Theorem mndlrinv 33000

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 32999	. . 3 ⊢ (𝜑 → ((𝑀 + 𝑋) + 𝑁) = (𝑀 + (𝑋 + 𝑁)))
8		mndlrinv.1	. . . 4 ⊢ (𝜑 → (𝑀 + 𝑋) = 0 )
9	8	oveq1d 7361	. . 3 ⊢ (𝜑 → ((𝑀 + 𝑋) + 𝑁) = ( 0 + 𝑁))
10		mndlrinv.2	. . . 4 ⊢ (𝜑 → (𝑋 + 𝑁) = 0 )
11	10	oveq2d 7362	. . 3 ⊢ (𝜑 → (𝑀 + (𝑋 + 𝑁)) = (𝑀 + 0 ))
12	7, 9, 11	3eqtr3rd 2775	. 2 ⊢ (𝜑 → (𝑀 + 0 ) = ( 0 + 𝑁))
13		mndlrinv.z	. . . 4 ⊢ 0 = (0g‘𝐸)
14	1, 2, 13	mndrid 18660	. . 3 ⊢ ((𝐸 ∈ Mnd ∧ 𝑀 ∈ 𝐵) → (𝑀 + 0 ) = 𝑀)
15	3, 4, 14	syl2anc 584	. 2 ⊢ (𝜑 → (𝑀 + 0 ) = 𝑀)
16	1, 2, 13	mndlid 18659	. . 3 ⊢ ((𝐸 ∈ Mnd ∧ 𝑁 ∈ 𝐵) → ( 0 + 𝑁) = 𝑁)
17	3, 6, 16	syl2anc 584	. 2 ⊢ (𝜑 → ( 0 + 𝑁) = 𝑁)
18	12, 15, 17	3eqtr3d 2774	1 ⊢ (𝜑 → 𝑀 = 𝑁)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  ‘cfv 6481  (class class class)co 7346  Basecbs 17117  +_gcplusg 17158  0_gc0g 17340  Mndcmnd 18639

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-iota 6437  df-fun 6483  df-fv 6489  df-riota 7303  df-ov 7349  df-0g 17342  df-mgm 18545  df-sgrp 18624  df-mnd 18640

This theorem is referenced by:  mndlrinvb  33001  mndlactf1o  33006