Theorem mndmolinv 42465

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 1916	. . . . . 6 ⊢ Ⅎ𝑦(𝐴(+g‘𝑀)𝑥) = (0g‘𝑀)
3		nfv 1916	. . . . . 6 ⊢ Ⅎ𝑥(𝐴(+g‘𝑀)𝑦) = (0g‘𝑀)
4		oveq2 7376	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴(+g‘𝑀)𝑥) = (𝐴(+g‘𝑀)𝑦))
5	4	eqeq1d 2739	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴(+g‘𝑀)𝑥) = (0g‘𝑀) ↔ (𝐴(+g‘𝑀)𝑦) = (0g‘𝑀)))
6	2, 3, 5	cbvrexw 3281	. . . . 5 ⊢ (∃𝑥 ∈ 𝐵 (𝐴(+g‘𝑀)𝑥) = (0g‘𝑀) ↔ ∃𝑦 ∈ 𝐵 (𝐴(+g‘𝑀)𝑦) = (0g‘𝑀))
7	6	biimpi 216	. . . 4 ⊢ (∃𝑥 ∈ 𝐵 (𝐴(+g‘𝑀)𝑥) = (0g‘𝑀) → ∃𝑦 ∈ 𝐵 (𝐴(+g‘𝑀)𝑦) = (0g‘𝑀))
8	1, 7	syl 17	. . 3 ⊢ (𝜑 → ∃𝑦 ∈ 𝐵 (𝐴(+g‘𝑀)𝑦) = (0g‘𝑀))
9		mndmolinv.2	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ Mnd)
10	9	ad4antr 733	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝐴(+g‘𝑀)𝑦) = (0g‘𝑀)) ∧ 𝑥 ∈ 𝐵) ∧ (𝑥(+g‘𝑀)𝐴) = (0g‘𝑀)) → 𝑀 ∈ Mnd)
11		simplr 769	. . . . . . . . . . 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 2737	. . . . . . . . . . . 12 ⊢ (+g‘𝑀) = (+g‘𝑀)
14		eqid 2737	. . . . . . . . . . . 12 ⊢ (0g‘𝑀) = (0g‘𝑀)
15	12, 13, 14	mndrid 18692	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → (𝑥(+g‘𝑀)(0g‘𝑀)) = 𝑥)
16	10, 11, 15	syl2anc 585	. . . . . . . . . 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 2743	. . . . . . . . 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 776	. . . . . . . . . . 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 2743	. . . . . . . . . 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 7384	. . . . . . . . 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 2772	. . . . . . . 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 733	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝐴(+g‘𝑀)𝑦) = (0g‘𝑀)) ∧ 𝑥 ∈ 𝐵) ∧ (𝑥(+g‘𝑀)𝐴) = (0g‘𝑀)) → 𝐴 ∈ 𝐵)
24		simp-4r 784	. . . . . . . . . . 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 1129	. . . . . . . . . 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 18680	. . . . . . . . . 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 585	. . . . . . . . 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 2743	. . . . . . . 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 484	. . . . . . . . . 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 7383	. . . . . . . . 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 18691	. . . . . . . . . 10 ⊢ ((𝑀 ∈ Mnd ∧ 𝑦 ∈ 𝐵) → ((0g‘𝑀)(+g‘𝑀)𝑦) = 𝑦)
32	10, 24, 31	syl2anc 585	. . . . . . . . 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 2772	. . . . . . . 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 2776	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝐴(+g‘𝑀)𝑦) = (0g‘𝑀)) ∧ 𝑥 ∈ 𝐵) ∧ (𝑥(+g‘𝑀)𝐴) = (0g‘𝑀)) → 𝑥 = 𝑦)
35	34	ex 412	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝐴(+g‘𝑀)𝑦) = (0g‘𝑀)) ∧ 𝑥 ∈ 𝐵) → ((𝑥(+g‘𝑀)𝐴) = (0g‘𝑀) → 𝑥 = 𝑦))
36	35	ralrimiva 3130	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝐴(+g‘𝑀)𝑦) = (0g‘𝑀)) → ∀𝑥 ∈ 𝐵 ((𝑥(+g‘𝑀)𝐴) = (0g‘𝑀) → 𝑥 = 𝑦))
37	36	ex 412	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ((𝐴(+g‘𝑀)𝑦) = (0g‘𝑀) → ∀𝑥 ∈ 𝐵 ((𝑥(+g‘𝑀)𝐴) = (0g‘𝑀) → 𝑥 = 𝑦)))
38	37	reximdva 3151	. . 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 15	. 2 ⊢ (𝜑 → ∃𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑥(+g‘𝑀)𝐴) = (0g‘𝑀) → 𝑥 = 𝑦))
40		nfv 1916	. . 3 ⊢ Ⅎ𝑦(𝑥(+g‘𝑀)𝐴) = (0g‘𝑀)
41	40	rmo2i 3840	. 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 17	1 ⊢ (𝜑 → ∃*𝑥 ∈ 𝐵 (𝑥(+g‘𝑀)𝐴) = (0g‘𝑀))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  ∃*wrmo 3351  ‘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:  primrootsunit1  42467