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Theorem dgrco 26240

Description: The degree of a composition of two polynomials is the product of the degrees. (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))

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

Proof of Theorem dgrco Dummy variables 𝑓 𝑥 𝑦 𝑑 𝑔 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		plyssc 26165	. . 3 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
2		dgrco.3	. . 3 ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))
3	1, 2	sselid 3919	. 2 ⊢ (𝜑 → 𝐹 ∈ (Poly‘ℂ))
4		dgrco.1	. . . 4 ⊢ 𝑀 = (deg‘𝐹)
5		dgrcl 26198	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ₀)
6	2, 5	syl 17	. . . 4 ⊢ (𝜑 → (deg‘𝐹) ∈ ℕ₀)
7	4, 6	eqeltrid 2840	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
8		breq2 5089	. . . . . . 7 ⊢ (𝑥 = 0 → ((deg‘𝑓) ≤ 𝑥 ↔ (deg‘𝑓) ≤ 0))
9	8	imbi1d 341	. . . . . 6 ⊢ (𝑥 = 0 → (((deg‘𝑓) ≤ 𝑥 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁)) ↔ ((deg‘𝑓) ≤ 0 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁))))
10	9	ralbidv 3160	. . . . 5 ⊢ (𝑥 = 0 → (∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 𝑥 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁)) ↔ ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 0 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁))))
11	10	imbi2d 340	. . . 4 ⊢ (𝑥 = 0 → ((𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 𝑥 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁))) ↔ (𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 0 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁)))))
12		breq2 5089	. . . . . . 7 ⊢ (𝑥 = 𝑑 → ((deg‘𝑓) ≤ 𝑥 ↔ (deg‘𝑓) ≤ 𝑑))
13	12	imbi1d 341	. . . . . 6 ⊢ (𝑥 = 𝑑 → (((deg‘𝑓) ≤ 𝑥 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁)) ↔ ((deg‘𝑓) ≤ 𝑑 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁))))
14	13	ralbidv 3160	. . . . 5 ⊢ (𝑥 = 𝑑 → (∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 𝑥 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁)) ↔ ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 𝑑 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁))))
15	14	imbi2d 340	. . . 4 ⊢ (𝑥 = 𝑑 → ((𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 𝑥 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁))) ↔ (𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 𝑑 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁)))))
16		breq2 5089	. . . . . . 7 ⊢ (𝑥 = (𝑑 + 1) → ((deg‘𝑓) ≤ 𝑥 ↔ (deg‘𝑓) ≤ (𝑑 + 1)))
17	16	imbi1d 341	. . . . . 6 ⊢ (𝑥 = (𝑑 + 1) → (((deg‘𝑓) ≤ 𝑥 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁)) ↔ ((deg‘𝑓) ≤ (𝑑 + 1) → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁))))
18	17	ralbidv 3160	. . . . 5 ⊢ (𝑥 = (𝑑 + 1) → (∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 𝑥 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁)) ↔ ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ (𝑑 + 1) → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁))))
19	18	imbi2d 340	. . . 4 ⊢ (𝑥 = (𝑑 + 1) → ((𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 𝑥 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁))) ↔ (𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ (𝑑 + 1) → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁)))))
20		breq2 5089	. . . . . . 7 ⊢ (𝑥 = 𝑀 → ((deg‘𝑓) ≤ 𝑥 ↔ (deg‘𝑓) ≤ 𝑀))
21	20	imbi1d 341	. . . . . 6 ⊢ (𝑥 = 𝑀 → (((deg‘𝑓) ≤ 𝑥 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁)) ↔ ((deg‘𝑓) ≤ 𝑀 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁))))
