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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  plymulx0 Structured version   Visualization version   GIF version
Theorem plymulx0 33857
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 4126 . . . . 5 (𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) β†’ 𝐹 ∈ (Polyβ€˜β„))
2 ax-resscn 11170 . . . . . . 7 ℝ βŠ† β„‚
3 1re 11219 . . . . . . 7 1 ∈ ℝ
4 plyid 25959 . . . . . . 7 ((ℝ βŠ† β„‚ ∧ 1 ∈ ℝ) β†’ Xp ∈ (Polyβ€˜β„))
52, 3, 4mp2an 689 . . . . . 6 Xp ∈ (Polyβ€˜β„)
65a1i 11 . . . . 5 (𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) β†’ Xp ∈ (Polyβ€˜β„))
7 simprl 768 . . . . . 6 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ (π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ)) β†’ π‘₯ ∈ ℝ)
8 simprr 770 . . . . . 6 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ (π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ)) β†’ 𝑦 ∈ ℝ)
97, 8readdcld 11248 . . . . 5 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ (π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ)) β†’ (π‘₯ + 𝑦) ∈ ℝ)
107, 8remulcld 11249 . . . . 5 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ (π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ)) β†’ (π‘₯ Β· 𝑦) ∈ ℝ)
111, 6, 9, 10plymul 25968 . . . 4 (𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) β†’ (𝐹 ∘f Β· Xp) ∈ (Polyβ€˜β„))
12 0re 11221 . . . 4 0 ∈ ℝ
13 eqid 2731 . . . . 5 (coeffβ€˜(𝐹 ∘f Β· Xp)) = (coeffβ€˜(𝐹 ∘f Β· Xp))
1413coef2 25981 . . . 4 (((𝐹 ∘f Β· Xp) ∈ (Polyβ€˜β„) ∧ 0 ∈ ℝ) β†’ (coeffβ€˜(𝐹 ∘f Β· Xp)):β„•0βŸΆβ„)
1511, 12, 14sylancl 585 . . 3 (𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) β†’ (coeffβ€˜(𝐹 ∘f Β· Xp)):β„•0βŸΆβ„)
1615feqmptd 6960 . 2 (𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) β†’ (coeffβ€˜(𝐹 ∘f Β· Xp)) = (𝑛 ∈ β„•0 ↦ ((coeffβ€˜(𝐹 ∘f Β· Xp))β€˜π‘›)))
17 cnex 11194 . . . . . . . . 9 β„‚ ∈ V
1817a1i 11 . . . . . . . 8 (𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) β†’ β„‚ ∈ V)
19 plyf 25948 . . . . . . . . 9 (𝐹 ∈ (Polyβ€˜β„) β†’ 𝐹:β„‚βŸΆβ„‚)
201, 19syl 17 . . . . . . . 8 (𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) β†’ 𝐹:β„‚βŸΆβ„‚)
21 plyf 25948 . . . . . . . . . 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 768 . . . . . . . . 9 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ (π‘₯ ∈ β„‚ ∧ 𝑦 ∈ β„‚)) β†’ π‘₯ ∈ β„‚)
25 simprr 770 . . . . . . . . 9 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ (π‘₯ ∈ β„‚ ∧ 𝑦 ∈ β„‚)) β†’ 𝑦 ∈ β„‚)
2624, 25mulcomd 11240 . . . . . . . 8 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ (π‘₯ ∈ β„‚ ∧ 𝑦 ∈ β„‚)) β†’ (π‘₯ Β· 𝑦) = (𝑦 Β· π‘₯))
2718, 20, 23, 26caofcom 7708 . . . . . . 7 (𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) β†’ (𝐹 ∘f Β· Xp) = (Xp ∘f Β· 𝐹))
2827fveq2d 6895 . . . . . 6 (𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) β†’ (coeffβ€˜(𝐹 ∘f Β· Xp)) = (coeffβ€˜(Xp ∘f Β· 𝐹)))
2928fveq1d 6893 . . . . 5 (𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) β†’ ((coeffβ€˜(𝐹 ∘f Β· Xp))β€˜π‘›) = ((coeffβ€˜(Xp ∘f Β· 𝐹))β€˜π‘›))
3029adantr 480 . . . 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 480 . . . . . 6 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) β†’ 𝐹 ∈ (Polyβ€˜β„))
33 simpr 484 . . . . . 6 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) β†’ 𝑛 ∈ β„•0)
34 eqid 2731 . . . . . . 7 (coeffβ€˜Xp) = (coeffβ€˜Xp)
