Theorem clmgrp 24975

Description: A subcomplex module is an additive group. (Contributed by Mario Carneiro, 16-Oct-2015.)

Assertion

Ref	Expression
clmgrp	⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ Grp)

Proof of Theorem clmgrp

Step	Hyp	Ref	Expression
1		clmlmod 24974	. 2 ⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ LMod)
2		lmodgrp 20780	. 2 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp)
3	1, 2	syl 17	1 ⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2109  Grpcgrp 18872  LModclmod 20773  ℂModcclm 24969

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 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-nul 5264

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-iota 6467  df-fv 6522  df-ov 7393  df-lmod 20775  df-clm 24970

This theorem is referenced by:  clmmulg  25008  clmvsrinv  25014  clmvslinv  25015  clmvz  25018  ttgcontlem1  28819