Theorem plymulx 34563

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 11212	. . . . . . 7 ⊢ ℝ ⊆ ℂ
2		1re 11261	. . . . . . 7 ⊢ 1 ∈ ℝ
3		plyid 26248	. . . . . . 7 ⊢ ((ℝ ⊆ ℂ ∧ 1 ∈ ℝ) → Xp ∈ (Poly‘ℝ))
4	1, 2, 3	mp2an 692	. . . . . 6 ⊢ Xp ∈ (Poly‘ℝ)
5		plymul02 34561	. . . . . . 7 ⊢ (Xp ∈ (Poly‘ℝ) → (0𝑝 ∘f · Xp) = 0𝑝)
6	5	fveq2d 6910	. . . . . 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 5747	. . . . . 6 ⊢ (ℕ0 × {0}) = (𝑛 ∈ ℕ0 ↦ 0)
9		coe0 26295	. . . . . 6 ⊢ (coeff‘0𝑝) = (ℕ0 × {0})
10		eqidd 2738	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑛 = 0) → 0 = 0)
11		elnnne0 12540	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ ↔ (𝑛 ∈ ℕ0 ∧ 𝑛 ≠ 0))
12		df-ne 2941	. . . . . . . . . . . 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 12567	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ0)
16	14, 15	sylbi 217	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0 ∧ ¬ 𝑛 = 0) → (𝑛 − 1) ∈ ℕ0)
17		eqidd 2738	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 − 1) → 0 = 0)
18		fconstmpt 5747	. . . . . . . . . . 11 ⊢ (ℕ0 × {0}) = (𝑚 ∈ ℕ0 ↦ 0)
19	9, 18	eqtri 2765	. . . . . . . . . 10 ⊢ (coeff‘0𝑝) = (𝑚 ∈ ℕ0 ↦ 0)
20		c0ex 11255	. . . . . . . . . 10 ⊢ 0 ∈ V
21	17, 19, 20	fvmpt 7016	. . . . . . . . 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 4562	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0 → if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1))) = 0)
24	23	mpteq2ia 5245	. . . . . 6 ⊢ (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1)))) = (𝑛 ∈ ℕ0 ↦ 0)
25	8, 9, 24	3eqtr4ri 2776	. . . . 5 ⊢ (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1)))) = (coeff‘0𝑝)
26	7, 25	eqtr4i 2768	. . . 4 ⊢ (coeff‘(0𝑝 ∘f · Xp)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1))))
27		fvoveq1 7454	. . . 4 ⊢ (𝐹 = 0𝑝 → (coeff‘(𝐹 ∘f · Xp)) = (coeff‘(0𝑝 ∘f · Xp)))
28		simpl 482	. . . . . . . 8 ⊢ ((𝐹 = 0𝑝 ∧ 𝑛 ∈ ℕ0) → 𝐹 = 0𝑝)
29	28	fveq2d 6910	. . . . . . 7 ⊢ ((𝐹 = 0𝑝 ∧ 𝑛 ∈ ℕ0) → (coeff‘𝐹) = (coeff‘0𝑝))
30	29	fveq1d 6908	. . . . . 6 ⊢ ((𝐹 = 0𝑝 ∧ 𝑛 ∈ ℕ0) → ((coeff‘𝐹)‘(𝑛 − 1)) = ((coeff‘0𝑝)‘(𝑛 − 1)))
31	30	ifeq2d 4546	. . . . 5 ⊢ ((𝐹 = 0𝑝 ∧ 𝑛 ∈ ℕ0) → if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1))) = if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1))))
32	31	mpteq2dva 5242	. . . 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 2803	. . 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 4640	. . . . . 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 3962	. . 3 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ ¬ 𝐹 = 0𝑝) → 𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}))
40		plymulx0 34562	. . 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 395   = wceq 1540   ∈ wcel 2108   ≠ wne 2940   ∖ cdif 3948   ⊆ wss 3951  ifcif 4525  {csn 4626   ↦ cmpt 5225   × cxp 5683  ‘cfv 6561  (class class class)co 7431   ∘_f cof 7695  ℂcc 11153  ℝcr 11154  0cc0 11155  1c1 11156   · cmul 11160   − cmin 11492  ℕcn 12266  ℕ₀cn0 12526  0_𝑝c0p 25704  Polycply 26223  X_pcidp 26224  coeffccoe 26225

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-rp 13035  df-fz 13548  df-fzo 13695  df-fl 13832  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-rlim 15525  df-sum 15723  df-0p 25705  df-ply 26227  df-idp 26228  df-coe 26229  df-dgr 26230

This theorem is referenced by: (None)