35 eqid 2731 . . . . . . 7 (coeffβ€˜πΉ) = (coeffβ€˜πΉ)
3634, 35coemul 26002 . . . . . 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 1370 . . . . 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 13599 . . . . . . . . . 10 (𝑖 ∈ (0...𝑛) β†’ 𝑖 ∈ β„•0)
39 coeidp 26014 . . . . . . . . . 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 7427 . . . . . . . 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 7509 . . . . . . . 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 2787 . . . . . . 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 481 . . . . . 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 15654 . . . . 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 4644 . . . . . . . . . 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 25981 . . . . . . . . . . . . 13 ((𝐹 ∈ (Polyβ€˜β„) ∧ 0 ∈ ℝ) β†’ (coeffβ€˜πΉ):β„•0βŸΆβ„)
501, 12, 49sylancl 585 . . . . . . . . . . . 12 (𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) β†’ (coeffβ€˜πΉ):β„•0βŸΆβ„)
5150ad2antrr 723 . . . . . . . . . . 11 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) ∧ 𝑖 ∈ (0...𝑛)) β†’ (coeffβ€˜πΉ):β„•0βŸΆβ„)
52 fznn0sub 13538 . . . . . . . . . . . 12 (𝑖 ∈ (0...𝑛) β†’ (𝑛 βˆ’ 𝑖) ∈ β„•0)
5352adantl 481 . . . . . . . . . . 11 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) ∧ 𝑖 ∈ (0...𝑛)) β†’ (𝑛 βˆ’ 𝑖) ∈ β„•0)
5451, 53ffvelcdmd 7087 . . . . . . . . . 10 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) ∧ 𝑖 ∈ (0...𝑛)) β†’ ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)) ∈ ℝ)
5554recnd 11247 . . . . . . . . 9 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) ∧ 𝑖 ∈ (0...𝑛)) β†’ ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)) ∈ β„‚)
5655mullidd 11237 . . . . . . . 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 11417 . . . . . . . 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 4556 . . . . . . 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 15654 . . . . . 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 2743 . . . . . . 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 2743 . . . . . . 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 7420 . . . . . . . . . . 11 (𝑛 = 0 β†’ (0...𝑛) = (0...0))
63 0z 12574 . . . . . . . . . . . 12 0 ∈ β„€
64 fzsn 13548 . . . . . . . . . . . 12 (0 ∈ β„€ β†’ (0...0) = {0})
6563, 64ax-mp 5 . . . . . . . . . . 11 (0...0) = {0}
6662, 65eqtrdi 2787 . . . . . . . . . 10 (𝑛 = 0 β†’ (0...𝑛) = {0})
67 elsni 4645 . . . . . . . . . . . . 13 (𝑖 ∈ {0} β†’ 𝑖 = 0)
6867adantl 481 . . . . . . . . . . . 12 ((𝑛 = 0 ∧ 𝑖 ∈ {0}) β†’ 𝑖 = 0)
69 ax-1ne0 11182 . . . . . . . . . . . . . 14 1 β‰  0
7069nesymi 2997 . . . . . . . . . . . . 13 Β¬ 0 = 1
71 eqeq1 2735 . . . . . . . . . . . . 13 (𝑖 = 0 β†’ (𝑖 = 1 ↔ 0 = 1))
7270, 71mtbiri 327 . . . . . . . . . . . 12 (𝑖 = 0 β†’ Β¬ 𝑖 = 1)
7368, 72syl 17 . . . . . . . . . . 11 ((𝑛 = 0 ∧ 𝑖 ∈ {0}) β†’ Β¬ 𝑖 = 1)
7447notbii 320 . . . . . . . . . . . 12 (Β¬ 𝑖 = 1 ↔ Β¬ 𝑖 ∈ {1})
7574biimpi 215 . . . . . . . . . . 11 (Β¬ 𝑖 = 1 β†’ Β¬ 𝑖 ∈ {1})
76 iffalse 4537 . . . . . . . . . . 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 15658 . . . . . . . . 9 (𝑛 = 0 β†’ Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)), 0) = Σ𝑖 ∈ {0}0)
79 snfi 9047 . . . . . . . . . . 11 {0} ∈ Fin
8079olci 863 . . . . . . . . . 10 ({0} βŠ† (β„€β‰₯β€˜0) ∨ {0} ∈ Fin)
