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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  plymulx0 Structured version   Visualization version   GIF version
Theorem plymulx0 33856
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 4125 . . . . 5 (𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) β†’ 𝐹 ∈ (Polyβ€˜β„))
2 ax-resscn 11169 . . . . . . 7 ℝ βŠ† β„‚
3 1re 11218 . . . . . . 7 1 ∈ ℝ
4 plyid 25958 . . . . . . 7 ((ℝ βŠ† β„‚ ∧ 1 ∈ ℝ) β†’ Xp ∈ (Polyβ€˜β„))
52, 3, 4mp2an 688 . . . . . 6 Xp ∈ (Polyβ€˜β„)
65a1i 11 . . . . 5 (𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) β†’ Xp ∈ (Polyβ€˜β„))
7 simprl 767 . . . . . 6 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ (π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ)) β†’ π‘₯ ∈ ℝ)
8 simprr 769 . . . . . 6 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ (π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ)) β†’ 𝑦 ∈ ℝ)
97, 8readdcld 11247 . . . . 5 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ (π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ)) β†’ (π‘₯ + 𝑦) ∈ ℝ)
107, 8remulcld 11248 . . . . 5 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ (π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ)) β†’ (π‘₯ Β· 𝑦) ∈ ℝ)
111, 6, 9, 10plymul 25967 . . . 4 (𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) β†’ (𝐹 ∘f Β· Xp) ∈ (Polyβ€˜β„))
12 0re 11220 . . . 4 0 ∈ ℝ
13 eqid 2730 . . . . 5 (coeffβ€˜(𝐹 ∘f Β· Xp)) = (coeffβ€˜(𝐹 ∘f Β· Xp))
1413coef2 25980 . . . 4 (((𝐹 ∘f Β· Xp) ∈ (Polyβ€˜β„) ∧ 0 ∈ ℝ) β†’ (coeffβ€˜(𝐹 ∘f Β· Xp)):β„•0βŸΆβ„)
1511, 12, 14sylancl 584 . . 3 (𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) β†’ (coeffβ€˜(𝐹 ∘f Β· Xp)):β„•0βŸΆβ„)
1615feqmptd 6959 . 2 (𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) β†’ (coeffβ€˜(𝐹 ∘f Β· Xp)) = (𝑛 ∈ β„•0 ↦ ((coeffβ€˜(𝐹 ∘f Β· Xp))β€˜π‘›)))
17 cnex 11193 . . . . . . . . 9 β„‚ ∈ V
1817a1i 11 . . . . . . . 8 (𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) β†’ β„‚ ∈ V)
19 plyf 25947 . . . . . . . . 9 (𝐹 ∈ (Polyβ€˜β„) β†’ 𝐹:β„‚βŸΆβ„‚)
201, 19syl 17 . . . . . . . 8 (𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) β†’ 𝐹:β„‚βŸΆβ„‚)
21 plyf 25947 . . . . . . . . . 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 767 . . . . . . . . 9 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ (π‘₯ ∈ β„‚ ∧ 𝑦 ∈ β„‚)) β†’ π‘₯ ∈ β„‚)
25 simprr 769 . . . . . . . . 9 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ (π‘₯ ∈ β„‚ ∧ 𝑦 ∈ β„‚)) β†’ 𝑦 ∈ β„‚)
2624, 25mulcomd 11239 . . . . . . . 8 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ (π‘₯ ∈ β„‚ ∧ 𝑦 ∈ β„‚)) β†’ (π‘₯ Β· 𝑦) = (𝑦 Β· π‘₯))
2718, 20, 23, 26caofcom 7707 . . . . . . 7 (𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) β†’ (𝐹 ∘f Β· Xp) = (Xp ∘f Β· 𝐹))
2827fveq2d 6894 . . . . . 6 (𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) β†’ (coeffβ€˜(𝐹 ∘f Β· Xp)) = (coeffβ€˜(Xp ∘f Β· 𝐹)))
2928fveq1d 6892 . . . . 5 (𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) β†’ ((coeffβ€˜(𝐹 ∘f Β· Xp))β€˜π‘›) = ((coeffβ€˜(Xp ∘f Β· 𝐹))β€˜π‘›))
3029adantr 479 . . . 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 479 . . . . . 6 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) β†’ 𝐹 ∈ (Polyβ€˜β„))
33 simpr 483 . . . . . 6 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) β†’ 𝑛 ∈ β„•0)
34 eqid 2730 . . . . . . 7 (coeffβ€˜Xp) = (coeffβ€˜Xp)
35 eqid 2730 . . . . . . 7 (coeffβ€˜πΉ) = (coeffβ€˜πΉ)
3634, 35coemul 26001 . . . . . 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 1369 . . . . 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 13598 . . . . . . . . . 10 (𝑖 ∈ (0...𝑛) β†’ 𝑖 ∈ β„•0)
39 coeidp 26013 . . . . . . . . . 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 7426 . . . . . . . 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 7508 . . . . . . . 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 2786 . . . . . . 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 480 . . . . . 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 15653 . . . . 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 4643 . . . . . . . . . 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 25980 . . . . . . . . . . . . 13 ((𝐹 ∈ (Polyβ€˜β„) ∧ 0 ∈ ℝ) β†’ (coeffβ€˜πΉ):β„•0βŸΆβ„)
