Theorem dgrcolem2 25787

Description: Lemma for dgrco 25788. (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 25711	. . . . . . . . . . 11 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ)
3	1, 2	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺:ℂ⟶ℂ)
4	3	ffvelcdmda 7086	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝐺‘𝑥) ∈ ℂ)
5		dgrco.3	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))
6		plyf 25711	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
7	5, 6	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:ℂ⟶ℂ)
8	7	ffvelcdmda 7086	. . . . . . . . 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 25745	. . . . . . . . . . . 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 25746	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
15	5, 14	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (deg‘𝐹) ∈ ℕ0)
16	13, 15	eqeltrid 2837	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℕ0)
17	12, 16	ffvelcdmd 7087	. . . . . . . . . 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 14110	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((𝐺‘𝑥)↑𝑀) ∈ ℂ)
21	18, 20	mulcld 11233	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)) ∈ ℂ)
22	9, 21	npcand 11574	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))) + ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))) = (𝐹‘(𝐺‘𝑥)))
23	22	mpteq2dva 5248	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ (((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))) + ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))) = (𝑥 ∈ ℂ ↦ (𝐹‘(𝐺‘𝑥))))
24		cnex 11190	. . . . . . . 8 ⊢ ℂ ∈ V
25	24	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℂ ∈ V)
26	9, 21	subcld 11570	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))) ∈ ℂ)
27		eqidd 2733	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))) = (𝑥 ∈ ℂ ↦ ((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))))
28		eqidd 2733	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))) = (𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))))
29	25, 26, 21, 27, 28	offval2 7689	. . . . . 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 6960	. . . . . . 7 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ ℂ ↦ (𝐺‘𝑥)))
31	7	feqmptd 6960	. . . . . . 7 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ ℂ ↦ (𝐹‘𝑦)))
32		fveq2 6891	. . . . . . 7 ⊢ (𝑦 = (𝐺‘𝑥) → (𝐹‘𝑦) = (𝐹‘(𝐺‘𝑥)))
33	4, 30, 31, 32	fmptco 7126	. . . . . 6 ⊢ (𝜑 → (𝐹 ∘ 𝐺) = (𝑥 ∈ ℂ ↦ (𝐹‘(𝐺‘𝑥))))
34	23, 29, 33	3eqtr4rd 2783	. . . . 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 6895	. . . 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 7689	. . . . . 6 ⊢ (𝜑 → ((𝐹 ∘ 𝐺) ∘f − (𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))) = (𝑥 ∈ ℂ ↦ ((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))))
38		plyssc 25713	. . . . . . . . 9 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
39	38, 5	sselid 3980	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (Poly‘ℂ))
40	38, 1	sselid 3980	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ (Poly‘ℂ))
41		addcl 11191	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (𝑧 + 𝑤) ∈ ℂ)
42	41	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ)) → (𝑧 + 𝑤) ∈ ℂ)
43		mulcl 11193	. . . . . . . . 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 25754	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∘ 𝐺) ∈ (Poly‘ℂ))
46		eqidd 2733	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))) = (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))
47		oveq1 7415	. . . . . . . . . 10 ⊢ (𝑦 = (𝐺‘𝑥) → (𝑦↑𝑀) = ((𝐺‘𝑥)↑𝑀))
48	47	oveq2d 7424	. . . . . . . . 9 ⊢ (𝑦 = (𝐺‘𝑥) → ((𝐴‘𝑀) · (𝑦↑𝑀)) = ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))
49	4, 30, 46, 48	fmptco 7126	. . . . . . . 8 ⊢ (𝜑 → ((𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))) ∘ 𝐺) = (𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))))
