Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 23822 | . 2 ⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ LMod) | |
2 | lmodgrp 19763 | . 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 2114 Grpcgrp 18222 LModclmod 19756 ℂModcclm 23817 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-nul 5175 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3401 df-sbc 3682 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-nul 4213 df-sn 4518 df-pr 4520 df-op 4524 df-uni 4798 df-br 5032 df-iota 6298 df-fv 6348 df-ov 7176 df-lmod 19758 df-clm 23818 |
This theorem is referenced by: clmmulg 23856 clmvsrinv 23862 clmvslinv 23863 clmvz 23866 ttgcontlem1 26834 |
Copyright terms: Public domain | W3C validator |