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Theorem dgrcolem2 25340

Description: Lemma for dgrco 25341. (Contributed by Mario Carneiro, 15-Sep-2014.)

**Hypotheses**
Ref	Expression
dgrco.1	⊢ 𝑀 = (deg‘𝐹)
dgrco.2	⊢ 𝑁 = (deg‘𝐺)
dgrco.3	⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))
dgrco.4	⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆))
dgrco.5	⊢ 𝐴 = (coeff‘𝐹)
dgrco.6	⊢ (𝜑 → 𝐷 ∈ ℕ₀)
dgrco.7	⊢ (𝜑 → 𝑀 = (𝐷 + 1))
dgrco.8	⊢ (𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 𝐷 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁)))

**Assertion**
Ref	Expression
dgrcolem2	⊢ (𝜑 → (deg‘(𝐹 ∘ 𝐺)) = (𝑀 · 𝑁))

Distinct variable groups: 𝐴,𝑓 𝑓,𝐹 𝑓,𝑀 𝑓,𝑁 𝐷,𝑓 𝑓,𝐺 𝜑,𝑓 Allowed substitution hint: 𝑆(𝑓)

Proof of Theorem dgrcolem2 Dummy variables 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dgrco.4	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆))
2		plyf 25264	. . . . . . . . . . 11 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ)
3	1, 2	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺:ℂ⟶ℂ)
4	3	ffvelrnda 6943	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝐺‘𝑥) ∈ ℂ)
5		dgrco.3	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))
6		plyf 25264	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
7	5, 6	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:ℂ⟶ℂ)
8	7	ffvelrnda 6943	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐺‘𝑥) ∈ ℂ) → (𝐹‘(𝐺‘𝑥)) ∈ ℂ)
9	4, 8	syldan 590	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝐹‘(𝐺‘𝑥)) ∈ ℂ)
10		dgrco.5	. . . . . . . . . . . . 13 ⊢ 𝐴 = (coeff‘𝐹)
11	10	coef3 25298	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ₀⟶ℂ)
12	5, 11	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴:ℕ₀⟶ℂ)
13		dgrco.1	. . . . . . . . . . . 12 ⊢ 𝑀 = (deg‘𝐹)
14		dgrcl 25299	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ₀)
15	5, 14	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (deg‘𝐹) ∈ ℕ₀)
16	13, 15	eqeltrid 2843	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
17	12, 16	ffvelrnd 6944	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴‘𝑀) ∈ ℂ)
18	17	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝐴‘𝑀) ∈ ℂ)
19	16	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝑀 ∈ ℕ₀)
20	4, 19	expcld 13792	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((𝐺‘𝑥)↑𝑀) ∈ ℂ)
21	18, 20	mulcld 10926	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)) ∈ ℂ)
22	9, 21	npcand 11266	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))) + ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))) = (𝐹‘(𝐺‘𝑥)))
23	22	mpteq2dva 5170	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ (((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))) + ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))) = (𝑥 ∈ ℂ ↦ (𝐹‘(𝐺‘𝑥))))
24		cnex 10883	. . . . . . . 8 ⊢ ℂ ∈ V
25	24	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℂ ∈ V)
26	9, 21	subcld 11262	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))) ∈ ℂ)
27		eqidd 2739	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))) = (𝑥 ∈ ℂ ↦ ((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))))
28		eqidd 2739	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))) = (𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))))
29	25, 26, 21, 27, 28	offval2 7531	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ ((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))) ∘_f + (𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))) = (𝑥 ∈ ℂ ↦ (((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))) + ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))))
30	3	feqmptd 6819	. . . . . . 7 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ ℂ ↦ (𝐺‘𝑥)))
