Users' Mathboxes Mathbox for metakunt < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mndmolinv Structured version   Visualization version   GIF version

Theorem mndmolinv 42096
Description: An element of a monoid that has a right inverse has at most one left inverse. (Contributed by metakunt, 25-Apr-2025.)
Hypotheses
Ref Expression
mndmolinv.1 𝐵 = (Base‘𝑀)
mndmolinv.2 (𝜑𝑀 ∈ Mnd)
mndmolinv.3 (𝜑𝐴𝐵)
mndmolinv.4 (𝜑 → ∃𝑥𝐵 (𝐴(+g𝑀)𝑥) = (0g𝑀))
Assertion
Ref Expression
mndmolinv (𝜑 → ∃*𝑥𝐵 (𝑥(+g𝑀)𝐴) = (0g𝑀))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑀   𝜑,𝑥

Proof of Theorem mndmolinv
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 mndmolinv.4 . . . 4 (𝜑 → ∃𝑥𝐵 (𝐴(+g𝑀)𝑥) = (0g𝑀))
2 nfv 1914 . . . . . 6 𝑦(𝐴(+g𝑀)𝑥) = (0g𝑀)
3 nfv 1914 . . . . . 6 𝑥(𝐴(+g𝑀)𝑦) = (0g𝑀)
4 oveq2 7439 . . . . . . 7 (𝑥 = 𝑦 → (𝐴(+g𝑀)𝑥) = (𝐴(+g𝑀)𝑦))
54eqeq1d 2739 . . . . . 6 (𝑥 = 𝑦 → ((𝐴(+g𝑀)𝑥) = (0g𝑀) ↔ (𝐴(+g𝑀)𝑦) = (0g𝑀)))
62, 3, 5cbvrexw 3307 . . . . 5 (∃𝑥𝐵 (𝐴(+g𝑀)𝑥) = (0g𝑀) ↔ ∃𝑦𝐵 (𝐴(+g𝑀)𝑦) = (0g𝑀))
76biimpi 216 . . . 4 (∃𝑥𝐵 (𝐴(+g𝑀)𝑥) = (0g𝑀) → ∃𝑦𝐵 (𝐴(+g𝑀)𝑦) = (0g𝑀))
81, 7syl 17 . . 3 (𝜑 → ∃𝑦𝐵 (𝐴(+g𝑀)𝑦) = (0g𝑀))
9 mndmolinv.2 . . . . . . . . . . . 12 (𝜑𝑀 ∈ Mnd)
109ad4antr 732 . . . . . . . . . . 11 (((((𝜑𝑦𝐵) ∧ (𝐴(+g𝑀)𝑦) = (0g𝑀)) ∧ 𝑥𝐵) ∧ (𝑥(+g𝑀)𝐴) = (0g𝑀)) → 𝑀 ∈ Mnd)
11 simplr 769 . . . . . . . . . . 11 (((((𝜑𝑦𝐵) ∧ (𝐴(+g𝑀)𝑦) = (0g𝑀)) ∧ 𝑥𝐵) ∧ (𝑥(+g𝑀)𝐴) = (0g𝑀)) → 𝑥𝐵)
12 mndmolinv.1 . . . . . . . . . . . 12 𝐵 = (Base‘𝑀)
13 eqid 2737 . . . . . . . . . . . 12 (+g𝑀) = (+g𝑀)
14 eqid 2737 . . . . . . . . . . . 12 (0g𝑀) = (0g𝑀)
