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Theorem dgrcolem2 26322

Description: Lemma for dgrco 26323. (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 26246	. . . . . . . . . . 11 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ)
3	1, 2	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺:ℂ⟶ℂ)
4	3	ffvelcdmda 7060	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝐺‘𝑥) ∈ ℂ)
5		dgrco.3	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))
6		plyf 26246	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
7	5, 6	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:ℂ⟶ℂ)
8	7	ffvelcdmda 7060	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐺‘𝑥) ∈ ℂ) → (𝐹‘(𝐺‘𝑥)) ∈ ℂ)
9	4, 8	syldan 600	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝐹‘(𝐺‘𝑥)) ∈ ℂ)
10		dgrco.5	. . . . . . . . . . . . 13 ⊢ 𝐴 = (coeff‘𝐹)
11	10	coef3 26280	. . . . . . . . . . . 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 26281	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ₀)
15	5, 14	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (deg‘𝐹) ∈ ℕ₀)
16	13, 15	eqeltrid 2865	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
17	12, 16	ffvelcdmd 7061	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴‘𝑀) ∈ ℂ)
18	17	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝐴‘𝑀) ∈ ℂ)
19	16	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝑀 ∈ ℕ₀)
20	4, 19	expcld 14153	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((𝐺‘𝑥)↑𝑀) ∈ ℂ)
21	18, 20	mulcld 11196	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)) ∈ ℂ)
22	9, 21	npcand 11540	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))) + ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))) = (𝐹‘(𝐺‘𝑥)))
23	22	mpteq2dva 5190	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ (((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))) + ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))) = (𝑥 ∈ ℂ ↦ (𝐹‘(𝐺‘𝑥))))
24		cnex 11148	. . . . . . . 8 ⊢ ℂ ∈ V
25	24	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℂ ∈ V)
26	9, 21	subcld 11536	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))) ∈ ℂ)
27		eqidd 2762	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))) = (𝑥 ∈ ℂ ↦ ((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))))
28		eqidd 2762	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))) = (𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))))
29	25, 26, 21, 27, 28	offval2 7675	. . . . . 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 6930	. . . . . . 7 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ ℂ ↦ (𝐺‘𝑥)))
31	7	feqmptd 6930	. . . . . . 7 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ ℂ ↦ (𝐹‘𝑦)))
32		fveq2 6862	. . . . . . 7 ⊢ (𝑦 = (𝐺‘𝑥) → (𝐹‘𝑦) = (𝐹‘(𝐺‘𝑥)))
33	4, 30, 31, 32	fmptco 7106	. . . . . 6 ⊢ (𝜑 → (𝐹 ∘ 𝐺) = (𝑥 ∈ ℂ ↦ (𝐹‘(𝐺‘𝑥))))
34	23, 29, 33	3eqtr4rd 2807	. . . . 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 6866	. . . 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 484	. . 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 7675	. . . . . 6 ⊢ (𝜑 → ((𝐹 ∘ 𝐺) ∘_f − (𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))) = (𝑥 ∈ ℂ ↦ ((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))))
38		plyssc 26248	. . . . . . . . 9 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
39	38, 5	sselid 3932	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (Poly‘ℂ))
40	38, 1	sselid 3932	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ (Poly‘ℂ))
41		addcl 11149	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (𝑧 + 𝑤) ∈ ℂ)
42	41	adantl 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ)) → (𝑧 + 𝑤) ∈ ℂ)
