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Theorem plymulx0 33159
Description: Coefficients of a polynomial multiplied by Xp. (Contributed by Thierry Arnoux, 25-Sep-2018.)
Assertion
Ref Expression
plymulx0 (𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) β†’ (coeffβ€˜(𝐹 ∘f Β· Xp)) = (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, 0, ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 1)))))
Distinct variable group:   𝑛,𝐹

Proof of Theorem plymulx0
Dummy variables 𝑖 π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldifi 4086 . . . . 5 (𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) β†’ 𝐹 ∈ (Polyβ€˜β„))
2 ax-resscn 11108 . . . . . . 7 ℝ βŠ† β„‚
3 1re 11155 . . . . . . 7 1 ∈ ℝ
4 plyid 25570 . . . . . . 7 ((ℝ βŠ† β„‚ ∧ 1 ∈ ℝ) β†’ Xp ∈ (Polyβ€˜β„))
52, 3, 4mp2an 690 . . . . . 6 Xp ∈ (Polyβ€˜β„)
65a1i 11 . . . . 5 (𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) β†’ Xp ∈ (Polyβ€˜β„))
7 simprl 769 . . . . . 6 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ (π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ)) β†’ π‘₯ ∈ ℝ)
8 simprr 771 . . . . . 6 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ (π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ)) β†’ 𝑦 ∈ ℝ)
97, 8readdcld 11184 . . . . 5 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ (π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ)) β†’ (π‘₯ + 𝑦) ∈ ℝ)
107, 8remulcld 11185 . . . . 5 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ (π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ)) β†’ (π‘₯ Β· 𝑦) ∈ ℝ)
111, 6, 9, 10plymul 25579 . . . 4 (𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) β†’ (𝐹 ∘f Β· Xp) ∈ (Polyβ€˜β„))
12 0re 11157 . . . 4 0 ∈ ℝ
13 eqid 2736 . . . . 5 (coeffβ€˜(𝐹 ∘f Β· Xp)) = (coeffβ€˜(𝐹 ∘f Β· Xp))
1413coef2 25592 . . . 4 (((𝐹 ∘f Β· Xp) ∈ (Polyβ€˜β„) ∧ 0 ∈ ℝ) β†’ (coeffβ€˜(𝐹 ∘f Β· Xp)):β„•0βŸΆβ„)
1511, 12, 14sylancl 586 . . 3 (𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) β†’ (coeffβ€˜(𝐹 ∘f Β· Xp)):β„•0βŸΆβ„)
1615feqmptd 6910 . 2 (𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) β†’ (coeffβ€˜(𝐹 ∘f Β· Xp)) = (𝑛 ∈ β„•0 ↦ ((coeffβ€˜(𝐹 ∘f Β· Xp))β€˜π‘›)))
17 cnex 11132 . . . . . . . . 9 β„‚ ∈ V
1817a1i 11 . . . . . . . 8 (𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) β†’ β„‚ ∈ V)
19 plyf 25559 . . . . . . . . 9 (𝐹 ∈ (Polyβ€˜β„) β†’ 𝐹:β„‚βŸΆβ„‚)
201, 19syl 17 . . . . . . . 8 (𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) β†’ 𝐹:β„‚βŸΆβ„‚)
21 plyf 25559 . . . . . . . . . 10 (Xp ∈ (Polyβ€˜β„) β†’ Xp:β„‚βŸΆβ„‚)
225, 21ax-mp 5 . . . . . . . . 9 Xp:β„‚βŸΆβ„‚
2322a1i 11 . . . . . . . 8 (𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) β†’ Xp:β„‚βŸΆβ„‚)
24 simprl 769 . . . . . . . . 9 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ (π‘₯ ∈ β„‚ ∧ 𝑦 ∈ β„‚)) β†’ π‘₯ ∈ β„‚)
25 simprr 771 . . . . . . . . 9 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ (π‘₯ ∈ β„‚ ∧ 𝑦 ∈ β„‚)) β†’ 𝑦 ∈ β„‚)
2624, 25mulcomd 11176 . . . . . . . 8 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ (π‘₯ ∈ β„‚ ∧ 𝑦 ∈ β„‚)) β†’ (π‘₯ Β· 𝑦) = (𝑦 Β· π‘₯))
2718, 20, 23, 26caofcom 7652 . . . . . . 7 (𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) β†’ (𝐹 ∘f Β· Xp) = (Xp ∘f Β· 𝐹))
2827fveq2d 6846 . . . . . 6 (𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) β†’ (coeffβ€˜(𝐹 ∘f Β· Xp)) = (coeffβ€˜(Xp ∘f Β· 𝐹)))
2928fveq1d 6844 . . . . 5 (𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) β†’ ((coeffβ€˜(𝐹 ∘f Β· Xp))β€˜π‘›) = ((coeffβ€˜(Xp ∘f Β· 𝐹))β€˜π‘›))
3029adantr 481 . . . 4 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) β†’ ((coeffβ€˜(𝐹 ∘f Β· Xp))β€˜π‘›) = ((coeffβ€˜(Xp ∘f Β· 𝐹))β€˜π‘›))