22	21	ralbidv 3160	. . . . 5 ⊢ (𝑥 = 𝑀 → (∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 𝑥 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁)) ↔ ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 𝑀 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁))))
23	22	imbi2d 340	. . . 4 ⊢ (𝑥 = 𝑀 → ((𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 𝑥 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁))) ↔ (𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 𝑀 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁)))))
24		dgrco.2	. . . . . . . . . . . 12 ⊢ 𝑁 = (deg‘𝐺)
25		dgrco.4	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆))
26		dgrcl 26198	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ₀)
27	25, 26	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (deg‘𝐺) ∈ ℕ₀)
28	24, 27	eqeltrid 2840	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
29	28	nn0cnd 12500	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℂ)
30	29	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝑁 ∈ ℂ)
31	30	mul02d 11344	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (0 · 𝑁) = 0)
32		simprr 773	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘𝑓) ≤ 0)
33		dgrcl 26198	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (Poly‘ℂ) → (deg‘𝑓) ∈ ℕ₀)
34	33	ad2antrl 729	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘𝑓) ∈ ℕ₀)
35	34	nn0ge0d 12501	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 0 ≤ (deg‘𝑓))
36	34	nn0red 12499	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘𝑓) ∈ ℝ)
37		0re 11146	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
38		letri3 11231	. . . . . . . . . . 11 ⊢ (((deg‘𝑓) ∈ ℝ ∧ 0 ∈ ℝ) → ((deg‘𝑓) = 0 ↔ ((deg‘𝑓) ≤ 0 ∧ 0 ≤ (deg‘𝑓))))
39	36, 37, 38	sylancl 587	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → ((deg‘𝑓) = 0 ↔ ((deg‘𝑓) ≤ 0 ∧ 0 ≤ (deg‘𝑓))))
40	32, 35, 39	mpbir2and 714	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘𝑓) = 0)
41	40	oveq1d 7382	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → ((deg‘𝑓) · 𝑁) = (0 · 𝑁))
42	31, 41, 40	3eqtr4d 2781	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → ((deg‘𝑓) · 𝑁) = (deg‘𝑓))
43		fconstmpt 5693	. . . . . . . . 9 ⊢ (ℂ × {(𝑓‘0)}) = (𝑦 ∈ ℂ ↦ (𝑓‘0))
44		0dgrb 26211	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (Poly‘ℂ) → ((deg‘𝑓) = 0 ↔ 𝑓 = (ℂ × {(𝑓‘0)})))
45	44	ad2antrl 729	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → ((deg‘𝑓) = 0 ↔ 𝑓 = (ℂ × {(𝑓‘0)})))
46	40, 45	mpbid 232	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝑓 = (ℂ × {(𝑓‘0)}))
47	25	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝐺 ∈ (Poly‘𝑆))
48		plyf 26163	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ)
49	47, 48	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝐺:ℂ⟶ℂ)
50	49	ffvelcdmda 7036	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) ∧ 𝑦 ∈ ℂ) → (𝐺‘𝑦) ∈ ℂ)
51	49	feqmptd 6908	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝐺 = (𝑦 ∈ ℂ ↦ (𝐺‘𝑦)))
52		fconstmpt 5693	. . . . . . . . . . 11 ⊢ (ℂ × {(𝑓‘0)}) = (𝑥 ∈ ℂ ↦ (𝑓‘0))
53	46, 52	eqtrdi 2787	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝑓 = (𝑥 ∈ ℂ ↦ (𝑓‘0)))
54		eqidd 2737	. . . . . . . . . 10 ⊢ (𝑥 = (𝐺‘𝑦) → (𝑓‘0) = (𝑓‘0))
55	50, 51, 53, 54	fmptco 7082	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (𝑓 ∘ 𝐺) = (𝑦 ∈ ℂ ↦ (𝑓‘0)))
56	43, 46, 55	3eqtr4a 2797	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝑓 = (𝑓 ∘ 𝐺))
57	56	fveq2d 6844	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘𝑓) = (deg‘(𝑓 ∘ 𝐺)))
58	42, 57	eqtr2d 2772	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁))
59	58	expr 456	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (Poly‘ℂ)) → ((deg‘𝑓) ≤ 0 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁)))
