Theorem plymulx 34712

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 11090	. . . . . . 7 ⊢ ℝ ⊆ ℂ
2		1re 11139	. . . . . . 7 ⊢ 1 ∈ ℝ
3		plyid 26188	. . . . . . 7 ⊢ ((ℝ ⊆ ℂ ∧ 1 ∈ ℝ) → Xp ∈ (Poly‘ℝ))
4	1, 2, 3	mp2an 693	. . . . . 6 ⊢ Xp ∈ (Poly‘ℝ)
5		plymul02 34710	. . . . . . 7 ⊢ (Xp ∈ (Poly‘ℝ) → (0𝑝 ∘f · Xp) = 0𝑝)
6	5	fveq2d 6840	. . . . . 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 5688	. . . . . 6 ⊢ (ℕ0 × {0}) = (𝑛 ∈ ℕ0 ↦ 0)
9		coe0 26235	. . . . . 6 ⊢ (coeff‘0𝑝) = (ℕ0 × {0})
10		eqidd 2738	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑛 = 0) → 0 = 0)
11		elnnne0 12446	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ ↔ (𝑛 ∈ ℕ0 ∧ 𝑛 ≠ 0))
12		df-ne 2934	. . . . . . . . . . . 12 ⊢ (𝑛 ≠ 0 ↔ ¬ 𝑛 = 0)
13	12	anbi2i 624	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑛 ≠ 0) ↔ (𝑛 ∈ ℕ0 ∧ ¬ 𝑛 = 0))
14	11, 13	bitr2i 276	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ0 ∧ ¬ 𝑛 = 0) ↔ 𝑛 ∈ ℕ)
15		nnm1nn0 12473	. . . . . . . . . 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 5688	. . . . . . . . . . 11 ⊢ (ℕ0 × {0}) = (𝑚 ∈ ℕ0 ↦ 0)
19	9, 18	eqtri 2760	. . . . . . . . . 10 ⊢ (coeff‘0𝑝) = (𝑚 ∈ ℕ0 ↦ 0)
20		c0ex 11133	. . . . . . . . . 10 ⊢ 0 ∈ V
21	17, 19, 20	fvmpt 6943	. . . . . . . . 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 4504	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0 → if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1))) = 0)
24	23	mpteq2ia 5181	. . . . . 6 ⊢ (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1)))) = (𝑛 ∈ ℕ0 ↦ 0)
25	8, 9, 24	3eqtr4ri 2771	. . . . 5 ⊢ (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1)))) = (coeff‘0𝑝)
26	7, 25	eqtr4i 2763	. . . 4 ⊢ (coeff‘(0𝑝 ∘f · Xp)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1))))
27		fvoveq1 7385	. . . 4 ⊢ (𝐹 = 0𝑝 → (coeff‘(𝐹 ∘f · Xp)) = (coeff‘(0𝑝 ∘f · Xp)))
28		simpl 482	. . . . . . . 8 ⊢ ((𝐹 = 0𝑝 ∧ 𝑛 ∈ ℕ0) → 𝐹 = 0𝑝)
29	28	fveq2d 6840	. . . . . . 7 ⊢ ((𝐹 = 0𝑝 ∧ 𝑛 ∈ ℕ0) → (coeff‘𝐹) = (coeff‘0𝑝))
30	29	fveq1d 6838	. . . . . 6 ⊢ ((𝐹 = 0𝑝 ∧ 𝑛 ∈ ℕ0) → ((coeff‘𝐹)‘(𝑛 − 1)) = ((coeff‘0𝑝)‘(𝑛 − 1)))
31	30	ifeq2d 4488	. . . . 5 ⊢ ((𝐹 = 0𝑝 ∧ 𝑛 ∈ ℕ0) → if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1))) = if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1))))
32	31	mpteq2dva 5179	. . . 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 2798	. . 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 4582	. . . . . 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 3901	. . 3 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ ¬ 𝐹 = 0𝑝) → 𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}))
40		plymulx0 34711	. . 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 1542   ∈ wcel 2114   ≠ wne 2933   ∖ cdif 3887   ⊆ wss 3890  ifcif 4467  {csn 4568   ↦ cmpt 5167   × cxp 5624  ‘cfv 6494  (class class class)co 7362   ∘_f cof 7624  ℂcc 11031  ℝcr 11032  0cc0 11033  1c1 11034   · cmul 11038   − cmin 11372  ℕcn 12169  ℕ₀cn0 12432  0_𝑝c0p 25650  Polycply 26163  X_pcidp 26164  coeffccoe 26165

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684  ax-inf2 9557  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110  ax-pre-sup 11111

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-se 5580  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-isom 6503  df-riota 7319  df-ov 7365  df-oprab 7366  df-mpo 7367  df-of 7626  df-om 7813  df-1st 7937  df-2nd 7938  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-1o 8400  df-er 8638  df-map 8770  df-pm 8771  df-en 8889  df-dom 8890  df-sdom 8891  df-fin 8892  df-sup 9350  df-inf 9351  df-oi 9420  df-card 9858  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-div 11803  df-nn 12170  df-2 12239  df-3 12240  df-n0 12433  df-z 12520  df-uz 12784  df-rp 12938  df-fz 13457  df-fzo 13604  df-fl 13746  df-seq 13959  df-exp 14019  df-hash 14288  df-cj 15056  df-re 15057  df-im 15058  df-sqrt 15192  df-abs 15193  df-clim 15445  df-rlim 15446  df-sum 15644  df-0p 25651  df-ply 26167  df-idp 26168  df-coe 26169  df-dgr 26170

This theorem is referenced by: (None)