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Theorem dgrcolem1 26483

Description: The degree of a composition of a monomial with a polynomial. (Contributed by Mario Carneiro, 15-Sep-2014.)

**Hypotheses**
Ref	Expression
dgrcolem1.1	⊢ 𝑁 = (deg‘𝐺)
dgrcolem1.2	⊢ (𝜑 → 𝑀 ∈ ℕ)
dgrcolem1.3	⊢ (𝜑 → 𝑁 ∈ ℕ)
dgrcolem1.4	⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆))

**Assertion**
Ref	Expression
dgrcolem1	⊢ (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀))) = (𝑀 · 𝑁))

Distinct variable groups: 𝑥,𝐺 𝑥,𝑀 𝜑,𝑥 Allowed substitution hints: 𝑆(𝑥) 𝑁(𝑥)

Proof of Theorem dgrcolem1 Dummy variables 𝑤 𝑑 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dgrcolem1.2	. 2 ⊢ (𝜑 → 𝑀 ∈ ℕ)
2		oveq2 7427	. . . . . . 7 ⊢ (𝑦 = 1 → ((𝐺‘𝑥)↑𝑦) = ((𝐺‘𝑥)↑1))
3	2	mpteq2dv 5207	. . . . . 6 ⊢ (𝑦 = 1 → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦)) = (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1)))
4	3	fveq2d 6889	. . . . 5 ⊢ (𝑦 = 1 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1))))
5		oveq1 7426	. . . . 5 ⊢ (𝑦 = 1 → (𝑦 · 𝑁) = (1 · 𝑁))
6	4, 5	eqeq12d 2781	. . . 4 ⊢ (𝑦 = 1 → ((deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (𝑦 · 𝑁) ↔ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1))) = (1 · 𝑁)))
7	6	imbi2d 343	. . 3 ⊢ (𝑦 = 1 → ((𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (𝑦 · 𝑁)) ↔ (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1))) = (1 · 𝑁))))
8		oveq2 7427	. . . . . . 7 ⊢ (𝑦 = 𝑑 → ((𝐺‘𝑥)↑𝑦) = ((𝐺‘𝑥)↑𝑑))
9	8	mpteq2dv 5207	. . . . . 6 ⊢ (𝑦 = 𝑑 → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦)) = (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)))
10	9	fveq2d 6889	. . . . 5 ⊢ (𝑦 = 𝑑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))))
11		oveq1 7426	. . . . 5 ⊢ (𝑦 = 𝑑 → (𝑦 · 𝑁) = (𝑑 · 𝑁))
12	10, 11	eqeq12d 2781	. . . 4 ⊢ (𝑦 = 𝑑 → ((deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (𝑦 · 𝑁) ↔ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)))
13	12	imbi2d 343	. . 3 ⊢ (𝑦 = 𝑑 → ((𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (𝑦 · 𝑁)) ↔ (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁))))
14		oveq2 7427	. . . . . . 7 ⊢ (𝑦 = (𝑑 + 1) → ((𝐺‘𝑥)↑𝑦) = ((𝐺‘𝑥)↑(𝑑 + 1)))
15	14	mpteq2dv 5207	. . . . . 6 ⊢ (𝑦 = (𝑑 + 1) → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦)) = (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1))))
16	15	fveq2d 6889	. . . . 5 ⊢ (𝑦 = (𝑑 + 1) → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1)))))
17		oveq1 7426	. . . . 5 ⊢ (𝑦 = (𝑑 + 1) → (𝑦 · 𝑁) = ((𝑑 + 1) · 𝑁))
18	16, 17	eqeq12d 2781	. . . 4 ⊢ (𝑦 = (𝑑 + 1) → ((deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (𝑦 · 𝑁) ↔ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1)))) = ((𝑑 + 1) · 𝑁)))
19	18	imbi2d 343	. . 3 ⊢ (𝑦 = (𝑑 + 1) → ((𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (𝑦 · 𝑁)) ↔ (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1)))) = ((𝑑 + 1) · 𝑁))))
20		oveq2 7427	. . . . . . 7 ⊢ (𝑦 = 𝑀 → ((𝐺‘𝑥)↑𝑦) = ((𝐺‘𝑥)↑𝑀))
21	20	mpteq2dv 5207	. . . . . 6 ⊢ (𝑦 = 𝑀 → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦)) = (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀)))
22	21	fveq2d 6889	. . . . 5 ⊢ (𝑦 = 𝑀 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀))))
23		oveq1 7426	. . . . 5 ⊢ (𝑦 = 𝑀 → (𝑦 · 𝑁) = (𝑀 · 𝑁))
24	22, 23	eqeq12d 2781	. . . 4 ⊢ (𝑦 = 𝑀 → ((deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (𝑦 · 𝑁) ↔ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀))) = (𝑀 · 𝑁)))
25	24	imbi2d 343	. . 3 ⊢ (𝑦 = 𝑀 → ((𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (𝑦 · 𝑁)) ↔ (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀))) = (𝑀 · 𝑁))))
26		dgrcolem1.4	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆))
27		plyf 26408	. . . . . . . . . . 11 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ)
28	26, 27	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺:ℂ⟶ℂ)
29	28	ffvelcdmda 7083	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝐺‘𝑥) ∈ ℂ)
30	29	exp1d 14197	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((𝐺‘𝑥)↑1) = (𝐺‘𝑥))
31	30	mpteq2dva 5206	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1)) = (𝑥 ∈ ℂ ↦ (𝐺‘𝑥)))
