Theorem mndinvmod 13741

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 13732	. . . . . . . 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 2244	. . . . . 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 6087	. . . . . . . . 9 ⊢ ( 0 = (𝐴 + 𝑣) → (𝑤 + 0 ) = (𝑤 + (𝐴 + 𝑣)))
11	10	eqcoms 2241	. . . . . . . 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 13720	. . . . . . . . 9 ⊢ ((𝐺 ∈ Mnd ∧ (𝑤 ∈ 𝐵 ∧ 𝐴 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → ((𝑤 + 𝐴) + 𝑣) = (𝑤 + (𝐴 + 𝑣)))
22	21	eqcomd 2244	. . . . . . . 8 ⊢ ((𝐺 ∈ Mnd ∧ (𝑤 ∈ 𝐵 ∧ 𝐴 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑤 + (𝐴 + 𝑣)) = ((𝑤 + 𝐴) + 𝑣))
23	15, 16, 18, 20, 22	syl13anc 1280	. . . . . . 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 6086	. . . . . . . . 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 13731	. . . . . . . 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 2275	. . . . 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 2275	. . . 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 2632	. 2 ⊢ (𝜑 → ∀𝑤 ∈ 𝐵 ∀𝑣 ∈ 𝐵 ((((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 )) → 𝑤 = 𝑣))
36		oveq1 6086	. . . . 5 ⊢ (𝑤 = 𝑣 → (𝑤 + 𝐴) = (𝑣 + 𝐴))
37	36	eqeq1d 2247	. . . 4 ⊢ (𝑤 = 𝑣 → ((𝑤 + 𝐴) = 0 ↔ (𝑣 + 𝐴) = 0 ))
38		oveq2 6087	. . . . 5 ⊢ (𝑤 = 𝑣 → (𝐴 + 𝑤) = (𝐴 + 𝑣))
39	38	eqeq1d 2247	. . . 4 ⊢ (𝑤 = 𝑣 → ((𝐴 + 𝑤) = 0 ↔ (𝐴 + 𝑣) = 0 ))
40	37, 39	anbi12d 477	. . 3 ⊢ (𝑤 = 𝑣 → (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ↔ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 )))
41	40	rmo4 3019	. 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 1009   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ∃*wrmo 2531  ‘cfv 5375  (class class class)co 6079  Basecbs 13335  +_gcplusg 13414  0_gc0g 13593  Mndcmnd 13712

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-cnex 8264  ax-resscn 8265  ax-1re 8267  ax-addrcl 8270

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-iota 5335  df-fun 5377  df-fn 5378  df-fv 5383  df-riota 6032  df-ov 6082  df-inn 9288  df-2 9346  df-ndx 13338  df-slot 13339  df-base 13341  df-plusg 13427  df-0g 13595  df-mgm 13659  df-sgrp 13700  df-mnd 13713

This theorem is referenced by:  rinvmod  14096