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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  plymulx0 Structured version   Visualization version   GIF version
Theorem plymulx0 33547
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 11164 . . . . . . 7 ℝ βŠ† β„‚
3 1re 11211 . . . . . . 7 1 ∈ ℝ
4 plyid 25715 . . . . . . 7 ((ℝ βŠ† β„‚ ∧ 1 ∈ ℝ) β†’ Xp ∈ (Polyβ€˜β„))
52, 3, 4mp2an 691 . . . . . 6 Xp ∈ (Polyβ€˜β„)
65a1i 11 . . . . 5 (𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) β†’ Xp ∈ (Polyβ€˜β„))
7 simprl 770 . . . . . 6 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ (π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ)) β†’ π‘₯ ∈ ℝ)
8 simprr 772 . . . . . 6 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ (π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ)) β†’ 𝑦 ∈ ℝ)
97, 8readdcld 11240 . . . . 5 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ (π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ)) β†’ (π‘₯ + 𝑦) ∈ ℝ)
107, 8remulcld 11241 . . . . 5 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ (π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ)) β†’ (π‘₯ Β· 𝑦) ∈ ℝ)
111, 6, 9, 10plymul 25724 . . . 4 (𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) β†’ (𝐹 ∘f Β· Xp) ∈ (Polyβ€˜β„))
12 0re 11213 . . . 4 0 ∈ ℝ
13 eqid 2733 . . . . 5 (coeffβ€˜(𝐹 ∘f Β· Xp)) = (coeffβ€˜(𝐹 ∘f Β· Xp))
1413coef2 25737 . . . 4 (((𝐹 ∘f Β· Xp) ∈ (Polyβ€˜β„) ∧ 0 ∈ ℝ) β†’ (coeffβ€˜(𝐹 ∘f Β· Xp)):β„•0βŸΆβ„)
1511, 12, 14sylancl 587 . . 3 (𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) β†’ (coeffβ€˜(𝐹 ∘f Β· Xp)):β„•0βŸΆβ„)
1615feqmptd 6958 . 2 (𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) β†’ (coeffβ€˜(𝐹 ∘f Β· Xp)) = (𝑛 ∈ β„•0 ↦ ((coeffβ€˜(𝐹 ∘f Β· Xp))β€˜π‘›)))
17 cnex 11188 . . . . . . . . 9 β„‚ ∈ V
1817a1i 11 . . . . . . . 8 (𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) β†’ β„‚ ∈ V)
19 plyf 25704 . . . . . . . . 9 (𝐹 ∈ (Polyβ€˜β„) β†’ 𝐹:β„‚βŸΆβ„‚)
201, 19syl 17 . . . . . . . 8 (𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) β†’ 𝐹:β„‚βŸΆβ„‚)
21 plyf 25704 . . . . . . . . . 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 770 . . . . . . . . 9 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ (π‘₯ ∈ β„‚ ∧ 𝑦 ∈ β„‚)) β†’ π‘₯ ∈ β„‚)
25 simprr 772 . . . . . . . . 9 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ (π‘₯ ∈ β„‚ ∧ 𝑦 ∈ β„‚)) β†’ 𝑦 ∈ β„‚)
2624, 25mulcomd 11232 . . . . . . . 8 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ (π‘₯ ∈ β„‚ ∧ 𝑦 ∈ β„‚)) β†’ (π‘₯ Β· 𝑦) = (𝑦 Β· π‘₯))
2718, 20, 23, 26caofcom 7702 . . . . . . 7 (𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) β†’ (𝐹 ∘f Β· Xp) = (Xp ∘f Β· 𝐹))
2827fveq2d 6893 . . . . . 6 (𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) β†’ (coeffβ€˜(𝐹 ∘f Β· Xp)) = (coeffβ€˜(Xp ∘f Β· 𝐹)))
2928fveq1d 6891 . . . . 5 (𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) β†’ ((coeffβ€˜(𝐹 ∘f Β· Xp))β€˜π‘›) = ((coeffβ€˜(Xp ∘f Β· 𝐹))β€˜π‘›))
3029adantr 482 . . . 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 482 . . . . . 6 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) β†’ 𝐹 ∈ (Polyβ€˜β„))
33 simpr 486 . . . . . 6 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) β†’ 𝑛 ∈ β„•0)
34 eqid 2733 . . . . . . 7 (coeffβ€˜Xp) = (coeffβ€˜Xp)
35 eqid 2733 . . . . . . 7 (coeffβ€˜πΉ) = (coeffβ€˜πΉ)
3634, 35coemul 25758 . . . . . 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 1372 . . . . 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 13591 . . . . . . . . . 10 (𝑖 ∈ (0...𝑛) β†’ 𝑖 ∈ β„•0)
39 coeidp 25769 . . . . . . . . . 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 7421 . . . . . . . 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 7503 . . . . . . . 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 2789 . . . . . . 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 483 . . . . . 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 15646 . . . . 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 25737 . . . . . . . . . . . . 13 ((𝐹 ∈ (Polyβ€˜β„) ∧ 0 ∈ ℝ) β†’ (coeffβ€˜πΉ):β„•0βŸΆβ„)
