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Theorem mndinvmod 18203
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 18194 . . . . . . . 8 ((𝐺 ∈ Mnd ∧ 𝑤𝐵) → (𝑤 + 0 ) = 𝑤)
71, 2, 6syl2an 599 . . . . . . 7 ((𝜑 ∧ (𝑤𝐵𝑣𝐵)) → (𝑤 + 0 ) = 𝑤)
87eqcomd 2743 . . . . . 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 7221 . . . . . . . . 9 ( 0 = (𝐴 + 𝑣) → (𝑤 + 0 ) = (𝑤 + (𝐴 + 𝑣)))
1110eqcoms 2745 . . . . . . . 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 18182 . . . . . . . . 9 ((𝐺 ∈ Mnd ∧ (𝑤𝐵𝐴𝐵𝑣𝐵)) → ((𝑤 + 𝐴) + 𝑣) = (𝑤 + (𝐴 + 𝑣)))
2221eqcomd 2743 . . . . . . . 8 ((𝐺 ∈ Mnd ∧ (𝑤𝐵𝐴𝐵𝑣𝐵)) → (𝑤 + (𝐴 + 𝑣)) = ((𝑤 + 𝐴) + 𝑣))
2315, 16, 18, 20, 22syl13anc 1374 . . . . . . 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 7220 . . . . . . . . 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 18193 . . . . . . . 8 ((𝐺 ∈ Mnd ∧ 𝑣𝐵) → ( 0 + 𝑣) = 𝑣)
301, 19, 29syl2an 599 . . . . . . 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 2781 . . . . 5 (((𝜑 ∧ (𝑤𝐵𝑣𝐵)) ∧ (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 ))) → (𝑤 + (𝐴 + 𝑣)) = 𝑣)
339, 14, 323eqtrd 2781 . . . 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 3112 . 2 (𝜑 → ∀𝑤𝐵𝑣𝐵 ((((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 )) → 𝑤 = 𝑣))
36 oveq1 7220 . . . . 5 (𝑤 = 𝑣 → (𝑤 + 𝐴) = (𝑣 + 𝐴))
3736eqeq1d 2739 . . . 4 (𝑤 = 𝑣 → ((𝑤 + 𝐴) = 0 ↔ (𝑣 + 𝐴) = 0 ))
38 oveq2 7221 . . . . 5 (𝑤 = 𝑣 → (𝐴 + 𝑤) = (𝐴 + 𝑣))
3938eqeq1d 2739 . . . 4 (𝑤 = 𝑣 → ((𝐴 + 𝑤) = 0 ↔ (𝐴 + 𝑣) = 0 ))
4037, 39anbi12d 634 . . 3 (𝑤 = 𝑣 → (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ↔ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 )))
4140rmo4 3643 . 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 1089   = wceq 1543  wcel 2110  wral 3061  ∃*wrmo 3064  cfv 6380  (class class class)co 7213  Basecbs 16760  +gcplusg 16802  0gc0g 16944  Mndcmnd 18173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-iota 6338  df-fun 6382  df-fv 6388  df-riota 7170  df-ov 7216  df-0g 16946  df-mgm 18114  df-sgrp 18163  df-mnd 18174
This theorem is referenced by:  rinvmod  19194
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