Theorem mndinvmod 18812

Description: Uniqueness of an inverse element in a monoid, if it exists. (Contributed by AV, 20-Jan-2024.)

Hypotheses

Ref	Expression
mndinvmod.b	⊢ 𝐵 = (Base‘𝐺)
mndinvmod.0	⊢ 0 = (0g‘𝐺)
mndinvmod.p	⊢ + = (+g‘𝐺)
mndinvmod.m	⊢ (𝜑 → 𝐺 ∈ Mnd)
mndinvmod.a	⊢ (𝜑 → 𝐴 ∈ 𝐵)

Assertion

Ref	Expression
mndinvmod	⊢ (𝜑 → ∃*𝑤 ∈ 𝐵 ((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ))

Distinct variable groups:   𝑤,𝐴   𝑤,𝐵   𝑤, 0   𝑤, +   𝜑,𝑤

Allowed substitution hint:   𝐺(𝑤)

Proof of Theorem mndinvmod

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mndinvmod.m	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ Mnd)
2		simpl 487	. . . . . . . 8 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) → 𝑤 ∈ 𝐵)
3		mndinvmod.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐺)
4		mndinvmod.p	. . . . . . . . 9 ⊢ + = (+g‘𝐺)
5		mndinvmod.0	. . . . . . . . 9 ⊢ 0 = (0g‘𝐺)
6	3, 4, 5	mndrid 18803	. . . . . . . 8 ⊢ ((𝐺 ∈ Mnd ∧ 𝑤 ∈ 𝐵) → (𝑤 + 0 ) = 𝑤)
7	1, 2, 6	syl2an 607	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑤 + 0 ) = 𝑤)
8	7	eqcomd 2771	. . . . . 6 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝑤 = (𝑤 + 0 ))
9	8	adantr 485	. . . . 5 ⊢ (((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 ))) → 𝑤 = (𝑤 + 0 ))
10		oveq2 7408	. . . . . . . . 9 ⊢ ( 0 = (𝐴 + 𝑣) → (𝑤 + 0 ) = (𝑤 + (𝐴 + 𝑣)))
11	10	eqcoms 2773	. . . . . . . 8 ⊢ ((𝐴 + 𝑣) = 0 → (𝑤 + 0 ) = (𝑤 + (𝐴 + 𝑣)))
12	11	adantl 486	. . . . . . 7 ⊢ (((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 ) → (𝑤 + 0 ) = (𝑤 + (𝐴 + 𝑣)))
13	12	adantl 486	. . . . . 6 ⊢ ((((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 )) → (𝑤 + 0 ) = (𝑤 + (𝐴 + 𝑣)))
14	13	adantl 486	. . . . 5 ⊢ (((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 ))) → (𝑤 + 0 ) = (𝑤 + (𝐴 + 𝑣)))
15	1	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝐺 ∈ Mnd)
16	2	adantl 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝑤 ∈ 𝐵)
17		mndinvmod.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
18	17	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝐴 ∈ 𝐵)
19		simpr 489	. . . . . . . . 9 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) → 𝑣 ∈ 𝐵)
20	19	adantl 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝑣 ∈ 𝐵)
21	3, 4	mndass 18791	. . . . . . . . 9 ⊢ ((𝐺 ∈ Mnd ∧ (𝑤 ∈ 𝐵 ∧ 𝐴 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → ((𝑤 + 𝐴) + 𝑣) = (𝑤 + (𝐴 + 𝑣)))
22	21	eqcomd 2771	. . . . . . . 8 ⊢ ((𝐺 ∈ Mnd ∧ (𝑤 ∈ 𝐵 ∧ 𝐴 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑤 + (𝐴 + 𝑣)) = ((𝑤 + 𝐴) + 𝑣))
23	15, 16, 18, 20, 22	syl13anc 1395	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑤 + (𝐴 + 𝑣)) = ((𝑤 + 𝐴) + 𝑣))
24	23	adantr 485	. . . . . 6 ⊢ (((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 ))) → (𝑤 + (𝐴 + 𝑣)) = ((𝑤 + 𝐴) + 𝑣))
25		oveq1 7407	. . . . . . . . 9 ⊢ ((𝑤 + 𝐴) = 0 → ((𝑤 + 𝐴) + 𝑣) = ( 0 + 𝑣))
26	25	adantr 485	. . . . . . . 8 ⊢ (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) → ((𝑤 + 𝐴) + 𝑣) = ( 0 + 𝑣))
27	26	adantr 485	. . . . . . 7 ⊢ ((((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 )) → ((𝑤 + 𝐴) + 𝑣) = ( 0 + 𝑣))
28	27	adantl 486	. . . . . 6 ⊢ (((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 ))) → ((𝑤 + 𝐴) + 𝑣) = ( 0 + 𝑣))
29	3, 4, 5	mndlid 18802	. . . . . . . 8 ⊢ ((𝐺 ∈ Mnd ∧ 𝑣 ∈ 𝐵) → ( 0 + 𝑣) = 𝑣)
30	1, 19, 29	syl2an 607	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → ( 0 + 𝑣) = 𝑣)
31	30	adantr 485	. . . . . 6 ⊢ (((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 ))) → ( 0 + 𝑣) = 𝑣)
32	24, 28, 31	3eqtrd 2804	. . . . 5 ⊢ (((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 ))) → (𝑤 + (𝐴 + 𝑣)) = 𝑣)
33	9, 14, 32	3eqtrd 2804	. . . 4 ⊢ (((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 ))) → 𝑤 = 𝑣)
34	33	ex 417	. . 3 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → ((((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 )) → 𝑤 = 𝑣))
35	34	ralrimivva 3208	. 2 ⊢ (𝜑 → ∀𝑤 ∈ 𝐵 ∀𝑣 ∈ 𝐵 ((((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 )) → 𝑤 = 𝑣))
36		oveq1 7407	. . . . 5 ⊢ (𝑤 = 𝑣 → (𝑤 + 𝐴) = (𝑣 + 𝐴))
37	36	eqeq1d 2767	. . . 4 ⊢ (𝑤 = 𝑣 → ((𝑤 + 𝐴) = 0 ↔ (𝑣 + 𝐴) = 0 ))
38		oveq2 7408	. . . . 5 ⊢ (𝑤 = 𝑣 → (𝐴 + 𝑤) = (𝐴 + 𝑣))
39	38	eqeq1d 2767	. . . 4 ⊢ (𝑤 = 𝑣 → ((𝐴 + 𝑤) = 0 ↔ (𝐴 + 𝑣) = 0 ))
40	37, 39	anbi12d 643	. . 3 ⊢ (𝑤 = 𝑣 → (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ↔ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 )))
41	40	rmo4 3696	. 2 ⊢ (∃*𝑤 ∈ 𝐵 ((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ↔ ∀𝑤 ∈ 𝐵 ∀𝑣 ∈ 𝐵 ((((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 )) → 𝑤 = 𝑣))
42	35, 41	sylibr 237	1 ⊢ (𝜑 → ∃*𝑤 ∈ 𝐵 ((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  ∀wral 3079  ∃*wrmo 3369  ‘cfv 6525  (class class class)co 7400  Basecbs 17259  +_gcplusg 17300  0_gc0g 17482  Mndcmnd 18782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-iota 6481  df-fun 6527  df-fv 6533  df-riota 7357  df-ov 7403  df-0g 17484  df-mgm 18688  df-sgrp 18767  df-mnd 18783

This theorem is referenced by:  rinvmod  19867