32	28	feqmptd 6953	. . . . . . 7 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ ℂ ↦ (𝐺‘𝑥)))
33	31, 32	eqtr4d 2803	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1)) = 𝐺)
34	33	fveq2d 6889	. . . . 5 ⊢ (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1))) = (deg‘𝐺))
35		dgrcolem1.1	. . . . 5 ⊢ 𝑁 = (deg‘𝐺)
36	34, 35	eqtr4di 2818	. . . 4 ⊢ (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1))) = 𝑁)
37		dgrcolem1.3	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ)
38	37	nncnd 12266	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℂ)
39	38	mullidd 11244	. . . 4 ⊢ (𝜑 → (1 · 𝑁) = 𝑁)
40	36, 39	eqtr4d 2803	. . 3 ⊢ (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1))) = (1 · 𝑁))
41	29	adantlr 728	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ 𝑥 ∈ ℂ) → (𝐺‘𝑥) ∈ ℂ)
42		nnnn0 12528	. . . . . . . . . . . . . 14 ⊢ (𝑑 ∈ ℕ → 𝑑 ∈ ℕ₀)
43	42	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝑑 ∈ ℕ₀)
44	43	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ 𝑥 ∈ ℂ) → 𝑑 ∈ ℕ₀)
45	41, 44	expp1d 14203	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ 𝑥 ∈ ℂ) → ((𝐺‘𝑥)↑(𝑑 + 1)) = (((𝐺‘𝑥)↑𝑑) · (𝐺‘𝑥)))
46	45	mpteq2dva 5206	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1))) = (𝑥 ∈ ℂ ↦ (((𝐺‘𝑥)↑𝑑) · (𝐺‘𝑥))))
47		cnex 11198	. . . . . . . . . . . 12 ⊢ ℂ ∈ V
48	47	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ℂ ∈ V)
49		ovexd 7454	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ 𝑥 ∈ ℂ) → ((𝐺‘𝑥)↑𝑑) ∈ V)
50		eqidd 2766	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) = (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)))
51	32	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝐺 = (𝑥 ∈ ℂ ↦ (𝐺‘𝑥)))
52	48, 49, 41, 50, 51	offval2 7704	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ∘_f · 𝐺) = (𝑥 ∈ ℂ ↦ (((𝐺‘𝑥)↑𝑑) · (𝐺‘𝑥))))
53	46, 52	eqtr4d 2803	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1))) = ((𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ∘_f · 𝐺))
54	53	fveq2d 6889	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1)))) = (deg‘((𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ∘_f · 𝐺)))
55	54	adantr 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1)))) = (deg‘((𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ∘_f · 𝐺)))
56		oveq1 7426	. . . . . . . . 9 ⊢ ((deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁) → ((deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) + 𝑁) = ((𝑑 · 𝑁) + 𝑁))
57	56	adantl 487	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → ((deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) + 𝑁) = ((𝑑 · 𝑁) + 𝑁))
58		eqidd 2766	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑦 ∈ ℂ ↦ (𝑦↑𝑑)) = (𝑦 ∈ ℂ ↦ (𝑦↑𝑑)))
59		oveq1 7426	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐺‘𝑥) → (𝑦↑𝑑) = ((𝐺‘𝑥)↑𝑑))
60	41, 51, 58, 59	fmptco 7129	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((𝑦 ∈ ℂ ↦ (𝑦↑𝑑)) ∘ 𝐺) = (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)))
61		ssidd 3961	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ℂ ⊆ ℂ)
62		1cnd 11219	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 1 ∈ ℂ)
63		plypow 26415	. . . . . . . . . . . . 13 ⊢ ((ℂ ⊆ ℂ ∧ 1 ∈ ℂ ∧ 𝑑 ∈ ℕ₀) → (𝑦 ∈ ℂ ↦ (𝑦↑𝑑)) ∈ (Poly‘ℂ))
64	61, 62, 43, 63	syl3anc 1398	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑦 ∈ ℂ ↦ (𝑦↑𝑑)) ∈ (Poly‘ℂ))
65		plyssc 26410	. . . . . . . . . . . . 13 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
66	26	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝐺 ∈ (Poly‘𝑆))
67	65, 66	sselid 3936	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝐺 ∈ (Poly‘ℂ))
68		addcl 11199	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (𝑧 + 𝑤) ∈ ℂ)
69	68	adantl 487	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ)) → (𝑧 + 𝑤) ∈ ℂ)
70		mulcl 11201	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (𝑧 · 𝑤) ∈ ℂ)
71	70	adantl 487	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ)) → (𝑧 · 𝑤) ∈ ℂ)
72	64, 67, 69, 71	plyco 26451	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((𝑦 ∈ ℂ ↦ (𝑦↑𝑑)) ∘ 𝐺) ∈ (Poly‘ℂ))
73	60, 72	eqeltrrd 2866	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ∈ (Poly‘ℂ))
74	73	adantr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ∈ (Poly‘ℂ))