43		mulcl 11151	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (𝑧 · 𝑤) ∈ ℂ)
44	43	adantl 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ)) → (𝑧 · 𝑤) ∈ ℂ)
45	39, 40, 42, 44	plyco 26289	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∘ 𝐺) ∈ (Poly‘ℂ))
46		eqidd 2762	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))) = (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))
47		oveq1 7398	. . . . . . . . . 10 ⊢ (𝑦 = (𝐺‘𝑥) → (𝑦↑𝑀) = ((𝐺‘𝑥)↑𝑀))
48	47	oveq2d 7407	. . . . . . . . 9 ⊢ (𝑦 = (𝐺‘𝑥) → ((𝐴‘𝑀) · (𝑦↑𝑀)) = ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))
49	4, 30, 46, 48	fmptco 7106	. . . . . . . 8 ⊢ (𝜑 → ((𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))) ∘ 𝐺) = (𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))))
50		ssidd 3957	. . . . . . . . . 10 ⊢ (𝜑 → ℂ ⊆ ℂ)
51		eqid 2761	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))) = (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))
52	51	ply1term 26252	. . . . . . . . . 10 ⊢ ((ℂ ⊆ ℂ ∧ (𝐴‘𝑀) ∈ ℂ ∧ 𝑀 ∈ ℕ₀) → (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))) ∈ (Poly‘ℂ))
53	50, 17, 16, 52	syl3anc 1389	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))) ∈ (Poly‘ℂ))
54	53, 40, 42, 44	plyco 26289	. . . . . . . 8 ⊢ (𝜑 → ((𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))) ∘ 𝐺) ∈ (Poly‘ℂ))
55	49, 54	eqeltrrd 2862	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))) ∈ (Poly‘ℂ))
56		plysubcl 26270	. . . . . . 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 593	. . . . . 6 ⊢ (𝜑 → ((𝐹 ∘ 𝐺) ∘_f − (𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))) ∈ (Poly‘ℂ))
58	37, 57	eqeltrrd 2862	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))) ∈ (Poly‘ℂ))
59	58	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (𝑥 ∈ ℂ ↦ ((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))) ∈ (Poly‘ℂ))
60	55	adantr 484	. . . 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 12514	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ ℕ₀ → (𝐷 + 1) ∈ ℕ)
64	62, 63	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐷 + 1) ∈ ℕ)
65	61, 64	eqeltrd 2861	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℕ)
66	65	nngt0d 12256	. . . . . . . . 9 ⊢ (𝜑 → 0 < 𝑀)
67		fveq2 6862	. . . . . . . . . . 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 26310	. . . . . . . . . . 11 ⊢ (deg‘0_𝑝) = 0
69	67, 68	eqtrdi 2812	. . . . . . . . . 10 ⊢ ((𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) = 0_𝑝 → (deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) = 0)
70	69	breq1d 5107	. . . . . . . . 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 249	. . . . . . . 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 2761	. . . . . . . . . . . 12 ⊢ (deg‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) = (deg‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))
74	13, 73	dgrsub 26320	. . . . . . . . . . 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 593	. . . . . . . . . 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 12257	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ≠ 0)
77	13, 10	dgreq0 26313	. . . . . . . . . . . . . . . . 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 6862	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 = 0_𝑝 → (deg‘𝐹) = (deg‘0_𝑝))
80	79, 68	eqtrdi 2812	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 = 0_𝑝 → (deg‘𝐹) = 0)
81	13, 80	eqtrid 2808	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 = 0_𝑝 → 𝑀 = 0)
82	78, 81	biimtrrdi 256	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐴‘𝑀) = 0 → 𝑀 = 0))
83	82	necon3d 2977	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑀 ≠ 0 → (𝐴‘𝑀) ≠ 0))
84	76, 83	mpd 15	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴‘𝑀) ≠ 0)