501, 12, 49sylancl 584 . . . . . . . . . . . 12 (𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) β†’ (coeffβ€˜πΉ):β„•0βŸΆβ„)
5150ad2antrr 722 . . . . . . . . . . 11 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) ∧ 𝑖 ∈ (0...𝑛)) β†’ (coeffβ€˜πΉ):β„•0βŸΆβ„)
52 fznn0sub 13537 . . . . . . . . . . . 12 (𝑖 ∈ (0...𝑛) β†’ (𝑛 βˆ’ 𝑖) ∈ β„•0)
5352adantl 480 . . . . . . . . . . 11 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) ∧ 𝑖 ∈ (0...𝑛)) β†’ (𝑛 βˆ’ 𝑖) ∈ β„•0)
5451, 53ffvelcdmd 7086 . . . . . . . . . 10 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) ∧ 𝑖 ∈ (0...𝑛)) β†’ ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)) ∈ ℝ)
5554recnd 11246 . . . . . . . . 9 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) ∧ 𝑖 ∈ (0...𝑛)) β†’ ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)) ∈ β„‚)
5655mullidd 11236 . . . . . . . 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 11416 . . . . . . . 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 4555 . . . . . . 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 15653 . . . . . 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 2742 . . . . . . 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 2742 . . . . . . 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 7419 . . . . . . . . . . 11 (𝑛 = 0 β†’ (0...𝑛) = (0...0))
63 0z 12573 . . . . . . . . . . . 12 0 ∈ β„€
64 fzsn 13547 . . . . . . . . . . . 12 (0 ∈ β„€ β†’ (0...0) = {0})
6563, 64ax-mp 5 . . . . . . . . . . 11 (0...0) = {0}
6662, 65eqtrdi 2786 . . . . . . . . . 10 (𝑛 = 0 β†’ (0...𝑛) = {0})
67 elsni 4644 . . . . . . . . . . . . 13 (𝑖 ∈ {0} β†’ 𝑖 = 0)
6867adantl 480 . . . . . . . . . . . 12 ((𝑛 = 0 ∧ 𝑖 ∈ {0}) β†’ 𝑖 = 0)
69 ax-1ne0 11181 . . . . . . . . . . . . . 14 1 β‰  0
7069nesymi 2996 . . . . . . . . . . . . 13 Β¬ 0 = 1
71 eqeq1 2734 . . . . . . . . . . . . 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 4536 . . . . . . . . . . 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 15657 . . . . . . . . 9 (𝑛 = 0 β†’ Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)), 0) = Σ𝑖 ∈ {0}0)
79 snfi 9046 . . . . . . . . . . 11 {0} ∈ Fin
8079olci 862 . . . . . . . . . 10 ({0} βŠ† (β„€β‰₯β€˜0) ∨ {0} ∈ Fin)
81 sumz 15672 . . . . . . . . . 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 2786 . . . . . . . 8 (𝑛 = 0 β†’ Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)), 0) = 0)
8483adantl 480 . . . . . . 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 763 . . . . . . . 8 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) ∧ Β¬ 𝑛 = 0) β†’ 𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}))
8633adantr 479 . . . . . . . . 9 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) ∧ Β¬ 𝑛 = 0) β†’ 𝑛 ∈ β„•0)
87 simpr 483 . . . . . . . . . 10 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) ∧ Β¬ 𝑛 = 0) β†’ Β¬ 𝑛 = 0)
8887neqned 2945 . . . . . . . . 9 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) ∧ Β¬ 𝑛 = 0) β†’ 𝑛 β‰  0)
89 elnnne0 12490 . . . . . . . . 9 (𝑛 ∈ β„• ↔ (𝑛 ∈ β„•0 ∧ 𝑛 β‰  0))
9086, 88, 89sylanbrc 581 . . . . . . . 8 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) ∧ Β¬ 𝑛 = 0) β†’ 𝑛 ∈ β„•)
91 1nn0 12492 . . . . . . . . . . . . 13 1 ∈ β„•0
9291a1i 11 . . . . . . . . . . . 12 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) β†’ 1 ∈ β„•0)
93 simpr 483 . . . . . . . . . . . . 13 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) β†’ 𝑛 ∈ β„•)
9493nnnn0d 12536 . . . . . . . . . . . 12 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) β†’ 𝑛 ∈ β„•0)
9593nnge1d 12264 . . . . . . . . . . . 12 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) β†’ 1 ≀ 𝑛)
96 elfz2nn0 13596 . . . . . . . . . . . 12 (1 ∈ (0...𝑛) ↔ (1 ∈ β„•0 ∧ 𝑛 ∈ β„•0 ∧ 1 ≀ 𝑛))
9792, 94, 95, 96syl3anbrc 1341 . . . . . . . . . . 11 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) β†’ 1 ∈ (0...𝑛))
9897snssd 4811 . . . . . . . . . 10 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) β†’ {1} βŠ† (0...𝑛))
9950ad2antrr 722 . . . . . . . . . . . . 13 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) ∧ 𝑖 ∈ {1}) β†’ (coeffβ€˜πΉ):β„•0βŸΆβ„)
100 oveq2 7419 . . . . . . . . . . . . . . . 16 (𝑖 = 1 β†’ (𝑛 βˆ’ 𝑖) = (𝑛 βˆ’ 1))
