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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 6684 | . . . . 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 7434 | . . 3 ⊢ (𝜑 → ((𝑉 × {𝑋}) ∘f + (𝑉 × {𝑌})) = (𝑉 × {(𝑋 + 𝑌)})) |
7 | 6 | oveq2d 7172 | . 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 36230 | . 2 ⊢ (𝜑 → (𝐺 ∘f · ((𝑉 × {𝑋}) ∘f + (𝑉 × {𝑌}))) = ((𝐺 ∘f · (𝑉 × {𝑋})) ∘f + (𝐺 ∘f · (𝑉 × {𝑌})))) |
16 | 7, 15 | eqtr3d 2858 | 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 1537 ∈ wcel 2114 Vcvv 3494 {csn 4567 × cxp 5553 ‘cfv 6355 (class class class)co 7156 ∘f cof 7407 Basecbs 16483 +gcplusg 16565 .rcmulr 16566 Scalarcsca 16568 LModclmod 19634 LFnlclfn 36208 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-map 8408 df-ring 19299 df-lmod 19636 df-lfl 36209 |
This theorem is referenced by: ldualvsdi2 36295 |
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