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Theorem dgrco 26413

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 26338	. . 3 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
2		dgrco.3	. . 3 ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))
3	1, 2	sselid 3936	. 2 ⊢ (𝜑 → 𝐹 ∈ (Poly‘ℂ))
4		dgrco.1	. . . 4 ⊢ 𝑀 = (deg‘𝐹)
5		dgrcl 26371	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ₀)
6	2, 5	syl 18	. . . 4 ⊢ (𝜑 → (deg‘𝐹) ∈ ℕ₀)
7	4, 6	eqeltrid 2867	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
8		breq2 5114	. . . . . . 7 ⊢ (𝑥 = 0 → ((deg‘𝑓) ≤ 𝑥 ↔ (deg‘𝑓) ≤ 0))
9	8	imbi1d 344	. . . . . 6 ⊢ (𝑥 = 0 → (((deg‘𝑓) ≤ 𝑥 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁)) ↔ ((deg‘𝑓) ≤ 0 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁))))
10	9	ralbidv 3188	. . . . 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 343	. . . 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 5114	. . . . . . 7 ⊢ (𝑥 = 𝑑 → ((deg‘𝑓) ≤ 𝑥 ↔ (deg‘𝑓) ≤ 𝑑))
13	12	imbi1d 344	. . . . . 6 ⊢ (𝑥 = 𝑑 → (((deg‘𝑓) ≤ 𝑥 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁)) ↔ ((deg‘𝑓) ≤ 𝑑 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁))))
14	13	ralbidv 3188	. . . . 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 343	. . . 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 5114	. . . . . . 7 ⊢ (𝑥 = (𝑑 + 1) → ((deg‘𝑓) ≤ 𝑥 ↔ (deg‘𝑓) ≤ (𝑑 + 1)))
17	16	imbi1d 344	. . . . . 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 3188	. . . . 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 343	. . . 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 5114	. . . . . . 7 ⊢ (𝑥 = 𝑀 → ((deg‘𝑓) ≤ 𝑥 ↔ (deg‘𝑓) ≤ 𝑀))
21	20	imbi1d 344	. . . . . 6 ⊢ (𝑥 = 𝑀 → (((deg‘𝑓) ≤ 𝑥 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁)) ↔ ((deg‘𝑓) ≤ 𝑀 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁))))
22	21	ralbidv 3188	. . . . 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 343	. . . 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 26371	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ₀)
27	25, 26	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (deg‘𝐺) ∈ ℕ₀)
28	24, 27	eqeltrid 2867	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
29	28	nn0cnd 12568	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℂ)
30	29	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝑁 ∈ ℂ)
31	30	mul02d 11409	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (0 · 𝑁) = 0)
32		simprr 784	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘𝑓) ≤ 0)
33		dgrcl 26371	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (Poly‘ℂ) → (deg‘𝑓) ∈ ℕ₀)
34	33	ad2antrl 740	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘𝑓) ∈ ℕ₀)
35	34	nn0ge0d 12569	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 0 ≤ (deg‘𝑓))
36	34	nn0red 12567	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘𝑓) ∈ ℝ)
37		0re 11211	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
38		letri3 11296	. . . . . . . . . . 11 ⊢ (((deg‘𝑓) ∈ ℝ ∧ 0 ∈ ℝ) → ((deg‘𝑓) = 0 ↔ ((deg‘𝑓) ≤ 0 ∧ 0 ≤ (deg‘𝑓))))
39	36, 37, 38	sylancl 597	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → ((deg‘𝑓) = 0 ↔ ((deg‘𝑓) ≤ 0 ∧ 0 ≤ (deg‘𝑓))))
40	32, 35, 39	mpbir2and 725	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘𝑓) = 0)
41	40	oveq1d 7427	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → ((deg‘𝑓) · 𝑁) = (0 · 𝑁))
42	31, 41, 40	3eqtr4d 2808	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → ((deg‘𝑓) · 𝑁) = (deg‘𝑓))
43		fconstmpt 5725	. . . . . . . . 9 ⊢ (ℂ × {(𝑓‘0)}) = (𝑦 ∈ ℂ ↦ (𝑓‘0))
44		0dgrb 26384	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (Poly‘ℂ) → ((deg‘𝑓) = 0 ↔ 𝑓 = (ℂ × {(𝑓‘0)})))
45	44	ad2antrl 740	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → ((deg‘𝑓) = 0 ↔ 𝑓 = (ℂ × {(𝑓‘0)})))
46	40, 45	mpbid 235	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝑓 = (ℂ × {(𝑓‘0)}))
47	25	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝐺 ∈ (Poly‘𝑆))
48		plyf 26336	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ)
49	47, 48	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝐺:ℂ⟶ℂ)
50	49	ffvelcdmda 7081	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) ∧ 𝑦 ∈ ℂ) → (𝐺‘𝑦) ∈ ℂ)
51	49	feqmptd 6951	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝐺 = (𝑦 ∈ ℂ ↦ (𝐺‘𝑦)))
52		fconstmpt 5725	. . . . . . . . . . 11 ⊢ (ℂ × {(𝑓‘0)}) = (𝑥 ∈ ℂ ↦ (𝑓‘0))
