Theorem mndlrid 18787

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 18778	. 2 ⊢ (𝐺 ∈ Mnd → ∃𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥))
5	1, 2, 3, 4	mgmlrid 18701	1 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (( 0 + 𝑋) = 𝑋 ∧ (𝑋 + 0 ) = 𝑋))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1560   ∈ wcel 2142  ‘cfv 6521  (class class class)co 7396  Basecbs 17245  +_gcplusg 17286  0_gc0g 17468  Mndcmnd 18768

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-iota 6477  df-fun 6523  df-fv 6529  df-riota 7353  df-ov 7399  df-0g 17470  df-mgm 18674  df-sgrp 18753  df-mnd 18769

This theorem is referenced by:  mndlid  18788  mndrid  18789  gsumvallem2  18868  gsumsubm  18869  srgidmlem  20251  ringidmlem  20318  frlmgsum  21824