Theorem mndlrid 18779

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 ∈ Mnd ∧ 𝑋 ∈ 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 18770	. 2 ⊢ (𝐺 ∈ Mnd → ∃𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥))
5	1, 2, 3, 4	mgmlrid 18693	1 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (( 0 + 𝑋) = 𝑋 ∧ (𝑋 + 0 ) = 𝑋))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ‘cfv 6563  (class class class)co 7431  Basecbs 17245  +_gcplusg 17298  0_gc0g 17486  Mndcmnd 18760

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-riota 7388  df-ov 7434  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761

This theorem is referenced by:  mndlid  18780  mndrid  18781  gsumvallem2  18860  gsumsubm  18861  srgidmlem  20219  ringidmlem  20282  frlmgsum  21810