Theorem plymulx 32031

Description: Coefficients of a polynomial multiplied by X_p. (Contributed by Thierry Arnoux, 25-Sep-2018.)

Assertion

Ref	Expression
plymulx	⊢ (𝐹 ∈ (Poly‘ℝ) → (coeff‘(𝐹 ∘f · 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.

Step	Hyp	Ref	Expression
1		ax-resscn 10617	. . . . . . 7 ⊢ ℝ ⊆ ℂ
2		1re 10664	. . . . . . 7 ⊢ 1 ∈ ℝ
3		plyid 24890	. . . . . . 7 ⊢ ((ℝ ⊆ ℂ ∧ 1 ∈ ℝ) → Xp ∈ (Poly‘ℝ))
4	1, 2, 3	mp2an 692	. . . . . 6 ⊢ Xp ∈ (Poly‘ℝ)
5		plymul02 32029	. . . . . . 7 ⊢ (Xp ∈ (Poly‘ℝ) → (0𝑝 ∘f · Xp) = 0𝑝)
6	5	fveq2d 6655	. . . . . 6 ⊢ (Xp ∈ (Poly‘ℝ) → (coeff‘(0𝑝 ∘f · Xp)) = (coeff‘0𝑝))
7	4, 6	ax-mp 5	. . . . 5 ⊢ (coeff‘(0𝑝 ∘f · Xp)) = (coeff‘0𝑝)
8		fconstmpt 5576	. . . . . 6 ⊢ (ℕ0 × {0}) = (𝑛 ∈ ℕ0 ↦ 0)
9		coe0 24937	. . . . . 6 ⊢ (coeff‘0𝑝) = (ℕ0 × {0})
10		eqidd 2760	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑛 = 0) → 0 = 0)
11		elnnne0 11933	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ ↔ (𝑛 ∈ ℕ0 ∧ 𝑛 ≠ 0))
12		df-ne 2950	. . . . . . . . . . . 12 ⊢ (𝑛 ≠ 0 ↔ ¬ 𝑛 = 0)
13	12	anbi2i 626	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑛 ≠ 0) ↔ (𝑛 ∈ ℕ0 ∧ ¬ 𝑛 = 0))
14	11, 13	bitr2i 279	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ0 ∧ ¬ 𝑛 = 0) ↔ 𝑛 ∈ ℕ)
15		nnm1nn0 11960	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ0)
16	14, 15	sylbi 220	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0 ∧ ¬ 𝑛 = 0) → (𝑛 − 1) ∈ ℕ0)
17		eqidd 2760	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 − 1) → 0 = 0)
18		fconstmpt 5576	. . . . . . . . . . 11 ⊢ (ℕ0 × {0}) = (𝑚 ∈ ℕ0 ↦ 0)
19	9, 18	eqtri 2782	. . . . . . . . . 10 ⊢ (coeff‘0𝑝) = (𝑚 ∈ ℕ0 ↦ 0)
20		c0ex 10658	. . . . . . . . . 10 ⊢ 0 ∈ V
21	17, 19, 20	fvmpt 6752	. . . . . . . . 9 ⊢ ((𝑛 − 1) ∈ ℕ0 → ((coeff‘0𝑝)‘(𝑛 − 1)) = 0)
22	16, 21	syl 17	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0 ∧ ¬ 𝑛 = 0) → ((coeff‘0𝑝)‘(𝑛 − 1)) = 0)
23	10, 22	ifeqda 4449	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0 → if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1))) = 0)
24	23	mpteq2ia 5116	. . . . . 6 ⊢ (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1)))) = (𝑛 ∈ ℕ0 ↦ 0)
25	8, 9, 24	3eqtr4ri 2793	. . . . 5 ⊢ (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1)))) = (coeff‘0𝑝)
26	7, 25	eqtr4i 2785	. . . 4 ⊢ (coeff‘(0𝑝 ∘f · Xp)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1))))
27		fvoveq1 7166	. . . 4 ⊢ (𝐹 = 0𝑝 → (coeff‘(𝐹 ∘f · Xp)) = (coeff‘(0𝑝 ∘f · Xp)))
28		simpl 487	. . . . . . . 8 ⊢ ((𝐹 = 0𝑝 ∧ 𝑛 ∈ ℕ0) → 𝐹 = 0𝑝)
29	28	fveq2d 6655	. . . . . . 7 ⊢ ((𝐹 = 0𝑝 ∧ 𝑛 ∈ ℕ0) → (coeff‘𝐹) = (coeff‘0𝑝))
30	29	fveq1d 6653	. . . . . 6 ⊢ ((𝐹 = 0𝑝 ∧ 𝑛 ∈ ℕ0) → ((coeff‘𝐹)‘(𝑛 − 1)) = ((coeff‘0𝑝)‘(𝑛 − 1)))
31	30	ifeq2d 4433	. . . . 5 ⊢ ((𝐹 = 0𝑝 ∧ 𝑛 ∈ ℕ0) → if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1))) = if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1))))
32	31	mpteq2dva 5120	. . . 4 ⊢ (𝐹 = 0𝑝 → (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1)))) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1)))))
33	26, 27, 32	3eqtr4a 2820	. . 3 ⊢ (𝐹 = 0𝑝 → (coeff‘(𝐹 ∘f · Xp)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1)))))
34	33	adantl 486	. 2 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝐹 = 0𝑝) → (coeff‘(𝐹 ∘f · Xp)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1)))))
35		simpl 487	. . . 4 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ ¬ 𝐹 = 0𝑝) → 𝐹 ∈ (Poly‘ℝ))
36		elsng 4529	. . . . . 6 ⊢ (𝐹 ∈ (Poly‘ℝ) → (𝐹 ∈ {0𝑝} ↔ 𝐹 = 0𝑝))
37	36	notbid 322	. . . . 5 ⊢ (𝐹 ∈ (Poly‘ℝ) → (¬ 𝐹 ∈ {0𝑝} ↔ ¬ 𝐹 = 0𝑝))
38	37	biimpar 482	. . . 4 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ ¬ 𝐹 = 0𝑝) → ¬ 𝐹 ∈ {0𝑝})
39	35, 38	eldifd 3865	. . 3 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ ¬ 𝐹 = 0𝑝) → 𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}))
40		plymulx0 32030	. . 3 ⊢ (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (coeff‘(𝐹 ∘f · Xp)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1)))))
41	39, 40	syl 17	. 2 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ ¬ 𝐹 = 0𝑝) → (coeff‘(𝐹 ∘f · Xp)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1)))))
42	34, 41	pm2.61dan 813	1 ⊢ (𝐹 ∈ (Poly‘ℝ) → (coeff‘(𝐹 ∘f · Xp)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1)))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1539   ∈ wcel 2112   ≠ wne 2949   ∖ cdif 3851   ⊆ wss 3854  ifcif 4413  {csn 4515   ↦ cmpt 5105   × cxp 5515  ‘cfv 6328  (class class class)co 7143   ∘_f cof 7396  ℂcc 10558  ℝcr 10559  0cc0 10560  1c1 10561   · cmul 10565   − cmin 10893  ℕcn 11659  ℕ₀cn0 11919  0_𝑝c0p 24354  Polycply 24865  X_pcidp 24866  coeffccoe 24867

