Theorem rinvmod 14024

Description: Uniqueness of a right inverse element in a commutative monoid, if it exists. Corresponds to caovimo 6248. (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 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → 𝐺 ∈ CMnd)
3		simpr 110	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → 𝑤 ∈ 𝐵)
4		rinvmod.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
5	4	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → 𝐴 ∈ 𝐵)
6		rinvmod.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐺)
7		rinvmod.p	. . . . . . . . 9 ⊢ + = (+g‘𝐺)
8	6, 7	cmncom 14017	. . . . . . . 8 ⊢ ((𝐺 ∈ CMnd ∧ 𝑤 ∈ 𝐵 ∧ 𝐴 ∈ 𝐵) → (𝑤 + 𝐴) = (𝐴 + 𝑤))
9	2, 3, 5, 8	syl3anc 1274	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → (𝑤 + 𝐴) = (𝐴 + 𝑤))
10	9	adantr 276	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐵) ∧ (𝐴 + 𝑤) = 0 ) → (𝑤 + 𝐴) = (𝐴 + 𝑤))
11		simpr 110	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐵) ∧ (𝐴 + 𝑤) = 0 ) → (𝐴 + 𝑤) = 0 )
12	10, 11	eqtrd 2265	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐵) ∧ (𝐴 + 𝑤) = 0 ) → (𝑤 + 𝐴) = 0 )
13	12, 11	jca 306	. . . 4 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐵) ∧ (𝐴 + 𝑤) = 0 ) → ((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ))
14	13	ex 115	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → ((𝐴 + 𝑤) = 0 → ((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 )))
15	14	ralrimiva 2615	. 2 ⊢ (𝜑 → ∀𝑤 ∈ 𝐵 ((𝐴 + 𝑤) = 0 → ((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 )))
16		rinvmod.0	. . 3 ⊢ 0 = (0g‘𝐺)
17		cmnmnd 14016	. . . 4 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
18	1, 17	syl 14	. . 3 ⊢ (𝜑 → 𝐺 ∈ Mnd)
19	6, 16, 7, 18, 4	mndinvmod 13656	. 2 ⊢ (𝜑 → ∃*𝑤 ∈ 𝐵 ((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ))
20		rmoim 3018	. 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 62	1 ⊢ (𝜑 → ∃*𝑤 ∈ 𝐵 (𝐴 + 𝑤) = 0 )

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2203  ∀wral 2520  ∃*wrmo 2523  ‘cfv 5352  (class class class)co 6050  Basecbs 13210  +_gcplusg 13288  0_gc0g 13467  Mndcmnd 13627  CMndccmn 13999

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-cnex 8218  ax-resscn 8219  ax-1re 8221  ax-addrcl 8224

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-iota 5312  df-fun 5354  df-fn 5355  df-fv 5360  df-riota 6003  df-ov 6053  df-inn 9238  df-2 9296  df-ndx 13213  df-slot 13214  df-base 13216  df-plusg 13301  df-0g 13469  df-mgm 13567  df-sgrp 13613  df-mnd 13628  df-cmn 14001

This theorem is referenced by: (None)