10146, 100sylbi 216 . . . . . . . . . . . . . . 15 (𝑖 ∈ {1} β†’ (𝑛 βˆ’ 𝑖) = (𝑛 βˆ’ 1))
102101adantl 480 . . . . . . . . . . . . . 14 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) ∧ 𝑖 ∈ {1}) β†’ (𝑛 βˆ’ 𝑖) = (𝑛 βˆ’ 1))
103 nnm1nn0 12517 . . . . . . . . . . . . . . 15 (𝑛 ∈ β„• β†’ (𝑛 βˆ’ 1) ∈ β„•0)
104103ad2antlr 723 . . . . . . . . . . . . . 14 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) ∧ 𝑖 ∈ {1}) β†’ (𝑛 βˆ’ 1) ∈ β„•0)
105102, 104eqeltrd 2831 . . . . . . . . . . . . 13 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) ∧ 𝑖 ∈ {1}) β†’ (𝑛 βˆ’ 𝑖) ∈ β„•0)
10699, 105ffvelcdmd 7086 . . . . . . . . . . . 12 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) ∧ 𝑖 ∈ {1}) β†’ ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)) ∈ ℝ)
107106recnd 11246 . . . . . . . . . . 11 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) ∧ 𝑖 ∈ {1}) β†’ ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)) ∈ β„‚)
108107ralrimiva 3144 . . . . . . . . . 10 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) β†’ βˆ€π‘– ∈ {1} ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)) ∈ β„‚)
109 fzfi 13941 . . . . . . . . . . . 12 (0...𝑛) ∈ Fin
110109olci 862 . . . . . . . . . . 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 15676 . . . . . . . . . 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 834 . . . . . . . . 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 479 . . . . . . . . . . . 12 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) β†’ (coeffβ€˜πΉ):β„•0βŸΆβ„)
115103adantl 480 . . . . . . . . . . . 12 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) β†’ (𝑛 βˆ’ 1) ∈ β„•0)
116114, 115ffvelcdmd 7086 . . . . . . . . . . 11 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) β†’ ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 1)) ∈ ℝ)
117116recnd 11246 . . . . . . . . . 10 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) β†’ ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 1)) ∈ β„‚)
118100fveq2d 6894 . . . . . . . . . . 11 (𝑖 = 1 β†’ ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)) = ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 1)))
119118sumsn 15696 . . . . . . . . . 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 585 . . . . . . . . 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 2772 . . . . . . . 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 582 . . . . . . 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 4565 . . . . . 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 2770 . . . . 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 2774 . . . 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 2770 . . 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 5247 . 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 2770 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 394   ∨ wo 843   = wceq 1539   ∈ wcel 2104   β‰  wne 2938  βˆ€wral 3059  Vcvv 3472   βˆ– cdif 3944   βŠ† wss 3947  ifcif 4527  {csn 4627   class class class wbr 5147   ↦ cmpt 5230  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7411   ∘f cof 7670  Fincfn 8941  β„‚cc 11110  β„cr 11111  0cc0 11112  1c1 11113   Β· cmul 11117   ≀ cle 11253   βˆ’ cmin 11448  β„•cn 12216  β„•0cn0 12476  β„€cz 12562  β„€β‰₯cuz 12826  ...cfz 13488  Ξ£csu 15636  0𝑝c0p 25418  Polycply 25933  Xpcidp 25934  coeffccoe 25935
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 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-inf2 9638  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190
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-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7672  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-map 8824  df-pm 8825  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-sup 9439  df-inf 9440  df-oi 9507  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-n0 12477  df-z 12563  df-uz 12827  df-rp 12979  df-fz 13489  df-fzo 13632  df-fl 13761  df-seq 13971  df-exp 14032  df-hash 14295  df-cj 15050  df-re 15051  df-im 15052  df-sqrt 15186  df-abs 15187  df-clim 15436  df-rlim 15437  df-sum 15637  df-0p 25419  df-ply 25937  df-idp 25938  df-coe 25939  df-dgr 25940
This theorem is referenced by:  plymulx  33857
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