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Theorem slmdvscl 30769
Description: Closure of scalar product for a semiring left module. (hvmulcl 28717 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 262 . 2 (𝑊 ∈ SLMod ↔ 𝑊 ∈ SLMod)
2 pm4.24 564 . 2 (𝑅𝐾 ↔ (𝑅𝐾𝑅𝐾))
3 pm4.24 564 . 2 (𝑋𝑉 ↔ (𝑋𝑉𝑋𝑉))
4 slmdvscl.v . . . . 5 𝑉 = (Base‘𝑊)
5 eqid 2818 . . . . 5 (+g𝑊) = (+g𝑊)
6 slmdvscl.s . . . . 5 · = ( ·𝑠𝑊)
7 eqid 2818 . . . . 5 (0g𝑊) = (0g𝑊)
8 slmdvscl.f . . . . 5 𝐹 = (Scalar‘𝑊)
9 slmdvscl.k . . . . 5 𝐾 = (Base‘𝐹)
10 eqid 2818 . . . . 5 (+g𝐹) = (+g𝐹)
11 eqid 2818 . . . . 5 (.r𝐹) = (.r𝐹)
12 eqid 2818 . . . . 5 (1r𝐹) = (1r𝐹)
13 eqid 2818 . . . . 5 (0g𝐹) = (0g𝐹)
144, 5, 6, 7, 8, 9, 10, 11, 12, 13slmdlema 30758 . . . 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 495 . . 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 1134 . 2 ((𝑊 ∈ SLMod ∧ (𝑅𝐾𝑅𝐾) ∧ (𝑋𝑉𝑋𝑉)) → (𝑅 · 𝑋) ∈ 𝑉)
171, 2, 3, 16syl3anb 1153 1 ((𝑊 ∈ SLMod ∧ 𝑅𝐾𝑋𝑉) → (𝑅 · 𝑋) ∈ 𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  cfv 6348  (class class class)co 7145  Basecbs 16471  +gcplusg 16553  .rcmulr 16554  Scalarcsca 16556   ·𝑠 cvsca 16557  0gc0g 16701  1rcur 19180  SLModcslmd 30755
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-nul 5201
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-ov 7148  df-slmd 30756
This theorem is referenced by:  gsumvsca1  30781  gsumvsca2  30782  sitgaddlemb  31505
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