75		simpr 490	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁))
76		simpr 490	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝑑 ∈ ℕ)
77	37	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝑁 ∈ ℕ)
78	76, 77	nnmulcld 12306	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑑 · 𝑁) ∈ ℕ)
79	78	nnne0d 12303	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑑 · 𝑁) ≠ 0)
80	79	adantr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → (𝑑 · 𝑁) ≠ 0)
81	75, 80	eqnetrd 3027	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) ≠ 0)
82		fveq2 6885	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) = 0_𝑝 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (deg‘0_𝑝))
83		dgr0 26472	. . . . . . . . . . . 12 ⊢ (deg‘0_𝑝) = 0
84	82, 83	eqtrdi 2816	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) = 0_𝑝 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = 0)
85	84	necon3i 2992	. . . . . . . . . 10 ⊢ ((deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) ≠ 0 → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ≠ 0_𝑝)
86	81, 85	syl 18	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ≠ 0_𝑝)
87	67	adantr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → 𝐺 ∈ (Poly‘ℂ))
88	37	nnne0d 12303	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ≠ 0)
89		fveq2 6885	. . . . . . . . . . . . . . 15 ⊢ (𝐺 = 0_𝑝 → (deg‘𝐺) = (deg‘0_𝑝))
90	89, 83	eqtrdi 2816	. . . . . . . . . . . . . 14 ⊢ (𝐺 = 0_𝑝 → (deg‘𝐺) = 0)
91	35, 90	eqtrid 2812	. . . . . . . . . . . . 13 ⊢ (𝐺 = 0_𝑝 → 𝑁 = 0)
92	91	necon3i 2992	. . . . . . . . . . . 12 ⊢ (𝑁 ≠ 0 → 𝐺 ≠ 0_𝑝)
93	88, 92	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ≠ 0_𝑝)
94	93	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝐺 ≠ 0_𝑝)
95	94	adantr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → 𝐺 ≠ 0_𝑝)
96		eqid 2765	. . . . . . . . . 10 ⊢ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)))
97	96, 35	dgrmul 26480	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ∈ (Poly‘ℂ) ∧ (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ≠ 0_𝑝) ∧ (𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0_𝑝)) → (deg‘((𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ∘_f · 𝐺)) = ((deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) + 𝑁))
98	74, 86, 87, 95, 97	syl22anc 852	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → (deg‘((𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ∘_f · 𝐺)) = ((deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) + 𝑁))
99		nncn 12258	. . . . . . . . . . 11 ⊢ (𝑑 ∈ ℕ → 𝑑 ∈ ℂ)
100	99	adantl 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝑑 ∈ ℂ)
101	38	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝑁 ∈ ℂ)
102	100, 101	adddirp1d 11252	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((𝑑 + 1) · 𝑁) = ((𝑑 · 𝑁) + 𝑁))
103	102	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → ((𝑑 + 1) · 𝑁) = ((𝑑 · 𝑁) + 𝑁))
104	57, 98, 103	3eqtr4rd 2811	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → ((𝑑 + 1) · 𝑁) = (deg‘((𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ∘_f · 𝐺)))
105	55, 104	eqtr4d 2803	. . . . . 6 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1)))) = ((𝑑 + 1) · 𝑁))
106	105	ex 418	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁) → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1)))) = ((𝑑 + 1) · 𝑁)))
107	106	expcom 419	. . . 4 ⊢ (𝑑 ∈ ℕ → (𝜑 → ((deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁) → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1)))) = ((𝑑 + 1) · 𝑁))))
108	107	a2d 30	. . 3 ⊢ (𝑑 ∈ ℕ → ((𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1)))) = ((𝑑 + 1) · 𝑁))))
109	7, 13, 19, 25, 40, 108	nnind 12268	. 2 ⊢ (𝑀 ∈ ℕ → (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀))) = (𝑀 · 𝑁)))
110	1, 109	mpcom 39	1 ⊢ (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀))) = (𝑀 · 𝑁))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2146 ≠ wne 2960 Vcvv 3457 ⊆ wss 3906 ↦ cmpt 5194 ∘ ccom 5667 ⟶wf 6536 ‘cfv 6540 (class class class)co 7419 ∘_f cof 7682 ℂcc 11115 0cc0 11117 1c1 11118 + caddc 11120 · cmul 11122 ℕcn 12250 ℕ₀cn0 12521 ↑cexp 14117 0_𝑝c0p 25881 Polycply 26394 degcdgr 26397

