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Theorem mndinvmod 17936
 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 = (0g𝐺)
mndinvmod.p + = (+g𝐺)
mndinvmod.m (𝜑𝐺 ∈ Mnd)
mndinvmod.a (𝜑𝐴𝐵)
Assertion
Ref Expression
mndinvmod (𝜑 → ∃*𝑤𝐵 ((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.
StepHypRef Expression
1 mndinvmod.m . . . . . . . 8 (𝜑𝐺 ∈ Mnd)
2 simpl 486 . . . . . . . 8 ((𝑤𝐵𝑣𝐵) → 𝑤𝐵)
3 mndinvmod.b . . . . . . . . 9 𝐵 = (Base‘𝐺)
4 mndinvmod.p . . . . . . . . 9 + = (+g𝐺)
5 mndinvmod.0 . . . . . . . . 9 0 = (0g𝐺)
63, 4, 5mndrid 17927 . . . . . . . 8 ((𝐺 ∈ Mnd ∧ 𝑤𝐵) → (𝑤 + 0 ) = 𝑤)
71, 2, 6syl2an 598 . . . . . . 7 ((𝜑 ∧ (𝑤𝐵𝑣𝐵)) → (𝑤 + 0 ) = 𝑤)
87eqcomd 2804 . . . . . 6 ((𝜑 ∧ (𝑤𝐵𝑣𝐵)) → 𝑤 = (𝑤 + 0 ))
98adantr 484 . . . . 5 (((𝜑 ∧ (𝑤𝐵𝑣𝐵)) ∧ (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 ))) → 𝑤 = (𝑤 + 0 ))
10 oveq2 7144 . . . . . . . . 9 ( 0 = (𝐴 + 𝑣) → (𝑤 + 0 ) = (𝑤 + (𝐴 + 𝑣)))
1110eqcoms 2806 . . . . . . . 8 ((𝐴 + 𝑣) = 0 → (𝑤 + 0 ) = (𝑤 + (𝐴 + 𝑣)))
1211adantl 485 . . . . . . 7 (((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 ) → (𝑤 + 0 ) = (𝑤 + (𝐴 + 𝑣)))
1312adantl 485 . . . . . 6 ((((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 )) → (𝑤 + 0 ) = (𝑤 + (𝐴 + 𝑣)))
1413adantl 485 . . . . 5 (((𝜑 ∧ (𝑤𝐵𝑣𝐵)) ∧ (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 ))) → (𝑤 + 0 ) = (𝑤 + (𝐴 + 𝑣)))
151adantr 484 . . . . . . . 8 ((𝜑 ∧ (𝑤𝐵𝑣𝐵)) → 𝐺 ∈ Mnd)
162adantl 485 . . . . . . . 8 ((𝜑 ∧ (𝑤𝐵𝑣𝐵)) → 𝑤𝐵)
17 mndinvmod.a . . . . . . . . 9 (𝜑𝐴𝐵)
1817adantr 484 . . . . . . . 8 ((𝜑 ∧ (𝑤𝐵𝑣𝐵)) → 𝐴𝐵)
19 simpr 488 . . . . . . . . 9 ((𝑤𝐵𝑣𝐵) → 𝑣𝐵)
2019adantl 485 . . . . . . . 8 ((𝜑 ∧ (𝑤𝐵𝑣𝐵)) → 𝑣𝐵)
213, 4mndass 17915 . . . . . . . . 9 ((𝐺 ∈ Mnd ∧ (𝑤𝐵𝐴𝐵𝑣𝐵)) → ((𝑤 + 𝐴) + 𝑣) = (𝑤 + (𝐴 + 𝑣)))
2221eqcomd 2804 . . . . . . . 8 ((𝐺 ∈ Mnd ∧ (𝑤𝐵𝐴𝐵𝑣𝐵)) → (𝑤 + (𝐴 + 𝑣)) = ((𝑤 + 𝐴) + 𝑣))
2315, 16, 18, 20, 22syl13anc 1369 . . . . . . 7 ((𝜑 ∧ (𝑤𝐵𝑣𝐵)) → (𝑤 + (𝐴 + 𝑣)) = ((𝑤 + 𝐴) + 𝑣))
2423adantr 484 . . . . . 6 (((𝜑 ∧ (𝑤𝐵𝑣𝐵)) ∧ (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 ))) → (𝑤 + (𝐴 + 𝑣)) = ((𝑤 + 𝐴) + 𝑣))
25 oveq1 7143 . . . . . . . . 9 ((𝑤 + 𝐴) = 0 → ((𝑤 + 𝐴) + 𝑣) = ( 0 + 𝑣))
