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Theorem slmdvscl 30107
Description: Closure of scalar product for a semiring left module. (hvmulcl 28210 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 = (Base‘𝑊)
slmdvscl.f 𝐹 = (Scalar‘𝑊)
slmdvscl.s · = ( ·𝑠𝑊)
slmdvscl.k 𝐾 = (Base‘𝐹)
Assertion
Ref Expression
slmdvscl ((𝑊 ∈ SLMod ∧ 𝑅𝐾𝑋𝑉) → (𝑅 · 𝑋) ∈ 𝑉)

Proof of Theorem slmdvscl
StepHypRef Expression
1 biid 251 . 2 (𝑊 ∈ SLMod ↔ 𝑊 ∈ SLMod)
2 pm4.24 553 . 2 (𝑅𝐾 ↔ (𝑅𝐾𝑅𝐾))
3 pm4.24 553 . 2 (𝑋𝑉 ↔ (𝑋𝑉𝑋𝑉))
4 slmdvscl.v . . . . 5 𝑉 = (Base‘𝑊)
5 eqid 2771 . . . . 5 (+g𝑊) = (+g𝑊)
6 slmdvscl.s . . . . 5 · = ( ·𝑠𝑊)
7 eqid 2771 . . . . 5 (0g𝑊) = (0g𝑊)
8 slmdvscl.f . . . . 5 𝐹 = (Scalar‘𝑊)
9 slmdvscl.k . . . . 5 𝐾 = (Base‘𝐹)
10 eqid 2771 . . . . 5 (+g𝐹) = (+g𝐹)
11 eqid 2771 . . . . 5 (.r𝐹) = (.r𝐹)
12 eqid 2771 . . . . 5 (1r𝐹) = (1r𝐹)
13 eqid 2771 . . . . 5 (0g𝐹) = (0g𝐹)
144, 5, 6, 7, 8, 9, 10, 11, 12, 13slmdlema 30096 . . . 4 ((𝑊 ∈ SLMod ∧ (𝑅𝐾𝑅𝐾) ∧ (𝑋𝑉𝑋𝑉)) → (((𝑅 · 𝑋) ∈ 𝑉 ∧ (𝑅 · (𝑋(+g𝑊)𝑋)) = ((𝑅 · 𝑋)(+g𝑊)(𝑅 · 𝑋)) ∧ ((𝑅(+g𝐹)𝑅) · 𝑋) = ((𝑅 · 𝑋)(+g𝑊)(𝑅 · 𝑋))) ∧ (((𝑅(.r𝐹)𝑅) · 𝑋) = (𝑅 · (𝑅 · 𝑋)) ∧ ((1r𝐹) · 𝑋) = 𝑋 ∧ ((0g𝐹) · 𝑋) = (0g𝑊))))
1514simpld 482 . . 3 ((𝑊 ∈ SLMod ∧ (𝑅𝐾𝑅𝐾) ∧ (𝑋𝑉𝑋𝑉)) → ((𝑅 · 𝑋) ∈ 𝑉 ∧ (𝑅 · (𝑋(+g𝑊)𝑋)) = ((𝑅 · 𝑋)(+g𝑊)(𝑅 · 𝑋)) ∧ ((𝑅(+g𝐹)𝑅) · 𝑋) = ((𝑅 · 𝑋)(+g𝑊)(𝑅 · 𝑋))))
1615simp1d 1136 . 2 ((𝑊 ∈ SLMod ∧ (𝑅𝐾𝑅𝐾) ∧ (𝑋𝑉𝑋𝑉)) → (𝑅 · 𝑋) ∈ 𝑉)
171, 2, 3, 16syl3anb 1164 1 ((𝑊 ∈ SLMod ∧ 𝑅𝐾𝑋𝑉) → (𝑅 · 𝑋) ∈ 𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  cfv 6031  (class class class)co 6793  Basecbs 16064  +gcplusg 16149  .rcmulr 16150  Scalarcsca 16152   ·𝑠 cvsca 16153  0gc0g 16308  1rcur 18709  SLModcslmd 30093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-nul 4923
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-iota 5994  df-fv 6039  df-ov 6796  df-slmd 30094
This theorem is referenced by:  gsumvsca1  30122  gsumvsca2  30123  sitgaddlemb  30750
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