Theorem plymulx 34522

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 11066	. . . . . . 7 ⊢ ℝ ⊆ ℂ
2		1re 11115	. . . . . . 7 ⊢ 1 ∈ ℝ
3		plyid 26112	. . . . . . 7 ⊢ ((ℝ ⊆ ℂ ∧ 1 ∈ ℝ) → Xp ∈ (Poly‘ℝ))
4	1, 2, 3	mp2an 692	. . . . . 6 ⊢ Xp ∈ (Poly‘ℝ)
5		plymul02 34520	. . . . . . 7 ⊢ (Xp ∈ (Poly‘ℝ) → (0𝑝 ∘f · Xp) = 0𝑝)
6	5	fveq2d 6826	. . . . . 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 5681	. . . . . 6 ⊢ (ℕ0 × {0}) = (𝑛 ∈ ℕ0 ↦ 0)
9		coe0 26159	. . . . . 6 ⊢ (coeff‘0𝑝) = (ℕ0 × {0})
10		eqidd 2730	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑛 = 0) → 0 = 0)
11		elnnne0 12398	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ ↔ (𝑛 ∈ ℕ0 ∧ 𝑛 ≠ 0))
12		df-ne 2926	. . . . . . . . . . . 12 ⊢ (𝑛 ≠ 0 ↔ ¬ 𝑛 = 0)
13	12	anbi2i 623	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑛 ≠ 0) ↔ (𝑛 ∈ ℕ0 ∧ ¬ 𝑛 = 0))
14	11, 13	bitr2i 276	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ0 ∧ ¬ 𝑛 = 0) ↔ 𝑛 ∈ ℕ)
15		nnm1nn0 12425	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ0)
16	14, 15	sylbi 217	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0 ∧ ¬ 𝑛 = 0) → (𝑛 − 1) ∈ ℕ0)
17		eqidd 2730	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 − 1) → 0 = 0)
18		fconstmpt 5681	. . . . . . . . . . 11 ⊢ (ℕ0 × {0}) = (𝑚 ∈ ℕ0 ↦ 0)
19	9, 18	eqtri 2752	. . . . . . . . . 10 ⊢ (coeff‘0𝑝) = (𝑚 ∈ ℕ0 ↦ 0)
20		c0ex 11109	. . . . . . . . . 10 ⊢ 0 ∈ V
21	17, 19, 20	fvmpt 6930	. . . . . . . . 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 4513	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0 → if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1))) = 0)
24	23	mpteq2ia 5187	. . . . . 6 ⊢ (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1)))) = (𝑛 ∈ ℕ0 ↦ 0)
25	8, 9, 24	3eqtr4ri 2763	. . . . 5 ⊢ (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1)))) = (coeff‘0𝑝)
26	7, 25	eqtr4i 2755	. . . 4 ⊢ (coeff‘(0𝑝 ∘f · Xp)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1))))
27		fvoveq1 7372	. . . 4 ⊢ (𝐹 = 0𝑝 → (coeff‘(𝐹 ∘f · Xp)) = (coeff‘(0𝑝 ∘f · Xp)))
28		simpl 482	. . . . . . . 8 ⊢ ((𝐹 = 0𝑝 ∧ 𝑛 ∈ ℕ0) → 𝐹 = 0𝑝)
29	28	fveq2d 6826	. . . . . . 7 ⊢ ((𝐹 = 0𝑝 ∧ 𝑛 ∈ ℕ0) → (coeff‘𝐹) = (coeff‘0𝑝))
30	29	fveq1d 6824	. . . . . 6 ⊢ ((𝐹 = 0𝑝 ∧ 𝑛 ∈ ℕ0) → ((coeff‘𝐹)‘(𝑛 − 1)) = ((coeff‘0𝑝)‘(𝑛 − 1)))
31	30	ifeq2d 4497	. . . . 5 ⊢ ((𝐹 = 0𝑝 ∧ 𝑛 ∈ ℕ0) → if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1))) = if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1))))
32	31	mpteq2dva 5185	. . . 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 2790	. . 3 ⊢ (𝐹 = 0𝑝 → (coeff‘(𝐹 ∘f · Xp)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1)))))
34	33	adantl 481	. 2 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝐹 = 0𝑝) → (coeff‘(𝐹 ∘f · Xp)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1)))))
35		simpl 482	. . . 4 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ ¬ 𝐹 = 0𝑝) → 𝐹 ∈ (Poly‘ℝ))
36		elsng 4591	. . . . . 6 ⊢ (𝐹 ∈ (Poly‘ℝ) → (𝐹 ∈ {0𝑝} ↔ 𝐹 = 0𝑝))
37	36	notbid 318	. . . . 5 ⊢ (𝐹 ∈ (Poly‘ℝ) → (¬ 𝐹 ∈ {0𝑝} ↔ ¬ 𝐹 = 0𝑝))
38	37	biimpar 477	. . . 4 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ ¬ 𝐹 = 0𝑝) → ¬ 𝐹 ∈ {0𝑝})
39	35, 38	eldifd 3914	. . 3 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ ¬ 𝐹 = 0𝑝) → 𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}))
40		plymulx0 34521	. . 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 812	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 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925   ∖ cdif 3900   ⊆ wss 3903  ifcif 4476  {csn 4577   ↦ cmpt 5173   × cxp 5617  ‘cfv 6482  (class class class)co 7349   ∘_f cof 7611  ℂcc 11007  ℝcr 11008  0cc0 11009  1c1 11010   · cmul 11014   − cmin 11347  ℕcn 12128  ℕ₀cn0 12384  0_𝑝c0p 25568  Polycply 26087  X_pcidp 26088  coeffccoe 26089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-inf2 9537  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086  ax-pre-sup 11087

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-isom 6491  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-of 7613  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-er 8625  df-map 8755  df-pm 8756  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-sup 9332  df-inf 9333  df-oi 9402  df-card 9835  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-div 11778  df-nn 12129  df-2 12191  df-3 12192  df-n0 12385  df-z 12472  df-uz 12736  df-rp 12894  df-fz 13411  df-fzo 13558  df-fl 13696  df-seq 13909  df-exp 13969  df-hash 14238  df-cj 15006  df-re 15007  df-im 15008  df-sqrt 15142  df-abs 15143  df-clim 15395  df-rlim 15396  df-sum 15594  df-0p 25569  df-ply 26091  df-idp 26092  df-coe 26093  df-dgr 26094

This theorem is referenced by: (None)