Theorem slmdvscl 33299

Description: Closure of scalar product for a semiring left module. (hvmulcl 31106 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 263	. 2 ⊢ (𝑊 ∈ SLMod ↔ 𝑊 ∈ SLMod)
2		pm4.24 569	. 2 ⊢ (𝑅 ∈ 𝐾 ↔ (𝑅 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾))
3		pm4.24 569	. 2 ⊢ (𝑋 ∈ 𝑉 ↔ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉))
4		slmdvscl.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑊)
5		eqid 2741	. . . . 5 ⊢ (+g‘𝑊) = (+g‘𝑊)
6		slmdvscl.s	. . . . 5 ⊢ · = ( ·𝑠 ‘𝑊)
7		eqid 2741	. . . . 5 ⊢ (0g‘𝑊) = (0g‘𝑊)
8		slmdvscl.f	. . . . 5 ⊢ 𝐹 = (Scalar‘𝑊)
9		slmdvscl.k	. . . . 5 ⊢ 𝐾 = (Base‘𝐹)
10		eqid 2741	. . . . 5 ⊢ (+g‘𝐹) = (+g‘𝐹)
11		eqid 2741	. . . . 5 ⊢ (.r‘𝐹) = (.r‘𝐹)
12		eqid 2741	. . . . 5 ⊢ (1r‘𝐹) = (1r‘𝐹)
13		eqid 2741	. . . . 5 ⊢ (0g‘𝐹) = (0g‘𝐹)
14	4, 5, 6, 7, 8, 9, 10, 11, 12, 13	slmdlema 33288	. . . 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 496	. . 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 1149	. 2 ⊢ ((𝑊 ∈ SLMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → (𝑅 · 𝑋) ∈ 𝑉)
17	1, 2, 3, 16	syl3anb 1168	1 ⊢ ((𝑊 ∈ SLMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑅 · 𝑋) ∈ 𝑉)

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121  ‘cfv 6489  (class class class)co 7360  Basecbs 17174  +_gcplusg 17215  ._rcmulr 17216  Scalarcsca 17218   ·_𝑠 cvsca 17219  0_gc0g 17397  1_rcur 20157  SLModcslmd 33285

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-nul 5231

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ne 2937  df-ral 3056  df-rab 3394  df-v 3435  df-sbc 3726  df-dif 3888  df-un 3890  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-iota 6445  df-fv 6497  df-ov 7363  df-slmd 33286

This theorem is referenced by:  gsumvsca1  33311  gsumvsca2  33312  sitgaddlemb  34544