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Theorem mndmolinv 42077
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 1912 . . . . . 6 𝑦(𝐴(+g𝑀)𝑥) = (0g𝑀)
3 nfv 1912 . . . . . 6 𝑥(𝐴(+g𝑀)𝑦) = (0g𝑀)
4 oveq2 7439 . . . . . . 7 (𝑥 = 𝑦 → (𝐴(+g𝑀)𝑥) = (𝐴(+g𝑀)𝑦))
54eqeq1d 2737 . . . . . 6 (𝑥 = 𝑦 → ((𝐴(+g𝑀)𝑥) = (0g𝑀) ↔ (𝐴(+g𝑀)𝑦) = (0g𝑀)))
62, 3, 5cbvrexw 3305 . . . . 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 2735 . . . . . . . . . . . 12 (+g𝑀) = (+g𝑀)
14 eqid 2735 . . . . . . . . . . . 12 (0g𝑀) = (0g𝑀)
1512, 13, 14mndrid 18781 . . . . . . . . . . 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 2741 . . . . . . . . 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 2741 . . . . . . . . . 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 2775 . . . . . . . 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 1127 . . . . . . . . . 10 (((((𝜑𝑦𝐵) ∧ (𝐴(+g𝑀)𝑦) = (0g𝑀)) ∧ 𝑥𝐵) ∧ (𝑥(+g𝑀)𝐴) = (0g𝑀)) → (𝑥𝐵𝐴𝐵𝑦𝐵))
2612, 13mndass 18769 . . . . . . . . . 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 2741 . . . . . . . 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 18780 . . . . . . . . . 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 2775 . . . . . . . 8 (((((𝜑𝑦𝐵) ∧ (𝐴(+g𝑀)𝑦) = (0g𝑀)) ∧ 𝑥𝐵) ∧ (𝑥(+g𝑀)𝐴) = (0g𝑀)) → ((𝑥(+g𝑀)𝐴)(+g𝑀)𝑦) = 𝑦)
3421, 28, 333eqtrd 2779 . . . . . . 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 3144 . . . . 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 3166 . . 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 1912 . . 3 𝑦(𝑥(+g𝑀)𝐴) = (0g𝑀)
4140rmo2i 3897 . 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 1086   = wceq 1537  wcel 2106  wral 3059  wrex 3068  ∃*wrmo 3377  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  0gc0g 17486  Mndcmnd 18760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-riota 7388  df-ov 7434  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761
This theorem is referenced by:  primrootsunit1  42079
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