Theorem clmvscl 22888

Description: Closure of scalar product for a subcomplex module. Analogue of lmodvscl 18880. (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 22867	. 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 18880	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑄 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑄 · 𝑋) ∈ 𝑉)
7	1, 6	syl3an1 1359	1 ⊢ ((𝑊 ∈ ℂMod ∧ 𝑄 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑄 · 𝑋) ∈ 𝑉)

Syntax hints:   → wi 4   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  Scalarcsca 15944   ·_𝑠 cvsca 15945  LModclmod 18863  ℂModcclm 22862

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-lmod 18865  df-clm 22863

This theorem is referenced by:  clmpm1dir  22903  clmnegsubdi2  22905  clmsub4  22906  clmvsubval2  22910  clmvz  22911  nmoleub2lem3  22915  nmoleub3  22919  ncvspi  22956