81 sumz 15673 . . . . . . . . . 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 2787 . . . . . . . 8 (𝑛 = 0 β†’ Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)), 0) = 0)
8483adantl 481 . . . . . . 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 764 . . . . . . . 8 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) ∧ Β¬ 𝑛 = 0) β†’ 𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}))
8633adantr 480 . . . . . . . . 9 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) ∧ Β¬ 𝑛 = 0) β†’ 𝑛 ∈ β„•0)
87 simpr 484 . . . . . . . . . 10 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) ∧ Β¬ 𝑛 = 0) β†’ Β¬ 𝑛 = 0)
8887neqned 2946 . . . . . . . . 9 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) ∧ Β¬ 𝑛 = 0) β†’ 𝑛 β‰  0)
89 elnnne0 12491 . . . . . . . . 9 (𝑛 ∈ β„• ↔ (𝑛 ∈ β„•0 ∧ 𝑛 β‰  0))
9086, 88, 89sylanbrc 582 . . . . . . . 8 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) ∧ Β¬ 𝑛 = 0) β†’ 𝑛 ∈ β„•)
91 1nn0 12493 . . . . . . . . . . . . 13 1 ∈ β„•0
9291a1i 11 . . . . . . . . . . . 12 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) β†’ 1 ∈ β„•0)
93 simpr 484 . . . . . . . . . . . . 13 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) β†’ 𝑛 ∈ β„•)
9493nnnn0d 12537 . . . . . . . . . . . 12 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) β†’ 𝑛 ∈ β„•0)
9593nnge1d 12265 . . . . . . . . . . . 12 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) β†’ 1 ≀ 𝑛)
96 elfz2nn0 13597 . . . . . . . . . . . 12 (1 ∈ (0...𝑛) ↔ (1 ∈ β„•0 ∧ 𝑛 ∈ β„•0 ∧ 1 ≀ 𝑛))
9792, 94, 95, 96syl3anbrc 1342 . . . . . . . . . . 11 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) β†’ 1 ∈ (0...𝑛))
9897snssd 4812 . . . . . . . . . 10 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) β†’ {1} βŠ† (0...𝑛))
9950ad2antrr 723 . . . . . . . . . . . . 13 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) ∧ 𝑖 ∈ {1}) β†’ (coeffβ€˜πΉ):β„•0βŸΆβ„)
100 oveq2 7420 . . . . . . . . . . . . . . . 16 (𝑖 = 1 β†’ (𝑛 βˆ’ 𝑖) = (𝑛 βˆ’ 1))
10146, 100sylbi 216 . . . . . . . . . . . . . . 15 (𝑖 ∈ {1} β†’ (𝑛 βˆ’ 𝑖) = (𝑛 βˆ’ 1))
102101adantl 481 . . . . . . . . . . . . . 14 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) ∧ 𝑖 ∈ {1}) β†’ (𝑛 βˆ’ 𝑖) = (𝑛 βˆ’ 1))
103 nnm1nn0 12518 . . . . . . . . . . . . . . 15 (𝑛 ∈ β„• β†’ (𝑛 βˆ’ 1) ∈ β„•0)
104103ad2antlr 724 . . . . . . . . . . . . . 14 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) ∧ 𝑖 ∈ {1}) β†’ (𝑛 βˆ’ 1) ∈ β„•0)
105102, 104eqeltrd 2832 . . . . . . . . . . . . 13 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) ∧ 𝑖 ∈ {1}) β†’ (𝑛 βˆ’ 𝑖) ∈ β„•0)
10699, 105ffvelcdmd 7087 . . . . . . . . . . . 12 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) ∧ 𝑖 ∈ {1}) β†’ ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)) ∈ ℝ)
107106recnd 11247 . . . . . . . . . . 11 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) ∧ 𝑖 ∈ {1}) β†’ ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)) ∈ β„‚)
108107ralrimiva 3145 . . . . . . . . . 10 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) β†’ βˆ€π‘– ∈ {1} ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)) ∈ β„‚)
109 fzfi 13942 . . . . . . . . . . . 12 (0...𝑛) ∈ Fin
110109olci 863 . . . . . . . . . . 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 15677 . . . . . . . . . 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 835 . . . . . . . . 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 480 . . . . . . . . . . . 12 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) β†’ (coeffβ€˜πΉ):β„•0βŸΆβ„)
