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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 19623 | . . 3 ⊢ (𝐹 ∈ 𝑉 → 𝐴 = (𝑛 ∈ ℕ0 ↦ (𝐹‘(1𝑜 × {𝑛})))) |
3 | 2 | fveq1d 6231 | . 2 ⊢ (𝐹 ∈ 𝑉 → (𝐴‘𝑁) = ((𝑛 ∈ ℕ0 ↦ (𝐹‘(1𝑜 × {𝑛})))‘𝑁)) |
4 | sneq 4220 | . . . . 5 ⊢ (𝑛 = 𝑁 → {𝑛} = {𝑁}) | |
5 | 4 | xpeq2d 5173 | . . . 4 ⊢ (𝑛 = 𝑁 → (1𝑜 × {𝑛}) = (1𝑜 × {𝑁})) |
6 | 5 | fveq2d 6233 | . . 3 ⊢ (𝑛 = 𝑁 → (𝐹‘(1𝑜 × {𝑛})) = (𝐹‘(1𝑜 × {𝑁}))) |
7 | eqid 2651 | . . 3 ⊢ (𝑛 ∈ ℕ0 ↦ (𝐹‘(1𝑜 × {𝑛}))) = (𝑛 ∈ ℕ0 ↦ (𝐹‘(1𝑜 × {𝑛}))) | |
8 | fvex 6239 | . . 3 ⊢ (𝐹‘(1𝑜 × {𝑁})) ∈ V | |
9 | 6, 7, 8 | fvmpt 6321 | . 2 ⊢ (𝑁 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ (𝐹‘(1𝑜 × {𝑛})))‘𝑁) = (𝐹‘(1𝑜 × {𝑁}))) |
10 | 3, 9 | sylan9eq 2705 | 1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝐴‘𝑁) = (𝐹‘(1𝑜 × {𝑁}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 {csn 4210 ↦ cmpt 4762 × cxp 5141 ‘cfv 5926 1𝑜c1o 7598 ℕ0cn0 11330 coe1cco1 19596 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-i2m1 10042 ax-1ne0 10043 ax-rrecex 10046 ax-cnre 10047 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-ov 6693 df-om 7108 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-nn 11059 df-n0 11331 df-coe1 19601 |
This theorem is referenced by: fvcoe1 19625 coe1mul2 19687 deg1le0 23916 |
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