31	7	feqmptd 6819	. . . . . . 7 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ ℂ ↦ (𝐹‘𝑦)))
32		fveq2 6756	. . . . . . 7 ⊢ (𝑦 = (𝐺‘𝑥) → (𝐹‘𝑦) = (𝐹‘(𝐺‘𝑥)))
33	4, 30, 31, 32	fmptco 6983	. . . . . 6 ⊢ (𝜑 → (𝐹 ∘ 𝐺) = (𝑥 ∈ ℂ ↦ (𝐹‘(𝐺‘𝑥))))
34	23, 29, 33	3eqtr4rd 2789	. . . . 5 ⊢ (𝜑 → (𝐹 ∘ 𝐺) = ((𝑥 ∈ ℂ ↦ ((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))) ∘_f + (𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))))
35	34	fveq2d 6760	. . . 4 ⊢ (𝜑 → (deg‘(𝐹 ∘ 𝐺)) = (deg‘((𝑥 ∈ ℂ ↦ ((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))) ∘_f + (𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))))))
36	35	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (deg‘(𝐹 ∘ 𝐺)) = (deg‘((𝑥 ∈ ℂ ↦ ((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))) ∘_f + (𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))))))
37	25, 9, 21, 33, 28	offval2 7531	. . . . . 6 ⊢ (𝜑 → ((𝐹 ∘ 𝐺) ∘_f − (𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))) = (𝑥 ∈ ℂ ↦ ((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))))
38		plyssc 25266	. . . . . . . . 9 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
39	38, 5	sselid 3915	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (Poly‘ℂ))
40	38, 1	sselid 3915	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ (Poly‘ℂ))
41		addcl 10884	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (𝑧 + 𝑤) ∈ ℂ)
42	41	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ)) → (𝑧 + 𝑤) ∈ ℂ)
43		mulcl 10886	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (𝑧 · 𝑤) ∈ ℂ)
44	43	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ)) → (𝑧 · 𝑤) ∈ ℂ)
45	39, 40, 42, 44	plyco 25307	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∘ 𝐺) ∈ (Poly‘ℂ))
46		eqidd 2739	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))) = (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))
47		oveq1 7262	. . . . . . . . . 10 ⊢ (𝑦 = (𝐺‘𝑥) → (𝑦↑𝑀) = ((𝐺‘𝑥)↑𝑀))
48	47	oveq2d 7271	. . . . . . . . 9 ⊢ (𝑦 = (𝐺‘𝑥) → ((𝐴‘𝑀) · (𝑦↑𝑀)) = ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))
49	4, 30, 46, 48	fmptco 6983	. . . . . . . 8 ⊢ (𝜑 → ((𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))) ∘ 𝐺) = (𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))))
50		ssidd 3940	. . . . . . . . . 10 ⊢ (𝜑 → ℂ ⊆ ℂ)
51		eqid 2738	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))) = (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))
52	51	ply1term 25270	. . . . . . . . . 10 ⊢ ((ℂ ⊆ ℂ ∧ (𝐴‘𝑀) ∈ ℂ ∧ 𝑀 ∈ ℕ₀) → (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))) ∈ (Poly‘ℂ))
53	50, 17, 16, 52	syl3anc 1369	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))) ∈ (Poly‘ℂ))
54	53, 40, 42, 44	plyco 25307	. . . . . . . 8 ⊢ (𝜑 → ((𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))) ∘ 𝐺) ∈ (Poly‘ℂ))
55	49, 54	eqeltrrd 2840	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))) ∈ (Poly‘ℂ))
56		plysubcl 25288	. . . . . . 7 ⊢ (((𝐹 ∘ 𝐺) ∈ (Poly‘ℂ) ∧ (𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))) ∈ (Poly‘ℂ)) → ((𝐹 ∘ 𝐺) ∘_f − (𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))) ∈ (Poly‘ℂ))
57	45, 55, 56	syl2anc 583	. . . . . 6 ⊢ (𝜑 → ((𝐹 ∘ 𝐺) ∘_f − (𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))) ∈ (Poly‘ℂ))
58	37, 57	eqeltrrd 2840	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))) ∈ (Poly‘ℂ))
59	58	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (𝑥 ∈ ℂ ↦ ((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))) ∈ (Poly‘ℂ))
60	55	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))) ∈ (Poly‘ℂ))
61		dgrco.7	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 = (𝐷 + 1))
62		dgrco.6	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐷 ∈ ℕ₀)