115103adantl 481 . . . . . . . . . . . 12 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) β†’ (𝑛 βˆ’ 1) ∈ β„•0)
116114, 115ffvelcdmd 7087 . . . . . . . . . . 11 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) β†’ ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 1)) ∈ ℝ)
117116recnd 11247 . . . . . . . . . 10 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) β†’ ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 1)) ∈ β„‚)
118100fveq2d 6895 . . . . . . . . . . 11 (𝑖 = 1 β†’ ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)) = ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 1)))
119118sumsn 15697 . . . . . . . . . 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 586 . . . . . . . . 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 2773 . . . . . . . 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 583 . . . . . . 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 4566 . . . . . 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 2771 . . . . 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 2775 . . . 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 2771 . . 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 5248 . 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 2771 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 395   ∨ wo 844   = wceq 1540   ∈ wcel 2105   β‰  wne 2939  βˆ€wral 3060  Vcvv 3473   βˆ– cdif 3945   βŠ† wss 3948  ifcif 4528  {csn 4628   class class class wbr 5148   ↦ cmpt 5231  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7412   ∘f cof 7671  Fincfn 8942  β„‚cc 11111  β„cr 11112  0cc0 11113  1c1 11114   Β· cmul 11118   ≀ cle 11254   βˆ’ cmin 11449  β„•cn 12217  β„•0cn0 12477  β„€cz 12563  β„€β‰₯cuz 12827  ...cfz 13489  Ξ£csu 15637  0𝑝c0p 25419  Polycply 25934  Xpcidp 25935  coeffccoe 25936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7728  ax-inf2 9639  ax-cnex 11169  ax-resscn 11170  ax-1cn 11171  ax-icn 11172  ax-addcl 11173  ax-addrcl 11174  ax-mulcl 11175  ax-mulrcl 11176  ax-mulcom 11177  ax-addass 11178  ax-mulass 11179  ax-distr 11180  ax-i2m1 11181  ax-1ne0 11182  ax-1rid 11183  ax-rnegex 11184  ax-rrecex 11185  ax-cnre 11186  ax-pre-lttri 11187  ax-pre-lttrn 11188  ax-pre-ltadd 11189  ax-pre-mulgt0 11190  ax-pre-sup 11191
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  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-int 4951  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-se 5632  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 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7673  df-om 7859  df-1st 7978  df-2nd 7979  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-1o 8469  df-er 8706  df-map 8825  df-pm 8826  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-sup 9440  df-inf 9441  df-oi 9508  df-card 9937  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-div 11877  df-nn 12218  df-2 12280  df-3 12281  df-n0 12478  df-z 12564  df-uz 12828  df-rp 12980  df-fz 13490  df-fzo 13633  df-fl 13762  df-seq 13972  df-exp 14033  df-hash 14296  df-cj 15051  df-re 15052  df-im 15053  df-sqrt 15187  df-abs 15188  df-clim 15437  df-rlim 15438  df-sum 15638  df-0p 25420  df-ply 25938  df-idp 25939  df-coe 25940  df-dgr 25941
This theorem is referenced by:  plymulx  33858
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