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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lflvsdi1 | Structured version Visualization version GIF version |
Description: Distributive law for (right vector space) scalar product of functionals. (Contributed by NM, 19-Oct-2014.) |
Ref | Expression |
---|---|
lfldi.v | ⊢ 𝑉 = (Base‘𝑊) |
lfldi.r | ⊢ 𝑅 = (Scalar‘𝑊) |
lfldi.k | ⊢ 𝐾 = (Base‘𝑅) |
lfldi.p | ⊢ + = (+g‘𝑅) |
lfldi.t | ⊢ · = (.r‘𝑅) |
lfldi.f | ⊢ 𝐹 = (LFnl‘𝑊) |
lfldi.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lfldi.x | ⊢ (𝜑 → 𝑋 ∈ 𝐾) |
lfldi1.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
lfldi1.h | ⊢ (𝜑 → 𝐻 ∈ 𝐹) |
Ref | Expression |
---|---|
lflvsdi1 | ⊢ (𝜑 → ((𝐺 ∘f + 𝐻) ∘f · (𝑉 × {𝑋})) = ((𝐺 ∘f · (𝑉 × {𝑋})) ∘f + (𝐻 ∘f · (𝑉 × {𝑋})))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lfldi.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
2 | 1 | fvexi 6921 | . . 3 ⊢ 𝑉 ∈ V |
3 | 2 | a1i 11 | . 2 ⊢ (𝜑 → 𝑉 ∈ V) |
4 | lfldi.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐾) | |
5 | fconst6g 6798 | . . 3 ⊢ (𝑋 ∈ 𝐾 → (𝑉 × {𝑋}):𝑉⟶𝐾) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝜑 → (𝑉 × {𝑋}):𝑉⟶𝐾) |
7 | lfldi.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
8 | lfldi1.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
9 | lfldi.r | . . . 4 ⊢ 𝑅 = (Scalar‘𝑊) | |
10 | lfldi.k | . . . 4 ⊢ 𝐾 = (Base‘𝑅) | |
11 | lfldi.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑊) | |
12 | 9, 10, 1, 11 | lflf 39045 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶𝐾) |
13 | 7, 8, 12 | syl2anc 584 | . 2 ⊢ (𝜑 → 𝐺:𝑉⟶𝐾) |
14 | lfldi1.h | . . 3 ⊢ (𝜑 → 𝐻 ∈ 𝐹) | |
15 | 9, 10, 1, 11 | lflf 39045 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐻 ∈ 𝐹) → 𝐻:𝑉⟶𝐾) |
16 | 7, 14, 15 | syl2anc 584 | . 2 ⊢ (𝜑 → 𝐻:𝑉⟶𝐾) |
17 | 9 | lmodring 20883 | . . . 4 ⊢ (𝑊 ∈ LMod → 𝑅 ∈ Ring) |
18 | 7, 17 | syl 17 | . . 3 ⊢ (𝜑 → 𝑅 ∈ Ring) |
19 | lfldi.p | . . . 4 ⊢ + = (+g‘𝑅) | |
20 | lfldi.t | . . . 4 ⊢ · = (.r‘𝑅) | |
21 | 10, 19, 20 | ringdir 20279 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) |
22 | 18, 21 | sylan 580 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) |
23 | 3, 6, 13, 16, 22 | caofdir 7739 | 1 ⊢ (𝜑 → ((𝐺 ∘f + 𝐻) ∘f · (𝑉 × {𝑋})) = ((𝐺 ∘f · (𝑉 × {𝑋})) ∘f + (𝐻 ∘f · (𝑉 × {𝑋})))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 Vcvv 3478 {csn 4631 × cxp 5687 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ∘f cof 7695 Basecbs 17245 +gcplusg 17298 .rcmulr 17299 Scalarcsca 17301 Ringcrg 20251 LModclmod 20875 LFnlclfn 39039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-map 8867 df-ring 20253 df-lmod 20877 df-lfl 39040 |
This theorem is referenced by: ldualvsdi1 39125 |
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