63		nn0p1nn 12202	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ ℕ₀ → (𝐷 + 1) ∈ ℕ)
64	62, 63	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐷 + 1) ∈ ℕ)
65	61, 64	eqeltrd 2839	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℕ)
66	65	nngt0d 11952	. . . . . . . . 9 ⊢ (𝜑 → 0 < 𝑀)
67		fveq2 6756	. . . . . . . . . . 11 ⊢ ((𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) = 0_𝑝 → (deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) = (deg‘0_𝑝))
68		dgr0 25328	. . . . . . . . . . 11 ⊢ (deg‘0_𝑝) = 0
69	67, 68	eqtrdi 2795	. . . . . . . . . 10 ⊢ ((𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) = 0_𝑝 → (deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) = 0)
70	69	breq1d 5080	. . . . . . . . 9 ⊢ ((𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) = 0_𝑝 → ((deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) < 𝑀 ↔ 0 < 𝑀))
71	66, 70	syl5ibrcom 246	. . . . . . . 8 ⊢ (𝜑 → ((𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) = 0_𝑝 → (deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) < 𝑀))
72		idd 24	. . . . . . . 8 ⊢ (𝜑 → ((deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) < 𝑀 → (deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) < 𝑀))
73		eqid 2738	. . . . . . . . . . . 12 ⊢ (deg‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) = (deg‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))
74	13, 73	dgrsub 25338	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘ℂ) ∧ (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))) ∈ (Poly‘ℂ)) → (deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) ≤ if(𝑀 ≤ (deg‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))), (deg‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))), 𝑀))
75	39, 53, 74	syl2anc 583	. . . . . . . . . 10 ⊢ (𝜑 → (deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) ≤ if(𝑀 ≤ (deg‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))), (deg‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))), 𝑀))
76	65	nnne0d 11953	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ≠ 0)
77	13, 10	dgreq0 25331	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐹 = 0_𝑝 ↔ (𝐴‘𝑀) = 0))
78	5, 77	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐹 = 0_𝑝 ↔ (𝐴‘𝑀) = 0))
79		fveq2 6756	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 = 0_𝑝 → (deg‘𝐹) = (deg‘0_𝑝))
80	79, 68	eqtrdi 2795	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 = 0_𝑝 → (deg‘𝐹) = 0)
81	13, 80	syl5eq 2791	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 = 0_𝑝 → 𝑀 = 0)
82	78, 81	syl6bir 253	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐴‘𝑀) = 0 → 𝑀 = 0))
83	82	necon3d 2963	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑀 ≠ 0 → (𝐴‘𝑀) ≠ 0))
84	76, 83	mpd 15	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴‘𝑀) ≠ 0)
85	51	dgr1term 25326	. . . . . . . . . . . . 13 ⊢ (((𝐴‘𝑀) ∈ ℂ ∧ (𝐴‘𝑀) ≠ 0 ∧ 𝑀 ∈ ℕ₀) → (deg‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) = 𝑀)
86	17, 84, 16, 85	syl3anc 1369	. . . . . . . . . . . 12 ⊢ (𝜑 → (deg‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) = 𝑀)
87	86	ifeq1d 4475	. . . . . . . . . . 11 ⊢ (𝜑 → if(𝑀 ≤ (deg‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))), (deg‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))), 𝑀) = if(𝑀 ≤ (deg‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))), 𝑀, 𝑀))
88		ifid 4496	. . . . . . . . . . 11 ⊢ if(𝑀 ≤ (deg‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))), 𝑀, 𝑀) = 𝑀
89	87, 88	eqtrdi 2795	. . . . . . . . . 10 ⊢ (𝜑 → if(𝑀 ≤ (deg‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))), (deg‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))), 𝑀) = 𝑀)