2625adantr 484 . . . . . . . 8 (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) → ((𝑤 + 𝐴) + 𝑣) = ( 0 + 𝑣))
2726adantr 484 . . . . . . 7 ((((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 )) → ((𝑤 + 𝐴) + 𝑣) = ( 0 + 𝑣))
2827adantl 485 . . . . . 6 (((𝜑 ∧ (𝑤𝐵𝑣𝐵)) ∧ (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 ))) → ((𝑤 + 𝐴) + 𝑣) = ( 0 + 𝑣))
293, 4, 5mndlid 17926 . . . . . . . 8 ((𝐺 ∈ Mnd ∧ 𝑣𝐵) → ( 0 + 𝑣) = 𝑣)
301, 19, 29syl2an 598 . . . . . . 7 ((𝜑 ∧ (𝑤𝐵𝑣𝐵)) → ( 0 + 𝑣) = 𝑣)
3130adantr 484 . . . . . 6 (((𝜑 ∧ (𝑤𝐵𝑣𝐵)) ∧ (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 ))) → ( 0 + 𝑣) = 𝑣)
3224, 28, 313eqtrd 2837 . . . . 5 (((𝜑 ∧ (𝑤𝐵𝑣𝐵)) ∧ (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 ))) → (𝑤 + (𝐴 + 𝑣)) = 𝑣)
339, 14, 323eqtrd 2837 . . . 4 (((𝜑 ∧ (𝑤𝐵𝑣𝐵)) ∧ (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 ))) → 𝑤 = 𝑣)
3433ex 416 . . 3 ((𝜑 ∧ (𝑤𝐵𝑣𝐵)) → ((((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 )) → 𝑤 = 𝑣))
3534ralrimivva 3156 . 2 (𝜑 → ∀𝑤𝐵𝑣𝐵 ((((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 )) → 𝑤 = 𝑣))
36 oveq1 7143 . . . . 5 (𝑤 = 𝑣 → (𝑤 + 𝐴) = (𝑣 + 𝐴))
3736eqeq1d 2800 . . . 4 (𝑤 = 𝑣 → ((𝑤 + 𝐴) = 0 ↔ (𝑣 + 𝐴) = 0 ))
38 oveq2 7144 . . . . 5 (𝑤 = 𝑣 → (𝐴 + 𝑤) = (𝐴 + 𝑣))
3938eqeq1d 2800 . . . 4 (𝑤 = 𝑣 → ((𝐴 + 𝑤) = 0 ↔ (𝐴 + 𝑣) = 0 ))
4037, 39anbi12d 633 . . 3 (𝑤 = 𝑣 → (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ↔ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 )))
4140rmo4 3669 . 2 (∃*𝑤𝐵 ((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ↔ ∀𝑤𝐵𝑣𝐵 ((((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 )) → 𝑤 = 𝑣))
4235, 41sylibr 237 1 (𝜑 → ∃*𝑤𝐵 ((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃*wrmo 3109  ‘cfv 6325  (class class class)co 7136  Basecbs 16478  +gcplusg 16560  0gc0g 16708  Mndcmnd 17906 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-iota 6284  df-fun 6327  df-fv 6333  df-riota 7094  df-ov 7139  df-0g 16710  df-mgm 17847  df-sgrp 17896  df-mnd 17907 This theorem is referenced by:  rinvmod  18926
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