Theorem mndmolinv 43411

Description: An element of a monoid that has a right inverse has at most one left inverse. (Contributed by metakunt, 25-Apr-2025.)

Hypotheses

Ref	Expression
mndmolinv.1	⊢ 𝐵 = (Base‘𝑀)
mndmolinv.2	⊢ (𝜑 → 𝑀 ∈ Mnd)
mndmolinv.3	⊢ (𝜑 → 𝐴 ∈ 𝐵)
mndmolinv.4	⊢ (𝜑 → ∃𝑥 ∈ 𝐵 (𝐴(+g‘𝑀)𝑥) = (0g‘𝑀))

Assertion

Ref	Expression
mndmolinv	⊢ (𝜑 → ∃*𝑥 ∈ 𝐵 (𝑥(+g‘𝑀)𝐴) = (0g‘𝑀))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑀   𝜑,𝑥

Proof of Theorem mndmolinv

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mndmolinv.4	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 (𝐴(+g‘𝑀)𝑥) = (0g‘𝑀))
2		nfv 1949	. . . . . 6 ⊢ Ⅎ𝑦(𝐴(+g‘𝑀)𝑥) = (0g‘𝑀)
3		nfv 1949	. . . . . 6 ⊢ Ⅎ𝑥(𝐴(+g‘𝑀)𝑦) = (0g‘𝑀)
4		oveq2 7459	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴(+g‘𝑀)𝑥) = (𝐴(+g‘𝑀)𝑦))
5	4	eqeq1d 2765	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴(+g‘𝑀)𝑥) = (0g‘𝑀) ↔ (𝐴(+g‘𝑀)𝑦) = (0g‘𝑀)))
6	2, 3, 5	cbvrexw 3309	. . . . 5 ⊢ (∃𝑥 ∈ 𝐵 (𝐴(+g‘𝑀)𝑥) = (0g‘𝑀) ↔ ∃𝑦 ∈ 𝐵 (𝐴(+g‘𝑀)𝑦) = (0g‘𝑀))
7	6	biimpi 219	. . . 4 ⊢ (∃𝑥 ∈ 𝐵 (𝐴(+g‘𝑀)𝑥) = (0g‘𝑀) → ∃𝑦 ∈ 𝐵 (𝐴(+g‘𝑀)𝑦) = (0g‘𝑀))
8	1, 7	syl 18	. . 3 ⊢ (𝜑 → ∃𝑦 ∈ 𝐵 (𝐴(+g‘𝑀)𝑦) = (0g‘𝑀))
9		mndmolinv.2	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ Mnd)
10	9	ad4antr 745	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝐴(+g‘𝑀)𝑦) = (0g‘𝑀)) ∧ 𝑥 ∈ 𝐵) ∧ (𝑥(+g‘𝑀)𝐴) = (0g‘𝑀)) → 𝑀 ∈ Mnd)
11		simplr 781	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝐴(+g‘𝑀)𝑦) = (0g‘𝑀)) ∧ 𝑥 ∈ 𝐵) ∧ (𝑥(+g‘𝑀)𝐴) = (0g‘𝑀)) → 𝑥 ∈ 𝐵)
12		mndmolinv.1	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝑀)
13		eqid 2763	. . . . . . . . . . . 12 ⊢ (+g‘𝑀) = (+g‘𝑀)
14		eqid 2763	. . . . . . . . . . . 12 ⊢ (0g‘𝑀) = (0g‘𝑀)
15	12, 13, 14	mndrid 18995	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → (𝑥(+g‘𝑀)(0g‘𝑀)) = 𝑥)
16	10, 11, 15	syl2anc 596	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝐴(+g‘𝑀)𝑦) = (0g‘𝑀)) ∧ 𝑥 ∈ 𝐵) ∧ (𝑥(+g‘𝑀)𝐴) = (0g‘𝑀)) → (𝑥(+g‘𝑀)(0g‘𝑀)) = 𝑥)
