Theorem mndinvmod 18877

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 488	. . . . . . . 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 18864	. . . . . . . 8 ⊢ ((𝐺 ∈ Mnd ∧ 𝑤 ∈ 𝐵) → (𝑤 + 0 ) = 𝑤)
7	1, 2, 6	syl2an 608	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑤 + 0 ) = 𝑤)
8	7	eqcomd 2768	. . . . . 6 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝑤 = (𝑤 + 0 ))
9	8	adantr 486	. . . . 5 ⊢ (((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 ))) → 𝑤 = (𝑤 + 0 ))
10		oveq2 7425	. . . . . . . . 9 ⊢ ( 0 = (𝐴 + 𝑣) → (𝑤 + 0 ) = (𝑤 + (𝐴 + 𝑣)))
11	10	eqcoms 2770	. . . . . . . 8 ⊢ ((𝐴 + 𝑣) = 0 → (𝑤 + 0 ) = (𝑤 + (𝐴 + 𝑣)))
12	11	adantl 487	. . . . . . 7 ⊢ (((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 ) → (𝑤 + 0 ) = (𝑤 + (𝐴 + 𝑣)))
13	12	adantl 487	. . . . . 6 ⊢ ((((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 )) → (𝑤 + 0 ) = (𝑤 + (𝐴 + 𝑣)))
14	13	adantl 487	. . . . 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 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝐺 ∈ Mnd)
16	2	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝑤 ∈ 𝐵)
17		mndinvmod.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
18	17	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝐴 ∈ 𝐵)
19		simpr 490	. . . . . . . . 9 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) → 𝑣 ∈ 𝐵)
20	19	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝑣 ∈ 𝐵)
21	3, 4	mndass 18851	. . . . . . . . 9 ⊢ ((𝐺 ∈ Mnd ∧ (𝑤 ∈ 𝐵 ∧ 𝐴 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → ((𝑤 + 𝐴) + 𝑣) = (𝑤 + (𝐴 + 𝑣)))
22	21	eqcomd 2768	. . . . . . . 8 ⊢ ((𝐺 ∈ Mnd ∧ (𝑤 ∈ 𝐵 ∧ 𝐴 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑤 + (𝐴 + 𝑣)) = ((𝑤 + 𝐴) + 𝑣))
23	15, 16, 18, 20, 22	syl13anc 1399	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑤 + (𝐴 + 𝑣)) = ((𝑤 + 𝐴) + 𝑣))
24	23	adantr 486	. . . . . 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 7424	. . . . . . . . 9 ⊢ ((𝑤 + 𝐴) = 0 → ((𝑤 + 𝐴) + 𝑣) = ( 0 + 𝑣))
26	25	adantr 486	. . . . . . . 8 ⊢ (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) → ((𝑤 + 𝐴) + 𝑣) = ( 0 + 𝑣))
27	26	adantr 486	. . . . . . 7 ⊢ ((((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 )) → ((𝑤 + 𝐴) + 𝑣) = ( 0 + 𝑣))
28	27	adantl 487	. . . . . 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 18863	. . . . . . . 8 ⊢ ((𝐺 ∈ Mnd ∧ 𝑣 ∈ 𝐵) → ( 0 + 𝑣) = 𝑣)
30	1, 19, 29	syl2an 608	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → ( 0 + 𝑣) = 𝑣)
31	30	adantr 486	. . . . . 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 2801	. . . . 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 2801	. . . 4 ⊢ (((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 ))) → 𝑤 = 𝑣)
34	33	ex 418	. . 3 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → ((((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 )) → 𝑤 = 𝑣))
35	34	ralrimivva 3207	. 2 ⊢ (𝜑 → ∀𝑤 ∈ 𝐵 ∀𝑣 ∈ 𝐵 ((((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 )) → 𝑤 = 𝑣))
36		oveq1 7424	. . . . 5 ⊢ (𝑤 = 𝑣 → (𝑤 + 𝐴) = (𝑣 + 𝐴))
37	36	eqeq1d 2764	. . . 4 ⊢ (𝑤 = 𝑣 → ((𝑤 + 𝐴) = 0 ↔ (𝑣 + 𝐴) = 0 ))
38		oveq2 7425	. . . . 5 ⊢ (𝑤 = 𝑣 → (𝐴 + 𝑤) = (𝐴 + 𝑣))
39	38	eqeq1d 2764	. . . 4 ⊢ (𝑤 = 𝑣 → ((𝐴 + 𝑤) = 0 ↔ (𝐴 + 𝑣) = 0 ))
40	37, 39	anbi12d 644	. . 3 ⊢ (𝑤 = 𝑣 → (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ↔ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 )))
41	40	rmo4 3691	. 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 ))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3078  ∃*wrmo 3366  ‘cfv 6537  (class class class)co 7417  Basecbs 17307  +_gcplusg 17348  0_gc0g 17530  Mndcmnd 18842

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

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 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6493  df-fun 6539  df-fv 6545  df-riota 7374  df-ov 7420  df-0g 17532  df-mgm 18736  df-sgrp 18827  df-mnd 18843

This theorem is used by:  rinvmod  19939