90	75, 89	breqtrd 5096	. . . . . . . . 9 ⊢ (𝜑 → (deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) ≤ 𝑀)
91		eqid 2738	. . . . . . . . . . . . 13 ⊢ (coeff‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) = (coeff‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))
92	10, 91	coesub 25323	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Poly‘ℂ) ∧ (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))) ∈ (Poly‘ℂ)) → (coeff‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) = (𝐴 ∘_f − (coeff‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))))
93	39, 53, 92	syl2anc 583	. . . . . . . . . . 11 ⊢ (𝜑 → (coeff‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) = (𝐴 ∘_f − (coeff‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))))
94	93	fveq1d 6758	. . . . . . . . . 10 ⊢ (𝜑 → ((coeff‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))))‘𝑀) = ((𝐴 ∘_f − (coeff‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))))‘𝑀))
95	12	ffnd 6585	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 Fn ℕ₀)
96	91	coef3 25298	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))) ∈ (Poly‘ℂ) → (coeff‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))):ℕ₀⟶ℂ)
97	53, 96	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (coeff‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))):ℕ₀⟶ℂ)
98	97	ffnd 6585	. . . . . . . . . . . 12 ⊢ (𝜑 → (coeff‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) Fn ℕ₀)
99		nn0ex 12169	. . . . . . . . . . . . 13 ⊢ ℕ₀ ∈ V
100	99	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ℕ₀ ∈ V)
101		inidm 4149	. . . . . . . . . . . 12 ⊢ (ℕ₀ ∩ ℕ₀) = ℕ₀
102		eqidd 2739	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑀 ∈ ℕ₀) → (𝐴‘𝑀) = (𝐴‘𝑀))
103	51	coe1term 25325	. . . . . . . . . . . . . . 15 ⊢ (((𝐴‘𝑀) ∈ ℂ ∧ 𝑀 ∈ ℕ₀ ∧ 𝑀 ∈ ℕ₀) → ((coeff‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))‘𝑀) = if(𝑀 = 𝑀, (𝐴‘𝑀), 0))
104	17, 16, 16, 103	syl3anc 1369	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((coeff‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))‘𝑀) = if(𝑀 = 𝑀, (𝐴‘𝑀), 0))
105		eqid 2738	. . . . . . . . . . . . . . 15 ⊢ 𝑀 = 𝑀
106	105	iftruei 4463	. . . . . . . . . . . . . 14 ⊢ if(𝑀 = 𝑀, (𝐴‘𝑀), 0) = (𝐴‘𝑀)
107	104, 106	eqtrdi 2795	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((coeff‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))‘𝑀) = (𝐴‘𝑀))
108	107	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑀 ∈ ℕ₀) → ((coeff‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))‘𝑀) = (𝐴‘𝑀))
109	95, 98, 100, 100, 101, 102, 108	ofval 7522	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑀 ∈ ℕ₀) → ((𝐴 ∘_f − (coeff‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))))‘𝑀) = ((𝐴‘𝑀) − (𝐴‘𝑀)))
110	16, 109	mpdan 683	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 ∘_f − (coeff‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))))‘𝑀) = ((𝐴‘𝑀) − (𝐴‘𝑀)))
111	17	subidd 11250	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴‘𝑀) − (𝐴‘𝑀)) = 0)
112	94, 110, 111	3eqtrd 2782	. . . . . . . . 9 ⊢ (𝜑 → ((coeff‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))))‘𝑀) = 0)
113		plysubcl 25288	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘ℂ) ∧ (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))) ∈ (Poly‘ℂ)) → (𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) ∈ (Poly‘ℂ))
114	39, 53, 113	syl2anc 583	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) ∈ (Poly‘ℂ))
115		eqid 2738	. . . . . . . . . . 11 ⊢ (deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) = (deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))))