85	51	dgr1term 26308	. . . . . . . . . . . . 13 ⊢ (((𝐴‘𝑀) ∈ ℂ ∧ (𝐴‘𝑀) ≠ 0 ∧ 𝑀 ∈ ℕ₀) → (deg‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) = 𝑀)
86	17, 84, 16, 85	syl3anc 1389	. . . . . . . . . . . 12 ⊢ (𝜑 → (deg‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) = 𝑀)
87	86	ifeq1d 4497	. . . . . . . . . . 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 4518	. . . . . . . . . . 11 ⊢ if(𝑀 ≤ (deg‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))), 𝑀, 𝑀) = 𝑀
89	87, 88	eqtrdi 2812	. . . . . . . . . 10 ⊢ (𝜑 → if(𝑀 ≤ (deg‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))), (deg‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))), 𝑀) = 𝑀)
90	75, 89	breqtrd 5123	. . . . . . . . 9 ⊢ (𝜑 → (deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) ≤ 𝑀)
91		eqid 2761	. . . . . . . . . . . . 13 ⊢ (coeff‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) = (coeff‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))
92	10, 91	coesub 26305	. . . . . . . . . . . 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 593	. . . . . . . . . . 11 ⊢ (𝜑 → (coeff‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) = (𝐴 ∘_f − (coeff‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))))
94	93	fveq1d 6864	. . . . . . . . . 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 6687	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 Fn ℕ₀)
96	91	coef3 26280	. . . . . . . . . . . . . 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 6687	. . . . . . . . . . . 12 ⊢ (𝜑 → (coeff‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) Fn ℕ₀)
99		nn0ex 12481	. . . . . . . . . . . . 13 ⊢ ℕ₀ ∈ V
100	99	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ℕ₀ ∈ V)
101		inidm 4176	. . . . . . . . . . . 12 ⊢ (ℕ₀ ∩ ℕ₀) = ℕ₀
102		eqidd 2762	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑀 ∈ ℕ₀) → (𝐴‘𝑀) = (𝐴‘𝑀))
103	51	coe1term 26307	. . . . . . . . . . . . . . 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 1389	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((coeff‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))‘𝑀) = if(𝑀 = 𝑀, (𝐴‘𝑀), 0))
105		eqid 2761	. . . . . . . . . . . . . . 15 ⊢ 𝑀 = 𝑀
106	105	iftruei 4484	. . . . . . . . . . . . . 14 ⊢ if(𝑀 = 𝑀, (𝐴‘𝑀), 0) = (𝐴‘𝑀)
107	104, 106	eqtrdi 2812	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((coeff‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))‘𝑀) = (𝐴‘𝑀))
108	107	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑀 ∈ ℕ₀) → ((coeff‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))‘𝑀) = (𝐴‘𝑀))
109	95, 98, 100, 100, 101, 102, 108	ofval 7666	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑀 ∈ ℕ₀) → ((𝐴 ∘_f − (coeff‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))))‘𝑀) = ((𝐴‘𝑀) − (𝐴‘𝑀)))
110	16, 109	mpdan 697	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 ∘_f − (coeff‘(𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))))‘𝑀) = ((𝐴‘𝑀) − (𝐴‘𝑀)))
111	17	subidd 11524	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴‘𝑀) − (𝐴‘𝑀)) = 0)
112	94, 110, 111	3eqtrd 2800	. . . . . . . . 9 ⊢ (𝜑 → ((coeff‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))))‘𝑀) = 0)
113		plysubcl 26270	. . . . . . . . . . 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 593	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) ∈ (Poly‘ℂ))
115		eqid 2761	. . . . . . . . . . 11 ⊢ (deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) = (deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))))
116		eqid 2761	. . . . . . . . . . 11 ⊢ (coeff‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) = (coeff‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))))
