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Mirrors > Home > MPE Home > Th. List > lvecgrpd | Structured version Visualization version GIF version |
Description: A vector space is a group. (Contributed by SN, 16-May-2024.) |
Ref | Expression |
---|---|
lvecgrpd.1 | ⊢ (𝜑 → 𝑊 ∈ LVec) |
Ref | Expression |
---|---|
lvecgrpd | ⊢ (𝜑 → 𝑊 ∈ Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lvecgrpd.1 | . . 3 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
2 | 1 | lveclmodd 20945 | . 2 ⊢ (𝜑 → 𝑊 ∈ LMod) |
3 | 2 | lmodgrpd 20706 | 1 ⊢ (𝜑 → 𝑊 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 Grpcgrp 18853 LVecclvec 20940 |
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 2695 ax-nul 5296 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-ral 3054 df-rab 3425 df-v 3468 df-sbc 3770 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-iota 6485 df-fv 6541 df-ov 7404 df-lmod 20698 df-lvec 20941 |
This theorem is referenced by: dimkerim 33191 algextdeglem8 33260 |
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