dgrco - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > dgrco		Structured version Visualization version GIF version

Theorem dgrco 25636

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 25561	. . 3 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
2		dgrco.3	. . 3 ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))
3	1, 2	sselid 3942	. 2 ⊢ (𝜑 → 𝐹 ∈ (Poly‘ℂ))
4		dgrco.1	. . . 4 ⊢ 𝑀 = (deg‘𝐹)
5		dgrcl 25594	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ₀)
6	2, 5	syl 17	. . . 4 ⊢ (𝜑 → (deg‘𝐹) ∈ ℕ₀)
7	4, 6	eqeltrid 2842	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
8		breq2 5109	. . . . . . 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 3174	. . . . 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 5109	. . . . . . 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 3174	. . . . 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 5109	. . . . . . 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 3174	. . . . 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 5109	. . . . . . 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 3174	. . . . 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 25594	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ₀)
27	25, 26	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (deg‘𝐺) ∈ ℕ₀)
28	24, 27	eqeltrid 2842	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
29	28	nn0cnd 12475	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℂ)
30	29	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝑁 ∈ ℂ)
31	30	mul02d 11353	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (0 · 𝑁) = 0)
32		simprr 771	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘𝑓) ≤ 0)
33		dgrcl 25594	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (Poly‘ℂ) → (deg‘𝑓) ∈ ℕ₀)
34	33	ad2antrl 726	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘𝑓) ∈ ℕ₀)
35	34	nn0ge0d 12476	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 0 ≤ (deg‘𝑓))
36	34	nn0red 12474	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘𝑓) ∈ ℝ)
37		0re 11157	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
38		letri3 11240	. . . . . . . . . . 11 ⊢ (((deg‘𝑓) ∈ ℝ ∧ 0 ∈ ℝ) → ((deg‘𝑓) = 0 ↔ ((deg‘𝑓) ≤ 0 ∧ 0 ≤ (deg‘𝑓))))
39	36, 37, 38	sylancl 586	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → ((deg‘𝑓) = 0 ↔ ((deg‘𝑓) ≤ 0 ∧ 0 ≤ (deg‘𝑓))))
40	32, 35, 39	mpbir2and 711	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘𝑓) = 0)
41	40	oveq1d 7372	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → ((deg‘𝑓) · 𝑁) = (0 · 𝑁))
42	31, 41, 40	3eqtr4d 2786	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → ((deg‘𝑓) · 𝑁) = (deg‘𝑓))
43		fconstmpt 5694	. . . . . . . . 9 ⊢ (ℂ × {(𝑓‘0)}) = (𝑦 ∈ ℂ ↦ (𝑓‘0))
44		0dgrb 25607	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (Poly‘ℂ) → ((deg‘𝑓) = 0 ↔ 𝑓 = (ℂ × {(𝑓‘0)})))
45	44	ad2antrl 726	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → ((deg‘𝑓) = 0 ↔ 𝑓 = (ℂ × {(𝑓‘0)})))
46	40, 45	mpbid 231	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝑓 = (ℂ × {(𝑓‘0)}))
47	25	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝐺 ∈ (Poly‘𝑆))
48		plyf 25559	. . . . . . . . . . . 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 7035	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) ∧ 𝑦 ∈ ℂ) → (𝐺‘𝑦) ∈ ℂ)
51	49	feqmptd 6910	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝐺 = (𝑦 ∈ ℂ ↦ (𝐺‘𝑦)))
52		fconstmpt 5694	. . . . . . . . . . 11 ⊢ (ℂ × {(𝑓‘0)}) = (𝑥 ∈ ℂ ↦ (𝑓‘0))
53	46, 52	eqtrdi 2792	. . . . . . . . . 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 7075	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (𝑓 ∘ 𝐺) = (𝑦 ∈ ℂ ↦ (𝑓‘0)))
56	43, 46, 55	3eqtr4a 2802	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝑓 = (𝑓 ∘ 𝐺))
57	56	fveq2d 6846	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘𝑓) = (deg‘(𝑓 ∘ 𝐺)))
58	42, 57	eqtr2d 2777	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁))
59	58	expr 457	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (Poly‘ℂ)) → ((deg‘𝑓) ≤ 0 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁)))
60	59	ralrimiva 3143	. . . 4 ⊢ (𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 0 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁)))
61		fveq2 6842	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (deg‘𝑓) = (deg‘𝑔))
62	61	breq1d 5115	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → ((deg‘𝑓) ≤ 𝑑 ↔ (deg‘𝑔) ≤ 𝑑))
63		coeq1 5813	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → (𝑓 ∘ 𝐺) = (𝑔 ∘ 𝐺))
64	63	fveq2d 6846	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (deg‘(𝑓 ∘ 𝐺)) = (deg‘(𝑔 ∘ 𝐺)))
65	61	oveq1d 7372	. . . . . . . . . 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 3225	. . . . . . 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 726	. . . . . . . . . . . 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 12474	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔 ∘ 𝐺)) = ((deg‘𝑔) · 𝑁)))) → (deg‘𝑓) ∈ ℝ)
