MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  clmvscl Structured version   Visualization version   GIF version

Theorem clmvscl 24965
Description: Closure of scalar product for a subcomplex module. Analogue of lmodvscl 20721. (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 24944 . 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 20721 . 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 6536  (class class class)co 7404  Basecbs 17150  Scalarcsca 17206   ·𝑠 cvsca 17207  LModclmod 20703  β„‚Modcclm 24939
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 2697  ax-nul 5299
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-iota 6488  df-fv 6544  df-ov 7407  df-lmod 20705  df-clm 24940
This theorem is referenced by:  clmpm1dir  24980  clmnegsubdi2  24982  clmsub4  24983  clmvsubval2  24987  clmvz  24988  nmoleub2lem3  24992  nmoleub3  24996  ncvspi  25034
  Copyright terms: Public domain W3C validator