117	115, 116	dgrlt 26314	. . . . . . . . . 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 593	. . . . . . . . 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 723	. . . . . . . 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 871	. . . . . . 7 ⊢ (𝜑 → (deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) < 𝑀)
121	120	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) < 𝑀)
122		dgrcl 26281	. . . . . . . . . 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 12537	. . . . . . . 8 ⊢ (𝜑 → (deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) ∈ ℝ)
125	124	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) ∈ ℝ)
126	16	nn0red 12537	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℝ)
127	126	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝑀 ∈ ℝ)
128		nnre 12211	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
129	128	adantl 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℝ)
130		nngt0 12238	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁)
131	130	adantl 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 0 < 𝑁)
132		ltmul1 12035	. . . . . . 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 1392	. . . . . 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 234	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → ((deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) · 𝑁) < (𝑀 · 𝑁))
135	7	ffvelcdmda 7060	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝐹‘𝑦) ∈ ℂ)
136	17	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝐴‘𝑀) ∈ ℂ)
137		id 22	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℂ → 𝑦 ∈ ℂ)
138		expcl 14086	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℂ ∧ 𝑀 ∈ ℕ₀) → (𝑦↑𝑀) ∈ ℂ)
139	137, 16, 138	syl2anr 606	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑦↑𝑀) ∈ ℂ)
140	136, 139	mulcld 11196	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → ((𝐴‘𝑀) · (𝑦↑𝑀)) ∈ ℂ)
141	25, 135, 140, 31, 46	offval2 7675	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) = (𝑦 ∈ ℂ ↦ ((𝐹‘𝑦) − ((𝐴‘𝑀) · (𝑦↑𝑀)))))
142	32, 48	oveq12d 7409	. . . . . . . . 9 ⊢ (𝑦 = (𝐺‘𝑥) → ((𝐹‘𝑦) − ((𝐴‘𝑀) · (𝑦↑𝑀))) = ((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))))
143	4, 30, 141, 142	fmptco 7106	. . . . . . . 8 ⊢ (𝜑 → ((𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) ∘ 𝐺) = (𝑥 ∈ ℂ ↦ ((𝐹‘(𝐺‘𝑥)) − ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))))
144	143	fveq2d 6866	. . . . . . 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 5123	. . . . . . . . 9 ⊢ (𝜑 → (deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) < (𝐷 + 1))
146		nn0leltp1 12626	. . . . . . . . . 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 593	. . . . . . . . 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 259	. . . . . . . 8 ⊢ (𝜑 → (deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))) ≤ 𝐷)
149		fveq2 6862	. . . . . . . . . . 11 ⊢ (𝑓 = (𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) → (deg‘𝑓) = (deg‘(𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀))))))
150	149	breq1d 5107	. . . . . . . . . 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 5825	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) → (𝑓 ∘ 𝐺) = ((𝐹 ∘_f − (𝑦 ∈ ℂ ↦ ((𝐴‘𝑀) · (𝑦↑𝑀)))) ∘ 𝐺))
152	151	fveq2d 6866	. . . . . . . . . . 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 7406	. . . . . . . . . . 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 2777	. . . . . . . . . 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 346	. . . . . . . . 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 3581	. . . . . . . 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 2798	. . . . . 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 484	. . . . 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 5705	. . . . . . . . . . 11 ⊢ (ℂ × {(𝐴‘𝑀)}) = (𝑥 ∈ ℂ ↦ (𝐴‘𝑀))
