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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 23274 | . 2 ⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ LMod) | |
2 | lmodgrp 19262 | . 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 2107 Grpcgrp 17809 LModclmod 19255 ℂModcclm 23269 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-nul 5025 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-iota 6099 df-fv 6143 df-ov 6925 df-lmod 19257 df-clm 23270 |
This theorem is referenced by: clmmulg 23308 clmvsrinv 23314 clmvslinv 23315 clmvz 23318 ttgcontlem1 26234 |
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