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Theorem plymulx 31717
Description: Coefficients of a polynomial multiplied by Xp. (Contributed by Thierry Arnoux, 25-Sep-2018.)
Assertion
Ref Expression
plymulx (𝐹 ∈ (Poly‘ℝ) → (coeff‘(𝐹f · Xp)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1)))))
Distinct variable group:   𝑛,𝐹

Proof of Theorem plymulx
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 ax-resscn 10582 . . . . . . 7 ℝ ⊆ ℂ
2 1re 10629 . . . . . . 7 1 ∈ ℝ
3 plyid 24726 . . . . . . 7 ((ℝ ⊆ ℂ ∧ 1 ∈ ℝ) → Xp ∈ (Poly‘ℝ))
41, 2, 3mp2an 688 . . . . . 6 Xp ∈ (Poly‘ℝ)
5 plymul02 31715 . . . . . . 7 (Xp ∈ (Poly‘ℝ) → (0𝑝f · Xp) = 0𝑝)
65fveq2d 6667 . . . . . 6 (Xp ∈ (Poly‘ℝ) → (coeff‘(0𝑝f · Xp)) = (coeff‘0𝑝))
74, 6ax-mp 5 . . . . 5 (coeff‘(0𝑝f · Xp)) = (coeff‘0𝑝)
8 fconstmpt 5607 . . . . . 6 (ℕ0 × {0}) = (𝑛 ∈ ℕ0 ↦ 0)
9 coe0 24773 . . . . . 6 (coeff‘0𝑝) = (ℕ0 × {0})
10 eqidd 2819 . . . . . . . 8 ((𝑛 ∈ ℕ0𝑛 = 0) → 0 = 0)
11 elnnne0 11899 . . . . . . . . . . 11 (𝑛 ∈ ℕ ↔ (𝑛 ∈ ℕ0𝑛 ≠ 0))
12 df-ne 3014 . . . . . . . . . . . 12 (𝑛 ≠ 0 ↔ ¬ 𝑛 = 0)
1312anbi2i 622 . . . . . . . . . . 11 ((𝑛 ∈ ℕ0𝑛 ≠ 0) ↔ (𝑛 ∈ ℕ0 ∧ ¬ 𝑛 = 0))
1411, 13bitr2i 277 . . . . . . . . . 10 ((𝑛 ∈ ℕ0 ∧ ¬ 𝑛 = 0) ↔ 𝑛 ∈ ℕ)
15 nnm1nn0 11926 . . . . . . . . . 10 (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ0)
1614, 15sylbi 218 . . . . . . . . 9 ((𝑛 ∈ ℕ0 ∧ ¬ 𝑛 = 0) → (𝑛 − 1) ∈ ℕ0)
17 eqidd 2819 . . . . . . . . . 10 (𝑚 = (𝑛 − 1) → 0 = 0)
18 fconstmpt 5607 . . . . . . . . . . 11 (ℕ0 × {0}) = (𝑚 ∈ ℕ0 ↦ 0)
199, 18eqtri 2841 . . . . . . . . . 10 (coeff‘0𝑝) = (𝑚 ∈ ℕ0 ↦ 0)
20 c0ex 10623 . . . . . . . . . 10 0 ∈ V
2117, 19, 20fvmpt 6761 . . . . . . . . 9 ((𝑛 − 1) ∈ ℕ0 → ((coeff‘0𝑝)‘(𝑛 − 1)) = 0)
2216, 21syl 17 . . . . . . . 8 ((𝑛 ∈ ℕ0 ∧ ¬ 𝑛 = 0) → ((coeff‘0𝑝)‘(𝑛 − 1)) = 0)
2310, 22ifeqda 4498 . . . . . . 7 (𝑛 ∈ ℕ0 → if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1))) = 0)
2423mpteq2ia 5148 . . . . . 6 (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1)))) = (𝑛 ∈ ℕ0 ↦ 0)
258, 9, 243eqtr4ri 2852 . . . . 5 (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1)))) = (coeff‘0𝑝)
267, 25eqtr4i 2844 . . . 4 (coeff‘(0𝑝f · Xp)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1))))
27 fvoveq1 7168 . . . 4 (𝐹 = 0𝑝 → (coeff‘(𝐹f · Xp)) = (coeff‘(0𝑝f · Xp)))
28 simpl 483 . . . . . . . 8 ((𝐹 = 0𝑝𝑛 ∈ ℕ0) → 𝐹 = 0𝑝)