116		eqid 2738	. . . . . . . . . . 11 ⊢ (coeff‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) = (coeff‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))))
117	115, 116	dgrlt 25332	. . . . . . . . . 10 ⊢ (((𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) ∈ (Poly‘ℂ) ∧ 𝑀 ∈ ℕ₀) → (((𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) = 0_𝑝 ∨ (deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) < 𝑀) ↔ ((deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) ≤ 𝑀 ∧ ((coeff‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))))‘𝑀) = 0)))
118	114, 16, 117	syl2anc 583	. . . . . . . . 9 ⊢ (𝜑 → (((𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) = 0_𝑝 ∨ (deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) < 𝑀) ↔ ((deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) ≤ 𝑀 ∧ ((coeff‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))))‘𝑀) = 0)))
119	90, 112, 118	mpbir2and 709	. . . . . . . 8 ⊢ (𝜑 → ((𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) = 0_𝑝 ∨ (deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) < 𝑀))
120	71, 72, 119	mpjaod 856	. . . . . . 7 ⊢ (𝜑 → (deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) < 𝑀)
121	120	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) < 𝑀)
122		dgrcl 25299	. . . . . . . . . 10 ⊢ ((𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) ∈ (Poly‘ℂ) → (deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) ∈ ℕ₀)
123	114, 122	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) ∈ ℕ₀)
124	123	nn0red 12224	. . . . . . . 8 ⊢ (𝜑 → (deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) ∈ ℝ)
125	124	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) ∈ ℝ)
126	16	nn0red 12224	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℝ)
127	126	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝑀 ∈ ℝ)
128		nnre 11910	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
129	128	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℝ)
130		nngt0 11934	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁)
131	130	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 0 < 𝑁)
132		ltmul1 11755	. . . . . . 7 ⊢ (((deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → ((deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) < 𝑀 ↔ ((deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) · 𝑁) < (𝑀 · 𝑁)))
133	125, 127, 129, 131, 132	syl112anc 1372	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → ((deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) < 𝑀 ↔ ((deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) · 𝑁) < (𝑀 · 𝑁)))
134	121, 133	mpbid 231	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → ((deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) · 𝑁) < (𝑀 · 𝑁))
135	7	ffvelrnda 6943	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝐹‘𝑦) ∈ ℂ)
136	17	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝐴‘𝑀) ∈ ℂ)
137		id 22	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℂ → 𝑦 ∈ ℂ)
138		expcl 13728	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℂ ∧ 𝑀 ∈ ℕ₀) → (𝑦↑𝑀) ∈ ℂ)
139	137, 16, 138	syl2anr 596	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑦↑𝑀) ∈ ℂ)
140	136, 139	mulcld 10926	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → ((𝐴‘𝑀) · (𝑦↑𝑀)) ∈ ℂ)
141	25, 135, 140, 31, 46	offval2 7531	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) = (𝑦 ∈ ℂ ↦ ((𝐹‘𝑦) − ((𝐴‘𝑀) · (𝑦↑𝑀)))))
142	32, 48	oveq12d 7273	. . . . . . . . 9 ⊢ (𝑦 = (𝐺‘𝑥) → ((𝐹‘𝑦) − ((𝐴‘𝑀) · (𝑦↑𝑀))) = ((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))))
143	4, 30, 141, 142	fmptco 6983	. . . . . . . 8 ⊢ (𝜑 → ((𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) ∘ 𝐺) = (𝑥 ∈ ℂ ↦ ((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))))
