![]() |
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 20768. (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 25014 | . 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 20768 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑄 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑄 · 𝑋) ∈ 𝑉) |
7 | 1, 6 | syl3an1 1160 | 1 ⊢ ((𝑊 ∈ ℂMod ∧ 𝑄 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑄 · 𝑋) ∈ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ‘cfv 6553 (class class class)co 7426 Basecbs 17187 Scalarcsca 17243 ·𝑠 cvsca 17244 LModclmod 20750 ℂModcclm 25009 |
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 2699 ax-nul 5310 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rab 3431 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-iota 6505 df-fv 6561 df-ov 7429 df-lmod 20752 df-clm 25010 |
This theorem is referenced by: clmpm1dir 25050 clmnegsubdi2 25052 clmsub4 25053 clmvsubval2 25057 clmvz 25058 nmoleub2lem3 25062 nmoleub3 25066 ncvspi 25104 |
Copyright terms: Public domain | W3C validator |