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Theorem slmdvscl 32617
Description: Closure of scalar product for a semiring left module. (hvmulcl 30521 analog.) (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
Hypotheses
Ref Expression
slmdvscl.v 𝑉 = (Baseβ€˜π‘Š)
slmdvscl.f 𝐹 = (Scalarβ€˜π‘Š)
slmdvscl.s Β· = ( ·𝑠 β€˜π‘Š)
slmdvscl.k 𝐾 = (Baseβ€˜πΉ)
Assertion
Ref Expression
slmdvscl ((π‘Š ∈ SLMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) β†’ (𝑅 Β· 𝑋) ∈ 𝑉)
Proof of Theorem slmdvscl
StepHypRef Expression
1 biid 260 . 2 (π‘Š ∈ SLMod ↔ π‘Š ∈ SLMod)
2 pm4.24 564 . 2 (𝑅 ∈ 𝐾 ↔ (𝑅 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾))
3 pm4.24 564 . 2 (𝑋 ∈ 𝑉 ↔ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉))
4 slmdvscl.v . . . . 5 𝑉 = (Baseβ€˜π‘Š)
5 eqid 2732 . . . . 5 (+gβ€˜π‘Š) = (+gβ€˜π‘Š)
6 slmdvscl.s . . . . 5 Β· = ( ·𝑠 β€˜π‘Š)
7 eqid 2732 . . . . 5 (0gβ€˜π‘Š) = (0gβ€˜π‘Š)
8 slmdvscl.f . . . . 5 𝐹 = (Scalarβ€˜π‘Š)
9 slmdvscl.k . . . . 5 𝐾 = (Baseβ€˜πΉ)
10 eqid 2732 . . . . 5 (+gβ€˜πΉ) = (+gβ€˜πΉ)
11 eqid 2732 . . . . 5 (.rβ€˜πΉ) = (.rβ€˜πΉ)
12 eqid 2732 . . . . 5 (1rβ€˜πΉ) = (1rβ€˜πΉ)
13 eqid 2732 . . . . 5 (0gβ€˜πΉ) = (0gβ€˜πΉ)
144, 5, 6, 7, 8, 9, 10, 11, 12, 13slmdlema 32606 . . . 4 ((π‘Š ∈ SLMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) β†’ (((𝑅 Β· 𝑋) ∈ 𝑉 ∧ (𝑅 Β· (𝑋(+gβ€˜π‘Š)𝑋)) = ((𝑅 Β· 𝑋)(+gβ€˜π‘Š)(𝑅 Β· 𝑋)) ∧ ((𝑅(+gβ€˜πΉ)𝑅) Β· 𝑋) = ((𝑅 Β· 𝑋)(+gβ€˜π‘Š)(𝑅 Β· 𝑋))) ∧ (((𝑅(.rβ€˜πΉ)𝑅) Β· 𝑋) = (𝑅 Β· (𝑅 Β· 𝑋)) ∧ ((1rβ€˜πΉ) Β· 𝑋) = 𝑋 ∧ ((0gβ€˜πΉ) Β· 𝑋) = (0gβ€˜π‘Š))))
1514simpld 495 . . 3 ((π‘Š ∈ SLMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) β†’ ((𝑅 Β· 𝑋) ∈ 𝑉 ∧ (𝑅 Β· (𝑋(+gβ€˜π‘Š)𝑋)) = ((𝑅 Β· 𝑋)(+gβ€˜π‘Š)(𝑅 Β· 𝑋)) ∧ ((𝑅(+gβ€˜πΉ)𝑅) Β· 𝑋) = ((𝑅 Β· 𝑋)(+gβ€˜π‘Š)(𝑅 Β· 𝑋))))
1615simp1d 1142 . 2 ((π‘Š ∈ SLMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) β†’ (𝑅 Β· 𝑋) ∈ 𝑉)
171, 2, 3, 16syl3anb 1161 1 ((π‘Š ∈ SLMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) β†’ (𝑅 Β· 𝑋) ∈ 𝑉)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  β€˜cfv 6543  (class class class)co 7411  Basecbs 17148  +gcplusg 17201  .rcmulr 17202  Scalarcsca 17204   ·𝑠 cvsca 17205  0gc0g 17389  1rcur 20075  SLModcslmd 32603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7414  df-slmd 32604
This theorem is referenced by:  gsumvsca1  32629  gsumvsca2  32630  sitgaddlemb  33633
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