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Theorem plymulx 29785
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 9850 . . . . . . 7 ℝ ⊆ ℂ
2 1re 9896 . . . . . . 7 1 ∈ ℝ
3 plyid 23714 . . . . . . 7 ((ℝ ⊆ ℂ ∧ 1 ∈ ℝ) → Xp ∈ (Poly‘ℝ))
41, 2, 3mp2an 703 . . . . . 6 Xp ∈ (Poly‘ℝ)
5 plymul02 29783 . . . . . . 7 (Xp ∈ (Poly‘ℝ) → (0𝑝𝑓 · Xp) = 0𝑝)
65fveq2d 6092 . . . . . 6 (Xp ∈ (Poly‘ℝ) → (coeff‘(0𝑝𝑓 · Xp)) = (coeff‘0𝑝))
74, 6ax-mp 5 . . . . 5 (coeff‘(0𝑝𝑓 · Xp)) = (coeff‘0𝑝)
8 fconstmpt 5075 . . . . . 6 (ℕ0 × {0}) = (𝑛 ∈ ℕ0 ↦ 0)
9 coe0 23761 . . . . . 6 (coeff‘0𝑝) = (ℕ0 × {0})
10 eqidd 2610 . . . . . . . 8 ((𝑛 ∈ ℕ0𝑛 = 0) → 0 = 0)
11 elnnne0 11156 . . . . . . . . . . 11 (𝑛 ∈ ℕ ↔ (𝑛 ∈ ℕ0𝑛 ≠ 0))
12 df-ne 2781 . . . . . . . . . . . 12 (𝑛 ≠ 0 ↔ ¬ 𝑛 = 0)
1312anbi2i 725 . . . . . . . . . . 11 ((𝑛 ∈ ℕ0𝑛 ≠ 0) ↔ (𝑛 ∈ ℕ0 ∧ ¬ 𝑛 = 0))
1411, 13bitr2i 263 . . . . . . . . . 10 ((𝑛 ∈ ℕ0 ∧ ¬ 𝑛 = 0) ↔ 𝑛 ∈ ℕ)
15 nnm1nn0 11184 . . . . . . . . . 10 (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ0)
1614, 15sylbi 205 . . . . . . . . 9 ((𝑛 ∈ ℕ0 ∧ ¬ 𝑛 = 0) → (𝑛 − 1) ∈ ℕ0)
17 eqidd 2610 . . . . . . . . . 10 (𝑚 = (𝑛 − 1) → 0 = 0)
18 fconstmpt 5075 . . . . . . . . . . 11 (ℕ0 × {0}) = (𝑚 ∈ ℕ0 ↦ 0)
199, 18eqtri 2631 . . . . . . . . . 10 (coeff‘0𝑝) = (𝑚 ∈ ℕ0 ↦ 0)
20 c0ex 9891 . . . . . . . . . 10 0 ∈ V
2117, 19, 20fvmpt 6176 . . . . . . . . 9 ((𝑛 − 1) ∈ ℕ0 → ((coeff‘0𝑝)‘(𝑛 − 1)) = 0)
2216, 21syl 17 . . . . . . . 8 ((𝑛 ∈ ℕ0 ∧ ¬ 𝑛 = 0) → ((coeff‘0𝑝)‘(𝑛 − 1)) = 0)
2310, 22ifeqda 4070 . . . . . . 7 (𝑛 ∈ ℕ0 → if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1))) = 0)
2423mpteq2ia 4662 . . . . . 6 (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1)))) = (𝑛 ∈ ℕ0 ↦ 0)
258, 9, 243eqtr4ri 2642 . . . . 5 (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1)))) = (coeff‘0𝑝)
267, 25eqtr4i 2634 . . . 4 (coeff‘(0𝑝𝑓 · Xp)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1))))
27 oveq1 6534 . . . . 5 (𝐹 = 0𝑝 → (𝐹𝑓 · Xp) = (0𝑝𝑓 · Xp))
2827fveq2d 6092 . . . 4 (𝐹 = 0𝑝 → (coeff‘(𝐹𝑓 · Xp)) = (coeff‘(0𝑝𝑓 · Xp)))
29 simpl 471 . . . . . . . 8 ((𝐹 = 0𝑝𝑛 ∈ ℕ0) → 𝐹 = 0𝑝)
3029fveq2d 6092 . . . . . . 7 ((𝐹 = 0𝑝𝑛 ∈ ℕ0) → (coeff‘𝐹) = (coeff‘0𝑝))
3130fveq1d 6090 . . . . . 6 ((𝐹 = 0𝑝𝑛 ∈ ℕ0) → ((coeff‘𝐹)‘(𝑛 − 1)) = ((coeff‘0𝑝)‘(𝑛 − 1)))
3231ifeq2d 4054 . . . . 5 ((𝐹 = 0𝑝𝑛 ∈ ℕ0) → if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1))) = if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1))))