60	59	ralrimiva 3129	. . . 4 ⊢ (𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 0 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁)))
61		fveq2 6840	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (deg‘𝑓) = (deg‘𝑔))
62	61	breq1d 5095	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → ((deg‘𝑓) ≤ 𝑑 ↔ (deg‘𝑔) ≤ 𝑑))
63		coeq1 5812	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → (𝑓 ∘ 𝐺) = (𝑔 ∘ 𝐺))
64	63	fveq2d 6844	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (deg‘(𝑓 ∘ 𝐺)) = (deg‘(𝑔 ∘ 𝐺)))
65	61	oveq1d 7382	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → ((deg‘𝑓) · 𝑁) = ((deg‘𝑔) · 𝑁))
66	64, 65	eqeq12d 2752	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → ((deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁) ↔ (deg‘(𝑔 ∘ 𝐺)) = ((deg‘𝑔) · 𝑁)))
67	62, 66	imbi12d 344	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → (((deg‘𝑓) ≤ 𝑑 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁)) ↔ ((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔 ∘ 𝐺)) = ((deg‘𝑔) · 𝑁))))
68	67	cbvralvw 3215	. . . . . . 7 ⊢ (∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 𝑑 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁)) ↔ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔 ∘ 𝐺)) = ((deg‘𝑔) · 𝑁)))
69	33	ad2antrl 729	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔 ∘ 𝐺)) = ((deg‘𝑔) · 𝑁)))) → (deg‘𝑓) ∈ ℕ₀)
70	69	nn0red 12499	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔 ∘ 𝐺)) = ((deg‘𝑔) · 𝑁)))) → (deg‘𝑓) ∈ ℝ)
71		nn0p1nn 12476	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ ℕ₀ → (𝑑 + 1) ∈ ℕ)
72	71	ad2antlr 728	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔 ∘ 𝐺)) = ((deg‘𝑔) · 𝑁)))) → (𝑑 + 1) ∈ ℕ)
73	72	nnred 12189	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔 ∘ 𝐺)) = ((deg‘𝑔) · 𝑁)))) → (𝑑 + 1) ∈ ℝ)
74	70, 73	leloed 11289	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔 ∘ 𝐺)) = ((deg‘𝑔) · 𝑁)))) → ((deg‘𝑓) ≤ (𝑑 + 1) ↔ ((deg‘𝑓) < (𝑑 + 1) ∨ (deg‘𝑓) = (𝑑 + 1))))
75		simplr 769	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔 ∘ 𝐺)) = ((deg‘𝑔) · 𝑁)))) → 𝑑 ∈ ℕ₀)
76		nn0leltp1 12588	. . . . . . . . . . . . 13 ⊢ (((deg‘𝑓) ∈ ℕ₀ ∧ 𝑑 ∈ ℕ₀) → ((deg‘𝑓) ≤ 𝑑 ↔ (deg‘𝑓) < (𝑑 + 1)))
77	69, 75, 76	syl2anc 585	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔 ∘ 𝐺)) = ((deg‘𝑔) · 𝑁)))) → ((deg‘𝑓) ≤ 𝑑 ↔ (deg‘𝑓) < (𝑑 + 1)))
78		fveq2 6840	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = 𝑓 → (deg‘𝑔) = (deg‘𝑓))
79	78	breq1d 5095	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑓 → ((deg‘𝑔) ≤ 𝑑 ↔ (deg‘𝑓) ≤ 𝑑))
80		coeq1 5812	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = 𝑓 → (𝑔 ∘ 𝐺) = (𝑓 ∘ 𝐺))
81	80	fveq2d 6844	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = 𝑓 → (deg‘(𝑔 ∘ 𝐺)) = (deg‘(𝑓 ∘ 𝐺)))
82	78	oveq1d 7382	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = 𝑓 → ((deg‘𝑔) · 𝑁) = ((deg‘𝑓) · 𝑁))
83	81, 82	eqeq12d 2752	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑓 → ((deg‘(𝑔 ∘ 𝐺)) = ((deg‘𝑔) · 𝑁) ↔ (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁)))
84	79, 83	imbi12d 344	. . . . . . . . . . . . . 14 ⊢ (𝑔 = 𝑓 → (((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔 ∘ 𝐺)) = ((deg‘𝑔) · 𝑁)) ↔ ((deg‘𝑓) ≤ 𝑑 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁))))
85	84	rspcva 3562	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔 ∘ 𝐺)) = ((deg‘𝑔) · 𝑁))) → ((deg‘𝑓) ≤ 𝑑 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁)))
86	85	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔 ∘ 𝐺)) = ((deg‘𝑔) · 𝑁)))) → ((deg‘𝑓) ≤ 𝑑 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁)))