315a1i 11 . . . . . 6 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) β†’ Xp ∈ (Polyβ€˜β„))
321adantr 481 . . . . . 6 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) β†’ 𝐹 ∈ (Polyβ€˜β„))
33 simpr 485 . . . . . 6 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) β†’ 𝑛 ∈ β„•0)
34 eqid 2736 . . . . . . 7 (coeffβ€˜Xp) = (coeffβ€˜Xp)
35 eqid 2736 . . . . . . 7 (coeffβ€˜πΉ) = (coeffβ€˜πΉ)
3634, 35coemul 25613 . . . . . 6 ((Xp ∈ (Polyβ€˜β„) ∧ 𝐹 ∈ (Polyβ€˜β„) ∧ 𝑛 ∈ β„•0) β†’ ((coeffβ€˜(Xp ∘f Β· 𝐹))β€˜π‘›) = Σ𝑖 ∈ (0...𝑛)(((coeffβ€˜Xp)β€˜π‘–) Β· ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖))))
3731, 32, 33, 36syl3anc 1371 . . . . 5 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) β†’ ((coeffβ€˜(Xp ∘f Β· 𝐹))β€˜π‘›) = Σ𝑖 ∈ (0...𝑛)(((coeffβ€˜Xp)β€˜π‘–) Β· ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖))))
38 elfznn0 13534 . . . . . . . . . 10 (𝑖 ∈ (0...𝑛) β†’ 𝑖 ∈ β„•0)
39 coeidp 25624 . . . . . . . . . 10 (𝑖 ∈ β„•0 β†’ ((coeffβ€˜Xp)β€˜π‘–) = if(𝑖 = 1, 1, 0))
4038, 39syl 17 . . . . . . . . 9 (𝑖 ∈ (0...𝑛) β†’ ((coeffβ€˜Xp)β€˜π‘–) = if(𝑖 = 1, 1, 0))
4140oveq1d 7372 . . . . . . . 8 (𝑖 ∈ (0...𝑛) β†’ (((coeffβ€˜Xp)β€˜π‘–) Β· ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖))) = (if(𝑖 = 1, 1, 0) Β· ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖))))
42 ovif 7454 . . . . . . . 8 (if(𝑖 = 1, 1, 0) Β· ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖))) = if(𝑖 = 1, (1 Β· ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖))), (0 Β· ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖))))
4341, 42eqtrdi 2792 . . . . . . 7 (𝑖 ∈ (0...𝑛) β†’ (((coeffβ€˜Xp)β€˜π‘–) Β· ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖))) = if(𝑖 = 1, (1 Β· ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖))), (0 Β· ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)))))
4443adantl 482 . . . . . 6 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) ∧ 𝑖 ∈ (0...𝑛)) β†’ (((coeffβ€˜Xp)β€˜π‘–) Β· ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖))) = if(𝑖 = 1, (1 Β· ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖))), (0 Β· ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)))))
4544sumeq2dv 15588 . . . . 5 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) β†’ Σ𝑖 ∈ (0...𝑛)(((coeffβ€˜Xp)β€˜π‘–) Β· ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖))) = Σ𝑖 ∈ (0...𝑛)if(𝑖 = 1, (1 Β· ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖))), (0 Β· ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)))))
46 velsn 4602 . . . . . . . . . 10 (𝑖 ∈ {1} ↔ 𝑖 = 1)
4746bicomi 223 . . . . . . . . 9 (𝑖 = 1 ↔ 𝑖 ∈ {1})
4847a1i 11 . . . . . . . 8 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) ∧ 𝑖 ∈ (0...𝑛)) β†’ (𝑖 = 1 ↔ 𝑖 ∈ {1}))
4935coef2 25592 . . . . . . . . . . . . 13 ((𝐹 ∈ (Polyβ€˜β„) ∧ 0 ∈ ℝ) β†’ (coeffβ€˜πΉ):β„•0βŸΆβ„)
501, 12, 49sylancl 586 . . . . . . . . . . . 12 (𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) β†’ (coeffβ€˜πΉ):β„•0βŸΆβ„)
5150ad2antrr 724 . . . . . . . . . . 11 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) ∧ 𝑖 ∈ (0...𝑛)) β†’ (coeffβ€˜πΉ):β„•0βŸΆβ„)
52 fznn0sub 13473 . . . . . . . . . . . 12 (𝑖 ∈ (0...𝑛) β†’ (𝑛 βˆ’ 𝑖) ∈ β„•0)
5352adantl 482 . . . . . . . . . . 11 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) ∧ 𝑖 ∈ (0...𝑛)) β†’ (𝑛 βˆ’ 𝑖) ∈ β„•0)
5451, 53ffvelcdmd 7036 . . . . . . . . . 10 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) ∧ 𝑖 ∈ (0...𝑛)) β†’ ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)) ∈ ℝ)
5554recnd 11183 . . . . . . . . 9 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) ∧ 𝑖 ∈ (0...𝑛)) β†’ ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)) ∈ β„‚)
