Theorem mndinvmod 13086

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 109	. . . . . . . 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 13077	. . . . . . . 8 ⊢ ((𝐺 ∈ Mnd ∧ 𝑤 ∈ 𝐵) → (𝑤 + 0 ) = 𝑤)
7	1, 2, 6	syl2an 289	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑤 + 0 ) = 𝑤)
8	7	eqcomd 2202	. . . . . 6 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝑤 = (𝑤 + 0 ))
9	8	adantr 276	. . . . 5 ⊢ (((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 ))) → 𝑤 = (𝑤 + 0 ))
10		oveq2 5930	. . . . . . . . 9 ⊢ ( 0 = (𝐴 + 𝑣) → (𝑤 + 0 ) = (𝑤 + (𝐴 + 𝑣)))
11	10	eqcoms 2199	. . . . . . . 8 ⊢ ((𝐴 + 𝑣) = 0 → (𝑤 + 0 ) = (𝑤 + (𝐴 + 𝑣)))
12	11	adantl 277	. . . . . . 7 ⊢ (((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 ) → (𝑤 + 0 ) = (𝑤 + (𝐴 + 𝑣)))
13	12	adantl 277	. . . . . 6 ⊢ ((((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 )) → (𝑤 + 0 ) = (𝑤 + (𝐴 + 𝑣)))
14	13	adantl 277	. . . . 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 276	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝐺 ∈ Mnd)
16	2	adantl 277	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝑤 ∈ 𝐵)
17		mndinvmod.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
18	17	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝐴 ∈ 𝐵)
19		simpr 110	. . . . . . . . 9 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) → 𝑣 ∈ 𝐵)
20	19	adantl 277	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → 𝑣 ∈ 𝐵)
21	3, 4	mndass 13065	. . . . . . . . 9 ⊢ ((𝐺 ∈ Mnd ∧ (𝑤 ∈ 𝐵 ∧ 𝐴 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → ((𝑤 + 𝐴) + 𝑣) = (𝑤 + (𝐴 + 𝑣)))
22	21	eqcomd 2202	. . . . . . . 8 ⊢ ((𝐺 ∈ Mnd ∧ (𝑤 ∈ 𝐵 ∧ 𝐴 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑤 + (𝐴 + 𝑣)) = ((𝑤 + 𝐴) + 𝑣))
23	15, 16, 18, 20, 22	syl13anc 1251	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑤 + (𝐴 + 𝑣)) = ((𝑤 + 𝐴) + 𝑣))
24	23	adantr 276	. . . . . 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 5929	. . . . . . . . 9 ⊢ ((𝑤 + 𝐴) = 0 → ((𝑤 + 𝐴) + 𝑣) = ( 0 + 𝑣))
26	25	adantr 276	. . . . . . . 8 ⊢ (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) → ((𝑤 + 𝐴) + 𝑣) = ( 0 + 𝑣))
27	26	adantr 276	. . . . . . 7 ⊢ ((((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 )) → ((𝑤 + 𝐴) + 𝑣) = ( 0 + 𝑣))
28	27	adantl 277	. . . . . 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 13076	. . . . . . . 8 ⊢ ((𝐺 ∈ Mnd ∧ 𝑣 ∈ 𝐵) → ( 0 + 𝑣) = 𝑣)
30	1, 19, 29	syl2an 289	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → ( 0 + 𝑣) = 𝑣)
31	30	adantr 276	. . . . . 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 2233	. . . . 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 2233	. . . 4 ⊢ (((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 ))) → 𝑤 = 𝑣)
34	33	ex 115	. . 3 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → ((((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 )) → 𝑤 = 𝑣))
35	34	ralrimivva 2579	. 2 ⊢ (𝜑 → ∀𝑤 ∈ 𝐵 ∀𝑣 ∈ 𝐵 ((((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 )) → 𝑤 = 𝑣))
36		oveq1 5929	. . . . 5 ⊢ (𝑤 = 𝑣 → (𝑤 + 𝐴) = (𝑣 + 𝐴))
37	36	eqeq1d 2205	. . . 4 ⊢ (𝑤 = 𝑣 → ((𝑤 + 𝐴) = 0 ↔ (𝑣 + 𝐴) = 0 ))
38		oveq2 5930	. . . . 5 ⊢ (𝑤 = 𝑣 → (𝐴 + 𝑤) = (𝐴 + 𝑣))
39	38	eqeq1d 2205	. . . 4 ⊢ (𝑤 = 𝑣 → ((𝐴 + 𝑤) = 0 ↔ (𝐴 + 𝑣) = 0 ))
40	37, 39	anbi12d 473	. . 3 ⊢ (𝑤 = 𝑣 → (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ↔ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 )))
41	40	rmo4 2957	. 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 134	1 ⊢ (𝜑 → ∃*𝑤 ∈ 𝐵 ((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 980   = wceq 1364   ∈ wcel 2167  ∀wral 2475  ∃*wrmo 2478  ‘cfv 5258  (class class class)co 5922  Basecbs 12678  +_gcplusg 12755  0_gc0g 12927  Mndcmnd 13057

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-cnex 7970  ax-resscn 7971  ax-1re 7973  ax-addrcl 7976

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-riota 5877  df-ov 5925  df-inn 8991  df-2 9049  df-ndx 12681  df-slot 12682  df-base 12684  df-plusg 12768  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058

This theorem is referenced by:  rinvmod  13439