144	143	fveq2d 6760	. . . . . . 7 ⊢ (𝜑 → (deg‘((𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) ∘ 𝐺)) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))))))
145	120, 61	breqtrd 5096	. . . . . . . . 9 ⊢ (𝜑 → (deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) < (𝐷 + 1))
146		nn0leltp1 12309	. . . . . . . . . 10 ⊢ (((deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) ∈ ℕ₀ ∧ 𝐷 ∈ ℕ₀) → ((deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) ≤ 𝐷 ↔ (deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) < (𝐷 + 1)))
147	123, 62, 146	syl2anc 583	. . . . . . . . 9 ⊢ (𝜑 → ((deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) ≤ 𝐷 ↔ (deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) < (𝐷 + 1)))
148	145, 147	mpbird 256	. . . . . . . 8 ⊢ (𝜑 → (deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) ≤ 𝐷)
149		fveq2 6756	. . . . . . . . . . 11 ⊢ (𝑓 = (𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) → (deg‘𝑓) = (deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))))
150	149	breq1d 5080	. . . . . . . . . 10 ⊢ (𝑓 = (𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) → ((deg‘𝑓) ≤ 𝐷 ↔ (deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) ≤ 𝐷))
151		coeq1 5755	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) → (𝑓 ∘ 𝐺) = ((𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) ∘ 𝐺))
152	151	fveq2d 6760	. . . . . . . . . . 11 ⊢ (𝑓 = (𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) → (deg‘(𝑓 ∘ 𝐺)) = (deg‘((𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) ∘ 𝐺)))
153	149	oveq1d 7270	. . . . . . . . . . 11 ⊢ (𝑓 = (𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) → ((deg‘𝑓) · 𝑁) = ((deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) · 𝑁))
154	152, 153	eqeq12d 2754	. . . . . . . . . 10 ⊢ (𝑓 = (𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) → ((deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁) ↔ (deg‘((𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) ∘ 𝐺)) = ((deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) · 𝑁)))
155	150, 154	imbi12d 344	. . . . . . . . 9 ⊢ (𝑓 = (𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) → (((deg‘𝑓) ≤ 𝐷 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁)) ↔ ((deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) ≤ 𝐷 → (deg‘((𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) ∘ 𝐺)) = ((deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) · 𝑁))))
156		dgrco.8	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 𝐷 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁)))
157	155, 156, 114	rspcdva 3554	. . . . . . . 8 ⊢ (𝜑 → ((deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) ≤ 𝐷 → (deg‘((𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) ∘ 𝐺)) = ((deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) · 𝑁)))
158	148, 157	mpd 15	. . . . . . 7 ⊢ (𝜑 → (deg‘((𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) ∘ 𝐺)) = ((deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) · 𝑁))
159	144, 158	eqtr3d 2780	. . . . . 6 ⊢ (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))))) = ((deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) · 𝑁))
160	159	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (deg‘(𝑥 ∈ ℂ ↦ ((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))))) = ((deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) · 𝑁))
161		fconstmpt 5640	. . . . . . . . . . 11 ⊢ (ℂ × {(𝐴‘𝑀)}) = (𝑥 ∈ ℂ ↦ (𝐴‘𝑀))
162	161	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (ℂ × {(𝐴‘𝑀)}) = (𝑥 ∈ ℂ ↦ (𝐴‘𝑀)))
163		eqidd 2739	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀)) = (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀)))
