Theorem rinvmod 19747

Description: Uniqueness of a right inverse element in a commutative monoid, if it exists. Corresponds to caovmo 7605. (Contributed by AV, 31-Dec-2023.)

Hypotheses

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

Assertion

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

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

Allowed substitution hint:   𝐺(𝑤)

Proof of Theorem rinvmod

Step	Hyp	Ref	Expression
1		rinvmod.m	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ CMnd)
2	1	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → 𝐺 ∈ CMnd)
3		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → 𝑤 ∈ 𝐵)
4		rinvmod.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
5	4	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → 𝐴 ∈ 𝐵)
6		rinvmod.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐺)
7		rinvmod.p	. . . . . . . . 9 ⊢ + = (+g‘𝐺)
8	6, 7	cmncom 19739	. . . . . . . 8 ⊢ ((𝐺 ∈ CMnd ∧ 𝑤 ∈ 𝐵 ∧ 𝐴 ∈ 𝐵) → (𝑤 + 𝐴) = (𝐴 + 𝑤))
9	2, 3, 5, 8	syl3anc 1374	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → (𝑤 + 𝐴) = (𝐴 + 𝑤))
10	9	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐵) ∧ (𝐴 + 𝑤) = 0 ) → (𝑤 + 𝐴) = (𝐴 + 𝑤))
11		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐵) ∧ (𝐴 + 𝑤) = 0 ) → (𝐴 + 𝑤) = 0 )
12	10, 11	eqtrd 2772	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐵) ∧ (𝐴 + 𝑤) = 0 ) → (𝑤 + 𝐴) = 0 )
13	12, 11	jca 511	. . . 4 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐵) ∧ (𝐴 + 𝑤) = 0 ) → ((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ))
14	13	ex 412	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → ((𝐴 + 𝑤) = 0 → ((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 )))
15	14	ralrimiva 3130	. 2 ⊢ (𝜑 → ∀𝑤 ∈ 𝐵 ((𝐴 + 𝑤) = 0 → ((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 )))
16		rinvmod.0	. . 3 ⊢ 0 = (0g‘𝐺)
17		cmnmnd 19738	. . . 4 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
18	1, 17	syl 17	. . 3 ⊢ (𝜑 → 𝐺 ∈ Mnd)
19	6, 16, 7, 18, 4	mndinvmod 18701	. 2 ⊢ (𝜑 → ∃*𝑤 ∈ 𝐵 ((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ))
20		rmoim 3700	. 2 ⊢ (∀𝑤 ∈ 𝐵 ((𝐴 + 𝑤) = 0 → ((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 )) → (∃*𝑤 ∈ 𝐵 ((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) → ∃*𝑤 ∈ 𝐵 (𝐴 + 𝑤) = 0 ))
21	15, 19, 20	sylc 65	1 ⊢ (𝜑 → ∃*𝑤 ∈ 𝐵 (𝐴 + 𝑤) = 0 )

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃*wrmo 3351  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  +_gcplusg 17189  0_gc0g 17371  Mndcmnd 18671  CMndccmn 19721

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6456  df-fun 6502  df-fv 6508  df-riota 7325  df-ov 7371  df-0g 17373  df-mgm 18577  df-sgrp 18656  df-mnd 18672  df-cmn 19723

This theorem is referenced by: (None)