Theorem mndlrinv 33012

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 33011	. . 3 ⊢ (𝜑 → ((𝑀 + 𝑋) + 𝑁) = (𝑀 + (𝑋 + 𝑁)))
8		mndlrinv.1	. . . 4 ⊢ (𝜑 → (𝑀 + 𝑋) = 0 )
9	8	oveq1d 7465	. . 3 ⊢ (𝜑 → ((𝑀 + 𝑋) + 𝑁) = ( 0 + 𝑁))
10		mndlrinv.2	. . . 4 ⊢ (𝜑 → (𝑋 + 𝑁) = 0 )
11	10	oveq2d 7466	. . 3 ⊢ (𝜑 → (𝑀 + (𝑋 + 𝑁)) = (𝑀 + 0 ))
12	7, 9, 11	3eqtr3rd 2789	. 2 ⊢ (𝜑 → (𝑀 + 0 ) = ( 0 + 𝑁))
13		mndlrinv.z	. . . 4 ⊢ 0 = (0g‘𝐸)
14	1, 2, 13	mndrid 18795	. . 3 ⊢ ((𝐸 ∈ Mnd ∧ 𝑀 ∈ 𝐵) → (𝑀 + 0 ) = 𝑀)
15	3, 4, 14	syl2anc 583	. 2 ⊢ (𝜑 → (𝑀 + 0 ) = 𝑀)
16	1, 2, 13	mndlid 18794	. . 3 ⊢ ((𝐸 ∈ Mnd ∧ 𝑁 ∈ 𝐵) → ( 0 + 𝑁) = 𝑁)
17	3, 6, 16	syl2anc 583	. 2 ⊢ (𝜑 → ( 0 + 𝑁) = 𝑁)
18	12, 15, 17	3eqtr3d 2788	1 ⊢ (𝜑 → 𝑀 = 𝑁)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  ‘cfv 6575  (class class class)co 7450  Basecbs 17260  +_gcplusg 17313  0_gc0g 17501  Mndcmnd 18774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6527  df-fun 6577  df-fv 6583  df-riota 7406  df-ov 7453  df-0g 17503  df-mgm 18680  df-sgrp 18759  df-mnd 18775

This theorem is referenced by:  mndlrinvb  33013  mndlactf1o  33018