164	25, 18, 20, 162, 163	offval2 7531	. . . . . . . . 9 ⊢ (𝜑 → ((ℂ × {(𝐴‘𝑀)}) ∘_f · (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀))) = (𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))))
165	164	fveq2d 6760	. . . . . . . 8 ⊢ (𝜑 → (deg‘((ℂ × {(𝐴‘𝑀)}) ∘_f · (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀)))) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))))
166		eqidd 2739	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑦 ∈ ℂ ↦ (𝑦↑𝑀)) = (𝑦 ∈ ℂ ↦ (𝑦↑𝑀)))
167	4, 30, 166, 47	fmptco 6983	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑦 ∈ ℂ ↦ (𝑦↑𝑀)) ∘ 𝐺) = (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀)))
168		1cnd 10901	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℂ)
169		plypow 25271	. . . . . . . . . . . 12 ⊢ ((ℂ ⊆ ℂ ∧ 1 ∈ ℂ ∧ 𝑀 ∈ ℕ₀) → (𝑦 ∈ ℂ ↦ (𝑦↑𝑀)) ∈ (Poly‘ℂ))
170	50, 168, 16, 169	syl3anc 1369	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑦 ∈ ℂ ↦ (𝑦↑𝑀)) ∈ (Poly‘ℂ))
171	170, 40, 42, 44	plyco 25307	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑦 ∈ ℂ ↦ (𝑦↑𝑀)) ∘ 𝐺) ∈ (Poly‘ℂ))
172	167, 171	eqeltrrd 2840	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀)) ∈ (Poly‘ℂ))
173		dgrmulc 25337	. . . . . . . . 9 ⊢ (((𝐴‘𝑀) ∈ ℂ ∧ (𝐴‘𝑀) ≠ 0 ∧ (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀)) ∈ (Poly‘ℂ)) → (deg‘((ℂ × {(𝐴‘𝑀)}) ∘_f · (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀)))) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀))))
174	17, 84, 172, 173	syl3anc 1369	. . . . . . . 8 ⊢ (𝜑 → (deg‘((ℂ × {(𝐴‘𝑀)}) ∘_f · (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀)))) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀))))
175	165, 174	eqtr3d 2780	. . . . . . 7 ⊢ (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀))))
176	175	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (deg‘(𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀))))
177		dgrco.2	. . . . . . 7 ⊢ 𝑁 = (deg‘𝐺)
178	65	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝑀 ∈ ℕ)
179		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ)
180	1	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝐺 ∈ (Poly‘𝑆))
181	177, 178, 179, 180	dgrcolem1 25339	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀))) = (𝑀 · 𝑁))
182	176, 181	eqtrd 2778	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (deg‘(𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))) = (𝑀 · 𝑁))
183	134, 160, 182	3brtr4d 5102	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (deg‘(𝑥 ∈ ℂ ↦ ((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))))) < (deg‘(𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))))
184		eqid 2738	. . . . 5 ⊢ (deg‘(𝑥 ∈ ℂ ↦ ((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))))) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))))
185		eqid 2738	. . . . 5 ⊢ (deg‘(𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))))
186	184, 185	dgradd2 25334	. . . 4 ⊢ (((𝑥 ∈ ℂ ↦ ((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))) ∈ (Poly‘ℂ) ∧ (𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))) ∈ (Poly‘ℂ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))))) < (deg‘(𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))))) → (deg‘((𝑥 ∈ ℂ ↦ ((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))) ∘_f + (𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))))) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))))
187	59, 60, 183, 186	syl3anc 1369	. . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (deg‘((𝑥 ∈ ℂ ↦ ((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))) ∘_f + (𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))))) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))))
