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Theorem dgrcolem2 26187

Description: Lemma for dgrco 26188. (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 26110	. . . . . . . . . . 11 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ)
3	1, 2	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺:ℂ⟶ℂ)
4	3	ffvelcdmda 7059	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝐺‘𝑥) ∈ ℂ)
5		dgrco.3	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))
6		plyf 26110	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
7	5, 6	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:ℂ⟶ℂ)
8	7	ffvelcdmda 7059	. . . . . . . . 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 26144	. . . . . . . . . . . 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 26145	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ₀)
15	5, 14	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (deg‘𝐹) ∈ ℕ₀)
16	13, 15	eqeltrid 2833	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
17	12, 16	ffvelcdmd 7060	. . . . . . . . . 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 14118	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((𝐺‘𝑥)↑𝑀) ∈ ℂ)
21	18, 20	mulcld 11201	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)) ∈ ℂ)
22	9, 21	npcand 11544	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))) + ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))) = (𝐹‘(𝐺‘𝑥)))
23	22	mpteq2dva 5203	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ (((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))) + ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))) = (𝑥 ∈ ℂ ↦ (𝐹‘(𝐺‘𝑥))))
24		cnex 11156	. . . . . . . 8 ⊢ ℂ ∈ V
25	24	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℂ ∈ V)
26	9, 21	subcld 11540	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))) ∈ ℂ)
27		eqidd 2731	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))) = (𝑥 ∈ ℂ ↦ ((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))))
28		eqidd 2731	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))) = (𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))))
29	25, 26, 21, 27, 28	offval2 7676	. . . . . 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 6932	. . . . . . 7 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ ℂ ↦ (𝐺‘𝑥)))
31	7	feqmptd 6932	. . . . . . 7 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ ℂ ↦ (𝐹‘𝑦)))
32		fveq2 6861	. . . . . . 7 ⊢ (𝑦 = (𝐺‘𝑥) → (𝐹‘𝑦) = (𝐹‘(𝐺‘𝑥)))
33	4, 30, 31, 32	fmptco 7104	. . . . . 6 ⊢ (𝜑 → (𝐹 ∘ 𝐺) = (𝑥 ∈ ℂ ↦ (𝐹‘(𝐺‘𝑥))))
34	23, 29, 33	3eqtr4rd 2776	. . . . 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 6865	. . . 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 7676	. . . . . 6 ⊢ (𝜑 → ((𝐹 ∘ 𝐺) ∘_f − (𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))) = (𝑥 ∈ ℂ ↦ ((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))))
38		plyssc 26112	. . . . . . . . 9 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
39	38, 5	sselid 3947	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (Poly‘ℂ))
40	38, 1	sselid 3947	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ (Poly‘ℂ))
41		addcl 11157	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (𝑧 + 𝑤) ∈ ℂ)
42	41	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ)) → (𝑧 + 𝑤) ∈ ℂ)
43		mulcl 11159	. . . . . . . . 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 26153	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∘ 𝐺) ∈ (Poly‘ℂ))
46		eqidd 2731	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))) = (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))
47		oveq1 7397	. . . . . . . . . 10 ⊢ (𝑦 = (𝐺‘𝑥) → (𝑦↑𝑀) = ((𝐺‘𝑥)↑𝑀))
48	47	oveq2d 7406	. . . . . . . . 9 ⊢ (𝑦 = (𝐺‘𝑥) → ((𝐴‘𝑀) · (𝑦↑𝑀)) = ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))
49	4, 30, 46, 48	fmptco 7104	. . . . . . . 8 ⊢ (𝜑 → ((𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))) ∘ 𝐺) = (𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))))
50		ssidd 3973	. . . . . . . . . 10 ⊢ (𝜑 → ℂ ⊆ ℂ)
51		eqid 2730	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))) = (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))
52	51	ply1term 26116	. . . . . . . . . 10 ⊢ ((ℂ ⊆ ℂ ∧ (𝐴‘𝑀) ∈ ℂ ∧ 𝑀 ∈ ℕ₀) → (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))) ∈ (Poly‘ℂ))
53	50, 17, 16, 52	syl3anc 1373	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))) ∈ (Poly‘ℂ))
54	53, 40, 42, 44	plyco 26153	. . . . . . . 8 ⊢ (𝜑 → ((𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))) ∘ 𝐺) ∈ (Poly‘ℂ))