71		nn0p1nn 12452	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ ℕ₀ → (𝑑 + 1) ∈ ℕ)
72	71	ad2antlr 725	. . . . . . . . . . . 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 12168	. . . . . . . . . . 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 11298	. . . . . . . . . 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 767	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔 ∘ 𝐺)) = ((deg‘𝑔) · 𝑁)))) → 𝑑 ∈ ℕ₀)
76		nn0leltp1 12562	. . . . . . . . . . . . 13 ⊢ (((deg‘𝑓) ∈ ℕ₀ ∧ 𝑑 ∈ ℕ₀) → ((deg‘𝑓) ≤ 𝑑 ↔ (deg‘𝑓) < (𝑑 + 1)))
77	69, 75, 76	syl2anc 584	. . . . . . . . . . . 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 6842	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = 𝑓 → (deg‘𝑔) = (deg‘𝑓))
79	78	breq1d 5115	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑓 → ((deg‘𝑔) ≤ 𝑑 ↔ (deg‘𝑓) ≤ 𝑑))
80		coeq1 5813	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = 𝑓 → (𝑔 ∘ 𝐺) = (𝑓 ∘ 𝐺))
81	80	fveq2d 6846	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = 𝑓 → (deg‘(𝑔 ∘ 𝐺)) = (deg‘(𝑓 ∘ 𝐺)))
82	78	oveq1d 7372	. . . . . . . . . . . . . . . 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 3579	. . . . . . . . . . . . 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 482	. . . . . . . . . . . 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 259	. . . . . . . . . . 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 777	. . . . . . . . . . . . 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 3942	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺 ∈ (Poly‘ℂ))
91	90	ad2antrr 724	. . . . . . . . . . . . 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 767	. . . . . . . . . . . . 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 771	. . . . . . . . . . . . 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 778	. . . . . . . . . . . . . 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 6842	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = ℎ → (deg‘𝑔) = (deg‘ℎ))
97	96	breq1d 5115	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = ℎ → ((deg‘𝑔) ≤ 𝑑 ↔ (deg‘ℎ) ≤ 𝑑))
98		coeq1 5813	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔 = ℎ → (𝑔 ∘ 𝐺) = (ℎ ∘ 𝐺))
99	98	fveq2d 6846	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = ℎ → (deg‘(𝑔 ∘ 𝐺)) = (deg‘(ℎ ∘ 𝐺)))
100	96	oveq1d 7372	. . . . . . . . . . . . . . . . 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 3225	. . . . . . . . . . . . . 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 217	. . . . . . . . . . . . 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 25635	. . . . . . . . . . . 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 457	. . . . . . . . . . 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 857	. . . . . . . . . 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 239	. . . . . . . . 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 457	. . . . . . . 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 3151	. . . . . . 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 241	. . . . . 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 414	. . . . 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 12598	. . 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 12474	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℝ)
117	116	leidd 11721	. 2 ⊢ (𝜑 → 𝑀 ≤ 𝑀)
118		fveq2 6842	. . . . . 6 ⊢ (𝑓 = 𝐹 → (deg‘𝑓) = (deg‘𝐹))
119	118, 4	eqtr4di 2794	. . . . 5 ⊢ (𝑓 = 𝐹 → (deg‘𝑓) = 𝑀)
120	119	breq1d 5115	. . . 4 ⊢ (𝑓 = 𝐹 → ((deg‘𝑓) ≤ 𝑀 ↔ 𝑀 ≤ 𝑀))
121		coeq1 5813	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓 ∘ 𝐺) = (𝐹 ∘ 𝐺))
122	121	fveq2d 6846	. . . . 5 ⊢ (𝑓 = 𝐹 → (deg‘(𝑓 ∘ 𝐺)) = (deg‘(𝐹 ∘ 𝐺)))
123	119	oveq1d 7372	. . . . 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 3577	. 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 205 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ∀wral 3064 {csn 4586 class class class wbr 5105 ↦ cmpt 5188 × cxp 5631 ∘ ccom 5637 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 ℂcc 11049 ℝcr 11050 0cc0 11051 1c1 11052 + caddc 11054 · cmul 11056 < clt 11189 ≤ cle 11190 ℕcn 12153 ℕ₀cn0 12413 Polycply 25545 coeffccoe 25547 degcdgr 25548

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-pm 8768 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9378 df-inf 9379 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-n0 12414 df-z 12500 df-uz 12764 df-rp 12916 df-fz 13425 df-fzo 13568 df-fl 13697 df-seq 13907 df-exp 13968 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-clim 15370 df-rlim 15371 df-sum 15571 df-0p 25034 df-ply 25549 df-coe 25551 df-dgr 25552

This theorem is referenced by: taylply2 25727 ftalem7 26428

W3C validator