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Theorem mndractfo 33019
Description: An element 𝑋 of a monoid 𝐸 is right-invertible iff its right-translation 𝐺 is surjective. (Contributed by Thierry Arnoux, 3-Aug-2025.)
Hypotheses
Ref Expression
mndractfo.b 𝐵 = (Base‘𝐸)
mndractfo.z 0 = (0g𝐸)
mndractfo.p + = (+g𝐸)
mndractfo.f 𝐺 = (𝑎𝐵 ↦ (𝑎 + 𝑋))
mndractfo.e (𝜑𝐸 ∈ Mnd)
mndractfo.x (𝜑𝑋𝐵)
Assertion
Ref Expression
mndractfo (𝜑 → (𝐺:𝐵onto𝐵 ↔ ∃𝑦𝐵 (𝑦 + 𝑋) = 0 ))
Distinct variable groups:   + ,𝑎   0 ,𝑎,𝑦   𝐵,𝑎,𝑦   𝐺,𝑎,𝑦   𝑋,𝑎   𝜑,𝑎,𝑦
Allowed substitution hints:   + (𝑦)   𝐸(𝑦,𝑎)   𝑋(𝑦)

Proof of Theorem mndractfo
Dummy variables 𝑧 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 484 . . . 4 ((𝜑𝐺:𝐵onto𝐵) → 𝐺:𝐵onto𝐵)
2 mndractfo.e . . . . . 6 (𝜑𝐸 ∈ Mnd)
3 mndractfo.b . . . . . . 7 𝐵 = (Base‘𝐸)
4 mndractfo.z . . . . . . 7 0 = (0g𝐸)
53, 4mndidcl 18758 . . . . . 6 (𝐸 ∈ Mnd → 0𝐵)
62, 5syl 17 . . . . 5 (𝜑0𝐵)
76adantr 480 . . . 4 ((𝜑𝐺:𝐵onto𝐵) → 0𝐵)
8 foelcdmi 6968 . . . 4 ((𝐺:𝐵onto𝐵0𝐵) → ∃𝑦𝐵 (𝐺𝑦) = 0 )
91, 7, 8syl2anc 584 . . 3 ((𝜑𝐺:𝐵onto𝐵) → ∃𝑦𝐵 (𝐺𝑦) = 0 )
10 mndractfo.f . . . . . . 7 𝐺 = (𝑎𝐵 ↦ (𝑎 + 𝑋))
11 oveq1 7436 . . . . . . 7 (𝑎 = 𝑦 → (𝑎 + 𝑋) = (𝑦 + 𝑋))
12 simpr 484 . . . . . . 7 (((𝜑𝐺:𝐵onto𝐵) ∧ 𝑦𝐵) → 𝑦𝐵)
13 ovexd 7464 . . . . . . 7 (((𝜑𝐺:𝐵onto𝐵) ∧ 𝑦𝐵) → (𝑦 + 𝑋) ∈ V)
1410, 11, 12, 13fvmptd3 7037 . . . . . 6 (((𝜑𝐺:𝐵onto𝐵) ∧ 𝑦𝐵) → (𝐺𝑦) = (𝑦 + 𝑋))
1514eqeq1d 2738 . . . . 5 (((𝜑𝐺:𝐵onto𝐵) ∧ 𝑦𝐵) → ((𝐺𝑦) = 0 ↔ (𝑦 + 𝑋) = 0 ))
1615biimpd 229 . . . 4 (((𝜑𝐺:𝐵onto𝐵) ∧ 𝑦𝐵) → ((𝐺𝑦) = 0 → (𝑦 + 𝑋) = 0 ))
1716reximdva 3167 . . 3 ((𝜑𝐺:𝐵onto𝐵) → (∃𝑦𝐵 (𝐺𝑦) = 0 → ∃𝑦𝐵 (𝑦 + 𝑋) = 0 ))
189, 17mpd 15 . 2 ((𝜑𝐺:𝐵onto𝐵) → ∃𝑦𝐵 (𝑦 + 𝑋) = 0 )
19 mndractfo.p . . . . . . 7 + = (+g𝐸)
202adantr 480 . . . . . . 7 ((𝜑𝑎𝐵) → 𝐸 ∈ Mnd)
21 simpr 484 . . . . . . 7 ((𝜑𝑎𝐵) → 𝑎𝐵)
22 mndractfo.x . . . . . . . 8 (𝜑𝑋𝐵)
2322adantr 480 . . . . . . 7 ((𝜑𝑎𝐵) → 𝑋𝐵)
243, 19, 20, 21, 23mndcld 33012 . . . . . 6 ((𝜑𝑎𝐵) → (𝑎 + 𝑋) ∈ 𝐵)
2524, 10fmptd 7132 . . . . 5 (𝜑𝐺:𝐵𝐵)
2625ad2antrr 726 . . . 4 (((𝜑𝑦𝐵) ∧ (𝑦 + 𝑋) = 0 ) → 𝐺:𝐵𝐵)
27 fveq2 6904 . . . . . . 7 (𝑥 = (𝑧 + 𝑦) → (𝐺𝑥) = (𝐺‘(𝑧 + 𝑦)))
2827eqeq2d 2747 . . . . . 6 (𝑥 = (𝑧 + 𝑦) → (𝑧 = (𝐺𝑥) ↔ 𝑧 = (𝐺‘(𝑧 + 𝑦))))
292ad3antrrr 730 . . . . . . 7 ((((𝜑𝑦𝐵) ∧ (𝑦 + 𝑋) = 0 ) ∧ 𝑧𝐵) → 𝐸 ∈ Mnd)
30 simpr 484 . . . . . . 7 ((((𝜑𝑦𝐵) ∧ (𝑦 + 𝑋) = 0 ) ∧ 𝑧𝐵) → 𝑧𝐵)
