.  O  <->  (  .0.  e.  B  /\  A. y  e.  B  ( (  .0.  .+  y )  =  y  /\  ( y 
.+  .0.  )  =  y ) ) )
136, 8, 12sylanbrc 417 . . 3  |-  ( G  e.  Mnd  ->  .0.  e.  O )
1413snssd 3763 . 2  |-  ( G  e.  Mnd  ->  {  .0.  } 
C_  O )
155, 14eqssd 3196 1  |-  ( G  e.  Mnd  ->  O  =  {  .0.  } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2164   A.wral 2472   {crab 2476   {csn 3618   ` cfv 5246  (class class class)co 5910   Basecbs 12608   +g cplusg 12685   0gc0g 12857   Mndcmnd 12987
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-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4462  ax-cnex 7953  ax-resscn 7954  ax-1re 7956  ax-addrcl 7959
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4322  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-iota 5207  df-fun 5248  df-fn 5249  df-fv 5254  df-riota 5865  df-ov 5913  df-inn 8973  df-2 9031  df-ndx 12611  df-slot 12612  df-base 12614  df-plusg 12698  df-0g 12859  df-mgm 12929  df-sgrp 12975  df-mnd 12988
This theorem is referenced by: (None)
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