501, 12, 49sylancl 587 . . . . . . . . . . . 12 (𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) β†’ (coeffβ€˜πΉ):β„•0βŸΆβ„)
5150ad2antrr 725 . . . . . . . . . . 11 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) ∧ 𝑖 ∈ (0...𝑛)) β†’ (coeffβ€˜πΉ):β„•0βŸΆβ„)
52 fznn0sub 13530 . . . . . . . . . . . 12 (𝑖 ∈ (0...𝑛) β†’ (𝑛 βˆ’ 𝑖) ∈ β„•0)
5352adantl 483 . . . . . . . . . . 11 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) ∧ 𝑖 ∈ (0...𝑛)) β†’ (𝑛 βˆ’ 𝑖) ∈ β„•0)
5451, 53ffvelcdmd 7085 . . . . . . . . . 10 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) ∧ 𝑖 ∈ (0...𝑛)) β†’ ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)) ∈ ℝ)
5554recnd 11239 . . . . . . . . 9 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) ∧ 𝑖 ∈ (0...𝑛)) β†’ ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)) ∈ β„‚)
5655mullidd 11229 . . . . . . . 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 11409 . . . . . . . 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 15646 . . . . . 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 2745 . . . . . . 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 2745 . . . . . . 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 7414 . . . . . . . . . . 11 (𝑛 = 0 β†’ (0...𝑛) = (0...0))
63 0z 12566 . . . . . . . . . . . 12 0 ∈ β„€
64 fzsn 13540 . . . . . . . . . . . 12 (0 ∈ β„€ β†’ (0...0) = {0})
6563, 64ax-mp 5 . . . . . . . . . . 11 (0...0) = {0}
6662, 65eqtrdi 2789 . . . . . . . . . 10 (𝑛 = 0 β†’ (0...𝑛) = {0})
67 elsni 4645 . . . . . . . . . . . . 13 (𝑖 ∈ {0} β†’ 𝑖 = 0)
6867adantl 483 . . . . . . . . . . . 12 ((𝑛 = 0 ∧ 𝑖 ∈ {0}) β†’ 𝑖 = 0)
69 ax-1ne0 11176 . . . . . . . . . . . . . 14 1 β‰  0
7069nesymi 2999 . . . . . . . . . . . . 13 Β¬ 0 = 1
71 eqeq1 2737 . . . . . . . . . . . . 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 15650 . . . . . . . . 9 (𝑛 = 0 β†’ Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)), 0) = Σ𝑖 ∈ {0}0)
79 snfi 9041 . . . . . . . . . . 11 {0} ∈ Fin
8079olci 865 . . . . . . . . . 10 ({0} βŠ† (β„€β‰₯β€˜0) ∨ {0} ∈ Fin)
81 sumz 15665 . . . . . . . . . 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 2789 . . . . . . . 8 (𝑛 = 0 β†’ Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)), 0) = 0)
8483adantl 483 . . . . . . 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 766 . . . . . . . 8 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) ∧ Β¬ 𝑛 = 0) β†’ 𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}))
8633adantr 482 . . . . . . . . 9 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) ∧ Β¬ 𝑛 = 0) β†’ 𝑛 ∈ β„•0)
87 simpr 486 . . . . . . . . . 10 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) ∧ Β¬ 𝑛 = 0) β†’ Β¬ 𝑛 = 0)
8887neqned 2948 . . . . . . . . 9 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) ∧ Β¬ 𝑛 = 0) β†’ 𝑛 β‰  0)
89 elnnne0 12483 . . . . . . . . 9 (𝑛 ∈ β„• ↔ (𝑛 ∈ β„•0 ∧ 𝑛 β‰  0))
9086, 88, 89sylanbrc 584 . . . . . . . 8 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•0) ∧ Β¬ 𝑛 = 0) β†’ 𝑛 ∈ β„•)
91 1nn0 12485 . . . . . . . . . . . . 13 1 ∈ β„•0
9291a1i 11 . . . . . . . . . . . 12 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) β†’ 1 ∈ β„•0)
93 simpr 486 . . . . . . . . . . . . 13 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) β†’ 𝑛 ∈ β„•)
9493nnnn0d 12529 . . . . . . . . . . . 12 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) β†’ 𝑛 ∈ β„•0)
9593nnge1d 12257 . . . . . . . . . . . 12 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) β†’ 1 ≀ 𝑛)
96 elfz2nn0 13589 . . . . . . . . . . . 12 (1 ∈ (0...𝑛) ↔ (1 ∈ β„•0 ∧ 𝑛 ∈ β„•0 ∧ 1 ≀ 𝑛))
9792, 94, 95, 96syl3anbrc 1344 . . . . . . . . . . 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 725 . . . . . . . . . . . . 13 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) ∧ 𝑖 ∈ {1}) β†’ (coeffβ€˜πΉ):β„•0βŸΆβ„)
100 oveq2 7414 . . . . . . . . . . . . . . . 16 (𝑖 = 1 β†’ (𝑛 βˆ’ 𝑖) = (𝑛 βˆ’ 1))
10146, 100sylbi 216 . . . . . . . . . . . . . . 15 (𝑖 ∈ {1} β†’ (𝑛 βˆ’ 𝑖) = (𝑛 βˆ’ 1))
102101adantl 483 . . . . . . . . . . . . . 14 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) ∧ 𝑖 ∈ {1}) β†’ (𝑛 βˆ’ 𝑖) = (𝑛 βˆ’ 1))
