Theorem slmdvsdir 30894

Description: Distributive law for scalar product. (ax-hvdistr1 28791 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.)

Hypotheses

Ref	Expression
slmdvsdir.v	⊢ 𝑉 = (Base‘𝑊)
slmdvsdir.a	⊢ + = (+g‘𝑊)
slmdvsdir.f	⊢ 𝐹 = (Scalar‘𝑊)
slmdvsdir.s	⊢ · = ( ·𝑠 ‘𝑊)
slmdvsdir.k	⊢ 𝐾 = (Base‘𝐹)
slmdvsdir.p	⊢ ⨣ = (+g‘𝐹)

Assertion

Ref	Expression
slmdvsdir	⊢ ((𝑊 ∈ SLMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 ⨣ 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋)))

Proof of Theorem slmdvsdir

Step	Hyp	Ref	Expression
1		slmdvsdir.v	. . . . . . . 8 ⊢ 𝑉 = (Base‘𝑊)
2		slmdvsdir.a	. . . . . . . 8 ⊢ + = (+g‘𝑊)
3		slmdvsdir.s	. . . . . . . 8 ⊢ · = ( ·𝑠 ‘𝑊)
4		eqid 2798	. . . . . . . 8 ⊢ (0g‘𝑊) = (0g‘𝑊)
5		slmdvsdir.f	. . . . . . . 8 ⊢ 𝐹 = (Scalar‘𝑊)
6		slmdvsdir.k	. . . . . . . 8 ⊢ 𝐾 = (Base‘𝐹)
7		slmdvsdir.p	. . . . . . . 8 ⊢ ⨣ = (+g‘𝐹)
8		eqid 2798	. . . . . . . 8 ⊢ (.r‘𝐹) = (.r‘𝐹)
9		eqid 2798	. . . . . . . 8 ⊢ (1r‘𝐹) = (1r‘𝐹)
10		eqid 2798	. . . . . . . 8 ⊢ (0g‘𝐹) = (0g‘𝐹)
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	slmdlema 30881	. . . . . . 7 ⊢ ((𝑊 ∈ SLMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → (((𝑅 · 𝑋) ∈ 𝑉 ∧ (𝑅 · (𝑋 + 𝑋)) = ((𝑅 · 𝑋) + (𝑅 · 𝑋)) ∧ ((𝑄 ⨣ 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋))) ∧ (((𝑄(.r‘𝐹)𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)) ∧ ((1r‘𝐹) · 𝑋) = 𝑋 ∧ ((0g‘𝐹) · 𝑋) = (0g‘𝑊))))
12	11	simpld 498	. . . . . 6 ⊢ ((𝑊 ∈ SLMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → ((𝑅 · 𝑋) ∈ 𝑉 ∧ (𝑅 · (𝑋 + 𝑋)) = ((𝑅 · 𝑋) + (𝑅 · 𝑋)) ∧ ((𝑄 ⨣ 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋))))
13	12	simp3d 1141	. . . . 5 ⊢ ((𝑊 ∈ SLMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 ⨣ 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋)))
14	13	3expa 1115	. . . 4 ⊢ (((𝑊 ∈ SLMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 ⨣ 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋)))
15	14	anabsan2 673	. . 3 ⊢ (((𝑊 ∈ SLMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾)) ∧ 𝑋 ∈ 𝑉) → ((𝑄 ⨣ 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋)))
16	15	exp42 439	. 2 ⊢ (𝑊 ∈ SLMod → (𝑄 ∈ 𝐾 → (𝑅 ∈ 𝐾 → (𝑋 ∈ 𝑉 → ((𝑄 ⨣ 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋))))))
17	16	3imp2 1346	1 ⊢ ((𝑊 ∈ SLMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 ⨣ 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋)))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ‘cfv 6324  (class class class)co 7135  Basecbs 16475  +_gcplusg 16557  ._rcmulr 16558  Scalarcsca 16560   ·_𝑠 cvsca 16561  0_gc0g 16705  1_rcur 19244  SLModcslmd 30878

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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-nul 5174

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 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-iota 6283  df-fv 6332  df-ov 7138  df-slmd 30879

This theorem is referenced by:  gsumvsca2  30905