5655mulid2d 11173 . . . . . . . 8 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) ∧ 𝑖 ∈ (0...𝑛)) β†’ (1 Β· ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖))) = ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)))
5755mul02d 11353 . . . . . . . 8 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) ∧ 𝑖 ∈ (0...𝑛)) β†’ (0 Β· ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖))) = 0)
5848, 56, 57ifbieq12d 4514 . . . . . . 7 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) ∧ 𝑖 ∈ (0...𝑛)) β†’ if(𝑖 = 1, (1 Β· ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖))), (0 Β· ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)))) = if(𝑖 ∈ {1}, ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)), 0))
5958sumeq2dv 15588 . . . . . 6 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) β†’ Σ𝑖 ∈ (0...𝑛)if(𝑖 = 1, (1 Β· ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖))), (0 Β· ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)))) = Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)), 0))
60 eqeq2 2748 . . . . . . 7 (0 = if(𝑛 = 0, 0, ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 1))) β†’ (Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)), 0) = 0 ↔ Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)), 0) = if(𝑛 = 0, 0, ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 1)))))
61 eqeq2 2748 . . . . . . 7 (((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 1)) = if(𝑛 = 0, 0, ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 1))) β†’ (Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)), 0) = ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 1)) ↔ Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)), 0) = if(𝑛 = 0, 0, ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 1)))))
62 oveq2 7365 . . . . . . . . . . 11 (𝑛 = 0 β†’ (0...𝑛) = (0...0))
63 0z 12510 . . . . . . . . . . . 12 0 ∈ β„€
64 fzsn 13483 . . . . . . . . . . . 12 (0 ∈ β„€ β†’ (0...0) = {0})
6563, 64ax-mp 5 . . . . . . . . . . 11 (0...0) = {0}
6662, 65eqtrdi 2792 . . . . . . . . . 10 (𝑛 = 0 β†’ (0...𝑛) = {0})
67 elsni 4603 . . . . . . . . . . . . 13 (𝑖 ∈ {0} β†’ 𝑖 = 0)
6867adantl 482 . . . . . . . . . . . 12 ((𝑛 = 0 ∧ 𝑖 ∈ {0}) β†’ 𝑖 = 0)
69 ax-1ne0 11120 . . . . . . . . . . . . . 14 1 β‰  0
7069nesymi 3001 . . . . . . . . . . . . 13 Β¬ 0 = 1
71 eqeq1 2740 . . . . . . . . . . . . 13 (𝑖 = 0 β†’ (𝑖 = 1 ↔ 0 = 1))
7270, 71mtbiri 326 . . . . . . . . . . . 12 (𝑖 = 0 β†’ Β¬ 𝑖 = 1)
7368, 72syl 17 . . . . . . . . . . 11 ((𝑛 = 0 ∧ 𝑖 ∈ {0}) β†’ Β¬ 𝑖 = 1)
7447notbii 319 . . . . . . . . . . . 12 (Β¬ 𝑖 = 1 ↔ Β¬ 𝑖 ∈ {1})
7574biimpi 215 . . . . . . . . . . 11 (Β¬ 𝑖 = 1 β†’ Β¬ 𝑖 ∈ {1})
76 iffalse 4495 . . . . . . . . . . 11 (Β¬ 𝑖 ∈ {1} β†’ if(𝑖 ∈ {1}, ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)), 0) = 0)
7773, 75, 763syl 18 . . . . . . . . . 10 ((𝑛 = 0 ∧ 𝑖 ∈ {0}) β†’ if(𝑖 ∈ {1}, ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)), 0) = 0)
7866, 77sumeq12rdv 15592 . . . . . . . . 9 (𝑛 = 0 β†’ Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)), 0) = Σ𝑖 ∈ {0}0)
79 snfi 8988 . . . . . . . . . . 11 {0} ∈ Fin
8079olci 864 . . . . . . . . . 10 ({0} βŠ† (β„€β‰₯β€˜0) ∨ {0} ∈ Fin)
81 sumz 15607 . . . . . . . . . 10 (({0} βŠ† (β„€β‰₯β€˜0) ∨ {0} ∈ Fin) β†’ Σ𝑖 ∈ {0}0 = 0)
8280, 81ax-mp 5 . . . . . . . . 9 Σ𝑖 ∈ {0}0 = 0
8378, 82eqtrdi 2792 . . . . . . . 8 (𝑛 = 0 β†’ Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)), 0) = 0)
8483adantl 482 . . . . . . 7 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) ∧ 𝑛 = 0) β†’ Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)), 0) = 0)
85 simpll 765 . . . . . . . 8 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) ∧ Β¬ 𝑛 = 0) β†’ 𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}))
8633adantr 481 . . . . . . . . 9 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) ∧ Β¬ 𝑛 = 0) β†’ 𝑛 ∈ β„•0)