50		ssidd 4005	. . . . . . . . . 10 ⊢ (𝜑 → ℂ ⊆ ℂ)
51		eqid 2732	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))) = (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))
52	51	ply1term 25717	. . . . . . . . . 10 ⊢ ((ℂ ⊆ ℂ ∧ (𝐴‘𝑀) ∈ ℂ ∧ 𝑀 ∈ ℕ0) → (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))) ∈ (Poly‘ℂ))
53	50, 17, 16, 52	syl3anc 1371	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))) ∈ (Poly‘ℂ))
54	53, 40, 42, 44	plyco 25754	. . . . . . . 8 ⊢ (𝜑 → ((𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))) ∘ 𝐺) ∈ (Poly‘ℂ))
55	49, 54	eqeltrrd 2834	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))) ∈ (Poly‘ℂ))
56		plysubcl 25735	. . . . . . 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 2834	. . . . 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 12510	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ ℕ0 → (𝐷 + 1) ∈ ℕ)
64	62, 63	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐷 + 1) ∈ ℕ)
65	61, 64	eqeltrd 2833	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℕ)
66	65	nngt0d 12260	. . . . . . . . 9 ⊢ (𝜑 → 0 < 𝑀)
67		fveq2 6891	. . . . . . . . . . 11 ⊢ ((𝐹 ∘f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) = 0𝑝 → (deg‘(𝐹 ∘f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) = (deg‘0𝑝))
68		dgr0 25775	. . . . . . . . . . 11 ⊢ (deg‘0𝑝) = 0
69	67, 68	eqtrdi 2788	. . . . . . . . . 10 ⊢ ((𝐹 ∘f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) = 0𝑝 → (deg‘(𝐹 ∘f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) = 0)
70	69	breq1d 5158	. . . . . . . . 9 ⊢ ((𝐹 ∘f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) = 0𝑝 → ((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)))) = 0𝑝 → (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 2732	. . . . . . . . . . . 12 ⊢ (deg‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) = (deg‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))
74	13, 73	dgrsub 25785	. . . . . . . . . . 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 12261	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ≠ 0)
77	13, 10	dgreq0 25778	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐹 = 0𝑝 ↔ (𝐴‘𝑀) = 0))
78	5, 77	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐹 = 0𝑝 ↔ (𝐴‘𝑀) = 0))
79		fveq2 6891	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 = 0𝑝 → (deg‘𝐹) = (deg‘0𝑝))
80	79, 68	eqtrdi 2788	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 = 0𝑝 → (deg‘𝐹) = 0)
81	13, 80	eqtrid 2784	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 = 0𝑝 → 𝑀 = 0)
82	78, 81	syl6bir 253	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐴‘𝑀) = 0 → 𝑀 = 0))
83	82	necon3d 2961	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑀 ≠ 0 → (𝐴‘𝑀) ≠ 0))
84	76, 83	mpd 15	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴‘𝑀) ≠ 0)
85	51	dgr1term 25773	. . . . . . . . . . . . 13 ⊢ (((𝐴‘𝑀) ∈ ℂ ∧ (𝐴‘𝑀) ≠ 0 ∧ 𝑀 ∈ ℕ0) → (deg‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) = 𝑀)
86	17, 84, 16, 85	syl3anc 1371	. . . . . . . . . . . 12 ⊢ (𝜑 → (deg‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) = 𝑀)
87	86	ifeq1d 4547	. . . . . . . . . . 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 4568	. . . . . . . . . . 11 ⊢ if(𝑀 ≤ (deg‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))), 𝑀, 𝑀) = 𝑀
89	87, 88	eqtrdi 2788	. . . . . . . . . 10 ⊢ (𝜑 → if(𝑀 ≤ (deg‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))), (deg‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))), 𝑀) = 𝑀)
90	75, 89	breqtrd 5174	. . . . . . . . 9 ⊢ (𝜑 → (deg‘(𝐹 ∘f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) ≤ 𝑀)
91		eqid 2732	. . . . . . . . . . . . 13 ⊢ (coeff‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) = (coeff‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))
