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Theorem dgrcolem2 25435

Description: Lemma for dgrco 25436. (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 25359	. . . . . . . . . . 11 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ)
3	1, 2	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺:ℂ⟶ℂ)
4	3	ffvelrnda 6961	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝐺‘𝑥) ∈ ℂ)
5		dgrco.3	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))
6		plyf 25359	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
7	5, 6	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:ℂ⟶ℂ)
8	7	ffvelrnda 6961	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐺‘𝑥) ∈ ℂ) → (𝐹‘(𝐺‘𝑥)) ∈ ℂ)
9	4, 8	syldan 591	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝐹‘(𝐺‘𝑥)) ∈ ℂ)
10		dgrco.5	. . . . . . . . . . . . 13 ⊢ 𝐴 = (coeff‘𝐹)
11	10	coef3 25393	. . . . . . . . . . . 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 25394	. . . . . . . . . . . . 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 6962	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴‘𝑀) ∈ ℂ)
18	17	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝐴‘𝑀) ∈ ℂ)
19	16	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝑀 ∈ ℕ₀)
20	4, 19	expcld 13864	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((𝐺‘𝑥)↑𝑀) ∈ ℂ)
21	18, 20	mulcld 10995	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)) ∈ ℂ)
22	9, 21	npcand 11336	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))) + ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))) = (𝐹‘(𝐺‘𝑥)))
23	22	mpteq2dva 5174	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ (((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))) + ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))) = (𝑥 ∈ ℂ ↦ (𝐹‘(𝐺‘𝑥))))
24		cnex 10952	. . . . . . . 8 ⊢ ℂ ∈ V
25	24	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℂ ∈ V)
26	9, 21	subcld 11332	. . . . . . 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 7553	. . . . . 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 6837	. . . . . . 7 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ ℂ ↦ (𝐺‘𝑥)))
31	7	feqmptd 6837	. . . . . . 7 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ ℂ ↦ (𝐹‘𝑦)))
32		fveq2 6774	. . . . . . 7 ⊢ (𝑦 = (𝐺‘𝑥) → (𝐹‘𝑦) = (𝐹‘(𝐺‘𝑥)))
33	4, 30, 31, 32	fmptco 7001	. . . . . 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 6778	. . . 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 481	. . 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 7553	. . . . . 6 ⊢ (𝜑 → ((𝐹 ∘ 𝐺) ∘_f − (𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))) = (𝑥 ∈ ℂ ↦ ((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))))
38		plyssc 25361	. . . . . . . . 9 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
39	38, 5	sselid 3919	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (Poly‘ℂ))
40	38, 1	sselid 3919	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ (Poly‘ℂ))
41		addcl 10953	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (𝑧 + 𝑤) ∈ ℂ)
42	41	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ)) → (𝑧 + 𝑤) ∈ ℂ)
43		mulcl 10955	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (𝑧 · 𝑤) ∈ ℂ)
44	43	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ)) → (𝑧 · 𝑤) ∈ ℂ)
45	39, 40, 42, 44	plyco 25402	. . . . . . 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 7282	. . . . . . . . . 10 ⊢ (𝑦 = (𝐺‘𝑥) → (𝑦↑𝑀) = ((𝐺‘𝑥)↑𝑀))
48	47	oveq2d 7291	. . . . . . . . 9 ⊢ (𝑦 = (𝐺‘𝑥) → ((𝐴‘𝑀) · (𝑦↑𝑀)) = ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))
49	4, 30, 46, 48	fmptco 7001	. . . . . . . 8 ⊢ (𝜑 → ((𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))) ∘ 𝐺) = (𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))))
50		ssidd 3944	. . . . . . . . . 10 ⊢ (𝜑 → ℂ ⊆ ℂ)
51		eqid 2738	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))) = (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))
52	51	ply1term 25365	. . . . . . . . . 10 ⊢ ((ℂ ⊆ ℂ ∧ (𝐴‘𝑀) ∈ ℂ ∧ 𝑀 ∈ ℕ₀) → (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))) ∈ (Poly‘ℂ))
53	50, 17, 16, 52	syl3anc 1370	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))) ∈ (Poly‘ℂ))
