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Mirrors > Home > MPE Home > Th. List > clmgrp | Structured version Visualization version GIF version |
Description: A subcomplex module is an additive group. (Contributed by Mario Carneiro, 16-Oct-2015.) |
Ref | Expression |
---|---|
clmgrp | ⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clmlmod 25014 | . 2 ⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ LMod) | |
2 | lmodgrp 20757 | . 2 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 Grpcgrp 18897 LModclmod 20750 ℂModcclm 25009 |
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 2699 ax-nul 5310 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rab 3431 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-iota 6505 df-fv 6561 df-ov 7429 df-lmod 20752 df-clm 25010 |
This theorem is referenced by: clmmulg 25048 clmvsrinv 25054 clmvslinv 25055 clmvz 25058 ttgcontlem1 28715 |
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