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Mirrors > Home > MPE Home > Th. List > Mathboxes > lflvsdi2 | Structured version Visualization version GIF version |
Description: Reverse 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) |
lfldi2.y | ⊢ (𝜑 → 𝑌 ∈ 𝐾) |
lfldi2.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
Ref | Expression |
---|---|
lflvsdi2 | ⊢ (𝜑 → (𝐺 ∘f · ((𝑉 × {𝑋}) ∘f + (𝑉 × {𝑌}))) = ((𝐺 ∘f · (𝑉 × {𝑋})) ∘f + (𝐺 ∘f · (𝑉 × {𝑌})))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lfldi.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
2 | 1 | fvexi 6677 | . . 3 ⊢ 𝑉 ∈ V |
3 | 2 | a1i 11 | . 2 ⊢ (𝜑 → 𝑉 ∈ V) |
4 | lfldi.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
5 | lfldi2.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
6 | lfldi.r | . . . 4 ⊢ 𝑅 = (Scalar‘𝑊) | |
7 | lfldi.k | . . . 4 ⊢ 𝐾 = (Base‘𝑅) | |
8 | lfldi.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑊) | |
9 | 6, 7, 1, 8 | lflf 36079 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶𝐾) |
10 | 4, 5, 9 | syl2anc 584 | . 2 ⊢ (𝜑 → 𝐺:𝑉⟶𝐾) |
11 | lfldi.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐾) | |
12 | fconst6g 6561 | . . 3 ⊢ (𝑋 ∈ 𝐾 → (𝑉 × {𝑋}):𝑉⟶𝐾) | |
13 | 11, 12 | syl 17 | . 2 ⊢ (𝜑 → (𝑉 × {𝑋}):𝑉⟶𝐾) |
14 | lfldi2.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐾) | |
15 | fconst6g 6561 | . . 3 ⊢ (𝑌 ∈ 𝐾 → (𝑉 × {𝑌}):𝑉⟶𝐾) | |
16 | 14, 15 | syl 17 | . 2 ⊢ (𝜑 → (𝑉 × {𝑌}):𝑉⟶𝐾) |
17 | 6 | lmodring 19571 | . . . 4 ⊢ (𝑊 ∈ LMod → 𝑅 ∈ Ring) |
18 | 4, 17 | syl 17 | . . 3 ⊢ (𝜑 → 𝑅 ∈ Ring) |
19 | lfldi.p | . . . 4 ⊢ + = (+g‘𝑅) | |
20 | lfldi.t | . . . 4 ⊢ · = (.r‘𝑅) | |
21 | 7, 19, 20 | ringdi 19245 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧))) |
22 | 18, 21 | sylan 580 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧))) |
23 | 3, 10, 13, 16, 22 | caofdi 7434 | 1 ⊢ (𝜑 → (𝐺 ∘f · ((𝑉 × {𝑋}) ∘f + (𝑉 × {𝑌}))) = ((𝐺 ∘f · (𝑉 × {𝑋})) ∘f + (𝐺 ∘f · (𝑉 × {𝑌})))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 Vcvv 3492 {csn 4557 × cxp 5546 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ∘f cof 7396 Basecbs 16471 +gcplusg 16553 .rcmulr 16554 Scalarcsca 16556 Ringcrg 19226 LModclmod 19563 LFnlclfn 36073 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-map 8397 df-ring 19228 df-lmod 19565 df-lfl 36074 |
This theorem is referenced by: lflvsdi2a 36096 |
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