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Theorem plymulx 30934
Description: Coefficients of a polynomial multiplyed 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 10185 . . . . . . 7 ℝ ⊆ ℂ
2 1re 10231 . . . . . . 7 1 ∈ ℝ
3 plyid 24164 . . . . . . 7 ((ℝ ⊆ ℂ ∧ 1 ∈ ℝ) → Xp ∈ (Poly‘ℝ))
41, 2, 3mp2an 710 . . . . . 6 Xp ∈ (Poly‘ℝ)
5 plymul02 30932 . . . . . . 7 (Xp ∈ (Poly‘ℝ) → (0𝑝𝑓 · Xp) = 0𝑝)
65fveq2d 6356 . . . . . 6 (Xp ∈ (Poly‘ℝ) → (coeff‘(0𝑝𝑓 · Xp)) = (coeff‘0𝑝))
74, 6ax-mp 5 . . . . 5 (coeff‘(0𝑝𝑓 · Xp)) = (coeff‘0𝑝)
8 fconstmpt 5320 . . . . . 6 (ℕ0 × {0}) = (𝑛 ∈ ℕ0 ↦ 0)
9 coe0 24211 . . . . . 6 (coeff‘0𝑝) = (ℕ0 × {0})
10 eqidd 2761 . . . . . . . 8 ((𝑛 ∈ ℕ0𝑛 = 0) → 0 = 0)
11 elnnne0 11498 . . . . . . . . . . 11 (𝑛 ∈ ℕ ↔ (𝑛 ∈ ℕ0𝑛 ≠ 0))
12 df-ne 2933 . . . . . . . . . . . 12 (𝑛 ≠ 0 ↔ ¬ 𝑛 = 0)
1312anbi2i 732 . . . . . . . . . . 11 ((𝑛 ∈ ℕ0𝑛 ≠ 0) ↔ (𝑛 ∈ ℕ0 ∧ ¬ 𝑛 = 0))
1411, 13bitr2i 265 . . . . . . . . . 10 ((𝑛 ∈ ℕ0 ∧ ¬ 𝑛 = 0) ↔ 𝑛 ∈ ℕ)
15 nnm1nn0 11526 . . . . . . . . . 10 (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ0)
1614, 15sylbi 207 . . . . . . . . 9 ((𝑛 ∈ ℕ0 ∧ ¬ 𝑛 = 0) → (𝑛 − 1) ∈ ℕ0)
17 eqidd 2761 . . . . . . . . . 10 (𝑚 = (𝑛 − 1) → 0 = 0)
18 fconstmpt 5320 . . . . . . . . . . 11 (ℕ0 × {0}) = (𝑚 ∈ ℕ0 ↦ 0)
199, 18eqtri 2782 . . . . . . . . . 10 (coeff‘0𝑝) = (𝑚 ∈ ℕ0 ↦ 0)
20 c0ex 10226 . . . . . . . . . 10 0 ∈ V
2117, 19, 20fvmpt 6444 . . . . . . . . 9 ((𝑛 − 1) ∈ ℕ0 → ((coeff‘0𝑝)‘(𝑛 − 1)) = 0)
2216, 21syl 17 . . . . . . . 8 ((𝑛 ∈ ℕ0 ∧ ¬ 𝑛 = 0) → ((coeff‘0𝑝)‘(𝑛 − 1)) = 0)
2310, 22ifeqda 4265 . . . . . . 7 (𝑛 ∈ ℕ0 → if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1))) = 0)
2423mpteq2ia 4892 . . . . . 6 (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1)))) = (𝑛 ∈ ℕ0 ↦ 0)
258, 9, 243eqtr4ri 2793 . . . . 5 (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1)))) = (coeff‘0𝑝)
267, 25eqtr4i 2785 . . . 4 (coeff‘(0𝑝𝑓 · Xp)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1))))
27 oveq1 6820 . . . . 5 (𝐹 = 0𝑝 → (𝐹𝑓 · Xp) = (0𝑝𝑓 · Xp))
2827fveq2d 6356 . . . 4 (𝐹 = 0𝑝 → (coeff‘(𝐹𝑓 · Xp)) = (coeff‘(0𝑝𝑓 · Xp)))
29 simpl 474 . . . . . . . 8 ((𝐹 = 0𝑝𝑛 ∈ ℕ0) → 𝐹 = 0𝑝)
3029fveq2d 6356 . . . . . . 7 ((𝐹 = 0𝑝𝑛 ∈ ℕ0) → (coeff‘𝐹) = (coeff‘0𝑝))
3130fveq1d 6354 . . . . . 6 ((𝐹 = 0𝑝𝑛 ∈ ℕ0) → ((coeff‘𝐹)‘(𝑛 − 1)) = ((coeff‘0𝑝)‘(𝑛 − 1)))