54	53, 40, 42, 44	plyco 25402	. . . . . . . 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 25383	. . . . . . 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 584	. . . . . 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 481	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (𝑥 ∈ ℂ ↦ ((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))) ∈ (Poly‘ℂ))
60	55	adantr 481	. . . 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 12272	. . . . . . . . . . . 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 12022	. . . . . . . . 9 ⊢ (𝜑 → 0 < 𝑀)
67		fveq2 6774	. . . . . . . . . . 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 25423	. . . . . . . . . . 11 ⊢ (deg‘0_𝑝) = 0
69	67, 68	eqtrdi 2794	. . . . . . . . . 10 ⊢ ((𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) = 0_𝑝 → (deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) = 0)
70	69	breq1d 5084	. . . . . . . . 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 25433	. . . . . . . . . . 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 584	. . . . . . . . . 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 12023	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ≠ 0)
77	13, 10	dgreq0 25426	. . . . . . . . . . . . . . . . 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 6774	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 = 0_𝑝 → (deg‘𝐹) = (deg‘0_𝑝))
80	79, 68	eqtrdi 2794	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 = 0_𝑝 → (deg‘𝐹) = 0)
81	13, 80	eqtrid 2790	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 = 0_𝑝 → 𝑀 = 0)
82	78, 81	syl6bir 253	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐴‘𝑀) = 0 → 𝑀 = 0))
83	82	necon3d 2964	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑀 ≠ 0 → (𝐴‘𝑀) ≠ 0))
84	76, 83	mpd 15	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴‘𝑀) ≠ 0)
85	51	dgr1term 25421	. . . . . . . . . . . . 13 ⊢ (((𝐴‘𝑀) ∈ ℂ ∧ (𝐴‘𝑀) ≠ 0 ∧ 𝑀 ∈ ℕ₀) → (deg‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) = 𝑀)
86	17, 84, 16, 85	syl3anc 1370	. . . . . . . . . . . 12 ⊢ (𝜑 → (deg‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) = 𝑀)
87	86	ifeq1d 4478	. . . . . . . . . . 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 4499	. . . . . . . . . . 11 ⊢ if(𝑀 ≤ (deg‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))), 𝑀, 𝑀) = 𝑀
89	87, 88	eqtrdi 2794	. . . . . . . . . 10 ⊢ (𝜑 → if(𝑀 ≤ (deg‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))), (deg‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))), 𝑀) = 𝑀)
90	75, 89	breqtrd 5100	. . . . . . . . 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 25418	. . . . . . . . . . . 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 584	. . . . . . . . . . 11 ⊢ (𝜑 → (coeff‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) = (𝐴 ∘_f − (coeff‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))))
94	93	fveq1d 6776	. . . . . . . . . 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 6601	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 Fn ℕ₀)
96	91	coef3 25393	. . . . . . . . . . . . . 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 6601	. . . . . . . . . . . 12 ⊢ (𝜑 → (coeff‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) Fn ℕ₀)
99		nn0ex 12239	. . . . . . . . . . . . 13 ⊢ ℕ₀ ∈ V
100	99	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ℕ₀ ∈ V)
101		inidm 4152	. . . . . . . . . . . 12 ⊢ (ℕ₀ ∩ ℕ₀) = ℕ₀
102		eqidd 2739	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑀 ∈ ℕ₀) → (𝐴‘𝑀) = (𝐴‘𝑀))
103	51	coe1term 25420	. . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . 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 4466	. . . . . . . . . . . . . 14 ⊢ if(𝑀 = 𝑀, (𝐴‘𝑀), 0) = (𝐴‘𝑀)
107	104, 106	eqtrdi 2794	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((coeff‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))‘𝑀) = (𝐴‘𝑀))
108	107	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑀 ∈ ℕ₀) → ((coeff‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))‘𝑀) = (𝐴‘𝑀))
109	95, 98, 100, 100, 101, 102, 108	ofval 7544	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑀 ∈ ℕ₀) → ((𝐴 ∘_f − (coeff‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))))‘𝑀) = ((𝐴‘𝑀) − (𝐴‘𝑀)))
