Theorem mndid 18395

Description: A monoid has a two-sided identity element. (Contributed by NM, 16-Aug-2011.)

Hypotheses

Ref	Expression
mndcl.b	⊢ 𝐵 = (Base‘𝐺)
mndcl.p	⊢ + = (+g‘𝐺)

Assertion

Ref	Expression
mndid	⊢ (𝐺 ∈ Mnd → ∃𝑢 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥))

Distinct variable groups:   𝑥,𝐵   𝑥,𝐺   𝑥, + ,𝑢   𝑢,𝐵   𝑢,𝐺   𝑢, +

Proof of Theorem mndid

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mndcl.b	. . 3 ⊢ 𝐵 = (Base‘𝐺)
2		mndcl.p	. . 3 ⊢ + = (+g‘𝐺)
3	1, 2	ismnd 18388	. 2 ⊢ (𝐺 ∈ Mnd ↔ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) ∧ ∃𝑢 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥)))
4	3	simprbi 497	1 ⊢ (𝐺 ∈ Mnd → ∃𝑢 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  ‘cfv 6433  (class class class)co 7275  Basecbs 16912  +_gcplusg 16962  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-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-ne 2944  df-ral 3069  df-rex 3070  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-iota 6391  df-fv 6441  df-ov 7278  df-mgm 18326  df-sgrp 18375  df-mnd 18386

This theorem is referenced by:  mndideu  18396  mndidcl  18400  mndlrid  18404  prds0g  18419