3231ifeq2d 4249 . . . . 5 ((𝐹 = 0𝑝𝑛 ∈ ℕ0) → if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1))) = if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1))))
3332mpteq2dva 4896 . . . 4 (𝐹 = 0𝑝 → (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1)))) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1)))))
3426, 28, 333eqtr4a 2820 . . 3 (𝐹 = 0𝑝 → (coeff‘(𝐹𝑓 · Xp)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1)))))
3534adantl 473 . 2 ((𝐹 ∈ (Poly‘ℝ) ∧ 𝐹 = 0𝑝) → (coeff‘(𝐹𝑓 · Xp)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1)))))
36 simpl 474 . . . 4 ((𝐹 ∈ (Poly‘ℝ) ∧ ¬ 𝐹 = 0𝑝) → 𝐹 ∈ (Poly‘ℝ))
37 elsng 4335 . . . . . 6 (𝐹 ∈ (Poly‘ℝ) → (𝐹 ∈ {0𝑝} ↔ 𝐹 = 0𝑝))
3837notbid 307 . . . . 5 (𝐹 ∈ (Poly‘ℝ) → (¬ 𝐹 ∈ {0𝑝} ↔ ¬ 𝐹 = 0𝑝))
3938biimpar 503 . . . 4 ((𝐹 ∈ (Poly‘ℝ) ∧ ¬ 𝐹 = 0𝑝) → ¬ 𝐹 ∈ {0𝑝})
4036, 39eldifd 3726 . . 3 ((𝐹 ∈ (Poly‘ℝ) ∧ ¬ 𝐹 = 0𝑝) → 𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}))
41 plymulx0 30933 . . 3 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (coeff‘(𝐹𝑓 · Xp)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1)))))
4240, 41syl 17 . 2 ((𝐹 ∈ (Poly‘ℝ) ∧ ¬ 𝐹 = 0𝑝) → (coeff‘(𝐹𝑓 · Xp)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1)))))
4335, 42pm2.61dan 867 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 383   = wceq 1632  wcel 2139  wne 2932  cdif 3712  wss 3715  ifcif 4230  {csn 4321  cmpt 4881   × cxp 5264  cfv 6049  (class class class)co 6813  𝑓 cof 7060  cc 10126  cr 10127  0cc0 10128  1c1 10129   · cmul 10133  cmin 10458  cn 11212  0cn0 11484  0𝑝c0p 23635  Polycply 24139  Xpcidp 24140  coeffccoe 24141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205  ax-pre-sup 10206  ax-addf 10207
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-of 7062  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-map 8025  df-pm 8026  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-sup 8513  df-inf 8514  df-oi 8580  df-card 8955  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-div 10877  df-nn 11213  df-2 11271  df-3 11272  df-n0 11485  df-z 11570  df-uz 11880  df-rp 12026  df-fz 12520  df-fzo 12660  df-fl 12787  df-seq 12996  df-exp 13055  df-hash 13312  df-cj 14038  df-re 14039  df-im 14040  df-sqrt 14174  df-abs 14175  df-clim 14418  df-rlim 14419  df-sum 14616  df-0p 23636  df-ply 24143  df-idp 24144  df-coe 24145  df-dgr 24146
This theorem is referenced by: (None)
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