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| Mirrors > Home > MPE Home > Th. List > Mathboxes > lflvsdi2a | Structured version Visualization version GIF version | ||
| Description: Reverse distributive law for (right vector space) scalar product of functionals. (Contributed by NM, 21-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 | 
|---|---|
| lflvsdi2a | ⊢ (𝜑 → (𝐺 ∘f · (𝑉 × {(𝑋 + 𝑌)})) = ((𝐺 ∘f · (𝑉 × {𝑋})) ∘f + (𝐺 ∘f · (𝑉 × {𝑌})))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | lfldi.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑊) | |
| 2 | 1 | fvexi 6919 | . . . . 5 ⊢ 𝑉 ∈ V | 
| 3 | 2 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝑉 ∈ V) | 
| 4 | lfldi.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐾) | |
| 5 | lfldi2.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐾) | |
| 6 | 3, 4, 5 | ofc12 7728 | . . 3 ⊢ (𝜑 → ((𝑉 × {𝑋}) ∘f + (𝑉 × {𝑌})) = (𝑉 × {(𝑋 + 𝑌)})) | 
| 7 | 6 | oveq2d 7448 | . 2 ⊢ (𝜑 → (𝐺 ∘f · ((𝑉 × {𝑋}) ∘f + (𝑉 × {𝑌}))) = (𝐺 ∘f · (𝑉 × {(𝑋 + 𝑌)}))) | 
| 8 | lfldi.r | . . 3 ⊢ 𝑅 = (Scalar‘𝑊) | |
| 9 | lfldi.k | . . 3 ⊢ 𝐾 = (Base‘𝑅) | |
| 10 | lfldi.p | . . 3 ⊢ + = (+g‘𝑅) | |
| 11 | lfldi.t | . . 3 ⊢ · = (.r‘𝑅) | |
| 12 | lfldi.f | . . 3 ⊢ 𝐹 = (LFnl‘𝑊) | |
| 13 | lfldi.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
| 14 | lfldi2.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
| 15 | 1, 8, 9, 10, 11, 12, 13, 4, 5, 14 | lflvsdi2 39081 | . 2 ⊢ (𝜑 → (𝐺 ∘f · ((𝑉 × {𝑋}) ∘f + (𝑉 × {𝑌}))) = ((𝐺 ∘f · (𝑉 × {𝑋})) ∘f + (𝐺 ∘f · (𝑉 × {𝑌})))) | 
| 16 | 7, 15 | eqtr3d 2778 | 1 ⊢ (𝜑 → (𝐺 ∘f · (𝑉 × {(𝑋 + 𝑌)})) = ((𝐺 ∘f · (𝑉 × {𝑋})) ∘f + (𝐺 ∘f · (𝑉 × {𝑌})))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 Vcvv 3479 {csn 4625 × cxp 5682 ‘cfv 6560 (class class class)co 7432 ∘f cof 7696 Basecbs 17248 +gcplusg 17298 .rcmulr 17299 Scalarcsca 17301 LModclmod 20859 LFnlclfn 39059 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-map 8869 df-ring 20233 df-lmod 20861 df-lfl 39060 | 
| This theorem is referenced by: ldualvsdi2 39146 | 
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