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Theorem lmodvscld 20766
Description: Closure of scalar product for a left module. (Contributed by SN, 15-Mar-2025.)
Hypotheses
Ref Expression
lmodvscld.v 𝑉 = (Baseβ€˜π‘Š)
lmodvscld.f 𝐹 = (Scalarβ€˜π‘Š)
lmodvscld.s Β· = ( ·𝑠 β€˜π‘Š)
lmodvscld.k 𝐾 = (Baseβ€˜πΉ)
lmodvscld.w (πœ‘ β†’ π‘Š ∈ LMod)
lmodvscld.r (πœ‘ β†’ 𝑅 ∈ 𝐾)
lmodvscld.x (πœ‘ β†’ 𝑋 ∈ 𝑉)
Assertion
Ref Expression
lmodvscld (πœ‘ β†’ (𝑅 Β· 𝑋) ∈ 𝑉)

Proof of Theorem lmodvscld
StepHypRef Expression
1 lmodvscld.w . 2 (πœ‘ β†’ π‘Š ∈ LMod)
2 lmodvscld.r . 2 (πœ‘ β†’ 𝑅 ∈ 𝐾)
3 lmodvscld.x . 2 (πœ‘ β†’ 𝑋 ∈ 𝑉)
4 lmodvscld.v . . 3 𝑉 = (Baseβ€˜π‘Š)
5 lmodvscld.f . . 3 𝐹 = (Scalarβ€˜π‘Š)
6 lmodvscld.s . . 3 Β· = ( ·𝑠 β€˜π‘Š)
7 lmodvscld.k . . 3 𝐾 = (Baseβ€˜πΉ)
84, 5, 6, 7lmodvscl 20765 . 2 ((π‘Š ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) β†’ (𝑅 Β· 𝑋) ∈ 𝑉)
91, 2, 3, 8syl3anc 1368 1 (πœ‘ β†’ (𝑅 Β· 𝑋) ∈ 𝑉)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  β€˜cfv 6543  (class class class)co 7416  Basecbs 17179  Scalarcsca 17235   ·𝑠 cvsca 17236  LModclmod 20747
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 2696  ax-nul 5301
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 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rab 3420  df-v 3465  df-sbc 3769  df-dif 3942  df-un 3944  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-iota 6495  df-fv 6551  df-ov 7419  df-lmod 20749
This theorem is referenced by:  q1pvsca  33331  r1pvsca  33332  r1plmhm  33337  ply1degltdimlem  33377  algextdeglem8  33449  frlmsnic  41826  ply1vscl  41849  selvvvval  41883  prjspvs  42099  prjspeclsp  42101
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