Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > clmvscl | Structured version Visualization version GIF version |
Description: Closure of scalar product for a subcomplex module. Analogue of lmodvscl 20128. (Contributed by NM, 3-Nov-2006.) (Revised by AV, 28-Sep-2021.) |
Ref | Expression |
---|---|
clmvscl.v | ⊢ 𝑉 = (Base‘𝑊) |
clmvscl.f | ⊢ 𝐹 = (Scalar‘𝑊) |
clmvscl.s | ⊢ · = ( ·𝑠 ‘𝑊) |
clmvscl.k | ⊢ 𝐾 = (Base‘𝐹) |
Ref | Expression |
---|---|
clmvscl | ⊢ ((𝑊 ∈ ℂMod ∧ 𝑄 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑄 · 𝑋) ∈ 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clmlmod 24218 | . 2 ⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ LMod) | |
2 | clmvscl.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
3 | clmvscl.f | . . 3 ⊢ 𝐹 = (Scalar‘𝑊) | |
4 | clmvscl.s | . . 3 ⊢ · = ( ·𝑠 ‘𝑊) | |
5 | clmvscl.k | . . 3 ⊢ 𝐾 = (Base‘𝐹) | |
6 | 2, 3, 4, 5 | lmodvscl 20128 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑄 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑄 · 𝑋) ∈ 𝑉) |
7 | 1, 6 | syl3an1 1162 | 1 ⊢ ((𝑊 ∈ ℂMod ∧ 𝑄 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑄 · 𝑋) ∈ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ‘cfv 6427 (class class class)co 7268 Basecbs 16900 Scalarcsca 16953 ·𝑠 cvsca 16954 LModclmod 20111 ℂModcclm 24213 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-nul 5229 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3432 df-sbc 3717 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4258 df-if 4461 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-br 5075 df-iota 6385 df-fv 6435 df-ov 7271 df-lmod 20113 df-clm 24214 |
This theorem is referenced by: clmpm1dir 24254 clmnegsubdi2 24256 clmsub4 24257 clmvsubval2 24261 clmvz 24262 nmoleub2lem3 24266 nmoleub3 24270 ncvspi 24308 |
Copyright terms: Public domain | W3C validator |