87 simpr 485 . . . . . . . . . 10 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) ∧ Β¬ 𝑛 = 0) β†’ Β¬ 𝑛 = 0)
8887neqned 2950 . . . . . . . . 9 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) ∧ Β¬ 𝑛 = 0) β†’ 𝑛 β‰  0)
89 elnnne0 12427 . . . . . . . . 9 (𝑛 ∈ β„• ↔ (𝑛 ∈ β„•0 ∧ 𝑛 β‰  0))
9086, 88, 89sylanbrc 583 . . . . . . . 8 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) ∧ Β¬ 𝑛 = 0) β†’ 𝑛 ∈ β„•)
91 1nn0 12429 . . . . . . . . . . . . 13 1 ∈ β„•0
9291a1i 11 . . . . . . . . . . . 12 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) β†’ 1 ∈ β„•0)
93 simpr 485 . . . . . . . . . . . . 13 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) β†’ 𝑛 ∈ β„•)
9493nnnn0d 12473 . . . . . . . . . . . 12 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) β†’ 𝑛 ∈ β„•0)
9593nnge1d 12201 . . . . . . . . . . . 12 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) β†’ 1 ≀ 𝑛)
96 elfz2nn0 13532 . . . . . . . . . . . 12 (1 ∈ (0...𝑛) ↔ (1 ∈ β„•0 ∧ 𝑛 ∈ β„•0 ∧ 1 ≀ 𝑛))
9792, 94, 95, 96syl3anbrc 1343 . . . . . . . . . . 11 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) β†’ 1 ∈ (0...𝑛))
9897snssd 4769 . . . . . . . . . 10 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) β†’ {1} βŠ† (0...𝑛))
9950ad2antrr 724 . . . . . . . . . . . . 13 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) ∧ 𝑖 ∈ {1}) β†’ (coeffβ€˜πΉ):β„•0βŸΆβ„)
100 oveq2 7365 . . . . . . . . . . . . . . . 16 (𝑖 = 1 β†’ (𝑛 βˆ’ 𝑖) = (𝑛 βˆ’ 1))
10146, 100sylbi 216 . . . . . . . . . . . . . . 15 (𝑖 ∈ {1} β†’ (𝑛 βˆ’ 𝑖) = (𝑛 βˆ’ 1))
102101adantl 482 . . . . . . . . . . . . . 14 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) ∧ 𝑖 ∈ {1}) β†’ (𝑛 βˆ’ 𝑖) = (𝑛 βˆ’ 1))
103 nnm1nn0 12454 . . . . . . . . . . . . . . 15 (𝑛 ∈ β„• β†’ (𝑛 βˆ’ 1) ∈ β„•0)
104103ad2antlr 725 . . . . . . . . . . . . . 14 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) ∧ 𝑖 ∈ {1}) β†’ (𝑛 βˆ’ 1) ∈ β„•0)
105102, 104eqeltrd 2837 . . . . . . . . . . . . 13 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) ∧ 𝑖 ∈ {1}) β†’ (𝑛 βˆ’ 𝑖) ∈ β„•0)
10699, 105ffvelcdmd 7036 . . . . . . . . . . . 12 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) ∧ 𝑖 ∈ {1}) β†’ ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)) ∈ ℝ)
107106recnd 11183 . . . . . . . . . . 11 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) ∧ 𝑖 ∈ {1}) β†’ ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)) ∈ β„‚)
108107ralrimiva 3143 . . . . . . . . . 10 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) β†’ βˆ€π‘– ∈ {1} ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)) ∈ β„‚)
109 fzfi 13877 . . . . . . . . . . . 12 (0...𝑛) ∈ Fin
110109olci 864 . . . . . . . . . . 11 ((0...𝑛) βŠ† (β„€β‰₯β€˜0) ∨ (0...𝑛) ∈ Fin)
111110a1i 11 . . . . . . . . . 10 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) β†’ ((0...𝑛) βŠ† (β„€β‰₯β€˜0) ∨ (0...𝑛) ∈ Fin))
112 sumss2 15611 . . . . . . . . . 10 ((({1} βŠ† (0...𝑛) ∧ βˆ€π‘– ∈ {1} ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)) ∈ β„‚) ∧ ((0...𝑛) βŠ† (β„€β‰₯β€˜0) ∨ (0...𝑛) ∈ Fin)) β†’ Σ𝑖 ∈ {1} ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)) = Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)), 0))
11398, 108, 111, 112syl21anc 836 . . . . . . . . 9 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) β†’ Σ𝑖 ∈ {1} ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)) = Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)), 0))
11450adantr 481 . . . . . . . . . . . 12 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) β†’ (coeffβ€˜πΉ):β„•0βŸΆβ„)
115103adantl 482 . . . . . . . . . . . 12 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) β†’ (𝑛 βˆ’ 1) ∈ β„•0)