92	10, 91	coesub 25770	. . . . . . . . . . . 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 6893	. . . . . . . . . 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 6718	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 Fn ℕ0)
96	91	coef3 25745	. . . . . . . . . . . . . 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 6718	. . . . . . . . . . . 12 ⊢ (𝜑 → (coeff‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) Fn ℕ0)
99		nn0ex 12477	. . . . . . . . . . . . 13 ⊢ ℕ0 ∈ V
100	99	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ℕ0 ∈ V)
101		inidm 4218	. . . . . . . . . . . 12 ⊢ (ℕ0 ∩ ℕ0) = ℕ0
102		eqidd 2733	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑀 ∈ ℕ0) → (𝐴‘𝑀) = (𝐴‘𝑀))
103	51	coe1term 25772	. . . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((coeff‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))‘𝑀) = if(𝑀 = 𝑀, (𝐴‘𝑀), 0))
105		eqid 2732	. . . . . . . . . . . . . . 15 ⊢ 𝑀 = 𝑀
106	105	iftruei 4535	. . . . . . . . . . . . . 14 ⊢ if(𝑀 = 𝑀, (𝐴‘𝑀), 0) = (𝐴‘𝑀)
107	104, 106	eqtrdi 2788	. . . . . . . . . . . . 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 7680	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑀 ∈ ℕ0) → ((𝐴 ∘f − (coeff‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))))‘𝑀) = ((𝐴‘𝑀) − (𝐴‘𝑀)))
110	16, 109	mpdan 685	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 ∘f − (coeff‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))))‘𝑀) = ((𝐴‘𝑀) − (𝐴‘𝑀)))
111	17	subidd 11558	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴‘𝑀) − (𝐴‘𝑀)) = 0)
112	94, 110, 111	3eqtrd 2776	. . . . . . . . 9 ⊢ (𝜑 → ((coeff‘(𝐹 ∘f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))))‘𝑀) = 0)
113		plysubcl 25735	. . . . . . . . . . 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 2732	. . . . . . . . . . 11 ⊢ (deg‘(𝐹 ∘f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) = (deg‘(𝐹 ∘f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))))
116		eqid 2732	. . . . . . . . . . 11 ⊢ (coeff‘(𝐹 ∘f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) = (coeff‘(𝐹 ∘f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))))
117	115, 116	dgrlt 25779	. . . . . . . . . 10 ⊢ (((𝐹 ∘f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) ∈ (Poly‘ℂ) ∧ 𝑀 ∈ ℕ0) → (((𝐹 ∘f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) = 0𝑝 ∨ (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)))) = 0𝑝 ∨ (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 711	. . . . . . . 8 ⊢ (𝜑 → ((𝐹 ∘f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) = 0𝑝 ∨ (deg‘(𝐹 ∘f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) < 𝑀))
120	71, 72, 119	mpjaod 858	. . . . . . 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 25746	. . . . . . . . . 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 12532	. . . . . . . 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 12532	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℝ)
127	126	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝑀 ∈ ℝ)
128		nnre 12218	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
129	128	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℝ)
130		nngt0 12242	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁)
131	130	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 0 < 𝑁)
132		ltmul1 12063	. . . . . . 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 1374	. . . . . 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	ffvelcdmda 7086	. . . . . . . . . 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 14044	. . . . . . . . . . . 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 11233	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → ((𝐴‘𝑀) · (𝑦↑𝑀)) ∈ ℂ)