2928fveq2d 6667 . . . . . . 7 ((𝐹 = 0𝑝𝑛 ∈ ℕ0) → (coeff‘𝐹) = (coeff‘0𝑝))
3029fveq1d 6665 . . . . . 6 ((𝐹 = 0𝑝𝑛 ∈ ℕ0) → ((coeff‘𝐹)‘(𝑛 − 1)) = ((coeff‘0𝑝)‘(𝑛 − 1)))
3130ifeq2d 4482 . . . . 5 ((𝐹 = 0𝑝𝑛 ∈ ℕ0) → if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1))) = if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1))))
3231mpteq2dva 5152 . . . 4 (𝐹 = 0𝑝 → (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1)))) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1)))))
3326, 27, 323eqtr4a 2879 . . 3 (𝐹 = 0𝑝 → (coeff‘(𝐹f · Xp)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1)))))
3433adantl 482 . 2 ((𝐹 ∈ (Poly‘ℝ) ∧ 𝐹 = 0𝑝) → (coeff‘(𝐹f · Xp)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1)))))
35 simpl 483 . . . 4 ((𝐹 ∈ (Poly‘ℝ) ∧ ¬ 𝐹 = 0𝑝) → 𝐹 ∈ (Poly‘ℝ))
36 elsng 4571 . . . . . 6 (𝐹 ∈ (Poly‘ℝ) → (𝐹 ∈ {0𝑝} ↔ 𝐹 = 0𝑝))
3736notbid 319 . . . . 5 (𝐹 ∈ (Poly‘ℝ) → (¬ 𝐹 ∈ {0𝑝} ↔ ¬ 𝐹 = 0𝑝))
3837biimpar 478 . . . 4 ((𝐹 ∈ (Poly‘ℝ) ∧ ¬ 𝐹 = 0𝑝) → ¬ 𝐹 ∈ {0𝑝})
3935, 38eldifd 3944 . . 3 ((𝐹 ∈ (Poly‘ℝ) ∧ ¬ 𝐹 = 0𝑝) → 𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}))
40 plymulx0 31716 . . 3 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (coeff‘(𝐹f · Xp)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1)))))
4139, 40syl 17 . 2 ((𝐹 ∈ (Poly‘ℝ) ∧ ¬ 𝐹 = 0𝑝) → (coeff‘(𝐹f · Xp)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1)))))
4234, 41pm2.61dan 809 1 (𝐹 ∈ (Poly‘ℝ) → (coeff‘(𝐹f · Xp)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1528  wcel 2105  wne 3013  cdif 3930  wss 3933  ifcif 4463  {csn 4557  cmpt 5137   × cxp 5546  cfv 6348  (class class class)co 7145  f cof 7396  cc 10523  cr 10524  0cc0 10525  1c1 10526   · cmul 10530  cmin 10858  cn 11626  0cn0 11885  0𝑝c0p 24197  Polycply 24701  Xpcidp 24702  coeffccoe 24703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603  ax-addf 10604
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-pm 8398  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-sup 8894  df-inf 8895  df-oi 8962  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-fz 12881  df-fzo 13022  df-fl 13150  df-seq 13358  df-exp 13418  df-hash 13679  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-clim 14833  df-rlim 14834  df-sum 15031  df-0p 24198  df-ply 24705  df-idp 24706  df-coe 24707  df-dgr 24708
This theorem is referenced by: (None)
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