Theorem mndlrinv 33116

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 33115	. . 3 ⊢ (𝜑 → ((𝑀 + 𝑋) + 𝑁) = (𝑀 + (𝑋 + 𝑁)))
8		mndlrinv.1	. . . 4 ⊢ (𝜑 → (𝑀 + 𝑋) = 0 )
9	8	oveq1d 7383	. . 3 ⊢ (𝜑 → ((𝑀 + 𝑋) + 𝑁) = ( 0 + 𝑁))
10		mndlrinv.2	. . . 4 ⊢ (𝜑 → (𝑋 + 𝑁) = 0 )
11	10	oveq2d 7384	. . 3 ⊢ (𝜑 → (𝑀 + (𝑋 + 𝑁)) = (𝑀 + 0 ))
12	7, 9, 11	3eqtr3rd 2781	. 2 ⊢ (𝜑 → (𝑀 + 0 ) = ( 0 + 𝑁))
13		mndlrinv.z	. . . 4 ⊢ 0 = (0g‘𝐸)
14	1, 2, 13	mndrid 18692	. . 3 ⊢ ((𝐸 ∈ Mnd ∧ 𝑀 ∈ 𝐵) → (𝑀 + 0 ) = 𝑀)
15	3, 4, 14	syl2anc 585	. 2 ⊢ (𝜑 → (𝑀 + 0 ) = 𝑀)
16	1, 2, 13	mndlid 18691	. . 3 ⊢ ((𝐸 ∈ Mnd ∧ 𝑁 ∈ 𝐵) → ( 0 + 𝑁) = 𝑁)
17	3, 6, 16	syl2anc 585	. 2 ⊢ (𝜑 → ( 0 + 𝑁) = 𝑁)
18	12, 15, 17	3eqtr3d 2780	1 ⊢ (𝜑 → 𝑀 = 𝑁)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  +_gcplusg 17189  0_gc0g 17371  Mndcmnd 18671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6456  df-fun 6502  df-fv 6508  df-riota 7325  df-ov 7371  df-0g 17373  df-mgm 18577  df-sgrp 18656  df-mnd 18672

This theorem is referenced by:  mndlrinvb  33117  mndlactf1o  33122