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Theorem rinvmod 19825
Description: Uniqueness of a right inverse element in a commutative monoid, if it exists. Corresponds to caovmo 7671. (Contributed by AV, 31-Dec-2023.)
Hypotheses
Ref Expression
rinvmod.b 𝐵 = (Base‘𝐺)
rinvmod.0 0 = (0g𝐺)
rinvmod.p + = (+g𝐺)
rinvmod.m (𝜑𝐺 ∈ CMnd)
rinvmod.a (𝜑𝐴𝐵)
Assertion
Ref Expression
rinvmod (𝜑 → ∃*𝑤𝐵 (𝐴 + 𝑤) = 0 )
Distinct variable groups:   𝑤,𝐴   𝑤,𝐵   𝑤, 0   𝑤, +   𝜑,𝑤
Allowed substitution hint:   𝐺(𝑤)

Proof of Theorem rinvmod
StepHypRef Expression
1 rinvmod.m . . . . . . . . 9 (𝜑𝐺 ∈ CMnd)
21adantr 480 . . . . . . . 8 ((𝜑𝑤𝐵) → 𝐺 ∈ CMnd)
3 simpr 484 . . . . . . . 8 ((𝜑𝑤𝐵) → 𝑤𝐵)
4 rinvmod.a . . . . . . . . 9 (𝜑𝐴𝐵)
54adantr 480 . . . . . . . 8 ((𝜑𝑤𝐵) → 𝐴𝐵)
6 rinvmod.b . . . . . . . . 9 𝐵 = (Base‘𝐺)
7 rinvmod.p . . . . . . . . 9 + = (+g𝐺)
86, 7cmncom 19817 . . . . . . . 8 ((𝐺 ∈ CMnd ∧ 𝑤𝐵𝐴𝐵) → (𝑤 + 𝐴) = (𝐴 + 𝑤))
92, 3, 5, 8syl3anc 1372 . . . . . . 7 ((𝜑𝑤𝐵) → (𝑤 + 𝐴) = (𝐴 + 𝑤))
109adantr 480 . . . . . 6 (((𝜑𝑤𝐵) ∧ (𝐴 + 𝑤) = 0 ) → (𝑤 + 𝐴) = (𝐴 + 𝑤))
11 simpr 484 . . . . . 6 (((𝜑𝑤𝐵) ∧ (𝐴 + 𝑤) = 0 ) → (𝐴 + 𝑤) = 0 )
1210, 11eqtrd 2776 . . . . 5 (((𝜑𝑤𝐵) ∧ (𝐴 + 𝑤) = 0 ) → (𝑤 + 𝐴) = 0 )
1312, 11jca 511 . . . 4 (((𝜑𝑤𝐵) ∧ (𝐴 + 𝑤) = 0 ) → ((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ))
1413ex 412 . . 3 ((𝜑𝑤𝐵) → ((𝐴 + 𝑤) = 0 → ((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 )))
1514ralrimiva 3145 . 2 (𝜑 → ∀𝑤𝐵 ((𝐴 + 𝑤) = 0 → ((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 )))
16 rinvmod.0 . . 3 0 = (0g𝐺)
17 cmnmnd 19816 . . . 4 (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
181, 17syl 17 . . 3 (𝜑𝐺 ∈ Mnd)
196, 16, 7, 18, 4mndinvmod 18778 . 2 (𝜑 → ∃*𝑤𝐵 ((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ))
20 rmoim 3745 . 2 (∀𝑤𝐵 ((𝐴 + 𝑤) = 0 → ((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 )) → (∃*𝑤𝐵 ((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) → ∃*𝑤𝐵 (𝐴 + 𝑤) = 0 ))
2115, 19, 20sylc 65 1 (𝜑 → ∃*𝑤𝐵 (𝐴 + 𝑤) = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  wral 3060  ∃*wrmo 3378  cfv 6560  (class class class)co 7432  Basecbs 17248  +gcplusg 17298  0gc0g 17485  Mndcmnd 18748  CMndccmn 19799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fv 6568  df-riota 7389  df-ov 7435  df-0g 17487  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-cmn 19801
This theorem is referenced by: (None)
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