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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 | fvex 6113 | . . . 4 ⊢ (Base‘𝑊) ∈ V | |
3 | 1, 2 | eqeltri 2684 | . . 3 ⊢ 𝑉 ∈ V |
4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → 𝑉 ∈ V) |
5 | lfldi.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐾) | |
6 | fconst6g 6007 | . . 3 ⊢ (𝑋 ∈ 𝐾 → (𝑉 × {𝑋}):𝑉⟶𝐾) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (𝜑 → (𝑉 × {𝑋}):𝑉⟶𝐾) |
8 | lfldi.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
9 | lfldi1.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
10 | lfldi.r | . . . 4 ⊢ 𝑅 = (Scalar‘𝑊) | |
11 | lfldi.k | . . . 4 ⊢ 𝐾 = (Base‘𝑅) | |
12 | lfldi.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑊) | |
13 | 10, 11, 1, 12 | lflf 33368 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶𝐾) |
14 | 8, 9, 13 | syl2anc 691 | . 2 ⊢ (𝜑 → 𝐺:𝑉⟶𝐾) |
15 | lfldi1.h | . . 3 ⊢ (𝜑 → 𝐻 ∈ 𝐹) | |
16 | 10, 11, 1, 12 | lflf 33368 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐻 ∈ 𝐹) → 𝐻:𝑉⟶𝐾) |
17 | 8, 15, 16 | syl2anc 691 | . 2 ⊢ (𝜑 → 𝐻:𝑉⟶𝐾) |
18 | 10 | lmodring 18694 | . . . 4 ⊢ (𝑊 ∈ LMod → 𝑅 ∈ Ring) |
19 | 8, 18 | syl 17 | . . 3 ⊢ (𝜑 → 𝑅 ∈ Ring) |
20 | lfldi.p | . . . 4 ⊢ + = (+g‘𝑅) | |
21 | lfldi.t | . . . 4 ⊢ · = (.r‘𝑅) | |
22 | 11, 20, 21 | ringdir 18390 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) |
23 | 19, 22 | sylan 487 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) |
24 | 4, 7, 14, 17, 23 | caofdir 6832 | 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 1031 = wceq 1475 ∈ wcel 1977 Vcvv 3173 {csn 4125 × cxp 5036 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ∘𝑓 cof 6793 Basecbs 15695 +gcplusg 15768 .rcmulr 15769 Scalarcsca 15771 Ringcrg 18370 LModclmod 18686 LFnlclfn 33362 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-map 7746 df-ring 18372 df-lmod 18688 df-lfl 33363 |
This theorem is referenced by: ldualvsdi1 33448 |
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