55	49, 54	eqeltrrd 2830	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))) ∈ (Poly‘ℂ))
56		plysubcl 26134	. . . . . . 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 2830	. . . . 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 12488	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ ℕ₀ → (𝐷 + 1) ∈ ℕ)
64	62, 63	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐷 + 1) ∈ ℕ)
65	61, 64	eqeltrd 2829	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℕ)
66	65	nngt0d 12242	. . . . . . . . 9 ⊢ (𝜑 → 0 < 𝑀)
67		fveq2 6861	. . . . . . . . . . 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 26175	. . . . . . . . . . 11 ⊢ (deg‘0_𝑝) = 0
69	67, 68	eqtrdi 2781	. . . . . . . . . 10 ⊢ ((𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) = 0_𝑝 → (deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) = 0)
70	69	breq1d 5120	. . . . . . . . 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 247	. . . . . . . 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 2730	. . . . . . . . . . . 12 ⊢ (deg‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) = (deg‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))
74	13, 73	dgrsub 26185	. . . . . . . . . . 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 12243	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ≠ 0)
77	13, 10	dgreq0 26178	. . . . . . . . . . . . . . . . 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 6861	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 = 0_𝑝 → (deg‘𝐹) = (deg‘0_𝑝))
80	79, 68	eqtrdi 2781	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 = 0_𝑝 → (deg‘𝐹) = 0)
81	13, 80	eqtrid 2777	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 = 0_𝑝 → 𝑀 = 0)
82	78, 81	biimtrrdi 254	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐴‘𝑀) = 0 → 𝑀 = 0))
83	82	necon3d 2947	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑀 ≠ 0 → (𝐴‘𝑀) ≠ 0))
84	76, 83	mpd 15	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴‘𝑀) ≠ 0)
85	51	dgr1term 26172	. . . . . . . . . . . . 13 ⊢ (((𝐴‘𝑀) ∈ ℂ ∧ (𝐴‘𝑀) ≠ 0 ∧ 𝑀 ∈ ℕ₀) → (deg‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) = 𝑀)
86	17, 84, 16, 85	syl3anc 1373	. . . . . . . . . . . 12 ⊢ (𝜑 → (deg‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) = 𝑀)
87	86	ifeq1d 4511	. . . . . . . . . . 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 4532	. . . . . . . . . . 11 ⊢ if(𝑀 ≤ (deg‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))), 𝑀, 𝑀) = 𝑀
89	87, 88	eqtrdi 2781	. . . . . . . . . 10 ⊢ (𝜑 → if(𝑀 ≤ (deg‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))), (deg‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))), 𝑀) = 𝑀)
90	75, 89	breqtrd 5136	. . . . . . . . 9 ⊢ (𝜑 → (deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) ≤ 𝑀)
91		eqid 2730	. . . . . . . . . . . . 13 ⊢ (coeff‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) = (coeff‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))
92	10, 91	coesub 26169	. . . . . . . . . . . 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 6863	. . . . . . . . . 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 6692	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 Fn ℕ₀)
96	91	coef3 26144	. . . . . . . . . . . . . 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 6692	. . . . . . . . . . . 12 ⊢ (𝜑 → (coeff‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) Fn ℕ₀)
99		nn0ex 12455	. . . . . . . . . . . . 13 ⊢ ℕ₀ ∈ V
100	99	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ℕ₀ ∈ V)
101		inidm 4193	. . . . . . . . . . . 12 ⊢ (ℕ₀ ∩ ℕ₀) = ℕ₀
102		eqidd 2731	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑀 ∈ ℕ₀) → (𝐴‘𝑀) = (𝐴‘𝑀))
103	51	coe1term 26171	. . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((coeff‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))‘𝑀) = if(𝑀 = 𝑀, (𝐴‘𝑀), 0))
105		eqid 2730	. . . . . . . . . . . . . . 15 ⊢ 𝑀 = 𝑀
106	105	iftruei 4498	. . . . . . . . . . . . . 14 ⊢ if(𝑀 = 𝑀, (𝐴‘𝑀), 0) = (𝐴‘𝑀)
107	104, 106	eqtrdi 2781	. . . . . . . . . . . . 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 7667	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑀 ∈ ℕ₀) → ((𝐴 ∘_f − (coeff‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))))‘𝑀) = ((𝐴‘𝑀) − (𝐴‘𝑀)))
110	16, 109	mpdan 687	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 ∘_f − (coeff‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))))‘𝑀) = ((𝐴‘𝑀) − (𝐴‘𝑀)))