110	16, 109	mpdan 684	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 ∘_f − (coeff‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))))‘𝑀) = ((𝐴‘𝑀) − (𝐴‘𝑀)))
111	17	subidd 11320	. . . . . . . . . 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 25383	. . . . . . . . . . 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 584	. . . . . . . . . 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 25427	. . . . . . . . . 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 584	. . . . . . . . 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 710	. . . . . . . 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 857	. . . . . . 7 ⊢ (𝜑 → (deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) < 𝑀)
121	120	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) < 𝑀)
122		dgrcl 25394	. . . . . . . . . 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 12294	. . . . . . . 8 ⊢ (𝜑 → (deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) ∈ ℝ)
125	124	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) ∈ ℝ)
126	16	nn0red 12294	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℝ)
127	126	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝑀 ∈ ℝ)
128		nnre 11980	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
129	128	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℝ)
130		nngt0 12004	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁)
131	130	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 0 < 𝑁)
132		ltmul1 11825	. . . . . . 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 1373	. . . . . 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 6961	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝐹‘𝑦) ∈ ℂ)
136	17	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝐴‘𝑀) ∈ ℂ)
137		id 22	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℂ → 𝑦 ∈ ℂ)
138		expcl 13800	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℂ ∧ 𝑀 ∈ ℕ₀) → (𝑦↑𝑀) ∈ ℂ)
139	137, 16, 138	syl2anr 597	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑦↑𝑀) ∈ ℂ)
140	136, 139	mulcld 10995	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → ((𝐴‘𝑀) · (𝑦↑𝑀)) ∈ ℂ)
141	25, 135, 140, 31, 46	offval2 7553	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) = (𝑦 ∈ ℂ ↦ ((𝐹‘𝑦) − ((𝐴‘𝑀) · (𝑦↑𝑀)))))
142	32, 48	oveq12d 7293	. . . . . . . . 9 ⊢ (𝑦 = (𝐺‘𝑥) → ((𝐹‘𝑦) − ((𝐴‘𝑀) · (𝑦↑𝑀))) = ((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))))
143	4, 30, 141, 142	fmptco 7001	. . . . . . . 8 ⊢ (𝜑 → ((𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) ∘ 𝐺) = (𝑥 ∈ ℂ ↦ ((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))))
144	143	fveq2d 6778	. . . . . . 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 5100	. . . . . . . . 9 ⊢ (𝜑 → (deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) < (𝐷 + 1))
146		nn0leltp1 12379	. . . . . . . . . 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 584	. . . . . . . . 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 6774	. . . . . . . . . . 11 ⊢ (𝑓 = (𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) → (deg‘𝑓) = (deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))))
150	149	breq1d 5084	. . . . . . . . . 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 5766	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) → (𝑓 ∘ 𝐺) = ((𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) ∘ 𝐺))
152	151	fveq2d 6778	. . . . . . . . . . 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 7290	. . . . . . . . . . 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 345	. . . . . . . . 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 3562	. . . . . . . 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 481	. . . . 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 5649	. . . . . . . . . . 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 7553	. . . . . . . . 9 ⊢ (𝜑 → ((ℂ × {(𝐴‘𝑀)}) ∘_f · (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀))) = (𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))))
