Theorem plymulx 34546

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 11132	. . . . . . 7 ⊢ ℝ ⊆ ℂ
2		1re 11181	. . . . . . 7 ⊢ 1 ∈ ℝ
3		plyid 26121	. . . . . . 7 ⊢ ((ℝ ⊆ ℂ ∧ 1 ∈ ℝ) → Xp ∈ (Poly‘ℝ))
4	1, 2, 3	mp2an 692	. . . . . 6 ⊢ Xp ∈ (Poly‘ℝ)
5		plymul02 34544	. . . . . . 7 ⊢ (Xp ∈ (Poly‘ℝ) → (0𝑝 ∘f · Xp) = 0𝑝)
6	5	fveq2d 6865	. . . . . 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 5703	. . . . . 6 ⊢ (ℕ0 × {0}) = (𝑛 ∈ ℕ0 ↦ 0)
9		coe0 26168	. . . . . 6 ⊢ (coeff‘0𝑝) = (ℕ0 × {0})
10		eqidd 2731	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑛 = 0) → 0 = 0)
11		elnnne0 12463	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ ↔ (𝑛 ∈ ℕ0 ∧ 𝑛 ≠ 0))
12		df-ne 2927	. . . . . . . . . . . 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 12490	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ0)
16	14, 15	sylbi 217	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0 ∧ ¬ 𝑛 = 0) → (𝑛 − 1) ∈ ℕ0)
17		eqidd 2731	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 − 1) → 0 = 0)
18		fconstmpt 5703	. . . . . . . . . . 11 ⊢ (ℕ0 × {0}) = (𝑚 ∈ ℕ0 ↦ 0)
19	9, 18	eqtri 2753	. . . . . . . . . 10 ⊢ (coeff‘0𝑝) = (𝑚 ∈ ℕ0 ↦ 0)
20		c0ex 11175	. . . . . . . . . 10 ⊢ 0 ∈ V
21	17, 19, 20	fvmpt 6971	. . . . . . . . 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 4528	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0 → if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1))) = 0)
24	23	mpteq2ia 5205	. . . . . 6 ⊢ (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1)))) = (𝑛 ∈ ℕ0 ↦ 0)
25	8, 9, 24	3eqtr4ri 2764	. . . . 5 ⊢ (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1)))) = (coeff‘0𝑝)
26	7, 25	eqtr4i 2756	. . . 4 ⊢ (coeff‘(0𝑝 ∘f · Xp)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1))))
27		fvoveq1 7413	. . . 4 ⊢ (𝐹 = 0𝑝 → (coeff‘(𝐹 ∘f · Xp)) = (coeff‘(0𝑝 ∘f · Xp)))
28		simpl 482	. . . . . . . 8 ⊢ ((𝐹 = 0𝑝 ∧ 𝑛 ∈ ℕ0) → 𝐹 = 0𝑝)
29	28	fveq2d 6865	. . . . . . 7 ⊢ ((𝐹 = 0𝑝 ∧ 𝑛 ∈ ℕ0) → (coeff‘𝐹) = (coeff‘0𝑝))
30	29	fveq1d 6863	. . . . . 6 ⊢ ((𝐹 = 0𝑝 ∧ 𝑛 ∈ ℕ0) → ((coeff‘𝐹)‘(𝑛 − 1)) = ((coeff‘0𝑝)‘(𝑛 − 1)))
31	30	ifeq2d 4512	. . . . 5 ⊢ ((𝐹 = 0𝑝 ∧ 𝑛 ∈ ℕ0) → if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1))) = if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1))))
32	31	mpteq2dva 5203	. . . 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 2791	. . 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 4606	. . . . . 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 3928	. . 3 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ ¬ 𝐹 = 0𝑝) → 𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}))
40		plymulx0 34545	. . 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 2926   ∖ cdif 3914   ⊆ wss 3917  ifcif 4491  {csn 4592   ↦ cmpt 5191   × cxp 5639  ‘cfv 6514  (class class class)co 7390   ∘_f cof 7654  ℂcc 11073  ℝcr 11074  0cc0 11075  1c1 11076   · cmul 11080   − cmin 11412  ℕcn 12193  ℕ₀cn0 12449  0_𝑝c0p 25577  Polycply 26096  X_pcidp 26097  coeffccoe 26098

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-inf2 9601  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-isom 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-of 7656  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-er 8674  df-map 8804  df-pm 8805  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-sup 9400  df-inf 9401  df-oi 9470  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-div 11843  df-nn 12194  df-2 12256  df-3 12257  df-n0 12450  df-z 12537  df-uz 12801  df-rp 12959  df-fz 13476  df-fzo 13623  df-fl 13761  df-seq 13974  df-exp 14034  df-hash 14303  df-cj 15072  df-re 15073  df-im 15074  df-sqrt 15208  df-abs 15209  df-clim 15461  df-rlim 15462  df-sum 15660  df-0p 25578  df-ply 26100  df-idp 26101  df-coe 26102  df-dgr 26103

This theorem is referenced by: (None)