141	25, 135, 140, 31, 46	offval2 7689	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 ∘f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) = (𝑦 ∈ ℂ ↦ ((𝐹‘𝑦) − ((𝐴‘𝑀) · (𝑦↑𝑀)))))
142	32, 48	oveq12d 7426	. . . . . . . . 9 ⊢ (𝑦 = (𝐺‘𝑥) → ((𝐹‘𝑦) − ((𝐴‘𝑀) · (𝑦↑𝑀))) = ((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))))
143	4, 30, 141, 142	fmptco 7126	. . . . . . . 8 ⊢ (𝜑 → ((𝐹 ∘f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) ∘ 𝐺) = (𝑥 ∈ ℂ ↦ ((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))))
144	143	fveq2d 6895	. . . . . . 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 5174	. . . . . . . . 9 ⊢ (𝜑 → (deg‘(𝐹 ∘f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) < (𝐷 + 1))
146		nn0leltp1 12620	. . . . . . . . . 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 6891	. . . . . . . . . . 11 ⊢ (𝑓 = (𝐹 ∘f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) → (deg‘𝑓) = (deg‘(𝐹 ∘f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))))
150	149	breq1d 5158	. . . . . . . . . 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 5857	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝐹 ∘f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) → (𝑓 ∘ 𝐺) = ((𝐹 ∘f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) ∘ 𝐺))
152	151	fveq2d 6895	. . . . . . . . . . 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 7423	. . . . . . . . . . 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 2748	. . . . . . . . . 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 3613	. . . . . . . 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 2774	. . . . . 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 5738	. . . . . . . . . . 11 ⊢ (ℂ × {(𝐴‘𝑀)}) = (𝑥 ∈ ℂ ↦ (𝐴‘𝑀))
162	161	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (ℂ × {(𝐴‘𝑀)}) = (𝑥 ∈ ℂ ↦ (𝐴‘𝑀)))
163		eqidd 2733	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀)) = (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀)))
164	25, 18, 20, 162, 163	offval2 7689	. . . . . . . . 9 ⊢ (𝜑 → ((ℂ × {(𝐴‘𝑀)}) ∘f · (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀))) = (𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))))
165	164	fveq2d 6895	. . . . . . . 8 ⊢ (𝜑 → (deg‘((ℂ × {(𝐴‘𝑀)}) ∘f · (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀)))) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))))
166		eqidd 2733	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑦 ∈ ℂ ↦ (𝑦↑𝑀)) = (𝑦 ∈ ℂ ↦ (𝑦↑𝑀)))
167	4, 30, 166, 47	fmptco 7126	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑦 ∈ ℂ ↦ (𝑦↑𝑀)) ∘ 𝐺) = (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀)))
168		1cnd 11208	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℂ)
169		plypow 25718	. . . . . . . . . . . 12 ⊢ ((ℂ ⊆ ℂ ∧ 1 ∈ ℂ ∧ 𝑀 ∈ ℕ0) → (𝑦 ∈ ℂ ↦ (𝑦↑𝑀)) ∈ (Poly‘ℂ))
170	50, 168, 16, 169	syl3anc 1371	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑦 ∈ ℂ ↦ (𝑦↑𝑀)) ∈ (Poly‘ℂ))
171	170, 40, 42, 44	plyco 25754	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑦 ∈ ℂ ↦ (𝑦↑𝑀)) ∘ 𝐺) ∈ (Poly‘ℂ))
172	167, 171	eqeltrrd 2834	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀)) ∈ (Poly‘ℂ))
173		dgrmulc 25784	. . . . . . . . 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 1371	. . . . . . . 8 ⊢ (𝜑 → (deg‘((ℂ × {(𝐴‘𝑀)}) ∘f · (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀)))) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀))))
175	165, 174	eqtr3d 2774	. . . . . . 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 25786	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀))) = (𝑀 · 𝑁))
182	176, 181	eqtrd 2772	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (deg‘(𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))) = (𝑀 · 𝑁))
183	134, 160, 182	3brtr4d 5180	. . . 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 2732	. . . . 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 2732	. . . . 5 ⊢ (deg‘(𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))))
