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Theorem dgrco 26600

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 26524	. . 3 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
2		dgrco.3	. . 3 ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))
3	1, 2	sselid 3906	. 2 ⊢ (𝜑 → 𝐹 ∈ (Poly‘ℂ))
4		dgrco.1	. . . 4 ⊢ 𝑀 = (deg‘𝐹)
5		dgrcl 26558	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ₀)
6	2, 5	syl 18	. . . 4 ⊢ (𝜑 → (deg‘𝐹) ∈ ℕ₀)
7	4, 6	eqeltrid 2864	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
8		breq2 5106	. . . . . . 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 3185	. . . . 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 5106	. . . . . . 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 3185	. . . . 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 5106	. . . . . . 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 3185	. . . . 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 5106	. . . . . . 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 3185	. . . . 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 26558	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ₀)
27	25, 26	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (deg‘𝐺) ∈ ℕ₀)
28	24, 27	eqeltrid 2864	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
29	28	nn0cnd 12673	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℂ)
30	29	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝑁 ∈ ℂ)
31	30	mul02d 11512	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (0 · 𝑁) = 0)
32		simprr 785	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘𝑓) ≤ 0)
33		dgrcl 26558	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (Poly‘ℂ) → (deg‘𝑓) ∈ ℕ₀)
34	33	ad2antrl 741	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘𝑓) ∈ ℕ₀)
35	34	nn0ge0d 12674	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 0 ≤ (deg‘𝑓))
36	34	nn0red 12672	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘𝑓) ∈ ℝ)
37		0re 11314	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
38		letri3 11399	. . . . . . . . . . 11 ⊢ (((deg‘𝑓) ∈ ℝ ∧ 0 ∈ ℝ) → ((deg‘𝑓) = 0 ↔ ((deg‘𝑓) ≤ 0 ∧ 0 ≤ (deg‘𝑓))))
39	36, 37, 38	sylancl 598	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → ((deg‘𝑓) = 0 ↔ ((deg‘𝑓) ≤ 0 ∧ 0 ≤ (deg‘𝑓))))
40	32, 35, 39	mpbir2and 726	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘𝑓) = 0)
41	40	oveq1d 7436	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → ((deg‘𝑓) · 𝑁) = (0 · 𝑁))
42	31, 41, 40	3eqtr4d 2805	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → ((deg‘𝑓) · 𝑁) = (deg‘𝑓))
43		fconstmpt 5712	. . . . . . . . 9 ⊢ (ℂ × {(𝑓‘0)}) = (𝑦 ∈ ℂ ↦ (𝑓‘0))
44		0dgrb 26571	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (Poly‘ℂ) → ((deg‘𝑓) = 0 ↔ 𝑓 = (ℂ × {(𝑓‘0)})))
45	44	ad2antrl 741	. . . . . . . . . 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 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝐺 ∈ (Poly‘𝑆))
48		plyf 26522	. . . . . . . . . . . 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 7085	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) ∧ 𝑦 ∈ ℂ) → (𝐺‘𝑦) ∈ ℂ)
51	49	feqmptd 6954	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝐺 = (𝑦 ∈ ℂ ↦ (𝐺‘𝑦)))
52		fconstmpt 5712	. . . . . . . . . . 11 ⊢ (ℂ × {(𝑓‘0)}) = (𝑥 ∈ ℂ ↦ (𝑓‘0))
53	46, 52	eqtrdi 2811	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝑓 = (𝑥 ∈ ℂ ↦ (𝑓‘0)))
54		eqidd 2761	. . . . . . . . . 10 ⊢ (𝑥 = (𝐺‘𝑦) → (𝑓‘0) = (𝑓‘0))
55	50, 51, 53, 54	fmptco 7131	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (𝑓 ∘ 𝐺) = (𝑦 ∈ ℂ ↦ (𝑓‘0)))
56	43, 46, 55	3eqtr4a 2821	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝑓 = (𝑓 ∘ 𝐺))
57	56	fveq2d 6890	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘𝑓) = (deg‘(𝑓 ∘ 𝐺)))
58	42, 57	eqtr2d 2796	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁))
59	58	expr 462	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (Poly‘ℂ)) → ((deg‘𝑓) ≤ 0 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁)))
60	59	ralrimiva 3154	. . . 4 ⊢ (𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 0 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁)))
61		fveq2 6886	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (deg‘𝑓) = (deg‘𝑔))
62	61	breq1d 5112	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → ((deg‘𝑓) ≤ 𝑑 ↔ (deg‘𝑔) ≤ 𝑑))
63		coeq1 5834	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → (𝑓 ∘ 𝐺) = (𝑔 ∘ 𝐺))
64	63	fveq2d 6890	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (deg‘(𝑓 ∘ 𝐺)) = (deg‘(𝑔 ∘ 𝐺)))
65	61	oveq1d 7436	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → ((deg‘𝑓) · 𝑁) = ((deg‘𝑔) · 𝑁))