103 nnm1nn0 12510 . . . . . . . . . . . . . . 15 (𝑛 ∈ β„• β†’ (𝑛 βˆ’ 1) ∈ β„•0)
104103ad2antlr 726 . . . . . . . . . . . . . 14 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) ∧ 𝑖 ∈ {1}) β†’ (𝑛 βˆ’ 1) ∈ β„•0)
105102, 104eqeltrd 2834 . . . . . . . . . . . . 13 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) ∧ 𝑖 ∈ {1}) β†’ (𝑛 βˆ’ 𝑖) ∈ β„•0)
10699, 105ffvelcdmd 7085 . . . . . . . . . . . 12 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) ∧ 𝑖 ∈ {1}) β†’ ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)) ∈ ℝ)
107106recnd 11239 . . . . . . . . . . 11 (((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) ∧ 𝑖 ∈ {1}) β†’ ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)) ∈ β„‚)
108107ralrimiva 3147 . . . . . . . . . 10 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) β†’ βˆ€π‘– ∈ {1} ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)) ∈ β„‚)
109 fzfi 13934 . . . . . . . . . . . 12 (0...𝑛) ∈ Fin
110109olci 865 . . . . . . . . . . 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 15669 . . . . . . . . . 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 837 . . . . . . . . 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 482 . . . . . . . . . . . 12 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) β†’ (coeffβ€˜πΉ):β„•0βŸΆβ„)
115103adantl 483 . . . . . . . . . . . 12 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) β†’ (𝑛 βˆ’ 1) ∈ β„•0)
116114, 115ffvelcdmd 7085 . . . . . . . . . . 11 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) β†’ ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 1)) ∈ ℝ)
117116recnd 11239 . . . . . . . . . 10 ((𝐹 ∈ ((Polyβ€˜β„) βˆ– {0𝑝}) ∧ 𝑛 ∈ β„•) β†’ ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 1)) ∈ β„‚)
118100fveq2d 6893 . . . . . . . . . . 11 (𝑖 = 1 β†’ ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 𝑖)) = ((coeffβ€˜πΉ)β€˜(𝑛 βˆ’ 1)))
119118sumsn 15689 . . . . . . . . . 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 588 . . . . . . . . 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 2775 . . . . . . . 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 585 . . . . . . 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 2773 . . . . 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 2777 . . . 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 2773 . . 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 2773 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 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062  Vcvv 3475   βˆ– cdif 3945   βŠ† wss 3948  ifcif 4528  {csn 4628   class class class wbr 5148   ↦ cmpt 5231  βŸΆwf 6537  β€˜cfv 6541  (class class class)co 7406   ∘f cof 7665  Fincfn 8936  β„‚cc 11105  β„cr 11106  0cc0 11107  1c1 11108   Β· cmul 11112   ≀ cle 11246   βˆ’ cmin 11441  β„•cn 12209  β„•0cn0 12469  β„€cz 12555  β„€β‰₯cuz 12819  ...cfz 13481  Ξ£csu 15629  0𝑝c0p 25178  Polycply 25690  Xpcidp 25691  coeffccoe 25692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-inf2 9633  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184  ax-pre-sup 11185
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  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 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-isom 6550  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-of 7667  df-om 7853  df-1st 7972  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-1o 8463  df-er 8700  df-map 8819  df-pm 8820  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-sup 9434  df-inf 9435  df-oi 9502  df-card 9931  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-div 11869  df-nn 12210  df-2 12272  df-3 12273  df-n0 12470  df-z 12556  df-uz 12820  df-rp 12972  df-fz 13482  df-fzo 13625  df-fl 13754  df-seq 13964  df-exp 14025  df-hash 14288  df-cj 15043  df-re 15044  df-im 15045  df-sqrt 15179  df-abs 15180  df-clim 15429  df-rlim 15430  df-sum 15630  df-0p 25179  df-ply 25694  df-idp 25695  df-coe 25696  df-dgr 25697
This theorem is referenced by:  plymulx  33548
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