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Theorem mndlrid 16266
Description: A monoid's identity element is a two-sided identity. (Contributed by NM, 18-Aug-2011.)
Hypotheses
Ref Expression
mndlrid.b  |-  B  =  ( Base `  G
)
mndlrid.p  |-  .+  =  ( +g  `  G )
mndlrid.o  |-  .0.  =  ( 0g `  G )
Assertion
Ref Expression
mndlrid  |-  ( ( G  e.  Mnd  /\  X  e.  B )  ->  ( (  .0.  .+  X )  =  X  /\  ( X  .+  .0.  )  =  X
) )

Proof of Theorem mndlrid
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mndlrid.b . 2  |-  B  =  ( Base `  G
)
2 mndlrid.o . 2  |-  .0.  =  ( 0g `  G )
3 mndlrid.p . 2  |-  .+  =  ( +g  `  G )
41, 3mndid 16259 . 2  |-  ( G  e.  Mnd  ->  E. y  e.  B  A. x  e.  B  ( (
y  .+  x )  =  x  /\  (
x  .+  y )  =  x ) )
51, 2, 3, 4mgmlrid 16219 1  |-  ( ( G  e.  Mnd  /\  X  e.  B )  ->  ( (  .0.  .+  X )  =  X  /\  ( X  .+  .0.  )  =  X
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1407    e. wcel 1844   ` cfv 5571  (class class class)co 6280   Basecbs 14843   +g cplusg 14911   0gc0g 15056   Mndcmnd 16245
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-iota 5535  df-fun 5573  df-fv 5579  df-riota 6242  df-ov 6283  df-0g 15058  df-mgm 16198  df-sgrp 16237  df-mnd 16247
This theorem is referenced by:  mndlid  16267  mndrid  16268  gsumvallem2  16329  gsumsubm  16330  srgidmlem  17493  ringidmlem  17543  frlmgsumOLD  19099  frlmgsum  19100
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