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Theorem gsumvallem2 16205
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 16098 . 2  |-  ( G  e.  Mnd  ->  O  C_ 
{  .0.  } )
61, 2mndidcl 16140 . . . 4  |-  ( G  e.  Mnd  ->  .0.  e.  B )
71, 3, 2mndlrid 16142 . . . . 5  |-  ( ( G  e.  Mnd  /\  y  e.  B )  ->  ( (  .0.  .+  y )  =  y  /\  ( y  .+  .0.  )  =  y
) )
87ralrimiva 2868 . . . 4  |-  ( G  e.  Mnd  ->  A. y  e.  B  ( (  .0.  .+  y )  =  y  /\  ( y 
.+  .0.  )  =  y ) )
9 oveq1 6277 . . . . . . . 8  |-  ( x  =  .0.  ->  (
x  .+  y )  =  (  .0.  .+  y
) )
109eqeq1d 2456 . . . . . . 7  |-  ( x  =  .0.  ->  (
( x  .+  y
)  =  y  <->  (  .0.  .+  y )  =  y ) )
11 oveq2 6278 . . . . . . . 8  |-  ( x  =  .0.  ->  (
y  .+  x )  =  ( y  .+  .0.  ) )
1211eqeq1d 2456 . . . . . . 7  |-  ( x  =  .0.  ->  (
( y  .+  x
)  =  y  <->  ( y  .+  .0.  )  =  y ) )
1310, 12anbi12d 708 . . . . . 6  |-  ( x  =  .0.  ->  (
( ( x  .+  y )  =  y  /\  ( y  .+  x )  =  y )  <->  ( (  .0.  .+  y )  =  y  /\  ( y  .+  .0.  )  =  y
) ) )
1413ralbidv 2893 . . . . 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 3256 . . . 4  |-  (  .0. 
e.  O  <->  (  .0.  e.  B  /\  A. y  e.  B  ( (  .0.  .+  y )  =  y  /\  ( y 
.+  .0.  )  =  y ) ) )
166, 8, 15sylanbrc 662 . . 3  |-  ( G  e.  Mnd  ->  .0.  e.  O )
1716snssd 4161 . 2  |-  ( G  e.  Mnd  ->  {  .0.  } 
C_  O )
185, 17eqssd 3506 1  |-  ( G  e.  Mnd  ->  O  =  {  .0.  } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   {crab 2808   {csn 4016   ` cfv 5570  (class class class)co 6270   Basecbs 14719   +g cplusg 14787   0gc0g 14932   Mndcmnd 16121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-riota 6232  df-ov 6273  df-0g 14934  df-mgm 16074  df-sgrp 16113  df-mnd 16123
This theorem is referenced by:  gsumz  16207  gsumval3a  17107  gsumval3aOLD  17108
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