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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5149  ax-sep 5162  ax-nul 5169  ax-pow 5227  ax-pr 5291  ax-un 7452  ax-inf2 9122  ax-cnex 10616  ax-resscn 10617  ax-1cn 10618  ax-icn 10619  ax-addcl 10620  ax-addrcl 10621  ax-mulcl 10622  ax-mulrcl 10623  ax-mulcom 10624  ax-addass 10625  ax-mulass 10626  ax-distr 10627  ax-i2m1 10628  ax-1ne0 10629  ax-1rid 10630  ax-rnegex 10631  ax-rrecex 10632  ax-cnre 10633  ax-pre-lttri 10634  ax-pre-lttrn 10635  ax-pre-ltadd 10636  ax-pre-mulgt0 10637  ax-pre-sup 10638  ax-addf 10639

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2899  df-ne 2950  df-nel 3054  df-ral 3073  df-rex 3074  df-reu 3075  df-rmo 3076  df-rab 3077  df-v 3409  df-sbc 3694  df-csb 3802  df-dif 3857  df-un 3859  df-in 3861  df-ss 3871  df-pss 3873  df-nul 4222  df-if 4414  df-pw 4489  df-sn 4516  df-pr 4518  df-tp 4520  df-op 4522  df-uni 4792  df-int 4832  df-iun 4878  df-br 5026  df-opab 5088  df-mpt 5106  df-tr 5132  df-id 5423  df-eprel 5428  df-po 5436  df-so 5437  df-fr 5476  df-se 5477  df-we 5478  df-xp 5523  df-rel 5524  df-cnv 5525  df-co 5526  df-dm 5527  df-rn 5528  df-res 5529  df-ima 5530  df-pred 6119  df-ord 6165  df-on 6166  df-lim 6167  df-suc 6168  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7101  df-ov 7146  df-oprab 7147  df-mpo 7148  df-of 7398  df-om 7573  df-1st 7686  df-2nd 7687  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-1o 8105  df-oadd 8109  df-er 8292  df-map 8411  df-pm 8412  df-en 8521  df-dom 8522  df-sdom 8523  df-fin 8524  df-sup 8924  df-inf 8925  df-oi 8992  df-card 9386  df-pnf 10700  df-mnf 10701  df-xr 10702  df-ltxr 10703  df-le 10704  df-sub 10895  df-neg 10896  df-div 11321  df-nn 11660  df-2 11722  df-3 11723  df-n0 11920  df-z 12006  df-uz 12268  df-rp 12416  df-fz 12925  df-fzo 13068  df-fl 13196  df-seq 13404  df-exp 13465  df-hash 13726  df-cj 14491  df-re 14492  df-im 14493  df-sqrt 14627  df-abs 14628  df-clim 14878  df-rlim 14879  df-sum 15076  df-0p 24355  df-ply 24869  df-idp 24870  df-coe 24871  df-dgr 24872

This theorem is referenced by: (None)