17	16	eqcomd 2769	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝐴(+g‘𝑀)𝑦) = (0g‘𝑀)) ∧ 𝑥 ∈ 𝐵) ∧ (𝑥(+g‘𝑀)𝐴) = (0g‘𝑀)) → 𝑥 = (𝑥(+g‘𝑀)(0g‘𝑀)))
18		simpllr 788	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝐴(+g‘𝑀)𝑦) = (0g‘𝑀)) ∧ 𝑥 ∈ 𝐵) ∧ (𝑥(+g‘𝑀)𝐴) = (0g‘𝑀)) → (𝐴(+g‘𝑀)𝑦) = (0g‘𝑀))
19	18	eqcomd 2769	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝐴(+g‘𝑀)𝑦) = (0g‘𝑀)) ∧ 𝑥 ∈ 𝐵) ∧ (𝑥(+g‘𝑀)𝐴) = (0g‘𝑀)) → (0g‘𝑀) = (𝐴(+g‘𝑀)𝑦))
20	19	oveq2d 7467	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝐴(+g‘𝑀)𝑦) = (0g‘𝑀)) ∧ 𝑥 ∈ 𝐵) ∧ (𝑥(+g‘𝑀)𝐴) = (0g‘𝑀)) → (𝑥(+g‘𝑀)(0g‘𝑀)) = (𝑥(+g‘𝑀)(𝐴(+g‘𝑀)𝑦)))
21	17, 20	eqtrd 2798	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝐴(+g‘𝑀)𝑦) = (0g‘𝑀)) ∧ 𝑥 ∈ 𝐵) ∧ (𝑥(+g‘𝑀)𝐴) = (0g‘𝑀)) → 𝑥 = (𝑥(+g‘𝑀)(𝐴(+g‘𝑀)𝑦)))
22		mndmolinv.3	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
23	22	ad4antr 745	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝐴(+g‘𝑀)𝑦) = (0g‘𝑀)) ∧ 𝑥 ∈ 𝐵) ∧ (𝑥(+g‘𝑀)𝐴) = (0g‘𝑀)) → 𝐴 ∈ 𝐵)
24		simp-4r 796	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝐴(+g‘𝑀)𝑦) = (0g‘𝑀)) ∧ 𝑥 ∈ 𝐵) ∧ (𝑥(+g‘𝑀)𝐴) = (0g‘𝑀)) → 𝑦 ∈ 𝐵)
25	11, 23, 24	3jca 1146	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝐴(+g‘𝑀)𝑦) = (0g‘𝑀)) ∧ 𝑥 ∈ 𝐵) ∧ (𝑥(+g‘𝑀)𝐴) = (0g‘𝑀)) → (𝑥 ∈ 𝐵 ∧ 𝐴 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))
26	12, 13	mndass 18982	. . . . . . . . . 10 ⊢ ((𝑀 ∈ Mnd ∧ (𝑥 ∈ 𝐵 ∧ 𝐴 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥(+g‘𝑀)𝐴)(+g‘𝑀)𝑦) = (𝑥(+g‘𝑀)(𝐴(+g‘𝑀)𝑦)))
27	10, 25, 26	syl2anc 596	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝐴(+g‘𝑀)𝑦) = (0g‘𝑀)) ∧ 𝑥 ∈ 𝐵) ∧ (𝑥(+g‘𝑀)𝐴) = (0g‘𝑀)) → ((𝑥(+g‘𝑀)𝐴)(+g‘𝑀)𝑦) = (𝑥(+g‘𝑀)(𝐴(+g‘𝑀)𝑦)))
28	27	eqcomd 2769	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝐴(+g‘𝑀)𝑦) = (0g‘𝑀)) ∧ 𝑥 ∈ 𝐵) ∧ (𝑥(+g‘𝑀)𝐴) = (0g‘𝑀)) → (𝑥(+g‘𝑀)(𝐴(+g‘𝑀)𝑦)) = ((𝑥(+g‘𝑀)𝐴)(+g‘𝑀)𝑦))
29		simpr 490	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝐴(+g‘𝑀)𝑦) = (0g‘𝑀)) ∧ 𝑥 ∈ 𝐵) ∧ (𝑥(+g‘𝑀)𝐴) = (0g‘𝑀)) → (𝑥(+g‘𝑀)𝐴) = (0g‘𝑀))