162	161	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (ℂ × {(𝐴‘𝑀)}) = (𝑥 ∈ ℂ ↦ (𝐴‘𝑀)))
163		eqidd 2762	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀)) = (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀)))
164	25, 18, 20, 162, 163	offval2 7675	. . . . . . . . 9 ⊢ (𝜑 → ((ℂ × {(𝐴‘𝑀)}) ∘_f · (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀))) = (𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))))
165	164	fveq2d 6866	. . . . . . . 8 ⊢ (𝜑 → (deg‘((ℂ × {(𝐴‘𝑀)}) ∘_f · (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀)))) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))))
166		eqidd 2762	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑦 ∈ ℂ ↦ (𝑦↑𝑀)) = (𝑦 ∈ ℂ ↦ (𝑦↑𝑀)))
167	4, 30, 166, 47	fmptco 7106	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑦 ∈ ℂ ↦ (𝑦↑𝑀)) ∘ 𝐺) = (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀)))
168		1cnd 11169	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℂ)
169		plypow 26253	. . . . . . . . . . . 12 ⊢ ((ℂ ⊆ ℂ ∧ 1 ∈ ℂ ∧ 𝑀 ∈ ℕ₀) → (𝑦 ∈ ℂ ↦ (𝑦↑𝑀)) ∈ (Poly‘ℂ))
170	50, 168, 16, 169	syl3anc 1389	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑦 ∈ ℂ ↦ (𝑦↑𝑀)) ∈ (Poly‘ℂ))
171	170, 40, 42, 44	plyco 26289	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑦 ∈ ℂ ↦ (𝑦↑𝑀)) ∘ 𝐺) ∈ (Poly‘ℂ))
172	167, 171	eqeltrrd 2862	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀)) ∈ (Poly‘ℂ))
173		dgrmulc 26319	. . . . . . . . 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 1389	. . . . . . . 8 ⊢ (𝜑 → (deg‘((ℂ × {(𝐴‘𝑀)}) ∘_f · (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀)))) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀))))
175	165, 174	eqtr3d 2798	. . . . . . 7 ⊢ (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀))))
176	175	adantr 484	. . . . . 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 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝑀 ∈ ℕ)
179		simpr 488	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ)
180	1	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝐺 ∈ (Poly‘𝑆))
181	177, 178, 179, 180	dgrcolem1 26321	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀))) = (𝑀 · 𝑁))
182	176, 181	eqtrd 2796	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (deg‘(𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))) = (𝑀 · 𝑁))
183	134, 160, 182	3brtr4d 5129	. . . 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 2761	. . . . 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 2761	. . . . 5 ⊢ (deg‘(𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀)))) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐴‘𝑀) · ((𝐺‘𝑥)↑𝑀))))
186	184, 185	dgradd2 26316	. . . 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 1389	. . 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 2800	. 2 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (deg‘(𝐹 ∘ 𝐺)) = (𝑀 · 𝑁))
189		0cn 11165	. . . . . . . 8 ⊢ 0 ∈ ℂ
190		ffvelcdm 7057	. . . . . . . 8 ⊢ ((𝐺:ℂ⟶ℂ ∧ 0 ∈ ℂ) → (𝐺‘0) ∈ ℂ)
191	3, 189, 190	sylancl 595	. . . . . . 7 ⊢ (𝜑 → (𝐺‘0) ∈ ℂ)
192	7, 191	ffvelcdmd 7061	. . . . . 6 ⊢ (𝜑 → (𝐹‘(𝐺‘0)) ∈ ℂ)
193		0dgr 26293	. . . . . 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 12538	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℂ)
196	195	mul01d 11376	. . . . 5 ⊢ (𝜑 → (𝑀 · 0) = 0)
197	194, 196	eqtr4d 2799	. . . 4 ⊢ (𝜑 → (deg‘(ℂ × {(𝐹‘(𝐺‘0))})) = (𝑀 · 0))
198	197	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝑁 = 0) → (deg‘(ℂ × {(𝐹‘(𝐺‘0))})) = (𝑀 · 0))
199	191	ad2antrr 736	. . . . . 6 ⊢ (((𝜑 ∧ 𝑁 = 0) ∧ 𝑥 ∈ ℂ) → (𝐺‘0) ∈ ℂ)