66	64, 65	eqeq12d 2776	. . . . . . . . 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 3240	. . . . . . 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 741	. . . . . . . . . . . 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 12672	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔 ∘ 𝐺)) = ((deg‘𝑔) · 𝑁)))) → (deg‘𝑓) ∈ ℝ)
71		nn0p1nn 12649	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ ℕ₀ → (𝑑 + 1) ∈ ℕ)
72	71	ad2antlr 740	. . . . . . . . . . . 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 12354	. . . . . . . . . . 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 11457	. . . . . . . . . 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 781	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔 ∘ 𝐺)) = ((deg‘𝑔) · 𝑁)))) → 𝑑 ∈ ℕ₀)
76		nn0leltp1 12762	. . . . . . . . . . . . 13 ⊢ (((deg‘𝑓) ∈ ℕ₀ ∧ 𝑑 ∈ ℕ₀) → ((deg‘𝑓) ≤ 𝑑 ↔ (deg‘𝑓) < (𝑑 + 1)))
77	69, 75, 76	syl2anc 596	. . . . . . . . . . . 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 6886	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = 𝑓 → (deg‘𝑔) = (deg‘𝑓))
79	78	breq1d 5112	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑓 → ((deg‘𝑔) ≤ 𝑑 ↔ (deg‘𝑓) ≤ 𝑑))
80		coeq1 5834	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = 𝑓 → (𝑔 ∘ 𝐺) = (𝑓 ∘ 𝐺))
81	80	fveq2d 6890	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = 𝑓 → (deg‘(𝑔 ∘ 𝐺)) = (deg‘(𝑓 ∘ 𝐺)))
82	78	oveq1d 7436	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = 𝑓 → ((deg‘𝑔) · 𝑁) = ((deg‘𝑓) · 𝑁))
83	81, 82	eqeq12d 2776	. . . . . . . . . . . . . . 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 3574	. . . . . . . . . . . . 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 487	. . . . . . . . . . . 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 2760	. . . . . . . . . . . . 13 ⊢ (deg‘𝑓) = (deg‘𝑓)
89		simprll 791	. . . . . . . . . . . . 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 3906	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺 ∈ (Poly‘ℂ))
91	90	ad2antrr 739	. . . . . . . . . . . . 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 2760	. . . . . . . . . . . . 13 ⊢ (coeff‘𝑓) = (coeff‘𝑓)
93		simplr 781	. . . . . . . . . . . . 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 785	. . . . . . . . . . . . 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 792	. . . . . . . . . . . . . 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 6886	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = ℎ → (deg‘𝑔) = (deg‘ℎ))
97	96	breq1d 5112	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = ℎ → ((deg‘𝑔) ≤ 𝑑 ↔ (deg‘ℎ) ≤ 𝑑))
98		coeq1 5834	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔 = ℎ → (𝑔 ∘ 𝐺) = (ℎ ∘ 𝐺))
99	98	fveq2d 6890	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = ℎ → (deg‘(𝑔 ∘ 𝐺)) = (deg‘(ℎ ∘ 𝐺)))
100	96	oveq1d 7436	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = ℎ → ((deg‘𝑔) · 𝑁) = ((deg‘ℎ) · 𝑁))
101	99, 100	eqeq12d 2776	. . . . . . . . . . . . . . . 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 3240	. . . . . . . . . . . . . 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 26599	. . . . . . . . . . . 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 462	. . . . . . . . . . 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 873	. . . . . . . . . 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 462	. . . . . . . 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 3162	. . . . . . 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 419	. . . . 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 12798	. . 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 12672	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℝ)
117	116	leidd 11886	. 2 ⊢ (𝜑 → 𝑀 ≤ 𝑀)
118		fveq2 6886	. . . . . 6 ⊢ (𝑓 = 𝐹 → (deg‘𝑓) = (deg‘𝐹))
119	118, 4	eqtr4di 2813	. . . . 5 ⊢ (𝑓 = 𝐹 → (deg‘𝑓) = 𝑀)
120	119	breq1d 5112	. . . 4 ⊢ (𝑓 = 𝐹 → ((deg‘𝑓) ≤ 𝑀 ↔ 𝑀 ≤ 𝑀))
121		coeq1 5834	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓 ∘ 𝐺) = (𝐹 ∘ 𝐺))
122	121	fveq2d 6890	. . . . 5 ⊢ (𝑓 = 𝐹 → (deg‘(𝑓 ∘ 𝐺)) = (deg‘(𝐹 ∘ 𝐺)))
123	119	oveq1d 7436	. . . . 5 ⊢ (𝑓 = 𝐹 → ((deg‘𝑓) · 𝑁) = (𝑀 · 𝑁))
124	122, 123	eqeq12d 2776	. . . 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 3572	. 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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 = wceq 1569 ∈ wcel 2144 ∀wral 3076 {csn 4584 class class class wbr 5102 ↦ cmpt 5185 × cxp 5648 ∘ ccom 5654 ⟶wf 6535 ‘cfv 6539 (class class class)co 7421 ℂcc 11202 ℝcr 11203 0cc0 11204 1c1 11205 + caddc 11207 · cmul 11209 < clt 11347 ≤ cle 11348 ℕcn 12339 ℕ₀cn0 12610 Polycply 26508 coeffccoe 26510 degcdgr 26511

