Theorem slmdvscl 33185

Description: Closure of scalar product for a semiring left module. (hvmulcl 31036 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 261	. 2 ⊢ (𝑊 ∈ SLMod ↔ 𝑊 ∈ SLMod)
2		pm4.24 563	. 2 ⊢ (𝑅 ∈ 𝐾 ↔ (𝑅 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾))
3		pm4.24 563	. 2 ⊢ (𝑋 ∈ 𝑉 ↔ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉))
4		slmdvscl.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑊)
5		eqid 2734	. . . . 5 ⊢ (+g‘𝑊) = (+g‘𝑊)
6		slmdvscl.s	. . . . 5 ⊢ · = ( ·𝑠 ‘𝑊)
7		eqid 2734	. . . . 5 ⊢ (0g‘𝑊) = (0g‘𝑊)
8		slmdvscl.f	. . . . 5 ⊢ 𝐹 = (Scalar‘𝑊)
9		slmdvscl.k	. . . . 5 ⊢ 𝐾 = (Base‘𝐹)
10		eqid 2734	. . . . 5 ⊢ (+g‘𝐹) = (+g‘𝐹)
11		eqid 2734	. . . . 5 ⊢ (.r‘𝐹) = (.r‘𝐹)
12		eqid 2734	. . . . 5 ⊢ (1r‘𝐹) = (1r‘𝐹)
13		eqid 2734	. . . . 5 ⊢ (0g‘𝐹) = (0g‘𝐹)
14	4, 5, 6, 7, 8, 9, 10, 11, 12, 13	slmdlema 33174	. . . 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 494	. . 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 1142	. 2 ⊢ ((𝑊 ∈ SLMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → (𝑅 · 𝑋) ∈ 𝑉)
17	1, 2, 3, 16	syl3anb 1161	1 ⊢ ((𝑊 ∈ SLMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑅 · 𝑋) ∈ 𝑉)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2103  ‘cfv 6572  (class class class)co 7445  Basecbs 17253  +_gcplusg 17306  ._rcmulr 17307  Scalarcsca 17309   ·_𝑠 cvsca 17310  0_gc0g 17494  1_rcur 20203  SLModcslmd 33171

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-ext 2705  ax-nul 5327

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2943  df-ral 3064  df-rab 3439  df-v 3484  df-sbc 3799  df-dif 3973  df-un 3975  df-ss 3987  df-nul 4348  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5170  df-iota 6524  df-fv 6580  df-ov 7448  df-slmd 33172

This theorem is referenced by:  gsumvsca1  33197  gsumvsca2  33198  sitgaddlemb  34305