87	77, 86	sylbird 260	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔 ∘ 𝐺)) = ((deg‘𝑔) · 𝑁)))) → ((deg‘𝑓) < (𝑑 + 1) → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁)))
88		eqid 2736	. . . . . . . . . . . . 13 ⊢ (deg‘𝑓) = (deg‘𝑓)
89		simprll 779	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ ((𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔 ∘ 𝐺)) = ((deg‘𝑔) · 𝑁))) ∧ (deg‘𝑓) = (𝑑 + 1))) → 𝑓 ∈ (Poly‘ℂ))
90	1, 25	sselid 3919	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺 ∈ (Poly‘ℂ))
91	90	ad2antrr 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ ((𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔 ∘ 𝐺)) = ((deg‘𝑔) · 𝑁))) ∧ (deg‘𝑓) = (𝑑 + 1))) → 𝐺 ∈ (Poly‘ℂ))
92		eqid 2736	. . . . . . . . . . . . 13 ⊢ (coeff‘𝑓) = (coeff‘𝑓)
93		simplr 769	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ ((𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔 ∘ 𝐺)) = ((deg‘𝑔) · 𝑁))) ∧ (deg‘𝑓) = (𝑑 + 1))) → 𝑑 ∈ ℕ₀)
94		simprr 773	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ ((𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔 ∘ 𝐺)) = ((deg‘𝑔) · 𝑁))) ∧ (deg‘𝑓) = (𝑑 + 1))) → (deg‘𝑓) = (𝑑 + 1))
95		simprlr 780	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ ((𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔 ∘ 𝐺)) = ((deg‘𝑔) · 𝑁))) ∧ (deg‘𝑓) = (𝑑 + 1))) → ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔 ∘ 𝐺)) = ((deg‘𝑔) · 𝑁)))
96		fveq2 6840	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = ℎ → (deg‘𝑔) = (deg‘ℎ))
97	96	breq1d 5095	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = ℎ → ((deg‘𝑔) ≤ 𝑑 ↔ (deg‘ℎ) ≤ 𝑑))
98		coeq1 5812	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔 = ℎ → (𝑔 ∘ 𝐺) = (ℎ ∘ 𝐺))
99	98	fveq2d 6844	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = ℎ → (deg‘(𝑔 ∘ 𝐺)) = (deg‘(ℎ ∘ 𝐺)))
100	96	oveq1d 7382	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = ℎ → ((deg‘𝑔) · 𝑁) = ((deg‘ℎ) · 𝑁))
101	99, 100	eqeq12d 2752	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = ℎ → ((deg‘(𝑔 ∘ 𝐺)) = ((deg‘𝑔) · 𝑁) ↔ (deg‘(ℎ ∘ 𝐺)) = ((deg‘ℎ) · 𝑁)))
102	97, 101	imbi12d 344	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = ℎ → (((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔 ∘ 𝐺)) = ((deg‘𝑔) · 𝑁)) ↔ ((deg‘ℎ) ≤ 𝑑 → (deg‘(ℎ ∘ 𝐺)) = ((deg‘ℎ) · 𝑁))))
103	102	cbvralvw 3215	. . . . . . . . . . . . . 14 ⊢ (∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔 ∘ 𝐺)) = ((deg‘𝑔) · 𝑁)) ↔ ∀ℎ ∈ (Poly‘ℂ)((deg‘ℎ) ≤ 𝑑 → (deg‘(ℎ ∘ 𝐺)) = ((deg‘ℎ) · 𝑁)))
104	95, 103	sylib 218	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ ((𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔 ∘ 𝐺)) = ((deg‘𝑔) · 𝑁))) ∧ (deg‘𝑓) = (𝑑 + 1))) → ∀ℎ ∈ (Poly‘ℂ)((deg‘ℎ) ≤ 𝑑 → (deg‘(ℎ ∘ 𝐺)) = ((deg‘ℎ) · 𝑁)))
105	88, 24, 89, 91, 92, 93, 94, 104	dgrcolem2 26239	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ ((𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔 ∘ 𝐺)) = ((deg‘𝑔) · 𝑁))) ∧ (deg‘𝑓) = (𝑑 + 1))) → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁))
106	105	expr 456	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔 ∘ 𝐺)) = ((deg‘𝑔) · 𝑁)))) → ((deg‘𝑓) = (𝑑 + 1) → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁)))
107	87, 106	jaod 860	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔 ∘ 𝐺)) = ((deg‘𝑔) · 𝑁)))) → (((deg‘𝑓) < (𝑑 + 1) ∨ (deg‘𝑓) = (𝑑 + 1)) → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁)))
108	74, 107	sylbid 240	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔 ∘ 𝐺)) = ((deg‘𝑔) · 𝑁)))) → ((deg‘𝑓) ≤ (𝑑 + 1) → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁)))
