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Theorem lmodgrpd 39434
 Description: A left module is a group. (Contributed by SN, 16-May-2024.)
Hypothesis
Ref Expression
lmodgrpd.1 (𝜑𝑊 ∈ LMod)
Assertion
Ref Expression
lmodgrpd (𝜑𝑊 ∈ Grp)

Proof of Theorem lmodgrpd
StepHypRef Expression
1 lmodgrpd.1 . 2 (𝜑𝑊 ∈ LMod)
2 lmodgrp 19637 . 2 (𝑊 ∈ LMod → 𝑊 ∈ Grp)
31, 2syl 17 1 (𝜑𝑊 ∈ Grp)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2112  Grpcgrp 18098  LModclmod 19630 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-nul 5177 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-iota 6287  df-fv 6336  df-ov 7142  df-lmod 19632 This theorem is referenced by:  lvecgrpd  39437
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