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Theorem clmvscl 25035
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.)
Hypotheses
Ref Expression
clmvscl.v 𝑉 = (Baseβ€˜π‘Š)
clmvscl.f 𝐹 = (Scalarβ€˜π‘Š)
clmvscl.s Β· = ( ·𝑠 β€˜π‘Š)
clmvscl.k 𝐾 = (Baseβ€˜πΉ)
Assertion
Ref Expression
clmvscl ((π‘Š ∈ β„‚Mod ∧ 𝑄 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) β†’ (𝑄 Β· 𝑋) ∈ 𝑉)

Proof of Theorem clmvscl
StepHypRef Expression
1 clmlmod 25014 . 2 (π‘Š ∈ β„‚Mod β†’ π‘Š ∈ LMod)
2 clmvscl.v . . 3 𝑉 = (Baseβ€˜π‘Š)
3 clmvscl.f . . 3 𝐹 = (Scalarβ€˜π‘Š)
4 clmvscl.s . . 3 Β· = ( ·𝑠 β€˜π‘Š)
5 clmvscl.k . . 3 𝐾 = (Baseβ€˜πΉ)
62, 3, 4, 5lmodvscl 20768 . 2 ((π‘Š ∈ LMod ∧ 𝑄 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) β†’ (𝑄 Β· 𝑋) ∈ 𝑉)
71, 6syl3an1 1160 1 ((π‘Š ∈ β„‚Mod ∧ 𝑄 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) β†’ (𝑄 Β· 𝑋) ∈ 𝑉)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  β€˜cfv 6553  (class class class)co 7426  Basecbs 17187  Scalarcsca 17243   ·𝑠 cvsca 17244  LModclmod 20750  β„‚Modcclm 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
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