116114, 115ffvelcdmd 7036 . . . . . . . . . . 11 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) β†’ ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 1)) ∈ ℝ)
117116recnd 11183 . . . . . . . . . 10 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) β†’ ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 1)) ∈ β„‚)
118100fveq2d 6846 . . . . . . . . . . 11 (𝑖 = 1 β†’ ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)) = ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 1)))
119118sumsn 15631 . . . . . . . . . 10 ((1 ∈ ℝ ∧ ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 1)) ∈ β„‚) β†’ Σ𝑖 ∈ {1} ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)) = ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 1)))
1203, 117, 119sylancr 587 . . . . . . . . 9 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) β†’ Σ𝑖 ∈ {1} ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)) = ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 1)))
121113, 120eqtr3d 2778 . . . . . . . 8 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) β†’ Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)), 0) = ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 1)))
12285, 90, 121syl2anc 584 . . . . . . 7 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) ∧ Β¬ 𝑛 = 0) β†’ Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)), 0) = ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 1)))
12360, 61, 84, 122ifbothda 4524 . . . . . 6 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) β†’ Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)), 0) = if(𝑛 = 0, 0, ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 1))))
12459, 123eqtrd 2776 . . . . 5 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) β†’ Σ𝑖 ∈ (0...𝑛)if(𝑖 = 1, (1 Β· ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖))), (0 Β· ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)))) = if(𝑛 = 0, 0, ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 1))))
12537, 45, 1243eqtrd 2780 . . . 4 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) β†’ ((coeffβ€˜(Xp ∘f Β· 𝐹))β€˜π‘›) = if(𝑛 = 0, 0, ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 1))))
12630, 125eqtrd 2776 . . 3 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) β†’ ((coeffβ€˜(𝐹 ∘f Β· Xp))β€˜π‘›) = if(𝑛 = 0, 0, ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 1))))
127126mpteq2dva 5205 . 2 (𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) β†’ (𝑛 ∈ β„•0 ↦ ((coeffβ€˜(𝐹 ∘f Β· Xp))β€˜π‘›)) = (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, 0, ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 1)))))
12816, 127eqtrd 2776 1 (𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) β†’ (coeffβ€˜(𝐹 ∘f Β· Xp)) = (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, 0, ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 1)))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   = wceq 1541   ∈ wcel 2106   β‰  wne 2943  βˆ€wral 3064  Vcvv 3445   βˆ– cdif 3907   βŠ† wss 3910  ifcif 4486  {csn 4586   class class class wbr 5105   ↦ cmpt 5188  βŸΆwf 6492  β€˜cfv 6496  (class class class)co 7357   ∘f cof 7615  Fincfn 8883  β„‚cc 11049  β„cr 11050  0cc0 11051  1c1 11052   Β· cmul 11056   ≀ cle 11190   βˆ’ cmin 11385  β„•cn 12153  β„•0cn0 12413  β„€cz 12499  β„€β‰₯cuz 12763  ...cfz 13424  Ξ£csu 15570  0𝑝c0p 25033  Polycply 25545  Xpcidp 25546  coeffccoe 25547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-fz 13425  df-fzo 13568  df-fl 13697  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-rlim 15371  df-sum 15571  df-0p 25034  df-ply 25549  df-idp 25550  df-coe 25551  df-dgr 25552
This theorem is referenced by:  plymulx  33160
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