Theorem mndlrid 18404

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 18395	. 2 ⊢ (𝐺 ∈ Mnd → ∃𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥))
5	1, 2, 3, 4	mgmlrid 18351	1 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (( 0 + 𝑋) = 𝑋 ∧ (𝑋 + 0 ) = 𝑋))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ‘cfv 6433  (class class class)co 7275  Basecbs 16912  +_gcplusg 16962  0_gc0g 17150  Mndcmnd 18385

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-riota 7232  df-ov 7278  df-0g 17152  df-mgm 18326  df-sgrp 18375  df-mnd 18386

This theorem is referenced by:  mndlid  18405  mndrid  18406  gsumvallem2  18472  gsumsubm  18473  srgidmlem  19756  ringidmlem  19809  frlmgsum  20979