111	17	subidd 11528	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴‘𝑀) − (𝐴‘𝑀)) = 0)
112	94, 110, 111	3eqtrd 2769	. . . . . . . . 9 ⊢ (𝜑 → ((coeff‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))))‘𝑀) = 0)
113		plysubcl 26134	. . . . . . . . . . 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 2730	. . . . . . . . . . 11 ⊢ (deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) = (deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))))
116		eqid 2730	. . . . . . . . . . 11 ⊢ (coeff‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) = (coeff‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))))
117	115, 116	dgrlt 26179	. . . . . . . . . 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 713	. . . . . . . 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 860	. . . . . . 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 26145	. . . . . . . . . 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 12511	. . . . . . . 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 12511	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℝ)
127	126	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝑀 ∈ ℝ)
128		nnre 12200	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
129	128	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℝ)
130		nngt0 12224	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁)
131	130	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 0 < 𝑁)
132		ltmul1 12039	. . . . . . 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 1376	. . . . . 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 232	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → ((deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) · 𝑁) < (𝑀 · 𝑁))
135	7	ffvelcdmda 7059	. . . . . . . . . 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 14051	. . . . . . . . . . . 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 11201	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → ((𝐴‘𝑀) · (𝑦↑𝑀)) ∈ ℂ)
141	25, 135, 140, 31, 46	offval2 7676	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) = (𝑦 ∈ ℂ ↦ ((𝐹‘𝑦) − ((𝐴‘𝑀) · (𝑦↑𝑀)))))
142	32, 48	oveq12d 7408	. . . . . . . . 9 ⊢ (𝑦 = (𝐺‘𝑥) → ((𝐹‘𝑦) − ((𝐴‘𝑀) · (𝑦↑𝑀))) = ((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))))
143	4, 30, 141, 142	fmptco 7104	. . . . . . . 8 ⊢ (𝜑 → ((𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) ∘ 𝐺) = (𝑥 ∈ ℂ ↦ ((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))))
144	143	fveq2d 6865	. . . . . . 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 5136	. . . . . . . . 9 ⊢ (𝜑 → (deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) < (𝐷 + 1))
146		nn0leltp1 12600	. . . . . . . . . 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 257	. . . . . . . 8 ⊢ (𝜑 → (deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) ≤ 𝐷)
149		fveq2 6861	. . . . . . . . . . 11 ⊢ (𝑓 = (𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) → (deg‘𝑓) = (deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))))
150	149	breq1d 5120	. . . . . . . . . 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 5824	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) → (𝑓 ∘ 𝐺) = ((𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) ∘ 𝐺))
152	151	fveq2d 6865	. . . . . . . . . . 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 7405	. . . . . . . . . . 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 2746	. . . . . . . . . 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 3592	. . . . . . . 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 2767	. . . . . 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 5703	. . . . . . . . . . 11 ⊢ (ℂ × {(𝐴‘𝑀)}) = (𝑥 ∈ ℂ ↦ (𝐴‘𝑀))
162	161	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (ℂ × {(𝐴‘𝑀)}) = (𝑥 ∈ ℂ ↦ (𝐴‘𝑀)))
163		eqidd 2731	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀)) = (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀)))
164	25, 18, 20, 162, 163	offval2 7676	. . . . . . . . 9 ⊢ (𝜑 → ((ℂ × {(𝐴‘𝑀)}) ∘_f · (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀))) = (𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))))
165	164	fveq2d 6865	. . . . . . . 8 ⊢ (𝜑 → (deg‘((ℂ × {(𝐴‘𝑀)}) ∘_f · (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀)))) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))))
166		eqidd 2731	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑦 ∈ ℂ ↦ (𝑦↑𝑀)) = (𝑦 ∈ ℂ ↦ (𝑦↑𝑀)))