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-inf2 9617 ax-cnex 11173 ax-resscn 11174 ax-1cn 11175 ax-icn 11176 ax-addcl 11177 ax-addrcl 11178 ax-mulcl 11179 ax-mulrcl 11180 ax-mulcom 11181 ax-addass 11182 ax-mulass 11183 ax-distr 11184 ax-i2m1 11185 ax-1ne0 11186 ax-1rid 11187 ax-rnegex 11188 ax-rrecex 11189 ax-cnre 11190 ax-pre-lttri 11191 ax-pre-lttrn 11192 ax-pre-ltadd 11193 ax-pre-mulgt0 11194 ax-pre-sup 11195

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 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 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7684 df-om 7869 df-1st 7992 df-2nd 7993 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-er 8700 df-map 8832 df-pm 8833 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-sup 9409 df-inf 9410 df-oi 9479 df-card 9941 df-pnf 11262 df-mnf 11263 df-xr 11264 df-ltxr 11265 df-le 11266 df-sub 11460 df-neg 11461 df-div 11889 df-nn 12251 df-2 12320 df-3 12321 df-n0 12522 df-z 12609 df-uz 12881 df-rp 13035 df-fz 13554 df-fzo 13702 df-fl 13845 df-seq 14058 df-exp 14118 df-hash 14387 df-cj 15176 df-re 15177 df-im 15178 df-sqrt 15312 df-abs 15313 df-clim 15565 df-rlim 15566 df-sum 15764 df-0p 25882 df-ply 26398 df-coe 26400 df-dgr 26401

This theorem is used by: dgrcolem2 26484

W3C validator