MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mndlrid Structured version   Unicode version

Theorem mndlrid 14707
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 14689 . 2  |-  ( G  e.  Mnd  ->  E. y  e.  B  A. x  e.  B  ( (
y  .+  x )  =  x  /\  (
x  .+  y )  =  x ) )
51, 2, 3, 4mgmlrid 14704 1  |-  ( ( G  e.  Mnd  /\  X  e.  B )  ->  ( (  .0.  .+  X )  =  X  /\  ( X  .+  .0.  )  =  X
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   ` cfv 5446  (class class class)co 6073   Basecbs 13461   +g cplusg 13521   0gc0g 13715   Mndcmnd 14676
This theorem is referenced by:  mndlid  14708  mndrid  14709  gsumvallem2  14764  gsumsubm  14770  rngidmlem  15678  frlmgsum  27200
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-riota 6541  df-0g 13719  df-mnd 14682
  Copyright terms: Public domain W3C validator