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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 21950 | . . 3 ⊢ (𝐹 ∈ 𝑉 → 𝐴 = (𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛})))) |
3 | 2 | fveq1d 6894 | . 2 ⊢ (𝐹 ∈ 𝑉 → (𝐴‘𝑁) = ((𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛})))‘𝑁)) |
4 | sneq 4639 | . . . . 5 ⊢ (𝑛 = 𝑁 → {𝑛} = {𝑁}) | |
5 | 4 | xpeq2d 5707 | . . . 4 ⊢ (𝑛 = 𝑁 → (1o × {𝑛}) = (1o × {𝑁})) |
6 | 5 | fveq2d 6896 | . . 3 ⊢ (𝑛 = 𝑁 → (𝐹‘(1o × {𝑛})) = (𝐹‘(1o × {𝑁}))) |
7 | eqid 2730 | . . 3 ⊢ (𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛}))) = (𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛}))) | |
8 | fvex 6905 | . . 3 ⊢ (𝐹‘(1o × {𝑁})) ∈ V | |
9 | 6, 7, 8 | fvmpt 6999 | . 2 ⊢ (𝑁 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛})))‘𝑁) = (𝐹‘(1o × {𝑁}))) |
10 | 3, 9 | sylan9eq 2790 | 1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝐴‘𝑁) = (𝐹‘(1o × {𝑁}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1539 ∈ wcel 2104 {csn 4629 ↦ cmpt 5232 × cxp 5675 ‘cfv 6544 1oc1o 8463 ℕ0cn0 12478 coe1cco1 21923 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7729 ax-cnex 11170 ax-1cn 11172 ax-addcl 11174 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7416 df-om 7860 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-nn 12219 df-n0 12479 df-coe1 21928 |
This theorem is referenced by: fvcoe1 21952 coe1mul2 22013 deg1le0 25863 |
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