1512, 13, 14mndrid 18768 . . . . . . . . . . 11 ((𝑀 ∈ Mnd ∧ 𝑥𝐵) → (𝑥(+g𝑀)(0g𝑀)) = 𝑥)
1610, 11, 15syl2anc 584 . . . . . . . . . 10 (((((𝜑𝑦𝐵) ∧ (𝐴(+g𝑀)𝑦) = (0g𝑀)) ∧ 𝑥𝐵) ∧ (𝑥(+g𝑀)𝐴) = (0g𝑀)) → (𝑥(+g𝑀)(0g𝑀)) = 𝑥)
1716eqcomd 2743 . . . . . . . . 9 (((((𝜑𝑦𝐵) ∧ (𝐴(+g𝑀)𝑦) = (0g𝑀)) ∧ 𝑥𝐵) ∧ (𝑥(+g𝑀)𝐴) = (0g𝑀)) → 𝑥 = (𝑥(+g𝑀)(0g𝑀)))
18 simpllr 776 . . . . . . . . . . 11 (((((𝜑𝑦𝐵) ∧ (𝐴(+g𝑀)𝑦) = (0g𝑀)) ∧ 𝑥𝐵) ∧ (𝑥(+g𝑀)𝐴) = (0g𝑀)) → (𝐴(+g𝑀)𝑦) = (0g𝑀))
1918eqcomd 2743 . . . . . . . . . 10 (((((𝜑𝑦𝐵) ∧ (𝐴(+g𝑀)𝑦) = (0g𝑀)) ∧ 𝑥𝐵) ∧ (𝑥(+g𝑀)𝐴) = (0g𝑀)) → (0g𝑀) = (𝐴(+g𝑀)𝑦))
2019oveq2d 7447 . . . . . . . . 9 (((((𝜑𝑦𝐵) ∧ (𝐴(+g𝑀)𝑦) = (0g𝑀)) ∧ 𝑥𝐵) ∧ (𝑥(+g𝑀)𝐴) = (0g𝑀)) → (𝑥(+g𝑀)(0g𝑀)) = (𝑥(+g𝑀)(𝐴(+g𝑀)𝑦)))
2117, 20eqtrd 2777 . . . . . . . 8 (((((𝜑𝑦𝐵) ∧ (𝐴(+g𝑀)𝑦) = (0g𝑀)) ∧ 𝑥𝐵) ∧ (𝑥(+g𝑀)𝐴) = (0g𝑀)) → 𝑥 = (𝑥(+g𝑀)(𝐴(+g𝑀)𝑦)))
22 mndmolinv.3 . . . . . . . . . . . 12 (𝜑𝐴𝐵)
2322ad4antr 732 . . . . . . . . . . 11 (((((𝜑𝑦𝐵) ∧ (𝐴(+g𝑀)𝑦) = (0g𝑀)) ∧ 𝑥𝐵) ∧ (𝑥(+g𝑀)𝐴) = (0g𝑀)) → 𝐴𝐵)
24 simp-4r 784 . . . . . . . . . . 11 (((((𝜑𝑦𝐵) ∧ (𝐴(+g𝑀)𝑦) = (0g𝑀)) ∧ 𝑥𝐵) ∧ (𝑥(+g𝑀)𝐴) = (0g𝑀)) → 𝑦𝐵)
2511, 23, 243jca 1129 . . . . . . . . . 10 (((((𝜑𝑦𝐵) ∧ (𝐴(+g𝑀)𝑦) = (0g𝑀)) ∧ 𝑥𝐵) ∧ (𝑥(+g𝑀)𝐴) = (0g𝑀)) → (𝑥𝐵𝐴𝐵𝑦𝐵))
2612, 13mndass 18756 . . . . . . . . . 10 ((𝑀 ∈ Mnd ∧ (𝑥𝐵𝐴𝐵𝑦𝐵)) → ((𝑥(+g𝑀)𝐴)(+g𝑀)𝑦) = (𝑥(+g𝑀)(𝐴(+g𝑀)𝑦)))
2710, 25, 26syl2anc 584 . . . . . . . . 9 (((((𝜑𝑦𝐵) ∧ (𝐴(+g𝑀)𝑦) = (0g𝑀)) ∧ 𝑥𝐵) ∧ (𝑥(+g𝑀)𝐴) = (0g𝑀)) → ((𝑥(+g𝑀)𝐴)(+g𝑀)𝑦) = (𝑥(+g𝑀)(𝐴(+g𝑀)𝑦)))
2827eqcomd 2743 . . . . . . . 8 (((((𝜑𝑦𝐵) ∧ (𝐴(+g𝑀)𝑦) = (0g𝑀)) ∧ 𝑥𝐵) ∧ (𝑥(+g𝑀)𝐴) = (0g𝑀)) → (𝑥(+g𝑀)(𝐴(+g𝑀)𝑦)) = ((𝑥(+g𝑀)𝐴)(+g𝑀)𝑦))
29 simpr 484 . . . . . . . . . 10 (((((𝜑𝑦𝐵) ∧ (𝐴(+g𝑀)𝑦) = (0g𝑀)) ∧ 𝑥𝐵) ∧ (𝑥(+g𝑀)𝐴) = (0g𝑀)) → (𝑥(+g𝑀)𝐴) = (0g𝑀))
