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Mirrors > Home > MPE Home > Th. List > lmodvscld | Structured version Visualization version GIF version |
Description: Closure of scalar product for a left module. (Contributed by SN, 15-Mar-2025.) |
Ref | Expression |
---|---|
lmodvscld.v | ⊢ 𝑉 = (Base‘𝑊) |
lmodvscld.f | ⊢ 𝐹 = (Scalar‘𝑊) |
lmodvscld.s | ⊢ · = ( ·𝑠 ‘𝑊) |
lmodvscld.k | ⊢ 𝐾 = (Base‘𝐹) |
lmodvscld.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lmodvscld.r | ⊢ (𝜑 → 𝑅 ∈ 𝐾) |
lmodvscld.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
Ref | Expression |
---|---|
lmodvscld | ⊢ (𝜑 → (𝑅 · 𝑋) ∈ 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmodvscld.w | . 2 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
2 | lmodvscld.r | . 2 ⊢ (𝜑 → 𝑅 ∈ 𝐾) | |
3 | lmodvscld.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
4 | lmodvscld.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
5 | lmodvscld.f | . . 3 ⊢ 𝐹 = (Scalar‘𝑊) | |
6 | lmodvscld.s | . . 3 ⊢ · = ( ·𝑠 ‘𝑊) | |
7 | lmodvscld.k | . . 3 ⊢ 𝐾 = (Base‘𝐹) | |
8 | 4, 5, 6, 7 | lmodvscl 20709 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑅 · 𝑋) ∈ 𝑉) |
9 | 1, 2, 3, 8 | syl3anc 1368 | 1 ⊢ (𝜑 → (𝑅 · 𝑋) ∈ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ‘cfv 6533 (class class class)co 7401 Basecbs 17140 Scalarcsca 17196 ·𝑠 cvsca 17197 LModclmod 20691 |
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 20693 |
This theorem is referenced by: q1pvsca 33106 r1pvsca 33107 r1plmhm 33112 ply1degltdimlem 33152 algextdeglem8 33226 frlmsnic 41565 selvvvval 41612 prjspvs 41807 prjspeclsp 41809 |
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