Theorem mndinvmod 18730

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 483	. . . . . . . 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 18721	. . . . . . . 8 ⊢ ((𝐺 ∈ Mnd ∧ 𝑤 ∈ 𝐵) → (𝑤 + 0 ) = 𝑤)
7	1, 2, 6	syl2an 602	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑤 + 0 ) = 𝑤)
8	7	eqcomd 2746	. . . . . 6 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝑤 = (𝑤 + 0 ))
9	8	adantr 481	. . . . 5 ⊢ (((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 ))) → 𝑤 = (𝑤 + 0 ))
10		oveq2 7371	. . . . . . . . 9 ⊢ ( 0 = (𝐴 + 𝑣) → (𝑤 + 0 ) = (𝑤 + (𝐴 + 𝑣)))
11	10	eqcoms 2748	. . . . . . . 8 ⊢ ((𝐴 + 𝑣) = 0 → (𝑤 + 0 ) = (𝑤 + (𝐴 + 𝑣)))
12	11	adantl 482	. . . . . . 7 ⊢ (((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 ) → (𝑤 + 0 ) = (𝑤 + (𝐴 + 𝑣)))
13	12	adantl 482	. . . . . 6 ⊢ ((((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 )) → (𝑤 + 0 ) = (𝑤 + (𝐴 + 𝑣)))
14	13	adantl 482	. . . . 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 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝐺 ∈ Mnd)
16	2	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝑤 ∈ 𝐵)
17		mndinvmod.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
18	17	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝐴 ∈ 𝐵)
19		simpr 485	. . . . . . . . 9 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) → 𝑣 ∈ 𝐵)
20	19	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝑣 ∈ 𝐵)
21	3, 4	mndass 18709	. . . . . . . . 9 ⊢ ((𝐺 ∈ Mnd ∧ (𝑤 ∈ 𝐵 ∧ 𝐴 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → ((𝑤 + 𝐴) + 𝑣) = (𝑤 + (𝐴 + 𝑣)))
22	21	eqcomd 2746	. . . . . . . 8 ⊢ ((𝐺 ∈ Mnd ∧ (𝑤 ∈ 𝐵 ∧ 𝐴 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑤 + (𝐴 + 𝑣)) = ((𝑤 + 𝐴) + 𝑣))
23	15, 16, 18, 20, 22	syl13anc 1380	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑤 + (𝐴 + 𝑣)) = ((𝑤 + 𝐴) + 𝑣))
24	23	adantr 481	. . . . . 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 7370	. . . . . . . . 9 ⊢ ((𝑤 + 𝐴) = 0 → ((𝑤 + 𝐴) + 𝑣) = ( 0 + 𝑣))
26	25	adantr 481	. . . . . . . 8 ⊢ (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) → ((𝑤 + 𝐴) + 𝑣) = ( 0 + 𝑣))
27	26	adantr 481	. . . . . . 7 ⊢ ((((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 )) → ((𝑤 + 𝐴) + 𝑣) = ( 0 + 𝑣))
28	27	adantl 482	. . . . . 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 18720	. . . . . . . 8 ⊢ ((𝐺 ∈ Mnd ∧ 𝑣 ∈ 𝐵) → ( 0 + 𝑣) = 𝑣)
30	1, 19, 29	syl2an 602	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → ( 0 + 𝑣) = 𝑣)
31	30	adantr 481	. . . . . 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 2779	. . . . 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 2779	. . . 4 ⊢ (((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 ))) → 𝑤 = 𝑣)
34	33	ex 413	. . 3 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → ((((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 )) → 𝑤 = 𝑣))
35	34	ralrimivva 3183	. 2 ⊢ (𝜑 → ∀𝑤 ∈ 𝐵 ∀𝑣 ∈ 𝐵 ((((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 )) → 𝑤 = 𝑣))
36		oveq1 7370	. . . . 5 ⊢ (𝑤 = 𝑣 → (𝑤 + 𝐴) = (𝑣 + 𝐴))
37	36	eqeq1d 2742	. . . 4 ⊢ (𝑤 = 𝑣 → ((𝑤 + 𝐴) = 0 ↔ (𝑣 + 𝐴) = 0 ))
38		oveq2 7371	. . . . 5 ⊢ (𝑤 = 𝑣 → (𝐴 + 𝑤) = (𝐴 + 𝑣))
39	38	eqeq1d 2742	. . . 4 ⊢ (𝑤 = 𝑣 → ((𝐴 + 𝑤) = 0 ↔ (𝐴 + 𝑣) = 0 ))
40	37, 39	anbi12d 638	. . 3 ⊢ (𝑤 = 𝑣 → (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ↔ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 )))
41	40	rmo4 3678	. 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 235	1 ⊢ (𝜑 → ∃*𝑤 ∈ 𝐵 ((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3054  ∃*wrmo 3344  ‘cfv 6492  (class class class)co 7363  Basecbs 17177  +_gcplusg 17218  0_gc0g 17400  Mndcmnd 18700

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6448  df-fun 6494  df-fv 6500  df-riota 7320  df-ov 7366  df-0g 17402  df-mgm 18606  df-sgrp 18685  df-mnd 18701

This theorem is referenced by:  rinvmod  19779