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Mirrors > Home > MPE Home > Th. List > Mathboxes > lmodgrpd | Structured version Visualization version GIF version |
Description: A left module is a group. (Contributed by SN, 16-May-2024.) |
Ref | Expression |
---|---|
lmodgrpd.1 | ⊢ (𝜑 → 𝑊 ∈ LMod) |
Ref | Expression |
---|---|
lmodgrpd | ⊢ (𝜑 → 𝑊 ∈ Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmodgrpd.1 | . 2 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
2 | lmodgrp 20158 | . 2 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → 𝑊 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2101 Grpcgrp 18605 LModclmod 20151 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-ext 2704 ax-nul 5233 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-sb 2063 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2939 df-ral 3060 df-rab 3224 df-v 3436 df-sbc 3719 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4260 df-if 4463 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4842 df-br 5078 df-iota 6399 df-fv 6455 df-ov 7298 df-lmod 20153 |
This theorem is referenced by: lvecgrpd 40282 |
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