186	184, 185	dgradd2 25781	. . . 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 1371	. . 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 2776	. 2 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (deg‘(𝐹 ∘ 𝐺)) = (𝑀 · 𝑁))
189		0cn 11205	. . . . . . . 8 ⊢ 0 ∈ ℂ
190		ffvelcdm 7083	. . . . . . . 8 ⊢ ((𝐺:ℂ⟶ℂ ∧ 0 ∈ ℂ) → (𝐺‘0) ∈ ℂ)
191	3, 189, 190	sylancl 586	. . . . . . 7 ⊢ (𝜑 → (𝐺‘0) ∈ ℂ)
192	7, 191	ffvelcdmd 7087	. . . . . 6 ⊢ (𝜑 → (𝐹‘(𝐺‘0)) ∈ ℂ)
193		0dgr 25758	. . . . . 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 12533	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℂ)
196	195	mul01d 11412	. . . . 5 ⊢ (𝜑 → (𝑀 · 0) = 0)
197	194, 196	eqtr4d 2775	. . . 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 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑁 = 0) ∧ 𝑥 ∈ ℂ) → (𝐺‘0) ∈ ℂ)
200		simpr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 = 0) → 𝑁 = 0)
201	177, 200	eqtr3id 2786	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 = 0) → (deg‘𝐺) = 0)
202		0dgrb 25759	. . . . . . . . . 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 5738	. . . . . . 7 ⊢ (ℂ × {(𝐺‘0)}) = (𝑥 ∈ ℂ ↦ (𝐺‘0))
207	205, 206	eqtrdi 2788	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 = 0) → 𝐺 = (𝑥 ∈ ℂ ↦ (𝐺‘0)))
208	31	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 = 0) → 𝐹 = (𝑦 ∈ ℂ ↦ (𝐹‘𝑦)))
209		fveq2 6891	. . . . . 6 ⊢ (𝑦 = (𝐺‘0) → (𝐹‘𝑦) = (𝐹‘(𝐺‘0)))
210	199, 207, 208, 209	fmptco 7126	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 = 0) → (𝐹 ∘ 𝐺) = (𝑥 ∈ ℂ ↦ (𝐹‘(𝐺‘0))))
211		fconstmpt 5738	. . . . 5 ⊢ (ℂ × {(𝐹‘(𝐺‘0))}) = (𝑥 ∈ ℂ ↦ (𝐹‘(𝐺‘0)))
212	210, 211	eqtr4di 2790	. . . 4 ⊢ ((𝜑 ∧ 𝑁 = 0) → (𝐹 ∘ 𝐺) = (ℂ × {(𝐹‘(𝐺‘0))}))
213	212	fveq2d 6895	. . 3 ⊢ ((𝜑 ∧ 𝑁 = 0) → (deg‘(𝐹 ∘ 𝐺)) = (deg‘(ℂ × {(𝐹‘(𝐺‘0))})))
214	200	oveq2d 7424	. . 3 ⊢ ((𝜑 ∧ 𝑁 = 0) → (𝑀 · 𝑁) = (𝑀 · 0))
215	198, 213, 214	3eqtr4d 2782	. 2 ⊢ ((𝜑 ∧ 𝑁 = 0) → (deg‘(𝐹 ∘ 𝐺)) = (𝑀 · 𝑁))
216		dgrcl 25746	. . . . 5 ⊢ (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ0)
217	1, 216	syl 17	. . . 4 ⊢ (𝜑 → (deg‘𝐺) ∈ ℕ0)
218	177, 217	eqeltrid 2837	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
219		elnn0 12473	. . 3 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
220	218, 219	sylib 217	. 2 ⊢ (𝜑 → (𝑁 ∈ ℕ ∨ 𝑁 = 0))
221	188, 215, 220	mpjaodan 957	1 ⊢ (𝜑 → (deg‘(𝐹 ∘ 𝐺)) = (𝑀 · 𝑁))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   = wceq 1541   ∈ wcel 2106   ≠ wne 2940  ∀wral 3061  Vcvv 3474   ⊆ wss 3948  ifcif 4528  {csn 4628   class class class wbr 5148   ↦ cmpt 5231   × cxp 5674   ∘ ccom 5680  ⟶wf 6539  ‘cfv 6543  (class class class)co 7408   ∘_f cof 7667  ℂcc 11107  ℝcr 11108  0cc0 11109  1c1 11110   + caddc 11112   · cmul 11114   < clt 11247   ≤ cle 11248   − cmin 11443  ℕcn 12211  ℕ₀cn0 12471  ↑cexp 14026  0_𝑝c0p 25185  Polycply 25697  coeffccoe 25699  degcdgr 25700

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-inf2 9635  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186  ax-pre-sup 11187

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-of 7669  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-1o 8465  df-er 8702  df-map 8821  df-pm 8822  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-sup 9436  df-inf 9437  df-oi 9504  df-card 9933  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-div 11871  df-nn 12212  df-2 12274  df-3 12275  df-n0 12472  df-z 12558  df-uz 12822  df-rp 12974  df-fz 13484  df-fzo 13627  df-fl 13756  df-seq 13966  df-exp 14027  df-hash 14290  df-cj 15045  df-re 15046  df-im 15047  df-sqrt 15181  df-abs 15182  df-clim 15431  df-rlim 15432  df-sum 15632  df-0p 25186  df-ply 25701  df-coe 25703  df-dgr 25704

This theorem is referenced by:  dgrco  25788