188	36, 187, 182	3eqtrd 2782	. 2 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (deg‘(𝐹 ∘ 𝐺)) = (𝑀 · 𝑁))
189		0cn 10898	. . . . . . . 8 ⊢ 0 ∈ ℂ
190		ffvelrn 6941	. . . . . . . 8 ⊢ ((𝐺:ℂ⟶ℂ ∧ 0 ∈ ℂ) → (𝐺‘0) ∈ ℂ)
191	3, 189, 190	sylancl 585	. . . . . . 7 ⊢ (𝜑 → (𝐺‘0) ∈ ℂ)
192	7, 191	ffvelrnd 6944	. . . . . 6 ⊢ (𝜑 → (𝐹‘(𝐺‘0)) ∈ ℂ)
193		0dgr 25311	. . . . . 6 ⊢ ((𝐹‘(𝐺‘0)) ∈ ℂ → (deg‘(ℂ × {(𝐹‘(𝐺‘0))})) = 0)
194	192, 193	syl 17	. . . . 5 ⊢ (𝜑 → (deg‘(ℂ × {(𝐹‘(𝐺‘0))})) = 0)
195	16	nn0cnd 12225	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℂ)
196	195	mul01d 11104	. . . . 5 ⊢ (𝜑 → (𝑀 · 0) = 0)
197	194, 196	eqtr4d 2781	. . . 4 ⊢ (𝜑 → (deg‘(ℂ × {(𝐹‘(𝐺‘0))})) = (𝑀 · 0))
198	197	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑁 = 0) → (deg‘(ℂ × {(𝐹‘(𝐺‘0))})) = (𝑀 · 0))
199	191	ad2antrr 722	. . . . . 6 ⊢ (((𝜑 ∧ 𝑁 = 0) ∧ 𝑥 ∈ ℂ) → (𝐺‘0) ∈ ℂ)
200		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 = 0) → 𝑁 = 0)
201	177, 200	eqtr3id 2793	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 = 0) → (deg‘𝐺) = 0)
202		0dgrb 25312	. . . . . . . . . 10 ⊢ (𝐺 ∈ (Poly‘𝑆) → ((deg‘𝐺) = 0 ↔ 𝐺 = (ℂ × {(𝐺‘0)})))
203	1, 202	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((deg‘𝐺) = 0 ↔ 𝐺 = (ℂ × {(𝐺‘0)})))
204	203	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 = 0) → ((deg‘𝐺) = 0 ↔ 𝐺 = (ℂ × {(𝐺‘0)})))
205	201, 204	mpbid 231	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 = 0) → 𝐺 = (ℂ × {(𝐺‘0)}))
206		fconstmpt 5640	. . . . . . 7 ⊢ (ℂ × {(𝐺‘0)}) = (𝑥 ∈ ℂ ↦ (𝐺‘0))
207	205, 206	eqtrdi 2795	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 = 0) → 𝐺 = (𝑥 ∈ ℂ ↦ (𝐺‘0)))
208	31	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 = 0) → 𝐹 = (𝑦 ∈ ℂ ↦ (𝐹‘𝑦)))
209		fveq2 6756	. . . . . 6 ⊢ (𝑦 = (𝐺‘0) → (𝐹‘𝑦) = (𝐹‘(𝐺‘0)))
210	199, 207, 208, 209	fmptco 6983	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 = 0) → (𝐹 ∘ 𝐺) = (𝑥 ∈ ℂ ↦ (𝐹‘(𝐺‘0))))
211		fconstmpt 5640	. . . . 5 ⊢ (ℂ × {(𝐹‘(𝐺‘0))}) = (𝑥 ∈ ℂ ↦ (𝐹‘(𝐺‘0)))
212	210, 211	eqtr4di 2797	. . . 4 ⊢ ((𝜑 ∧ 𝑁 = 0) → (𝐹 ∘ 𝐺) = (ℂ × {(𝐹‘(𝐺‘0))}))
213	212	fveq2d 6760	. . 3 ⊢ ((𝜑 ∧ 𝑁 = 0) → (deg‘(𝐹 ∘ 𝐺)) = (deg‘(ℂ × {(𝐹‘(𝐺‘0))})))
214	200	oveq2d 7271	. . 3 ⊢ ((𝜑 ∧ 𝑁 = 0) → (𝑀 · 𝑁) = (𝑀 · 0))
215	198, 213, 214	3eqtr4d 2788	. 2 ⊢ ((𝜑 ∧ 𝑁 = 0) → (deg‘(𝐹 ∘ 𝐺)) = (𝑀 · 𝑁))
216		dgrcl 25299	. . . . 5 ⊢ (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ₀)
217	1, 216	syl 17	. . . 4 ⊢ (𝜑 → (deg‘𝐺) ∈ ℕ₀)
218	177, 217	eqeltrid 2843	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
219		elnn0 12165	. . 3 ⊢ (𝑁 ∈ ℕ₀ ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
220	218, 219	sylib 217	. 2 ⊢ (𝜑 → (𝑁 ∈ ℕ ∨ 𝑁 = 0))
221	188, 215, 220	mpjaodan 955	1 ⊢ (𝜑 → (deg‘(𝐹 ∘ 𝐺)) = (𝑀 · 𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 843 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∀wral 3063 Vcvv 3422 ⊆ wss 3883 ifcif 4456 {csn 4558 class class class wbr 5070 ↦ cmpt 5153 × cxp 5578 ∘ ccom 5584 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ∘_f cof 7509 ℂcc 10800 ℝcr 10801 0cc0 10802 1c1 10803 + caddc 10805 · cmul 10807 < clt 10940 ≤ cle 10941 − cmin 11135 ℕcn 11903 ℕ₀cn0 12163 ↑cexp 13710 0_𝑝c0p 24738 Polycply 25250 coeffccoe 25252 degcdgr 25253

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-pm 8576 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-sup 9131 df-inf 9132 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-n0 12164 df-z 12250 df-uz 12512 df-rp 12660 df-fz 13169 df-fzo 13312 df-fl 13440 df-seq 13650 df-exp 13711 df-hash 13973 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-clim 15125 df-rlim 15126 df-sum 15326 df-0p 24739 df-ply 25254 df-coe 25256 df-dgr 25257

This theorem is referenced by: dgrco 25341

W3C validator