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Theorem slmdvscl 30895
Description: Closure of scalar product for a semiring left module. (hvmulcl 28799 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 264 . 2 (𝑊 ∈ SLMod ↔ 𝑊 ∈ SLMod)
2 pm4.24 567 . 2 (𝑅𝐾 ↔ (𝑅𝐾𝑅𝐾))
3 pm4.24 567 . 2 (𝑋𝑉 ↔ (𝑋𝑉𝑋𝑉))
4 slmdvscl.v . . . . 5 𝑉 = (Base‘𝑊)
5 eqid 2801 . . . . 5 (+g𝑊) = (+g𝑊)
6 slmdvscl.s . . . . 5 · = ( ·𝑠𝑊)
7 eqid 2801 . . . . 5 (0g𝑊) = (0g𝑊)
8 slmdvscl.f . . . . 5 𝐹 = (Scalar‘𝑊)
9 slmdvscl.k . . . . 5 𝐾 = (Base‘𝐹)
10 eqid 2801 . . . . 5 (+g𝐹) = (+g𝐹)
11 eqid 2801 . . . . 5 (.r𝐹) = (.r𝐹)
12 eqid 2801 . . . . 5 (1r𝐹) = (1r𝐹)
13 eqid 2801 . . . . 5 (0g𝐹) = (0g𝐹)
144, 5, 6, 7, 8, 9, 10, 11, 12, 13slmdlema 30884 . . . 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 498 . . 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 1139 . 2 ((𝑊 ∈ SLMod ∧ (𝑅𝐾𝑅𝐾) ∧ (𝑋𝑉𝑋𝑉)) → (𝑅 · 𝑋) ∈ 𝑉)
171, 2, 3, 16syl3anb 1158 1 ((𝑊 ∈ SLMod ∧ 𝑅𝐾𝑋𝑉) → (𝑅 · 𝑋) ∈ 𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2112  cfv 6328  (class class class)co 7139  Basecbs 16478  +gcplusg 16560  .rcmulr 16561  Scalarcsca 16563   ·𝑠 cvsca 16564  0gc0g 16708  1rcur 19247  SLModcslmd 30881
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-nul 5177
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-iota 6287  df-fv 6336  df-ov 7142  df-slmd 30882
This theorem is referenced by:  gsumvsca1  30907  gsumvsca2  30908  sitgaddlemb  31714
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