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Theorem plymulx 29951
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 9872 . . . . . . 7 ℝ ⊆ ℂ
2 1re 9918 . . . . . . 7 1 ∈ ℝ
3 plyid 23769 . . . . . . 7 ((ℝ ⊆ ℂ ∧ 1 ∈ ℝ) → Xp ∈ (Poly‘ℝ))
41, 2, 3mp2an 704 . . . . . 6 Xp ∈ (Poly‘ℝ)
5 plymul02 29949 . . . . . . 7 (Xp ∈ (Poly‘ℝ) → (0𝑝𝑓 · Xp) = 0𝑝)
65fveq2d 6107 . . . . . 6 (Xp ∈ (Poly‘ℝ) → (coeff‘(0𝑝𝑓 · Xp)) = (coeff‘0𝑝))
74, 6ax-mp 5 . . . . 5 (coeff‘(0𝑝𝑓 · Xp)) = (coeff‘0𝑝)
8 fconstmpt 5085 . . . . . 6 (ℕ0 × {0}) = (𝑛 ∈ ℕ0 ↦ 0)
9 coe0 23816 . . . . . 6 (coeff‘0𝑝) = (ℕ0 × {0})
10 eqidd 2611 . . . . . . . 8 ((𝑛 ∈ ℕ0𝑛 = 0) → 0 = 0)
11 elnnne0 11183 . . . . . . . . . . 11 (𝑛 ∈ ℕ ↔ (𝑛 ∈ ℕ0𝑛 ≠ 0))
12 df-ne 2782 . . . . . . . . . . . 12 (𝑛 ≠ 0 ↔ ¬ 𝑛 = 0)
1312anbi2i 726 . . . . . . . . . . 11 ((𝑛 ∈ ℕ0𝑛 ≠ 0) ↔ (𝑛 ∈ ℕ0 ∧ ¬ 𝑛 = 0))
1411, 13bitr2i 264 . . . . . . . . . 10 ((𝑛 ∈ ℕ0 ∧ ¬ 𝑛 = 0) ↔ 𝑛 ∈ ℕ)
15 nnm1nn0 11211 . . . . . . . . . 10 (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ0)
1614, 15sylbi 206 . . . . . . . . 9 ((𝑛 ∈ ℕ0 ∧ ¬ 𝑛 = 0) → (𝑛 − 1) ∈ ℕ0)
17 eqidd 2611 . . . . . . . . . 10 (𝑚 = (𝑛 − 1) → 0 = 0)
18 fconstmpt 5085 . . . . . . . . . . 11 (ℕ0 × {0}) = (𝑚 ∈ ℕ0 ↦ 0)
199, 18eqtri 2632 . . . . . . . . . 10 (coeff‘0𝑝) = (𝑚 ∈ ℕ0 ↦ 0)
20 c0ex 9913 . . . . . . . . . 10 0 ∈ V
2117, 19, 20fvmpt 6191 . . . . . . . . 9 ((𝑛 − 1) ∈ ℕ0 → ((coeff‘0𝑝)‘(𝑛 − 1)) = 0)
2216, 21syl 17 . . . . . . . 8 ((𝑛 ∈ ℕ0 ∧ ¬ 𝑛 = 0) → ((coeff‘0𝑝)‘(𝑛 − 1)) = 0)
2310, 22ifeqda 4071 . . . . . . 7 (𝑛 ∈ ℕ0 → if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1))) = 0)
2423mpteq2ia 4668 . . . . . 6 (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1)))) = (𝑛 ∈ ℕ0 ↦ 0)
258, 9, 243eqtr4ri 2643 . . . . 5 (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1)))) = (coeff‘0𝑝)
267, 25eqtr4i 2635 . . . 4 (coeff‘(0𝑝𝑓 · Xp)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1))))
27 oveq1 6556 . . . . 5 (𝐹 = 0𝑝 → (𝐹𝑓 · Xp) = (0𝑝𝑓 · Xp))
2827fveq2d 6107 . . . 4 (𝐹 = 0𝑝 → (coeff‘(𝐹𝑓 · Xp)) = (coeff‘(0𝑝𝑓 · Xp)))
29 simpl 472 . . . . . . . 8 ((𝐹 = 0𝑝𝑛 ∈ ℕ0) → 𝐹 = 0𝑝)
3029fveq2d 6107 . . . . . . 7 ((𝐹 = 0𝑝𝑛 ∈ ℕ0) → (coeff‘𝐹) = (coeff‘0𝑝))
3130fveq1d 6105 . . . . . 6 ((𝐹 = 0𝑝𝑛 ∈ ℕ0) → ((coeff‘𝐹)‘(𝑛 − 1)) = ((coeff‘0𝑝)‘(𝑛 − 1)))
3231ifeq2d 4055 . . . . 5 ((𝐹 = 0𝑝𝑛 ∈ ℕ0) → if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1))) = if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1))))
3332mpteq2dva 4672 . . . 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 2670 . . 3 (𝐹 = 0𝑝 → (coeff‘(𝐹𝑓 · Xp)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1)))))
3534adantl 481 . 2 ((𝐹 ∈ (Poly‘ℝ) ∧ 𝐹 = 0𝑝) → (coeff‘(𝐹𝑓 · Xp)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1)))))
36 simpl 472 . . . 4 ((𝐹 ∈ (Poly‘ℝ) ∧ ¬ 𝐹 = 0𝑝) → 𝐹 ∈ (Poly‘ℝ))
37 elsng 4139 . . . . . 6 (𝐹 ∈ (Poly‘ℝ) → (𝐹 ∈ {0𝑝} ↔ 𝐹 = 0𝑝))
3837notbid 307 . . . . 5 (𝐹 ∈ (Poly‘ℝ) → (¬ 𝐹 ∈ {0𝑝} ↔ ¬ 𝐹 = 0𝑝))
3938biimpar 501 . . . 4 ((𝐹 ∈ (Poly‘ℝ) ∧ ¬ 𝐹 = 0𝑝) → ¬ 𝐹 ∈ {0𝑝})
4036, 39eldifd 3551 . . 3 ((𝐹 ∈ (Poly‘ℝ) ∧ ¬ 𝐹 = 0𝑝) → 𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}))
41 plymulx0 29950 . . 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 828 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 1475  wcel 1977  wne 2780  cdif 3537  wss 3540  ifcif 4036  {csn 4125  cmpt 4643   × cxp 5036  cfv 5804  (class class class)co 6549  𝑓 cof 6793  cc 9813  cr 9814  0cc0 9815  1c1 9816   · cmul 9820  cmin 10145  cn 10897  0cn0 11169  0𝑝c0p 23242  Polycply 23744  Xpcidp 23745  coeffccoe 23746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-rlim 14068  df-sum 14265  df-0p 23243  df-ply 23748  df-idp 23749  df-coe 23750  df-dgr 23751
This theorem is referenced by: (None)
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