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Mirrors > Home > MPE Home > Th. List > coe1fv | Structured version Visualization version GIF version |
Description: Value of an evaluated coefficient in a polynomial coefficient vector. (Contributed by Stefan O'Rear, 21-Mar-2015.) |
Ref | Expression |
---|---|
coe1fval.a | ⊢ 𝐴 = (coe1‘𝐹) |
Ref | Expression |
---|---|
coe1fv | ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝐴‘𝑁) = (𝐹‘(1o × {𝑁}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coe1fval.a | . . . 4 ⊢ 𝐴 = (coe1‘𝐹) | |
2 | 1 | coe1fval 21721 | . . 3 ⊢ (𝐹 ∈ 𝑉 → 𝐴 = (𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛})))) |
3 | 2 | fveq1d 6891 | . 2 ⊢ (𝐹 ∈ 𝑉 → (𝐴‘𝑁) = ((𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛})))‘𝑁)) |
4 | sneq 4638 | . . . . 5 ⊢ (𝑛 = 𝑁 → {𝑛} = {𝑁}) | |
5 | 4 | xpeq2d 5706 | . . . 4 ⊢ (𝑛 = 𝑁 → (1o × {𝑛}) = (1o × {𝑁})) |
6 | 5 | fveq2d 6893 | . . 3 ⊢ (𝑛 = 𝑁 → (𝐹‘(1o × {𝑛})) = (𝐹‘(1o × {𝑁}))) |
7 | eqid 2733 | . . 3 ⊢ (𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛}))) = (𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛}))) | |
8 | fvex 6902 | . . 3 ⊢ (𝐹‘(1o × {𝑁})) ∈ V | |
9 | 6, 7, 8 | fvmpt 6996 | . 2 ⊢ (𝑁 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛})))‘𝑁) = (𝐹‘(1o × {𝑁}))) |
10 | 3, 9 | sylan9eq 2793 | 1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝐴‘𝑁) = (𝐹‘(1o × {𝑁}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {csn 4628 ↦ cmpt 5231 × cxp 5674 ‘cfv 6541 1oc1o 8456 ℕ0cn0 12469 coe1cco1 21694 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-1cn 11165 ax-addcl 11167 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-ov 7409 df-om 7853 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-nn 12210 df-n0 12470 df-coe1 21699 |
This theorem is referenced by: fvcoe1 21723 coe1mul2 21783 deg1le0 25621 |
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