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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > slmdvsass | Structured version Visualization version GIF version |
Description: Associative law for scalar product. (ax-hvmulass 30940 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
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) |
Ref | Expression |
---|---|
slmdvsass | ⊢ ((𝑊 ∈ SLMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | slmdvsass.v | . . . . . . . 8 ⊢ 𝑉 = (Base‘𝑊) | |
2 | eqid 2726 | . . . . . . . 8 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
3 | slmdvsass.s | . . . . . . . 8 ⊢ · = ( ·𝑠 ‘𝑊) | |
4 | eqid 2726 | . . . . . . . 8 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
5 | slmdvsass.f | . . . . . . . 8 ⊢ 𝐹 = (Scalar‘𝑊) | |
6 | slmdvsass.k | . . . . . . . 8 ⊢ 𝐾 = (Base‘𝐹) | |
7 | eqid 2726 | . . . . . . . 8 ⊢ (+g‘𝐹) = (+g‘𝐹) | |
8 | slmdvsass.t | . . . . . . . 8 ⊢ × = (.r‘𝐹) | |
9 | eqid 2726 | . . . . . . . 8 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
10 | eqid 2726 | . . . . . . . 8 ⊢ (0g‘𝐹) = (0g‘𝐹) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | slmdlema 33067 | . . . . . . 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 494 | . . . . . 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 1139 | . . . . 5 ⊢ ((𝑊 ∈ SLMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋))) |
14 | 13 | 3expa 1115 | . . . 4 ⊢ (((𝑊 ∈ SLMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋))) |
15 | 14 | anabsan2 672 | . . 3 ⊢ (((𝑊 ∈ SLMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾)) ∧ 𝑋 ∈ 𝑉) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋))) |
16 | 15 | exp42 434 | . 2 ⊢ (𝑊 ∈ SLMod → (𝑄 ∈ 𝐾 → (𝑅 ∈ 𝐾 → (𝑋 ∈ 𝑉 → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)))))) |
17 | 16 | 3imp2 1346 | 1 ⊢ ((𝑊 ∈ SLMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ‘cfv 6554 (class class class)co 7424 Basecbs 17213 +gcplusg 17266 .rcmulr 17267 Scalarcsca 17269 ·𝑠 cvsca 17270 0gc0g 17454 1rcur 20164 SLModcslmd 33064 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 ax-nul 5311 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ne 2931 df-ral 3052 df-rab 3420 df-v 3464 df-sbc 3777 df-dif 3950 df-un 3952 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-iota 6506 df-fv 6562 df-ov 7427 df-slmd 33065 |
This theorem is referenced by: slmdvs0 33089 |
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