Theorem slmdvsdi 33174

Description: Distributive law for scalar product. (ax-hvdistr1 30943 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.)

Hypotheses

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

Assertion

Ref	Expression
slmdvsdi	⊢ ((𝑊 ∈ SLMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌)))

Proof of Theorem slmdvsdi

Step	Hyp	Ref	Expression
1		slmdvsdi.v	. . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑊)
2		slmdvsdi.a	. . . . . . . . 9 ⊢ + = (+g‘𝑊)
3		slmdvsdi.s	. . . . . . . . 9 ⊢ · = ( ·𝑠 ‘𝑊)
4		eqid 2730	. . . . . . . . 9 ⊢ (0g‘𝑊) = (0g‘𝑊)
5		slmdvsdi.f	. . . . . . . . 9 ⊢ 𝐹 = (Scalar‘𝑊)
6		slmdvsdi.k	. . . . . . . . 9 ⊢ 𝐾 = (Base‘𝐹)
7		eqid 2730	. . . . . . . . 9 ⊢ (+g‘𝐹) = (+g‘𝐹)
8		eqid 2730	. . . . . . . . 9 ⊢ (.r‘𝐹) = (.r‘𝐹)
9		eqid 2730	. . . . . . . . 9 ⊢ (1r‘𝐹) = (1r‘𝐹)
10		eqid 2730	. . . . . . . . 9 ⊢ (0g‘𝐹) = (0g‘𝐹)
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	slmdlema 33162	. . . . . . . 8 ⊢ ((𝑊 ∈ SLMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑌 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → (((𝑅 · 𝑋) ∈ 𝑉 ∧ (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌)) ∧ ((𝑅(+g‘𝐹)𝑅) · 𝑋) = ((𝑅 · 𝑋) + (𝑅 · 𝑋))) ∧ (((𝑅(.r‘𝐹)𝑅) · 𝑋) = (𝑅 · (𝑅 · 𝑋)) ∧ ((1r‘𝐹) · 𝑋) = 𝑋 ∧ ((0g‘𝐹) · 𝑋) = (0g‘𝑊))))
12	11	simpld 494	. . . . . . 7 ⊢ ((𝑊 ∈ SLMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑌 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → ((𝑅 · 𝑋) ∈ 𝑉 ∧ (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌)) ∧ ((𝑅(+g‘𝐹)𝑅) · 𝑋) = ((𝑅 · 𝑋) + (𝑅 · 𝑋))))
13	12	simp2d 1143	. . . . . 6 ⊢ ((𝑊 ∈ SLMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑌 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌)))
14	13	3expia 1121	. . . . 5 ⊢ ((𝑊 ∈ SLMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾)) → ((𝑌 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌))))
15	14	anabsan2 674	. . . 4 ⊢ ((𝑊 ∈ SLMod ∧ 𝑅 ∈ 𝐾) → ((𝑌 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌))))
16	15	exp4b 430	. . 3 ⊢ (𝑊 ∈ SLMod → (𝑅 ∈ 𝐾 → (𝑌 ∈ 𝑉 → (𝑋 ∈ 𝑉 → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌))))))
17	16	com34 91	. 2 ⊢ (𝑊 ∈ SLMod → (𝑅 ∈ 𝐾 → (𝑋 ∈ 𝑉 → (𝑌 ∈ 𝑉 → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌))))))
18	17	3imp2 1350	1 ⊢ ((𝑊 ∈ SLMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ‘cfv 6513  (class class class)co 7389  Basecbs 17185  +_gcplusg 17226  ._rcmulr 17227  Scalarcsca 17229   ·_𝑠 cvsca 17230  0_gc0g 17408  1_rcur 20096  SLModcslmd 33159

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-nul 5263

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rab 3409  df-v 3452  df-sbc 3756  df-dif 3919  df-un 3921  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-iota 6466  df-fv 6521  df-ov 7392  df-slmd 33160

This theorem is referenced by:  gsumvsca1  33185