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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 | fvex 6201 | . . . . . 6 ⊢ (Base‘𝑊) ∈ V | |
3 | 1, 2 | eqeltri 2697 | . . . . 5 ⊢ 𝑉 ∈ V |
4 | 3 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝑉 ∈ V) |
5 | lfldi.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐾) | |
6 | lfldi2.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐾) | |
7 | 4, 5, 6 | ofc12 6922 | . . 3 ⊢ (𝜑 → ((𝑉 × {𝑋}) ∘𝑓 + (𝑉 × {𝑌})) = (𝑉 × {(𝑋 + 𝑌)})) |
8 | 7 | oveq2d 6666 | . 2 ⊢ (𝜑 → (𝐺 ∘𝑓 · ((𝑉 × {𝑋}) ∘𝑓 + (𝑉 × {𝑌}))) = (𝐺 ∘𝑓 · (𝑉 × {(𝑋 + 𝑌)}))) |
9 | lfldi.r | . . 3 ⊢ 𝑅 = (Scalar‘𝑊) | |
10 | lfldi.k | . . 3 ⊢ 𝐾 = (Base‘𝑅) | |
11 | lfldi.p | . . 3 ⊢ + = (+g‘𝑅) | |
12 | lfldi.t | . . 3 ⊢ · = (.r‘𝑅) | |
13 | lfldi.f | . . 3 ⊢ 𝐹 = (LFnl‘𝑊) | |
14 | lfldi.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
15 | lfldi2.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
16 | 1, 9, 10, 11, 12, 13, 14, 5, 6, 15 | lflvsdi2 34366 | . 2 ⊢ (𝜑 → (𝐺 ∘𝑓 · ((𝑉 × {𝑋}) ∘𝑓 + (𝑉 × {𝑌}))) = ((𝐺 ∘𝑓 · (𝑉 × {𝑋})) ∘𝑓 + (𝐺 ∘𝑓 · (𝑉 × {𝑌})))) |
17 | 8, 16 | eqtr3d 2658 | 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 1483 ∈ wcel 1990 Vcvv 3200 {csn 4177 × cxp 5112 ‘cfv 5888 (class class class)co 6650 ∘𝑓 cof 6895 Basecbs 15857 +gcplusg 15941 .rcmulr 15942 Scalarcsca 15944 LModclmod 18863 LFnlclfn 34344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-map 7859 df-ring 18549 df-lmod 18865 df-lfl 34345 |
This theorem is referenced by: ldualvsdi2 34431 |
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