Theorem slmdvscl 29098

Description: Closure of scalar product for a semiring left module. (hvmulcl 27254 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	⊢ · = ( ·𝑠 ‘𝑊)
slmdvscl.k	⊢ 𝐾 = (Base‘𝐹)

Assertion

Ref	Expression
slmdvscl	⊢ ((𝑊 ∈ SLMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑅 · 𝑋) ∈ 𝑉)

Proof of Theorem slmdvscl

Step	Hyp	Ref	Expression
1		biid 250	. 2 ⊢ (𝑊 ∈ SLMod ↔ 𝑊 ∈ SLMod)
2		pm4.24 673	. 2 ⊢ (𝑅 ∈ 𝐾 ↔ (𝑅 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾))
3		pm4.24 673	. 2 ⊢ (𝑋 ∈ 𝑉 ↔ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉))
4		slmdvscl.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑊)
5		eqid 2610	. . . . 5 ⊢ (+g‘𝑊) = (+g‘𝑊)
6		slmdvscl.s	. . . . 5 ⊢ · = ( ·𝑠 ‘𝑊)
7		eqid 2610	. . . . 5 ⊢ (0g‘𝑊) = (0g‘𝑊)
8		slmdvscl.f	. . . . 5 ⊢ 𝐹 = (Scalar‘𝑊)
9		slmdvscl.k	. . . . 5 ⊢ 𝐾 = (Base‘𝐹)
10		eqid 2610	. . . . 5 ⊢ (+g‘𝐹) = (+g‘𝐹)
11		eqid 2610	. . . . 5 ⊢ (.r‘𝐹) = (.r‘𝐹)
12		eqid 2610	. . . . 5 ⊢ (1r‘𝐹) = (1r‘𝐹)
13		eqid 2610	. . . . 5 ⊢ (0g‘𝐹) = (0g‘𝐹)
14	4, 5, 6, 7, 8, 9, 10, 11, 12, 13	slmdlema 29087	. . . 4 ⊢ ((𝑊 ∈ SLMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → (((𝑅 · 𝑋) ∈ 𝑉 ∧ (𝑅 · (𝑋(+g‘𝑊)𝑋)) = ((𝑅 · 𝑋)(+g‘𝑊)(𝑅 · 𝑋)) ∧ ((𝑅(+g‘𝐹)𝑅) · 𝑋) = ((𝑅 · 𝑋)(+g‘𝑊)(𝑅 · 𝑋))) ∧ (((𝑅(.r‘𝐹)𝑅) · 𝑋) = (𝑅 · (𝑅 · 𝑋)) ∧ ((1r‘𝐹) · 𝑋) = 𝑋 ∧ ((0g‘𝐹) · 𝑋) = (0g‘𝑊))))
15	14	simpld 474	. . 3 ⊢ ((𝑊 ∈ SLMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → ((𝑅 · 𝑋) ∈ 𝑉 ∧ (𝑅 · (𝑋(+g‘𝑊)𝑋)) = ((𝑅 · 𝑋)(+g‘𝑊)(𝑅 · 𝑋)) ∧ ((𝑅(+g‘𝐹)𝑅) · 𝑋) = ((𝑅 · 𝑋)(+g‘𝑊)(𝑅 · 𝑋))))
16	15	simp1d 1066	. 2 ⊢ ((𝑊 ∈ SLMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → (𝑅 · 𝑋) ∈ 𝑉)
17	1, 2, 3, 16	syl3anb 1361	1 ⊢ ((𝑊 ∈ SLMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑅 · 𝑋) ∈ 𝑉)

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +_gcplusg 15768  ._rcmulr 15769  Scalarcsca 15771   ·_𝑠 cvsca 15772  0_gc0g 15923  1_rcur 18324  SLModcslmd 29084

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-slmd 29085

This theorem is referenced by:  gsumvsca1  29113  gsumvsca2  29114  sitgaddlemb  29737