167	4, 30, 166, 47	fmptco 7104	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑦 ∈ ℂ ↦ (𝑦↑𝑀)) ∘ 𝐺) = (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀)))
168		1cnd 11176	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℂ)
169		plypow 26117	. . . . . . . . . . . 12 ⊢ ((ℂ ⊆ ℂ ∧ 1 ∈ ℂ ∧ 𝑀 ∈ ℕ₀) → (𝑦 ∈ ℂ ↦ (𝑦↑𝑀)) ∈ (Poly‘ℂ))
170	50, 168, 16, 169	syl3anc 1373	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑦 ∈ ℂ ↦ (𝑦↑𝑀)) ∈ (Poly‘ℂ))
171	170, 40, 42, 44	plyco 26153	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑦 ∈ ℂ ↦ (𝑦↑𝑀)) ∘ 𝐺) ∈ (Poly‘ℂ))
172	167, 171	eqeltrrd 2830	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀)) ∈ (Poly‘ℂ))
173		dgrmulc 26184	. . . . . . . . 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 1373	. . . . . . . 8 ⊢ (𝜑 → (deg‘((ℂ × {(𝐴‘𝑀)}) ∘_f · (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀)))) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀))))
175	165, 174	eqtr3d 2767	. . . . . . 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 26186	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀))) = (𝑀 · 𝑁))
182	176, 181	eqtrd 2765	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (deg‘(𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))) = (𝑀 · 𝑁))
183	134, 160, 182	3brtr4d 5142	. . . 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 2730	. . . . 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 2730	. . . . 5 ⊢ (deg‘(𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))))
186	184, 185	dgradd2 26181	. . . 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 1373	. . 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 2769	. 2 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (deg‘(𝐹 ∘ 𝐺)) = (𝑀 · 𝑁))
189		0cn 11173	. . . . . . . 8 ⊢ 0 ∈ ℂ
190		ffvelcdm 7056	. . . . . . . 8 ⊢ ((𝐺:ℂ⟶ℂ ∧ 0 ∈ ℂ) → (𝐺‘0) ∈ ℂ)
191	3, 189, 190	sylancl 586	. . . . . . 7 ⊢ (𝜑 → (𝐺‘0) ∈ ℂ)
192	7, 191	ffvelcdmd 7060	. . . . . 6 ⊢ (𝜑 → (𝐹‘(𝐺‘0)) ∈ ℂ)
193		0dgr 26157	. . . . . 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 12512	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℂ)
196	195	mul01d 11380	. . . . 5 ⊢ (𝜑 → (𝑀 · 0) = 0)
197	194, 196	eqtr4d 2768	. . . 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 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑁 = 0) ∧ 𝑥 ∈ ℂ) → (𝐺‘0) ∈ ℂ)
200		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 = 0) → 𝑁 = 0)
201	177, 200	eqtr3id 2779	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 = 0) → (deg‘𝐺) = 0)
202		0dgrb 26158	. . . . . . . . . 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 232	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 = 0) → 𝐺 = (ℂ × {(𝐺‘0)}))
206		fconstmpt 5703	. . . . . . 7 ⊢ (ℂ × {(𝐺‘0)}) = (𝑥 ∈ ℂ ↦ (𝐺‘0))
207	205, 206	eqtrdi 2781	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 = 0) → 𝐺 = (𝑥 ∈ ℂ ↦ (𝐺‘0)))
208	31	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 = 0) → 𝐹 = (𝑦 ∈ ℂ ↦ (𝐹‘𝑦)))
209		fveq2 6861	. . . . . 6 ⊢ (𝑦 = (𝐺‘0) → (𝐹‘𝑦) = (𝐹‘(𝐺‘0)))
210	199, 207, 208, 209	fmptco 7104	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 = 0) → (𝐹 ∘ 𝐺) = (𝑥 ∈ ℂ ↦ (𝐹‘(𝐺‘0))))
211		fconstmpt 5703	. . . . 5 ⊢ (ℂ × {(𝐹‘(𝐺‘0))}) = (𝑥 ∈ ℂ ↦ (𝐹‘(𝐺‘0)))
212	210, 211	eqtr4di 2783	. . . 4 ⊢ ((𝜑 ∧ 𝑁 = 0) → (𝐹 ∘ 𝐺) = (ℂ × {(𝐹‘(𝐺‘0))}))
213	212	fveq2d 6865	. . 3 ⊢ ((𝜑 ∧ 𝑁 = 0) → (deg‘(𝐹 ∘ 𝐺)) = (deg‘(ℂ × {(𝐹‘(𝐺‘0))})))
214	200	oveq2d 7406	. . 3 ⊢ ((𝜑 ∧ 𝑁 = 0) → (𝑀 · 𝑁) = (𝑀 · 0))
215	198, 213, 214	3eqtr4d 2775	. 2 ⊢ ((𝜑 ∧ 𝑁 = 0) → (deg‘(𝐹 ∘ 𝐺)) = (𝑀 · 𝑁))
216		dgrcl 26145	. . . . 5 ⊢ (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ₀)
217	1, 216	syl 17	. . . 4 ⊢ (𝜑 → (deg‘𝐺) ∈ ℕ₀)
218	177, 217	eqeltrid 2833	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
219		elnn0 12451	. . 3 ⊢ (𝑁 ∈ ℕ₀ ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
220	218, 219	sylib 218	. 2 ⊢ (𝜑 → (𝑁 ∈ ℕ ∨ 𝑁 = 0))
221	188, 215, 220	mpjaodan 960	1 ⊢ (𝜑 → (deg‘(𝐹 ∘ 𝐺)) = (𝑀 · 𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2109 ≠ wne 2926 ∀wral 3045 Vcvv 3450 ⊆ wss 3917 ifcif 4491 {csn 4592 class class class wbr 5110 ↦ cmpt 5191 × cxp 5639 ∘ ccom 5645 ⟶wf 6510 ‘cfv 6514 (class class class)co 7390 ∘_f cof 7654 ℂcc 11073 ℝcr 11074 0cc0 11075 1c1 11076 + caddc 11078 · cmul 11080 < clt 11215 ≤ cle 11216 − cmin 11412 ℕcn 12193 ℕ₀cn0 12449 ↑cexp 14033 0_𝑝c0p 25577 Polycply 26096 coeffccoe 26098 degcdgr 26099

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-coe 26102 df-dgr 26103

This theorem is referenced by: dgrco 26188

W3C validator