53	46, 52	eqtrdi 2814	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝑓 = (𝑥 ∈ ℂ ↦ (𝑓‘0)))
54		eqidd 2764	. . . . . . . . . 10 ⊢ (𝑥 = (𝐺‘𝑦) → (𝑓‘0) = (𝑓‘0))
55	50, 51, 53, 54	fmptco 7127	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (𝑓 ∘ 𝐺) = (𝑦 ∈ ℂ ↦ (𝑓‘0)))
56	43, 46, 55	3eqtr4a 2824	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝑓 = (𝑓 ∘ 𝐺))
57	56	fveq2d 6887	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘𝑓) = (deg‘(𝑓 ∘ 𝐺)))
58	42, 57	eqtr2d 2799	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁))
59	58	expr 461	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (Poly‘ℂ)) → ((deg‘𝑓) ≤ 0 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁)))
60	59	ralrimiva 3157	. . . 4 ⊢ (𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 0 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁)))
61		fveq2 6883	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (deg‘𝑓) = (deg‘𝑔))
62	61	breq1d 5120	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → ((deg‘𝑓) ≤ 𝑑 ↔ (deg‘𝑔) ≤ 𝑑))
63		coeq1 5845	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → (𝑓 ∘ 𝐺) = (𝑔 ∘ 𝐺))
64	63	fveq2d 6887	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (deg‘(𝑓 ∘ 𝐺)) = (deg‘(𝑔 ∘ 𝐺)))
65	61	oveq1d 7427	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → ((deg‘𝑓) · 𝑁) = ((deg‘𝑔) · 𝑁))
66	64, 65	eqeq12d 2779	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → ((deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁) ↔ (deg‘(𝑔 ∘ 𝐺)) = ((deg‘𝑔) · 𝑁)))
67	62, 66	imbi12d 347	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → (((deg‘𝑓) ≤ 𝑑 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁)) ↔ ((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔 ∘ 𝐺)) = ((deg‘𝑔) · 𝑁))))
68	67	cbvralvw 3243	. . . . . . 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 740	. . . . . . . . . . . 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 12567	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔 ∘ 𝐺)) = ((deg‘𝑔) · 𝑁)))) → (deg‘𝑓) ∈ ℝ)
71		nn0p1nn 12544	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ ℕ₀ → (𝑑 + 1) ∈ ℕ)
72	71	ad2antlr 739	. . . . . . . . . . . 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 12249	. . . . . . . . . . 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 11354	. . . . . . . . . 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 780	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔 ∘ 𝐺)) = ((deg‘𝑔) · 𝑁)))) → 𝑑 ∈ ℕ₀)
76		nn0leltp1 12656	. . . . . . . . . . . . 13 ⊢ (((deg‘𝑓) ∈ ℕ₀ ∧ 𝑑 ∈ ℕ₀) → ((deg‘𝑓) ≤ 𝑑 ↔ (deg‘𝑓) < (𝑑 + 1)))
77	69, 75, 76	syl2anc 595	. . . . . . . . . . . 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 6883	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = 𝑓 → (deg‘𝑔) = (deg‘𝑓))
79	78	breq1d 5120	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑓 → ((deg‘𝑔) ≤ 𝑑 ↔ (deg‘𝑓) ≤ 𝑑))
80		coeq1 5845	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = 𝑓 → (𝑔 ∘ 𝐺) = (𝑓 ∘ 𝐺))
81	80	fveq2d 6887	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = 𝑓 → (deg‘(𝑔 ∘ 𝐺)) = (deg‘(𝑓 ∘ 𝐺)))
82	78	oveq1d 7427	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = 𝑓 → ((deg‘𝑔) · 𝑁) = ((deg‘𝑓) · 𝑁))
83	81, 82	eqeq12d 2779	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑓 → ((deg‘(𝑔 ∘ 𝐺)) = ((deg‘𝑔) · 𝑁) ↔ (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁)))
84	79, 83	imbi12d 347	. . . . . . . . . . . . . 14 ⊢ (𝑔 = 𝑓 → (((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔 ∘ 𝐺)) = ((deg‘𝑔) · 𝑁)) ↔ ((deg‘𝑓) ≤ 𝑑 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁))))
85	84	rspcva 3580	. . . . . . . . . . . . 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 486	. . . . . . . . . . . 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 263	. . . . . . . . . . 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 2763	. . . . . . . . . . . . 13 ⊢ (deg‘𝑓) = (deg‘𝑓)
89		simprll 790	. . . . . . . . . . . . 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 3936	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺 ∈ (Poly‘ℂ))
91	90	ad2antrr 738	. . . . . . . . . . . . 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 2763	. . . . . . . . . . . . 13 ⊢ (coeff‘𝑓) = (coeff‘𝑓)