165	164	fveq2d 6778	. . . . . . . 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 7001	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑦 ∈ ℂ ↦ (𝑦↑𝑀)) ∘ 𝐺) = (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀)))
168		1cnd 10970	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℂ)
169		plypow 25366	. . . . . . . . . . . 12 ⊢ ((ℂ ⊆ ℂ ∧ 1 ∈ ℂ ∧ 𝑀 ∈ ℕ₀) → (𝑦 ∈ ℂ ↦ (𝑦↑𝑀)) ∈ (Poly‘ℂ))
170	50, 168, 16, 169	syl3anc 1370	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑦 ∈ ℂ ↦ (𝑦↑𝑀)) ∈ (Poly‘ℂ))
171	170, 40, 42, 44	plyco 25402	. . . . . . . . . 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 25432	. . . . . . . . 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 1370	. . . . . . . 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 481	. . . . . 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 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝑀 ∈ ℕ)
179		simpr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ)
180	1	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝐺 ∈ (Poly‘𝑆))
181	177, 178, 179, 180	dgrcolem1 25434	. . . . . 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 5106	. . . 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 25429	. . . 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 1370	. . 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 10967	. . . . . . . 8 ⊢ 0 ∈ ℂ
190		ffvelrn 6959	. . . . . . . 8 ⊢ ((𝐺:ℂ⟶ℂ ∧ 0 ∈ ℂ) → (𝐺‘0) ∈ ℂ)
191	3, 189, 190	sylancl 586	. . . . . . 7 ⊢ (𝜑 → (𝐺‘0) ∈ ℂ)
192	7, 191	ffvelrnd 6962	. . . . . 6 ⊢ (𝜑 → (𝐹‘(𝐺‘0)) ∈ ℂ)
193		0dgr 25406	. . . . . 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 12295	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℂ)
196	195	mul01d 11174	. . . . 5 ⊢ (𝜑 → (𝑀 · 0) = 0)
197	194, 196	eqtr4d 2781	. . . 4 ⊢ (𝜑 → (deg‘(ℂ × {(𝐹‘(𝐺‘0))})) = (𝑀 · 0))
198	197	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑁 = 0) → (deg‘(ℂ × {(𝐹‘(𝐺‘0))})) = (𝑀 · 0))
199	191	ad2antrr 723	. . . . . 6 ⊢ (((𝜑 ∧ 𝑁 = 0) ∧ 𝑥 ∈ ℂ) → (𝐺‘0) ∈ ℂ)
200		simpr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 = 0) → 𝑁 = 0)
201	177, 200	eqtr3id 2792	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 = 0) → (deg‘𝐺) = 0)
202		0dgrb 25407	. . . . . . . . . 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 481	. . . . . . . 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 5649	. . . . . . 7 ⊢ (ℂ × {(𝐺‘0)}) = (𝑥 ∈ ℂ ↦ (𝐺‘0))
207	205, 206	eqtrdi 2794	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 = 0) → 𝐺 = (𝑥 ∈ ℂ ↦ (𝐺‘0)))
208	31	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 = 0) → 𝐹 = (𝑦 ∈ ℂ ↦ (𝐹‘𝑦)))
209		fveq2 6774	. . . . . 6 ⊢ (𝑦 = (𝐺‘0) → (𝐹‘𝑦) = (𝐹‘(𝐺‘0)))
210	199, 207, 208, 209	fmptco 7001	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 = 0) → (𝐹 ∘ 𝐺) = (𝑥 ∈ ℂ ↦ (𝐹‘(𝐺‘0))))
211		fconstmpt 5649	. . . . 5 ⊢ (ℂ × {(𝐹‘(𝐺‘0))}) = (𝑥 ∈ ℂ ↦ (𝐹‘(𝐺‘0)))
212	210, 211	eqtr4di 2796	. . . 4 ⊢ ((𝜑 ∧ 𝑁 = 0) → (𝐹 ∘ 𝐺) = (ℂ × {(𝐹‘(𝐺‘0))}))
213	212	fveq2d 6778	. . 3 ⊢ ((𝜑 ∧ 𝑁 = 0) → (deg‘(𝐹 ∘ 𝐺)) = (deg‘(ℂ × {(𝐹‘(𝐺‘0))})))
214	200	oveq2d 7291	. . 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 25394	. . . . 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 12235	. . 3 ⊢ (𝑁 ∈ ℕ₀ ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
220	218, 219	sylib 217	. 2 ⊢ (𝜑 → (𝑁 ∈ ℕ ∨ 𝑁 = 0))
221	188, 215, 220	mpjaodan 956	1 ⊢ (𝜑 → (deg‘(𝐹 ∘ 𝐺)) = (𝑀 · 𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 844 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 Vcvv 3432 ⊆ wss 3887 ifcif 4459 {csn 4561 class class class wbr 5074 ↦ cmpt 5157 × cxp 5587 ∘ ccom 5593 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ∘_f cof 7531 ℂcc 10869 ℝcr 10870 0cc0 10871 1c1 10872 + caddc 10874 · cmul 10876 < clt 11009 ≤ cle 11010 − cmin 11205 ℕcn 11973 ℕ₀cn0 12233 ↑cexp 13782 0_𝑝c0p 24833 Polycply 25345 coeffccoe 25347 degcdgr 25348

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-pm 8618 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12583 df-rp 12731 df-fz 13240 df-fzo 13383 df-fl 13512 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-rlim 15198 df-sum 15398 df-0p 24834 df-ply 25349 df-coe 25351 df-dgr 25352

This theorem is referenced by: dgrco 25436

W3C validator