30	29	oveq1d 7466	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝐴(+g‘𝑀)𝑦) = (0g‘𝑀)) ∧ 𝑥 ∈ 𝐵) ∧ (𝑥(+g‘𝑀)𝐴) = (0g‘𝑀)) → ((𝑥(+g‘𝑀)𝐴)(+g‘𝑀)𝑦) = ((0g‘𝑀)(+g‘𝑀)𝑦))
31	12, 13, 14	mndlid 18994	. . . . . . . . . 10 ⊢ ((𝑀 ∈ Mnd ∧ 𝑦 ∈ 𝐵) → ((0g‘𝑀)(+g‘𝑀)𝑦) = 𝑦)
32	10, 24, 31	syl2anc 596	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝐴(+g‘𝑀)𝑦) = (0g‘𝑀)) ∧ 𝑥 ∈ 𝐵) ∧ (𝑥(+g‘𝑀)𝐴) = (0g‘𝑀)) → ((0g‘𝑀)(+g‘𝑀)𝑦) = 𝑦)
33	30, 32	eqtrd 2798	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝐴(+g‘𝑀)𝑦) = (0g‘𝑀)) ∧ 𝑥 ∈ 𝐵) ∧ (𝑥(+g‘𝑀)𝐴) = (0g‘𝑀)) → ((𝑥(+g‘𝑀)𝐴)(+g‘𝑀)𝑦) = 𝑦)
34	21, 28, 33	3eqtrd 2802	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝐴(+g‘𝑀)𝑦) = (0g‘𝑀)) ∧ 𝑥 ∈ 𝐵) ∧ (𝑥(+g‘𝑀)𝐴) = (0g‘𝑀)) → 𝑥 = 𝑦)
35	34	ex 418	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝐴(+g‘𝑀)𝑦) = (0g‘𝑀)) ∧ 𝑥 ∈ 𝐵) → ((𝑥(+g‘𝑀)𝐴) = (0g‘𝑀) → 𝑥 = 𝑦))
36	35	ralrimiva 3158	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝐴(+g‘𝑀)𝑦) = (0g‘𝑀)) → ∀𝑥 ∈ 𝐵 ((𝑥(+g‘𝑀)𝐴) = (0g‘𝑀) → 𝑥 = 𝑦))
37	36	ex 418	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ((𝐴(+g‘𝑀)𝑦) = (0g‘𝑀) → ∀𝑥 ∈ 𝐵 ((𝑥(+g‘𝑀)𝐴) = (0g‘𝑀) → 𝑥 = 𝑦)))
38	37	reximdva 3179	. . 3 ⊢ (𝜑 → (∃𝑦 ∈ 𝐵 (𝐴(+g‘𝑀)𝑦) = (0g‘𝑀) → ∃𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑥(+g‘𝑀)𝐴) = (0g‘𝑀) → 𝑥 = 𝑦)))
39	8, 38	mpd 16	. 2 ⊢ (𝜑 → ∃𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑥(+g‘𝑀)𝐴) = (0g‘𝑀) → 𝑥 = 𝑦))
40		nfv 1949	. . 3 ⊢ Ⅎ𝑦(𝑥(+g‘𝑀)𝐴) = (0g‘𝑀)
41	40	rmo2i 3839	. 2 ⊢ (∃𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑥(+g‘𝑀)𝐴) = (0g‘𝑀) → 𝑥 = 𝑦) → ∃*𝑥 ∈ 𝐵 (𝑥(+g‘𝑀)𝐴) = (0g‘𝑀))
42	39, 41	syl 18	1 ⊢ (𝜑 → ∃*𝑥 ∈ 𝐵 (𝑥(+g‘𝑀)𝐴) = (0g‘𝑀))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  ∃*wrmo 3368  ‘cfv 6566  (class class class)co 7451  Basecbs 17431  +_gcplusg 17472  0_gc0g 17660  Mndcmnd 18973

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-iota 6520  df-fun 6568  df-fv 6574  df-riota 7408  df-ov 7454  df-0g 17662  df-mgm 18866  df-sgrp 18958  df-mnd 18974

This theorem is used by:  primrootsunit1  43413