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Theorem lvecgrpd 20946
Description: A vector space is a group. (Contributed by SN, 16-May-2024.)
Hypothesis
Ref Expression
lvecgrpd.1 (𝜑𝑊 ∈ LVec)
Assertion
Ref Expression
lvecgrpd (𝜑𝑊 ∈ Grp)

Proof of Theorem lvecgrpd
StepHypRef Expression
1 lvecgrpd.1 . . 3 (𝜑𝑊 ∈ LVec)
21lveclmodd 20945 . 2 (𝜑𝑊 ∈ LMod)
32lmodgrpd 20706 1 (𝜑𝑊 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  Grpcgrp 18853  LVecclvec 20940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-nul 5296
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rab 3425  df-v 3468  df-sbc 3770  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-iota 6485  df-fv 6541  df-ov 7404  df-lmod 20698  df-lvec 20941
This theorem is referenced by:  dimkerim  33191  algextdeglem8  33260
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