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Theorem mndinvmod 12845
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 109 . . . . . . . 8 ((𝑤𝐵𝑣𝐵) → 𝑤𝐵)
3 mndinvmod.b . . . . . . . . 9 𝐵 = (Base‘𝐺)
4 mndinvmod.p . . . . . . . . 9 + = (+g𝐺)
5 mndinvmod.0 . . . . . . . . 9 0 = (0g𝐺)
63, 4, 5mndrid 12836 . . . . . . . 8 ((𝐺 ∈ Mnd ∧ 𝑤𝐵) → (𝑤 + 0 ) = 𝑤)
71, 2, 6syl2an 289 . . . . . . 7 ((𝜑 ∧ (𝑤𝐵𝑣𝐵)) → (𝑤 + 0 ) = 𝑤)
87eqcomd 2183 . . . . . 6 ((𝜑 ∧ (𝑤𝐵𝑣𝐵)) → 𝑤 = (𝑤 + 0 ))
98adantr 276 . . . . 5 (((𝜑 ∧ (𝑤𝐵𝑣𝐵)) ∧ (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 ))) → 𝑤 = (𝑤 + 0 ))
10 oveq2 5882 . . . . . . . . 9 ( 0 = (𝐴 + 𝑣) → (𝑤 + 0 ) = (𝑤 + (𝐴 + 𝑣)))
1110eqcoms 2180 . . . . . . . 8 ((𝐴 + 𝑣) = 0 → (𝑤 + 0 ) = (𝑤 + (𝐴 + 𝑣)))
1211adantl 277 . . . . . . 7 (((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 ) → (𝑤 + 0 ) = (𝑤 + (𝐴 + 𝑣)))
1312adantl 277 . . . . . 6 ((((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 )) → (𝑤 + 0 ) = (𝑤 + (𝐴 + 𝑣)))
1413adantl 277 . . . . 5 (((𝜑 ∧ (𝑤𝐵𝑣𝐵)) ∧ (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 ))) → (𝑤 + 0 ) = (𝑤 + (𝐴 + 𝑣)))
151adantr 276 . . . . . . . 8 ((𝜑 ∧ (𝑤𝐵𝑣𝐵)) → 𝐺 ∈ Mnd)
162adantl 277 . . . . . . . 8 ((𝜑 ∧ (𝑤𝐵𝑣𝐵)) → 𝑤𝐵)
17 mndinvmod.a . . . . . . . . 9 (𝜑𝐴𝐵)
1817adantr 276 . . . . . . . 8 ((𝜑 ∧ (𝑤𝐵𝑣𝐵)) → 𝐴𝐵)
19 simpr 110 . . . . . . . . 9 ((𝑤𝐵𝑣𝐵) → 𝑣𝐵)
2019adantl 277 . . . . . . . 8 ((𝜑 ∧ (𝑤𝐵𝑣𝐵)) → 𝑣𝐵)
213, 4mndass 12824 . . . . . . . . 9 ((𝐺 ∈ Mnd ∧ (𝑤𝐵𝐴𝐵𝑣𝐵)) → ((𝑤 + 𝐴) + 𝑣) = (𝑤 + (𝐴 + 𝑣)))
2221eqcomd 2183 . . . . . . . 8 ((𝐺 ∈ Mnd ∧ (𝑤𝐵𝐴𝐵𝑣𝐵)) → (𝑤 + (𝐴 + 𝑣)) = ((𝑤 + 𝐴) + 𝑣))
2315, 16, 18, 20, 22syl13anc 1240 . . . . . . 7 ((𝜑 ∧ (𝑤𝐵𝑣𝐵)) → (𝑤 + (𝐴 + 𝑣)) = ((𝑤 + 𝐴) + 𝑣))
2423adantr 276 . . . . . 6 (((𝜑 ∧ (𝑤𝐵𝑣𝐵)) ∧ (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 ))) → (𝑤 + (𝐴 + 𝑣)) = ((𝑤 + 𝐴) + 𝑣))
25 oveq1 5881 . . . . . . . . 9 ((𝑤 + 𝐴) = 0 → ((𝑤 + 𝐴) + 𝑣) = ( 0 + 𝑣))
2625adantr 276 . . . . . . . 8 (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) → ((𝑤 + 𝐴) + 𝑣) = ( 0 + 𝑣))
2726adantr 276 . . . . . . 7 ((((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 )) → ((𝑤 + 𝐴) + 𝑣) = ( 0 + 𝑣))
2827adantl 277 . . . . . 6 (((𝜑 ∧ (𝑤𝐵𝑣𝐵)) ∧ (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 ))) → ((𝑤 + 𝐴) + 𝑣) = ( 0 + 𝑣))
293, 4, 5mndlid 12835 . . . . . . . 8 ((𝐺 ∈ Mnd ∧ 𝑣𝐵) → ( 0 + 𝑣) = 𝑣)
301, 19, 29syl2an 289 . . . . . . 7 ((𝜑 ∧ (𝑤𝐵𝑣𝐵)) → ( 0 + 𝑣) = 𝑣)
3130adantr 276 . . . . . 6 (((𝜑 ∧ (𝑤𝐵𝑣𝐵)) ∧ (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 ))) → ( 0 + 𝑣) = 𝑣)
3224, 28, 313eqtrd 2214 . . . . 5 (((𝜑 ∧ (𝑤𝐵𝑣𝐵)) ∧ (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 ))) → (𝑤 + (𝐴 + 𝑣)) = 𝑣)
339, 14, 323eqtrd 2214 . . . 4 (((𝜑 ∧ (𝑤𝐵𝑣𝐵)) ∧ (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 ))) → 𝑤 = 𝑣)
3433ex 115 . . 3 ((𝜑 ∧ (𝑤𝐵𝑣𝐵)) → ((((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 )) → 𝑤 = 𝑣))
3534ralrimivva 2559 . 2 (𝜑 → ∀𝑤𝐵𝑣𝐵 ((((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ∧ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 )) → 𝑤 = 𝑣))
36 oveq1 5881 . . . . 5 (𝑤 = 𝑣 → (𝑤 + 𝐴) = (𝑣 + 𝐴))
3736eqeq1d 2186 . . . 4 (𝑤 = 𝑣 → ((𝑤 + 𝐴) = 0 ↔ (𝑣 + 𝐴) = 0 ))
38 oveq2 5882 . . . . 5 (𝑤 = 𝑣 → (𝐴 + 𝑤) = (𝐴 + 𝑣))
3938eqeq1d 2186 . . . 4 (𝑤 = 𝑣 → ((𝐴 + 𝑤) = 0 ↔ (𝐴 + 𝑣) = 0 ))
4037, 39anbi12d 473 . . 3 (𝑤 = 𝑣 → (((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) ↔ ((𝑣 + 𝐴) = 0 ∧ (𝐴 + 𝑣) = 0 )))
4140rmo4 2930 . 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 134 1 (𝜑 → ∃*𝑤𝐵 ((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 978   = wceq 1353  wcel 2148  wral 2455  ∃*wrmo 2458  cfv 5216  (class class class)co 5874  Basecbs 12461  +gcplusg 12535  0gc0g 12704  Mndcmnd 12816
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-cnex 7901  ax-resscn 7902  ax-1re 7904  ax-addrcl 7907
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-iota 5178  df-fun 5218  df-fn 5219  df-fv 5224  df-riota 5830  df-ov 5877  df-inn 8919  df-2 8977  df-ndx 12464  df-slot 12465  df-base 12467  df-plusg 12548  df-0g 12706  df-mgm 12774  df-sgrp 12807  df-mnd 12817
This theorem is referenced by:  rinvmod  13110
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