MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mndinvmod Structured version   Visualization version   GIF version

Theorem mndinvmod 18578
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 483 . . . . . . . 8 ((𝑤𝐵𝑣𝐵) → 𝑤𝐵)
3 mndinvmod.b . . . . . . . . 9 𝐵 = (Base‘𝐺)
4 mndinvmod.p . . . . . . . . 9 + = (+g𝐺)
5 mndinvmod.0 . . . . . . . . 9 0 = (0g𝐺)
63, 4, 5mndrid 18569 . . . . . . . 8 ((𝐺 ∈ Mnd ∧ 𝑤𝐵) → (𝑤 + 0 ) = 𝑤)
71, 2, 6syl2an 596 . . . . . . 7 ((𝜑 ∧ (𝑤𝐵𝑣𝐵)) → (𝑤 + 0 ) = 𝑤)
87eqcomd 2742 . . . . . 6 ((𝜑 ∧ (𝑤𝐵𝑣𝐵)) → 𝑤 = (𝑤 + 0 ))
98adantr 481 . . . . 5 (((𝜑 ∧ (𝑤𝐵𝑣𝐵)) ∧ (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 ))) → 𝑤 = (𝑤 + 0 ))
10 oveq2 7361 . . . . . . . . 9 ( 0 = (𝐴 + 𝑣) → (𝑤 + 0 ) = (𝑤 + (𝐴 + 𝑣)))
1110eqcoms 2744 . . . . . . . 8 ((𝐴 + 𝑣) = 0 → (𝑤 + 0 ) = (𝑤 + (𝐴 + 𝑣)))
1211adantl 482 . . . . . . 7 (((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 ) → (𝑤 + 0 ) = (𝑤 + (𝐴 + 𝑣)))
1312adantl 482 . . . . . 6 ((((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 )) → (𝑤 + 0 ) = (𝑤 + (𝐴 + 𝑣)))
1413adantl 482 . . . . 5 (((𝜑 ∧ (𝑤𝐵𝑣𝐵)) ∧ (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 ))) → (𝑤 + 0 ) = (𝑤 + (𝐴 + 𝑣)))
151adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑤𝐵𝑣𝐵)) → 𝐺 ∈ Mnd)
162adantl 482 . . . . . . . 8 ((𝜑 ∧ (𝑤𝐵𝑣𝐵)) → 𝑤𝐵)
17 mndinvmod.a . . . . . . . . 9 (𝜑𝐴𝐵)
1817adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑤𝐵𝑣𝐵)) → 𝐴𝐵)
19 simpr 485 . . . . . . . . 9 ((𝑤𝐵𝑣𝐵) → 𝑣𝐵)
2019adantl 482 . . . . . . . 8 ((𝜑 ∧ (𝑤𝐵𝑣𝐵)) → 𝑣𝐵)
213, 4mndass 18557 . . . . . . . . 9 ((𝐺 ∈ Mnd ∧ (𝑤𝐵𝐴𝐵𝑣𝐵)) → ((𝑤 + 𝐴) + 𝑣) = (𝑤 + (𝐴 + 𝑣)))
2221eqcomd 2742 . . . . . . . 8 ((𝐺 ∈ Mnd ∧ (𝑤𝐵𝐴𝐵𝑣𝐵)) → (𝑤 + (𝐴 + 𝑣)) = ((𝑤 + 𝐴) + 𝑣))
2315, 16, 18, 20, 22syl13anc 1372 . . . . . . 7 ((𝜑 ∧ (𝑤𝐵𝑣𝐵)) → (𝑤 + (𝐴 + 𝑣)) = ((𝑤 + 𝐴) + 𝑣))
2423adantr 481 . . . . . 6 (((𝜑 ∧ (𝑤𝐵𝑣𝐵)) ∧ (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 ))) → (𝑤 + (𝐴 + 𝑣)) = ((𝑤 + 𝐴) + 𝑣))
25 oveq1 7360 . . . . . . . . 9 ((𝑤 + 𝐴) = 0 → ((𝑤 + 𝐴) + 𝑣) = ( 0 + 𝑣))
2625adantr 481 . . . . . . . 8 (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) → ((𝑤 + 𝐴) + 𝑣) = ( 0 + 𝑣))
2726adantr 481 . . . . . . 7 ((((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 )) → ((𝑤 + 𝐴) + 𝑣) = ( 0 + 𝑣))
2827adantl 482 . . . . . 6 (((𝜑 ∧ (𝑤𝐵𝑣𝐵)) ∧ (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 ))) → ((𝑤 + 𝐴) + 𝑣) = ( 0 + 𝑣))
293, 4, 5mndlid 18568 . . . . . . . 8 ((𝐺 ∈ Mnd ∧ 𝑣𝐵) → ( 0 + 𝑣) = 𝑣)
301, 19, 29syl2an 596 . . . . . . 7 ((𝜑 ∧ (𝑤𝐵𝑣𝐵)) → ( 0 + 𝑣) = 𝑣)
3130adantr 481 . . . . . 6 (((𝜑 ∧ (𝑤𝐵𝑣𝐵)) ∧ (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 ))) → ( 0 + 𝑣) = 𝑣)
3224, 28, 313eqtrd 2780 . . . . 5 (((𝜑 ∧ (𝑤𝐵𝑣𝐵)) ∧ (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 ))) → (𝑤 + (𝐴 + 𝑣)) = 𝑣)
339, 14, 323eqtrd 2780 . . . 4 (((𝜑 ∧ (𝑤𝐵𝑣𝐵)) ∧ (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 ))) → 𝑤 = 𝑣)
3433ex 413 . . 3 ((𝜑 ∧ (𝑤𝐵𝑣𝐵)) → ((((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 )) → 𝑤 = 𝑣))
3534ralrimivva 3195 . 2 (𝜑 → ∀𝑤𝐵𝑣𝐵 ((((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 )) → 𝑤 = 𝑣))
36 oveq1 7360 . . . . 5 (𝑤 = 𝑣 → (𝑤 + 𝐴) = (𝑣 + 𝐴))
3736eqeq1d 2738 . . . 4 (𝑤 = 𝑣 → ((𝑤 + 𝐴) = 0 ↔ (𝑣 + 𝐴) = 0 ))
38 oveq2 7361 . . . . 5 (𝑤 = 𝑣 → (𝐴 + 𝑤) = (𝐴 + 𝑣))
3938eqeq1d 2738 . . . 4 (𝑤 = 𝑣 → ((𝐴 + 𝑤) = 0 ↔ (𝐴 + 𝑣) = 0 ))
4037, 39anbi12d 631 . . 3 (𝑤 = 𝑣 → (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ↔ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 )))
4140rmo4 3686 . 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 233 1 (𝜑 → ∃*𝑤𝐵 ((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3062  ∃*wrmo 3350  cfv 6493  (class class class)co 7353  Basecbs 17075  +gcplusg 17125  0gc0g 17313  Mndcmnd 18548
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6445  df-fun 6495  df-fv 6501  df-riota 7309  df-ov 7356  df-0g 17315  df-mgm 18489  df-sgrp 18538  df-mnd 18549
This theorem is referenced by:  rinvmod  19578
  Copyright terms: Public domain W3C validator