93		simplr 780	. . . . . . . . . . . . 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 784	. . . . . . . . . . . . 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 791	. . . . . . . . . . . . . 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 6883	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = ℎ → (deg‘𝑔) = (deg‘ℎ))
97	96	breq1d 5120	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = ℎ → ((deg‘𝑔) ≤ 𝑑 ↔ (deg‘ℎ) ≤ 𝑑))
98		coeq1 5845	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔 = ℎ → (𝑔 ∘ 𝐺) = (ℎ ∘ 𝐺))
99	98	fveq2d 6887	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = ℎ → (deg‘(𝑔 ∘ 𝐺)) = (deg‘(ℎ ∘ 𝐺)))
100	96	oveq1d 7427	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = ℎ → ((deg‘𝑔) · 𝑁) = ((deg‘ℎ) · 𝑁))
101	99, 100	eqeq12d 2779	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = ℎ → ((deg‘(𝑔 ∘ 𝐺)) = ((deg‘𝑔) · 𝑁) ↔ (deg‘(ℎ ∘ 𝐺)) = ((deg‘ℎ) · 𝑁)))
102	97, 101	imbi12d 347	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = ℎ → (((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔 ∘ 𝐺)) = ((deg‘𝑔) · 𝑁)) ↔ ((deg‘ℎ) ≤ 𝑑 → (deg‘(ℎ ∘ 𝐺)) = ((deg‘ℎ) · 𝑁))))
103	102	cbvralvw 3243	. . . . . . . . . . . . . 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 221	. . . . . . . . . . . . 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 26412	. . . . . . . . . . . 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 461	. . . . . . . . . . 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 872	. . . . . . . . . 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 243	. . . . . . . . 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 461	. . . . . . . 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 3165	. . . . . . 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 245	. . . . . 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 418	. . . . 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 30	. . . 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 12692	. . 3 ⊢ (𝑀 ∈ ℕ₀ → (𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 𝑀 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁))))
115	7, 114	mpcom 39	. 2 ⊢ (𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 𝑀 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁)))
116	7	nn0red 12567	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℝ)
117	116	leidd 11781	. 2 ⊢ (𝜑 → 𝑀 ≤ 𝑀)
118		fveq2 6883	. . . . . 6 ⊢ (𝑓 = 𝐹 → (deg‘𝑓) = (deg‘𝐹))
119	118, 4	eqtr4di 2816	. . . . 5 ⊢ (𝑓 = 𝐹 → (deg‘𝑓) = 𝑀)
120	119	breq1d 5120	. . . 4 ⊢ (𝑓 = 𝐹 → ((deg‘𝑓) ≤ 𝑀 ↔ 𝑀 ≤ 𝑀))
121		coeq1 5845	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓 ∘ 𝐺) = (𝐹 ∘ 𝐺))
122	121	fveq2d 6887	. . . . 5 ⊢ (𝑓 = 𝐹 → (deg‘(𝑓 ∘ 𝐺)) = (deg‘(𝐹 ∘ 𝐺)))
123	119	oveq1d 7427	. . . . 5 ⊢ (𝑓 = 𝐹 → ((deg‘𝑓) · 𝑁) = (𝑀 · 𝑁))
124	122, 123	eqeq12d 2779	. . . 4 ⊢ (𝑓 = 𝐹 → ((deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁) ↔ (deg‘(𝐹 ∘ 𝐺)) = (𝑀 · 𝑁)))
125	120, 124	imbi12d 347	. . 3 ⊢ (𝑓 = 𝐹 → (((deg‘𝑓) ≤ 𝑀 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁)) ↔ (𝑀 ≤ 𝑀 → (deg‘(𝐹 ∘ 𝐺)) = (𝑀 · 𝑁))))
126	125	rspcv 3578	. 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 67	1 ⊢ (𝜑 → (deg‘(𝐹 ∘ 𝐺)) = (𝑀 · 𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 = wceq 1570 ∈ wcel 2143 ∀wral 3079 {csn 4590 class class class wbr 5110 ↦ cmpt 5193 × cxp 5661 ∘ ccom 5667 ⟶wf 6534 ‘cfv 6538 (class class class)co 7412 ℂcc 11099 ℝcr 11100 0cc0 11101 1c1 11102 + caddc 11104 · cmul 11106 < clt 11244 ≤ cle 11245 ℕcn 12234 ℕ₀cn0 12505 Polycply 26322 coeffccoe 26324 degcdgr 26325

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-inf2 9611 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178 ax-pre-sup 11179

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-int 4914 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7676 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-1o 8454 df-er 8695 df-map 8827 df-pm 8828 df-en 8945 df-dom 8946 df-sdom 8947 df-fin 8948 df-sup 9403 df-inf 9404 df-oi 9473 df-card 9926 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-div 11873 df-nn 12235 df-2 12304 df-3 12305 df-n0 12506 df-z 12593 df-uz 12864 df-rp 13018 df-fz 13537 df-fzo 13685 df-fl 13827 df-seq 14040 df-exp 14100 df-hash 14369 df-cj 15152 df-re 15153 df-im 15154 df-sqrt 15288 df-abs 15289 df-clim 15541 df-rlim 15542 df-sum 15740 df-0p 25810 df-ply 26326 df-coe 26328 df-dgr 26329

This theorem is referenced by: taylply2 26512 ftalem7 27224

W3C validator