31 simpllr 776 . . . . . . 7 ((((𝜑𝑦𝐵) ∧ (𝑦 + 𝑋) = 0 ) ∧ 𝑧𝐵) → 𝑦𝐵)
323, 19, 29, 30, 31mndcld 33012 . . . . . 6 ((((𝜑𝑦𝐵) ∧ (𝑦 + 𝑋) = 0 ) ∧ 𝑧𝐵) → (𝑧 + 𝑦) ∈ 𝐵)
3322ad3antrrr 730 . . . . . . . 8 ((((𝜑𝑦𝐵) ∧ (𝑦 + 𝑋) = 0 ) ∧ 𝑧𝐵) → 𝑋𝐵)
343, 19, 29, 30, 31, 33mndassd 33013 . . . . . . 7 ((((𝜑𝑦𝐵) ∧ (𝑦 + 𝑋) = 0 ) ∧ 𝑧𝐵) → ((𝑧 + 𝑦) + 𝑋) = (𝑧 + (𝑦 + 𝑋)))
35 oveq1 7436 . . . . . . . 8 (𝑎 = (𝑧 + 𝑦) → (𝑎 + 𝑋) = ((𝑧 + 𝑦) + 𝑋))
36 ovexd 7464 . . . . . . . 8 ((((𝜑𝑦𝐵) ∧ (𝑦 + 𝑋) = 0 ) ∧ 𝑧𝐵) → ((𝑧 + 𝑦) + 𝑋) ∈ V)
3710, 35, 32, 36fvmptd3 7037 . . . . . . 7 ((((𝜑𝑦𝐵) ∧ (𝑦 + 𝑋) = 0 ) ∧ 𝑧𝐵) → (𝐺‘(𝑧 + 𝑦)) = ((𝑧 + 𝑦) + 𝑋))
38 simplr 769 . . . . . . . . 9 ((((𝜑𝑦𝐵) ∧ (𝑦 + 𝑋) = 0 ) ∧ 𝑧𝐵) → (𝑦 + 𝑋) = 0 )
3938oveq2d 7445 . . . . . . . 8 ((((𝜑𝑦𝐵) ∧ (𝑦 + 𝑋) = 0 ) ∧ 𝑧𝐵) → (𝑧 + (𝑦 + 𝑋)) = (𝑧 + 0 ))
403, 19, 4mndrid 18764 . . . . . . . . 9 ((𝐸 ∈ Mnd ∧ 𝑧𝐵) → (𝑧 + 0 ) = 𝑧)
4129, 30, 40syl2anc 584 . . . . . . . 8 ((((𝜑𝑦𝐵) ∧ (𝑦 + 𝑋) = 0 ) ∧ 𝑧𝐵) → (𝑧 + 0 ) = 𝑧)
4239, 41eqtr2d 2777 . . . . . . 7 ((((𝜑𝑦𝐵) ∧ (𝑦 + 𝑋) = 0 ) ∧ 𝑧𝐵) → 𝑧 = (𝑧 + (𝑦 + 𝑋)))
4334, 37, 423eqtr4rd 2787 . . . . . 6 ((((𝜑𝑦𝐵) ∧ (𝑦 + 𝑋) = 0 ) ∧ 𝑧𝐵) → 𝑧 = (𝐺‘(𝑧 + 𝑦)))
4428, 32, 43rspcedvdw 3624 . . . . 5 ((((𝜑𝑦𝐵) ∧ (𝑦 + 𝑋) = 0 ) ∧ 𝑧𝐵) → ∃𝑥𝐵 𝑧 = (𝐺𝑥))
4544ralrimiva 3145 . . . 4 (((𝜑𝑦𝐵) ∧ (𝑦 + 𝑋) = 0 ) → ∀𝑧𝐵𝑥𝐵 𝑧 = (𝐺𝑥))
46 dffo3 7120 . . . 4 (𝐺:𝐵onto𝐵 ↔ (𝐺:𝐵𝐵 ∧ ∀𝑧𝐵𝑥𝐵 𝑧 = (𝐺𝑥)))
4726, 45, 46sylanbrc 583 . . 3 (((𝜑𝑦𝐵) ∧ (𝑦 + 𝑋) = 0 ) → 𝐺:𝐵onto𝐵)
4847r19.29an 3157 . 2 ((𝜑 ∧ ∃𝑦𝐵 (𝑦 + 𝑋) = 0 ) → 𝐺:𝐵onto𝐵)
4918, 48impbida 801 1 (𝜑 → (𝐺:𝐵onto𝐵 ↔ ∃𝑦𝐵 (𝑦 + 𝑋) = 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3060  wrex 3069  Vcvv 3479  cmpt 5223  wf 6555  ontowfo 6557  cfv 6559  (class class class)co 7429  Basecbs 17243  +gcplusg 17293  0gc0g 17480  Mndcmnd 18743
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 2707  ax-sep 5294  ax-nul 5304  ax-pr 5430
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 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-br 5142  df-opab 5204  df-mpt 5224  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-iota 6512  df-fun 6561  df-fn 6562  df-f 6563  df-fo 6565  df-fv 6567  df-riota 7386  df-ov 7432  df-0g 17482  df-mgm 18649  df-sgrp 18728  df-mnd 18744
This theorem is referenced by:  mndractf1o  33021
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