Theorem mndmolinv 42128

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 1915	. . . . . 6 ⊢ Ⅎ𝑦(𝐴(+g‘𝑀)𝑥) = (0g‘𝑀)
3		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑥(𝐴(+g‘𝑀)𝑦) = (0g‘𝑀)
4		oveq2 7349	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴(+g‘𝑀)𝑥) = (𝐴(+g‘𝑀)𝑦))
5	4	eqeq1d 2733	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴(+g‘𝑀)𝑥) = (0g‘𝑀) ↔ (𝐴(+g‘𝑀)𝑦) = (0g‘𝑀)))
6	2, 3, 5	cbvrexw 3275	. . . . 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 732	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝐴(+g‘𝑀)𝑦) = (0g‘𝑀)) ∧ 𝑥 ∈ 𝐵) ∧ (𝑥(+g‘𝑀)𝐴) = (0g‘𝑀)) → 𝑀 ∈ Mnd)
11		simplr 768	. . . . . . . . . . 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 2731	. . . . . . . . . . . 12 ⊢ (+g‘𝑀) = (+g‘𝑀)
14		eqid 2731	. . . . . . . . . . . 12 ⊢ (0g‘𝑀) = (0g‘𝑀)
15	12, 13, 14	mndrid 18658	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → (𝑥(+g‘𝑀)(0g‘𝑀)) = 𝑥)
16	10, 11, 15	syl2anc 584	. . . . . . . . . 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 2737	. . . . . . . . 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 775	. . . . . . . . . . 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 2737	. . . . . . . . . 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 7357	. . . . . . . . 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 2766	. . . . . . . 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 732	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝐴(+g‘𝑀)𝑦) = (0g‘𝑀)) ∧ 𝑥 ∈ 𝐵) ∧ (𝑥(+g‘𝑀)𝐴) = (0g‘𝑀)) → 𝐴 ∈ 𝐵)
24		simp-4r 783	. . . . . . . . . . 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 1128	. . . . . . . . . 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 18646	. . . . . . . . . 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 584	. . . . . . . . 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 2737	. . . . . . . 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 7356	. . . . . . . . 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 18657	. . . . . . . . . 10 ⊢ ((𝑀 ∈ Mnd ∧ 𝑦 ∈ 𝐵) → ((0g‘𝑀)(+g‘𝑀)𝑦) = 𝑦)
32	10, 24, 31	syl2anc 584	. . . . . . . . 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 2766	. . . . . . . 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 2770	. . . . . . 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 3124	. . . . 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 3145	. . 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 1915	. . 3 ⊢ Ⅎ𝑦(𝑥(+g‘𝑀)𝐴) = (0g‘𝑀)
41	40	rmo2i 3834	. 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 1086   = wceq 1541   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056  ∃*wrmo 3345  ‘cfv 6476  (class class class)co 7341  Basecbs 17115  +_gcplusg 17156  0_gc0g 17338  Mndcmnd 18637

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 5229  ax-nul 5239  ax-pr 5365

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 3737  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-iota 6432  df-fun 6478  df-fv 6484  df-riota 7298  df-ov 7344  df-0g 17340  df-mgm 18543  df-sgrp 18622  df-mnd 18638

This theorem is referenced by:  primrootsunit1  42130