![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
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 6460 | . . 3 ⊢ 𝑉 ∈ V |
3 | 2 | a1i 11 | . 2 ⊢ (𝜑 → 𝑉 ∈ V) |
4 | lfldi.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐾) | |
5 | fconst6g 6344 | . . 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 35201 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶𝐾) |
13 | 7, 8, 12 | syl2anc 579 | . 2 ⊢ (𝜑 → 𝐺:𝑉⟶𝐾) |
14 | lfldi1.h | . . 3 ⊢ (𝜑 → 𝐻 ∈ 𝐹) | |
15 | 9, 10, 1, 11 | lflf 35201 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐻 ∈ 𝐹) → 𝐻:𝑉⟶𝐾) |
16 | 7, 14, 15 | syl2anc 579 | . 2 ⊢ (𝜑 → 𝐻:𝑉⟶𝐾) |
17 | 9 | lmodring 19263 | . . . 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 18954 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) |
22 | 18, 21 | sylan 575 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) |
23 | 3, 6, 13, 16, 22 | caofdir 7211 | 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 1071 = wceq 1601 ∈ wcel 2106 Vcvv 3397 {csn 4397 × cxp 5353 ⟶wf 6131 ‘cfv 6135 (class class class)co 6922 ∘𝑓 cof 7172 Basecbs 16255 +gcplusg 16338 .rcmulr 16339 Scalarcsca 16341 Ringcrg 18934 LModclmod 19255 LFnlclfn 35195 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-map 8142 df-ring 18936 df-lmod 19257 df-lfl 35196 |
This theorem is referenced by: ldualvsdi1 35281 |
Copyright terms: Public domain | W3C validator |