200		simpr 488	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 = 0) → 𝑁 = 0)
201	177, 200	eqtr3id 2810	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 = 0) → (deg‘𝐺) = 0)
202		0dgrb 26294	. . . . . . . . . 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 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 = 0) → ((deg‘𝐺) = 0 ↔ 𝐺 = (ℂ × {(𝐺‘0)})))
205	201, 204	mpbid 234	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 = 0) → 𝐺 = (ℂ × {(𝐺‘0)}))
206		fconstmpt 5705	. . . . . . 7 ⊢ (ℂ × {(𝐺‘0)}) = (𝑥 ∈ ℂ ↦ (𝐺‘0))
207	205, 206	eqtrdi 2812	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 = 0) → 𝐺 = (𝑥 ∈ ℂ ↦ (𝐺‘0)))
208	31	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 = 0) → 𝐹 = (𝑦 ∈ ℂ ↦ (𝐹‘𝑦)))
209		fveq2 6862	. . . . . 6 ⊢ (𝑦 = (𝐺‘0) → (𝐹‘𝑦) = (𝐹‘(𝐺‘0)))
210	199, 207, 208, 209	fmptco 7106	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 = 0) → (𝐹 ∘ 𝐺) = (𝑥 ∈ ℂ ↦ (𝐹‘(𝐺‘0))))
211		fconstmpt 5705	. . . . 5 ⊢ (ℂ × {(𝐹‘(𝐺‘0))}) = (𝑥 ∈ ℂ ↦ (𝐹‘(𝐺‘0)))
212	210, 211	eqtr4di 2814	. . . 4 ⊢ ((𝜑 ∧ 𝑁 = 0) → (𝐹 ∘ 𝐺) = (ℂ × {(𝐹‘(𝐺‘0))}))
213	212	fveq2d 6866	. . 3 ⊢ ((𝜑 ∧ 𝑁 = 0) → (deg‘(𝐹 ∘ 𝐺)) = (deg‘(ℂ × {(𝐹‘(𝐺‘0))})))
214	200	oveq2d 7407	. . 3 ⊢ ((𝜑 ∧ 𝑁 = 0) → (𝑀 · 𝑁) = (𝑀 · 0))
215	198, 213, 214	3eqtr4d 2806	. 2 ⊢ ((𝜑 ∧ 𝑁 = 0) → (deg‘(𝐹 ∘ 𝐺)) = (𝑀 · 𝑁))
216		dgrcl 26281	. . . . 5 ⊢ (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ₀)
217	1, 216	syl 17	. . . 4 ⊢ (𝜑 → (deg‘𝐺) ∈ ℕ₀)
218	177, 217	eqeltrid 2865	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
219		elnn0 12477	. . 3 ⊢ (𝑁 ∈ ℕ₀ ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
220	218, 219	sylib 220	. 2 ⊢ (𝜑 → (𝑁 ∈ ℕ ∨ 𝑁 = 0))
221	188, 215, 220	mpjaodan 971	1 ⊢ (𝜑 → (deg‘(𝐹 ∘ 𝐺)) = (𝑀 · 𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 ∨ wo 858 = wceq 1559 ∈ wcel 2141 ≠ wne 2956 ∀wral 3075 Vcvv 3453 ⊆ wss 3902 ifcif 4477 {csn 4579 class class class wbr 5097 ↦ cmpt 5178 × cxp 5641 ∘ ccom 5647 ⟶wf 6512 ‘cfv 6516 (class class class)co 7391 ∘_f cof 7653 ℂcc 11065 ℝcr 11066 0cc0 11067 1c1 11068 + caddc 11070 · cmul 11072 < clt 11210 ≤ cle 11211 − cmin 11408 ℕcn 12204 ℕ₀cn0 12475 ↑cexp 14068 0_𝑝c0p 25719 Polycply 26232 coeffccoe 26234 degcdgr 26235

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7713 ax-inf2 9590 ax-cnex 11123 ax-resscn 11124 ax-1cn 11125 ax-icn 11126 ax-addcl 11127 ax-addrcl 11128 ax-mulcl 11129 ax-mulrcl 11130 ax-mulcom 11131 ax-addass 11132 ax-mulass 11133 ax-distr 11134 ax-i2m1 11135 ax-1ne0 11136 ax-1rid 11137 ax-rnegex 11138 ax-rrecex 11139 ax-cnre 11140 ax-pre-lttri 11141 ax-pre-lttrn 11142 ax-pre-ltadd 11143 ax-pre-mulgt0 11144 ax-pre-sup 11145

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-int 4903 df-iun 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-se 5597 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6283 df-ord 6344 df-on 6345 df-lim 6346 df-suc 6347 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-isom 6525 df-riota 7348 df-ov 7394 df-oprab 7395 df-mpo 7396 df-of 7655 df-om 7842 df-1st 7965 df-2nd 7966 df-frecs 8256 df-wrecs 8287 df-recs 8336 df-rdg 8375 df-1o 8431 df-er 8672 df-map 8804 df-pm 8805 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-sup 9382 df-inf 9383 df-oi 9452 df-card 9891 df-pnf 11212 df-mnf 11213 df-xr 11214 df-ltxr 11215 df-le 11216 df-sub 11410 df-neg 11411 df-div 11839 df-nn 12205 df-2 12274 df-3 12275 df-n0 12476 df-z 12563 df-uz 12834 df-rp 12988 df-fz 13507 df-fzo 13654 df-fl 13796 df-seq 14009 df-exp 14069 df-hash 14338 df-cj 15117 df-re 15118 df-im 15119 df-sqrt 15253 df-abs 15254 df-clim 15506 df-rlim 15507 df-sum 15705 df-0p 25720 df-ply 26236 df-coe 26238 df-dgr 26239

This theorem is referenced by: dgrco 26323

W3C validator