109	108	expr 456	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑓 ∈ (Poly‘ℂ)) → (∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔 ∘ 𝐺)) = ((deg‘𝑔) · 𝑁)) → ((deg‘𝑓) ≤ (𝑑 + 1) → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁))))
110	109	ralrimdva 3137	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → (∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔 ∘ 𝐺)) = ((deg‘𝑔) · 𝑁)) → ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ (𝑑 + 1) → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁))))
111	68, 110	biimtrid 242	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → (∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 𝑑 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁)) → ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ (𝑑 + 1) → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁))))
112	111	expcom 413	. . . . 5 ⊢ (𝑑 ∈ ℕ₀ → (𝜑 → (∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 𝑑 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁)) → ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ (𝑑 + 1) → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁)))))
113	112	a2d 29	. . . 4 ⊢ (𝑑 ∈ ℕ₀ → ((𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 𝑑 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁))) → (𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ (𝑑 + 1) → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁)))))
114	11, 15, 19, 23, 60, 113	nn0ind 12624	. . 3 ⊢ (𝑀 ∈ ℕ₀ → (𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 𝑀 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁))))
115	7, 114	mpcom 38	. 2 ⊢ (𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 𝑀 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁)))
116	7	nn0red 12499	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℝ)
117	116	leidd 11716	. 2 ⊢ (𝜑 → 𝑀 ≤ 𝑀)
118		fveq2 6840	. . . . . 6 ⊢ (𝑓 = 𝐹 → (deg‘𝑓) = (deg‘𝐹))
119	118, 4	eqtr4di 2789	. . . . 5 ⊢ (𝑓 = 𝐹 → (deg‘𝑓) = 𝑀)
120	119	breq1d 5095	. . . 4 ⊢ (𝑓 = 𝐹 → ((deg‘𝑓) ≤ 𝑀 ↔ 𝑀 ≤ 𝑀))
121		coeq1 5812	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓 ∘ 𝐺) = (𝐹 ∘ 𝐺))
122	121	fveq2d 6844	. . . . 5 ⊢ (𝑓 = 𝐹 → (deg‘(𝑓 ∘ 𝐺)) = (deg‘(𝐹 ∘ 𝐺)))
123	119	oveq1d 7382	. . . . 5 ⊢ (𝑓 = 𝐹 → ((deg‘𝑓) · 𝑁) = (𝑀 · 𝑁))
124	122, 123	eqeq12d 2752	. . . 4 ⊢ (𝑓 = 𝐹 → ((deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁) ↔ (deg‘(𝐹 ∘ 𝐺)) = (𝑀 · 𝑁)))
125	120, 124	imbi12d 344	. . 3 ⊢ (𝑓 = 𝐹 → (((deg‘𝑓) ≤ 𝑀 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁)) ↔ (𝑀 ≤ 𝑀 → (deg‘(𝐹 ∘ 𝐺)) = (𝑀 · 𝑁))))
126	125	rspcv 3560	. 2 ⊢ (𝐹 ∈ (Poly‘ℂ) → (∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 𝑀 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁)) → (𝑀 ≤ 𝑀 → (deg‘(𝐹 ∘ 𝐺)) = (𝑀 · 𝑁))))
127	3, 115, 117, 126	syl3c 66	1 ⊢ (𝜑 → (deg‘(𝐹 ∘ 𝐺)) = (𝑀 · 𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1542 ∈ wcel 2114 ∀wral 3051 {csn 4567 class class class wbr 5085 ↦ cmpt 5166 × cxp 5629 ∘ ccom 5635 ⟶wf 6494 ‘cfv 6498 (class class class)co 7367 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 + caddc 11041 · cmul 11043 < clt 11179 ≤ cle 11180 ℕcn 12174 ℕ₀cn0 12437 Polycply 26149 coeffccoe 26151 degcdgr 26152

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-er 8643 df-map 8775 df-pm 8776 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-sup 9355 df-inf 9356 df-oi 9425 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-n0 12438 df-z 12525 df-uz 12789 df-rp 12943 df-fz 13462 df-fzo 13609 df-fl 13751 df-seq 13964 df-exp 14024 df-hash 14293 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-clim 15450 df-rlim 15451 df-sum 15649 df-0p 25637 df-ply 26153 df-coe 26155 df-dgr 26156

This theorem is referenced by: taylply2 26333 ftalem7 27042

W3C validator