Theorem mndlrinv 33044

Description: In a monoid, if an element 𝑋 has both a left inverse 𝑀 and a right inverse 𝑁, they are equal. (Contributed by Thierry Arnoux, 3-Aug-2025.)

Hypotheses

Ref	Expression
mndlrinv.b	⊢ 𝐵 = (Base‘𝐸)
mndlrinv.z	⊢ 0 = (0g‘𝐸)
mndlrinv.p	⊢ + = (+g‘𝐸)
mndlrinv.e	⊢ (𝜑 → 𝐸 ∈ Mnd)
mndlrinv.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
mndlrinv.m	⊢ (𝜑 → 𝑀 ∈ 𝐵)
mndlrinv.n	⊢ (𝜑 → 𝑁 ∈ 𝐵)
mndlrinv.1	⊢ (𝜑 → (𝑀 + 𝑋) = 0 )
mndlrinv.2	⊢ (𝜑 → (𝑋 + 𝑁) = 0 )

Assertion

Ref	Expression
mndlrinv	⊢ (𝜑 → 𝑀 = 𝑁)

Proof of Theorem mndlrinv

Step	Hyp	Ref	Expression
1		mndlrinv.b	. . . 4 ⊢ 𝐵 = (Base‘𝐸)
2		mndlrinv.p	. . . 4 ⊢ + = (+g‘𝐸)
3		mndlrinv.e	. . . 4 ⊢ (𝜑 → 𝐸 ∈ Mnd)
4		mndlrinv.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝐵)
5		mndlrinv.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
6		mndlrinv.n	. . . 4 ⊢ (𝜑 → 𝑁 ∈ 𝐵)
7	1, 2, 3, 4, 5, 6	mndassd 33043	. . 3 ⊢ (𝜑 → ((𝑀 + 𝑋) + 𝑁) = (𝑀 + (𝑋 + 𝑁)))
8		mndlrinv.1	. . . 4 ⊢ (𝜑 → (𝑀 + 𝑋) = 0 )
9	8	oveq1d 7453	. . 3 ⊢ (𝜑 → ((𝑀 + 𝑋) + 𝑁) = ( 0 + 𝑁))
10		mndlrinv.2	. . . 4 ⊢ (𝜑 → (𝑋 + 𝑁) = 0 )
11	10	oveq2d 7454	. . 3 ⊢ (𝜑 → (𝑀 + (𝑋 + 𝑁)) = (𝑀 + 0 ))
12	7, 9, 11	3eqtr3rd 2786	. 2 ⊢ (𝜑 → (𝑀 + 0 ) = ( 0 + 𝑁))
13		mndlrinv.z	. . . 4 ⊢ 0 = (0g‘𝐸)
14	1, 2, 13	mndrid 18790	. . 3 ⊢ ((𝐸 ∈ Mnd ∧ 𝑀 ∈ 𝐵) → (𝑀 + 0 ) = 𝑀)
15	3, 4, 14	syl2anc 584	. 2 ⊢ (𝜑 → (𝑀 + 0 ) = 𝑀)
16	1, 2, 13	mndlid 18789	. . 3 ⊢ ((𝐸 ∈ Mnd ∧ 𝑁 ∈ 𝐵) → ( 0 + 𝑁) = 𝑁)
17	3, 6, 16	syl2anc 584	. 2 ⊢ (𝜑 → ( 0 + 𝑁) = 𝑁)
18	12, 15, 17	3eqtr3d 2785	1 ⊢ (𝜑 → 𝑀 = 𝑁)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  ‘cfv 6569  (class class class)co 7438  Basecbs 17254  +_gcplusg 17307  0_gc0g 17495  Mndcmnd 18769

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5305  ax-nul 5315  ax-pr 5441

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3483  df-sbc 3795  df-dif 3969  df-un 3971  df-ss 3983  df-nul 4343  df-if 4535  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-br 5152  df-opab 5214  df-mpt 5235  df-id 5587  df-xp 5699  df-rel 5700  df-cnv 5701  df-co 5702  df-dm 5703  df-iota 6522  df-fun 6571  df-fv 6577  df-riota 7395  df-ov 7441  df-0g 17497  df-mgm 18675  df-sgrp 18754  df-mnd 18770

This theorem is referenced by:  mndlrinvb  33045  mndlactf1o  33050