Theorem mndmolinv 42090

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 1914	. . . . . 6 ⊢ Ⅎ𝑦(𝐴(+g‘𝑀)𝑥) = (0g‘𝑀)
3		nfv 1914	. . . . . 6 ⊢ Ⅎ𝑥(𝐴(+g‘𝑀)𝑦) = (0g‘𝑀)
4		oveq2 7398	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴(+g‘𝑀)𝑥) = (𝐴(+g‘𝑀)𝑦))
5	4	eqeq1d 2732	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴(+g‘𝑀)𝑥) = (0g‘𝑀) ↔ (𝐴(+g‘𝑀)𝑦) = (0g‘𝑀)))
6	2, 3, 5	cbvrexw 3283	. . . . 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 2730	. . . . . . . . . . . 12 ⊢ (+g‘𝑀) = (+g‘𝑀)
14		eqid 2730	. . . . . . . . . . . 12 ⊢ (0g‘𝑀) = (0g‘𝑀)
15	12, 13, 14	mndrid 18689	. . . . . . . . . . 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 2736	. . . . . . . . 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 2736	. . . . . . . . . 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 7406	. . . . . . . . 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 2765	. . . . . . . 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 18677	. . . . . . . . . 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 2736	. . . . . . . 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 7405	. . . . . . . . 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 18688	. . . . . . . . . 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 2765	. . . . . . . 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 2769	. . . . . . 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 3126	. . . . 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 3147	. . 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 1914	. . 3 ⊢ Ⅎ𝑦(𝑥(+g‘𝑀)𝐴) = (0g‘𝑀)
41	40	rmo2i 3854	. 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 1540   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054  ∃*wrmo 3355  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  +_gcplusg 17227  0_gc0g 17409  Mndcmnd 18668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-iota 6467  df-fun 6516  df-fv 6522  df-riota 7347  df-ov 7393  df-0g 17411  df-mgm 18574  df-sgrp 18653  df-mnd 18669

This theorem is referenced by:  primrootsunit1  42092