3332mpteq2dva 4666 . . . 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 2669 . . 3 (𝐹 = 0𝑝 → (coeff‘(𝐹𝑓 · Xp)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1)))))
3534adantl 480 . 2 ((𝐹 ∈ (Poly‘ℝ) ∧ 𝐹 = 0𝑝) → (coeff‘(𝐹𝑓 · Xp)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1)))))
36 simpl 471 . . . 4 ((𝐹 ∈ (Poly‘ℝ) ∧ ¬ 𝐹 = 0𝑝) → 𝐹 ∈ (Poly‘ℝ))
37 elsng 4138 . . . . . 6 (𝐹 ∈ (Poly‘ℝ) → (𝐹 ∈ {0𝑝} ↔ 𝐹 = 0𝑝))
3837notbid 306 . . . . 5 (𝐹 ∈ (Poly‘ℝ) → (¬ 𝐹 ∈ {0𝑝} ↔ ¬ 𝐹 = 0𝑝))
3938biimpar 500 . . . 4 ((𝐹 ∈ (Poly‘ℝ) ∧ ¬ 𝐹 = 0𝑝) → ¬ 𝐹 ∈ {0𝑝})
4036, 39eldifd 3550 . . 3 ((𝐹 ∈ (Poly‘ℝ) ∧ ¬ 𝐹 = 0𝑝) → 𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}))
41 plymulx0 29784 . . 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 827 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 382   = wceq 1474  wcel 1976  wne 2779  cdif 3536  wss 3539  ifcif 4035  {csn 4124  cmpt 4637   × cxp 5026  cfv 5790  (class class class)co 6527  𝑓 cof 6771  cc 9791  cr 9792  0cc0 9793  1c1 9794   · cmul 9798  cmin 10118  cn 10870  0cn0 11142  0𝑝c0p 23187  Polycply 23689  Xpcidp 23690  coeffccoe 23691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-inf2 8399  ax-cnex 9849  ax-resscn 9850  ax-1cn 9851  ax-icn 9852  ax-addcl 9853  ax-addrcl 9854  ax-mulcl 9855  ax-mulrcl 9856  ax-mulcom 9857  ax-addass 9858  ax-mulass 9859  ax-distr 9860  ax-i2m1 9861  ax-1ne0 9862  ax-1rid 9863  ax-rnegex 9864  ax-rrecex 9865  ax-cnre 9866  ax-pre-lttri 9867  ax-pre-lttrn 9868  ax-pre-ltadd 9869  ax-pre-mulgt0 9870  ax-pre-sup 9871  ax-addf 9872
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-se 4988  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-isom 5799  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-of 6773  df-om 6936  df-1st 7037  df-2nd 7038  df-wrecs 7272  df-recs 7333  df-rdg 7371  df-1o 7425  df-oadd 7429  df-er 7607  df-map 7724  df-pm 7725  df-en 7820  df-dom 7821  df-sdom 7822  df-fin 7823  df-sup 8209  df-inf 8210  df-oi 8276  df-card 8626  df-pnf 9933  df-mnf 9934  df-xr 9935  df-ltxr 9936  df-le 9937  df-sub 10120  df-neg 10121  df-div 10537  df-nn 10871  df-2 10929  df-3 10930  df-n0 11143  df-z 11214  df-uz 11523  df-rp 11668  df-fz 12156  df-fzo 12293  df-fl 12413  df-seq 12622  df-exp 12681  df-hash 12938  df-cj 13636  df-re 13637  df-im 13638  df-sqrt 13772  df-abs 13773  df-clim 14016  df-rlim 14017  df-sum 14214  df-0p 23188  df-ply 23693  df-idp 23694  df-coe 23695  df-dgr 23696
This theorem is referenced by: (None)
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