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Theorem lmodgrpd 40279
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 20158 . 2 (𝑊 ∈ LMod → 𝑊 ∈ Grp)
31, 2syl 17 1 (𝜑𝑊 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2101  Grpcgrp 18605  LModclmod 20151
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-ext 2704  ax-nul 5233
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2063  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2939  df-ral 3060  df-rab 3224  df-v 3436  df-sbc 3719  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4260  df-if 4463  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4842  df-br 5078  df-iota 6399  df-fv 6455  df-ov 7298  df-lmod 20153
This theorem is referenced by:  lvecgrpd  40282
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