3029oveq1d 7446 . . . . . . . . 9 (((((𝜑𝑦𝐵) ∧ (𝐴(+g𝑀)𝑦) = (0g𝑀)) ∧ 𝑥𝐵) ∧ (𝑥(+g𝑀)𝐴) = (0g𝑀)) → ((𝑥(+g𝑀)𝐴)(+g𝑀)𝑦) = ((0g𝑀)(+g𝑀)𝑦))
3112, 13, 14mndlid 18767 . . . . . . . . . 10 ((𝑀 ∈ Mnd ∧ 𝑦𝐵) → ((0g𝑀)(+g𝑀)𝑦) = 𝑦)
3210, 24, 31syl2anc 584 . . . . . . . . 9 (((((𝜑𝑦𝐵) ∧ (𝐴(+g𝑀)𝑦) = (0g𝑀)) ∧ 𝑥𝐵) ∧ (𝑥(+g𝑀)𝐴) = (0g𝑀)) → ((0g𝑀)(+g𝑀)𝑦) = 𝑦)
3330, 32eqtrd 2777 . . . . . . . 8 (((((𝜑𝑦𝐵) ∧ (𝐴(+g𝑀)𝑦) = (0g𝑀)) ∧ 𝑥𝐵) ∧ (𝑥(+g𝑀)𝐴) = (0g𝑀)) → ((𝑥(+g𝑀)𝐴)(+g𝑀)𝑦) = 𝑦)
3421, 28, 333eqtrd 2781 . . . . . . 7 (((((𝜑𝑦𝐵) ∧ (𝐴(+g𝑀)𝑦) = (0g𝑀)) ∧ 𝑥𝐵) ∧ (𝑥(+g𝑀)𝐴) = (0g𝑀)) → 𝑥 = 𝑦)
3534ex 412 . . . . . 6 ((((𝜑𝑦𝐵) ∧ (𝐴(+g𝑀)𝑦) = (0g𝑀)) ∧ 𝑥𝐵) → ((𝑥(+g𝑀)𝐴) = (0g𝑀) → 𝑥 = 𝑦))
3635ralrimiva 3146 . . . . 5 (((𝜑𝑦𝐵) ∧ (𝐴(+g𝑀)𝑦) = (0g𝑀)) → ∀𝑥𝐵 ((𝑥(+g𝑀)𝐴) = (0g𝑀) → 𝑥 = 𝑦))
3736ex 412 . . . 4 ((𝜑𝑦𝐵) → ((𝐴(+g𝑀)𝑦) = (0g𝑀) → ∀𝑥𝐵 ((𝑥(+g𝑀)𝐴) = (0g𝑀) → 𝑥 = 𝑦)))
3837reximdva 3168 . . 3 (𝜑 → (∃𝑦𝐵 (𝐴(+g𝑀)𝑦) = (0g𝑀) → ∃𝑦𝐵𝑥𝐵 ((𝑥(+g𝑀)𝐴) = (0g𝑀) → 𝑥 = 𝑦)))
398, 38mpd 15 . 2 (𝜑 → ∃𝑦𝐵𝑥𝐵 ((𝑥(+g𝑀)𝐴) = (0g𝑀) → 𝑥 = 𝑦))
40 nfv 1914 . . 3 𝑦(𝑥(+g𝑀)𝐴) = (0g𝑀)
4140rmo2i 3888 . 2 (∃𝑦𝐵𝑥𝐵 ((𝑥(+g𝑀)𝐴) = (0g𝑀) → 𝑥 = 𝑦) → ∃*𝑥𝐵 (𝑥(+g𝑀)𝐴) = (0g𝑀))
4239, 41syl 17 1 (𝜑 → ∃*𝑥𝐵 (𝑥(+g𝑀)𝐴) = (0g𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070  ∃*wrmo 3379  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  0gc0g 17484  Mndcmnd 18747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-riota 7388  df-ov 7434  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748
This theorem is referenced by:  primrootsunit1  42098
  Copyright terms: Public domain W3C validator