Theorem clmvscl 25140

Description: Closure of scalar product for a subcomplex module. Analogue of lmodvscl 20898. (Contributed by NM, 3-Nov-2006.) (Revised by AV, 28-Sep-2021.)

Hypotheses

Ref	Expression
clmvscl.v	⊢ 𝑉 = (Base‘𝑊)
clmvscl.f	⊢ 𝐹 = (Scalar‘𝑊)
clmvscl.s	⊢ · = ( ·𝑠 ‘𝑊)
clmvscl.k	⊢ 𝐾 = (Base‘𝐹)

Assertion

Ref	Expression
clmvscl	⊢ ((𝑊 ∈ ℂMod ∧ 𝑄 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑄 · 𝑋) ∈ 𝑉)

Proof of Theorem clmvscl

Step	Hyp	Ref	Expression
1		clmlmod 25119	. 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 20898	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑄 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑄 · 𝑋) ∈ 𝑉)
7	1, 6	syl3an1 1163	1 ⊢ ((𝑊 ∈ ℂMod ∧ 𝑄 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑄 · 𝑋) ∈ 𝑉)

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ‘cfv 6573  (class class class)co 7448  Basecbs 17258  Scalarcsca 17314   ·_𝑠 cvsca 17315  LModclmod 20880  ℂModcclm 25114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-nul 5324

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6525  df-fv 6581  df-ov 7451  df-lmod 20882  df-clm 25115

This theorem is referenced by:  clmpm1dir  25155  clmnegsubdi2  25157  clmsub4  25158  clmvsubval2  25162  clmvz  25163  nmoleub2lem3  25167  nmoleub3  25171  ncvspi  25209