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7752 ax-inf2 9646 ax-cnex 11260 ax-resscn 11261 ax-1cn 11262 ax-icn 11263 ax-addcl 11264 ax-addrcl 11265 ax-mulcl 11266 ax-mulrcl 11267 ax-mulcom 11268 ax-addass 11269 ax-mulass 11270 ax-distr 11271 ax-i2m1 11272 ax-1ne0 11273 ax-1rid 11274 ax-rnegex 11275 ax-rrecex 11276 ax-cnre 11277 ax-pre-lttri 11278 ax-pre-lttrn 11279 ax-pre-ltadd 11280 ax-pre-mulgt0 11281 ax-pre-sup 11282

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-isom 6548 df-riota 7378 df-ov 7424 df-oprab 7425 df-mpo 7426 df-of 7694 df-om 7879 df-1st 8002 df-2nd 8003 df-frecs 8303 df-wrecs 8334 df-recs 8383 df-rdg 8422 df-1o 8480 df-er 8721 df-map 8853 df-pm 8854 df-en 8978 df-dom 8979 df-sdom 8980 df-fin 8981 df-sup 9438 df-inf 9439 df-oi 9508 df-card 10024 df-pnf 11349 df-mnf 11350 df-xr 11351 df-ltxr 11352 df-le 11353 df-sub 11547 df-neg 11548 df-div 11978 df-nn 12340 df-2 12409 df-3 12410 df-n0 12611 df-z 12698 df-uz 12970 df-rp 13125 df-fz 13644 df-fzo 13793 df-fl 13936 df-seq 14149 df-exp 14209 df-hash 14479 df-cj 15272 df-re 15273 df-im 15274 df-sqrt 15408 df-abs 15409 df-clim 15661 df-rlim 15662 df-sum 15860 df-0p 25997 df-ply 26512 df-coe 26514 df-dgr 26515

This theorem is used by: taylply2 26701 ftalem7 27412 sqrtnpoly 48037

W3C validator