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Theorem gsumvallem2 13101
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 13003 . 2  |-  ( G  e.  Mnd  ->  O  C_ 
{  .0.  } )
61, 2mndidcl 13047 . . . 4  |-  ( G  e.  Mnd  ->  .0.  e.  B )
71, 3, 2mndlrid 13051 . . . . 5  |-  ( ( G  e.  Mnd  /\  y  e.  B )  ->  ( (  .0.  .+  y )  =  y  /\  ( y  .+  .0.  )  =  y
) )
87ralrimiva 2570 . . . 4  |-  ( G  e.  Mnd  ->  A. y  e.  B  ( (  .0.  .+  y )  =  y  /\  ( y 
.+  .0.  )  =  y ) )
9 oveq1 5929 . . . . . . 7  |-  ( x  =  .0.  ->  (
x  .+  y )  =  (  .0.  .+  y
) )
109eqeq1d 2205 . . . . . 6  |-  ( x  =  .0.  ->  (
( x  .+  y
)  =  y  <->  (  .0.  .+  y )  =  y ) )
1110ovanraleqv 5946 . . . . 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 2923 . . . 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 3767 . 2  |-  ( G  e.  Mnd  ->  {  .0.  } 
C_  O )
155, 14eqssd 3200 1  |-  ( G  e.  Mnd  ->  O  =  {  .0.  } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2167   A.wral 2475   {crab 2479   {csn 3622   ` cfv 5258  (class class class)co 5922   Basecbs 12654   +g cplusg 12731   0gc0g 12903   Mndcmnd 13033
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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-cnex 7968  ax-resscn 7969  ax-1re 7971  ax-addrcl 7974
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-riota 5877  df-ov 5925  df-inn 8988  df-2 9046  df-ndx 12657  df-slot 12658  df-base 12660  df-plusg 12744  df-0g 12905  df-mgm 12975  df-sgrp 13021  df-mnd 13034
This theorem is referenced by: (None)
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