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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 23598 | . 2 ⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ LMod) | |
2 | lmodgrp 19570 | . 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 2105 Grpcgrp 18041 LModclmod 19563 ℂModcclm 23593 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-nul 5201 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 df-ov 7148 df-lmod 19565 df-clm 23594 |
This theorem is referenced by: clmmulg 23632 clmvsrinv 23638 clmvslinv 23639 clmvz 23642 ttgcontlem1 26598 |
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