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Mirrors > Home > MPE Home > Th. List > Mathboxes > lvecring | Structured version Visualization version GIF version |
Description: The scalar component of a left vector is a ring. (Contributed by Steven Nguyen, 28-May-2023.) |
Ref | Expression |
---|---|
lvecring.1 | ⊢ 𝐹 = (Scalar‘𝑊) |
Ref | Expression |
---|---|
lvecring | ⊢ (𝑊 ∈ LVec → 𝐹 ∈ Ring) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lveclmod 19871 | . 2 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
2 | lvecring.1 | . . 3 ⊢ 𝐹 = (Scalar‘𝑊) | |
3 | 2 | lmodring 19635 | . 2 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring) |
4 | 1, 3 | syl 17 | 1 ⊢ (𝑊 ∈ LVec → 𝐹 ∈ Ring) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ‘cfv 6348 Scalarcsca 16561 Ringcrg 19290 LModclmod 19627 LVecclvec 19867 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-nul 5203 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-iota 6307 df-fv 6356 df-ov 7152 df-lmod 19629 df-lvec 19868 |
This theorem is referenced by: (None) |
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