Theorem slmdvsass 33305

Description: Associative law for scalar product. (ax-hvmulass 31103 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.)

Hypotheses

Ref	Expression
slmdvsass.v	⊢ 𝑉 = (Base‘𝑊)
slmdvsass.f	⊢ 𝐹 = (Scalar‘𝑊)
slmdvsass.s	⊢ · = ( ·𝑠 ‘𝑊)
slmdvsass.k	⊢ 𝐾 = (Base‘𝐹)
slmdvsass.t	⊢ × = (.r‘𝐹)

Assertion

Ref	Expression
slmdvsass	⊢ ((𝑊 ∈ SLMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)))

Proof of Theorem slmdvsass

Step	Hyp	Ref	Expression
1		slmdvsass.v	. . . . . . . 8 ⊢ 𝑉 = (Base‘𝑊)
2		eqid 2740	. . . . . . . 8 ⊢ (+g‘𝑊) = (+g‘𝑊)
3		slmdvsass.s	. . . . . . . 8 ⊢ · = ( ·𝑠 ‘𝑊)
4		eqid 2740	. . . . . . . 8 ⊢ (0g‘𝑊) = (0g‘𝑊)
5		slmdvsass.f	. . . . . . . 8 ⊢ 𝐹 = (Scalar‘𝑊)
6		slmdvsass.k	. . . . . . . 8 ⊢ 𝐾 = (Base‘𝐹)
7		eqid 2740	. . . . . . . 8 ⊢ (+g‘𝐹) = (+g‘𝐹)
8		slmdvsass.t	. . . . . . . 8 ⊢ × = (.r‘𝐹)
9		eqid 2740	. . . . . . . 8 ⊢ (1r‘𝐹) = (1r‘𝐹)
10		eqid 2740	. . . . . . . 8 ⊢ (0g‘𝐹) = (0g‘𝐹)
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	slmdlema 33291	. . . . . . 7 ⊢ ((𝑊 ∈ SLMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → (((𝑅 · 𝑋) ∈ 𝑉 ∧ (𝑅 · (𝑋(+g‘𝑊)𝑋)) = ((𝑅 · 𝑋)(+g‘𝑊)(𝑅 · 𝑋)) ∧ ((𝑄(+g‘𝐹)𝑅) · 𝑋) = ((𝑄 · 𝑋)(+g‘𝑊)(𝑅 · 𝑋))) ∧ (((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)) ∧ ((1r‘𝐹) · 𝑋) = 𝑋 ∧ ((0g‘𝐹) · 𝑋) = (0g‘𝑊))))
12	11	simprd 496	. . . . . 6 ⊢ ((𝑊 ∈ SLMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → (((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)) ∧ ((1r‘𝐹) · 𝑋) = 𝑋 ∧ ((0g‘𝐹) · 𝑋) = (0g‘𝑊)))
13	12	simp1d 1148	. . . . 5 ⊢ ((𝑊 ∈ SLMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)))
14	13	3expa 1124	. . . 4 ⊢ (((𝑊 ∈ SLMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)))
15	14	anabsan2 680	. . 3 ⊢ (((𝑊 ∈ SLMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾)) ∧ 𝑋 ∈ 𝑉) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)))
16	15	exp42 436	. 2 ⊢ (𝑊 ∈ SLMod → (𝑄 ∈ 𝐾 → (𝑅 ∈ 𝐾 → (𝑋 ∈ 𝑉 → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋))))))
17	16	3imp2 1356	1 ⊢ ((𝑊 ∈ SLMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ‘cfv 6492  (class class class)co 7363  Basecbs 17177  +_gcplusg 17218  ._rcmulr 17219  Scalarcsca 17221   ·_𝑠 cvsca 17222  0_gc0g 17400  1_rcur 20160  SLModcslmd 33288

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-nul 5235

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-ral 3055  df-rab 3393  df-v 3434  df-sbc 3731  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-iota 6448  df-fv 6500  df-ov 7366  df-slmd 33289

This theorem is referenced by:  slmdvs0  33313