Theorem mndlrid 17310

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.

Step	Hyp	Ref	Expression
1		mndlrid.b	. 2 |- B = ( Base ` G )
2		mndlrid.o	. 2 |- .0. = ( 0g ` G )
3		mndlrid.p	. 2 |- .+ = ( +g ` G )
4	1, 3	mndid 17303	. 2 |- ( G e. Mnd -> E. y e. B A. x e. B ( ( y .+ x ) = x /\ ( x .+ y ) = x ) )
5	1, 2, 3, 4	mgmlrid 17266	1 |- ( ( G e. Mnd /\ X e. B ) -> ( ( .0. .+ X ) = X /\ ( X .+ .0. ) = X ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   0gc0g 16100   Mndcmnd 17294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-riota 6611  df-ov 6653  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295

This theorem is referenced by:  mndlid  17311  mndrid  17312  gsumvallem2  17372  gsumsubm  17373  srgidmlem  18520  ringidmlem  18570  frlmgsum  20111