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Theorem dgrcolem1 25022

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 7178	. . . . . . 7 ⊢ (𝑦 = 1 → ((𝐺‘𝑥)↑𝑦) = ((𝐺‘𝑥)↑1))
3	2	mpteq2dv 5126	. . . . . 6 ⊢ (𝑦 = 1 → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦)) = (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1)))
4	3	fveq2d 6678	. . . . 5 ⊢ (𝑦 = 1 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1))))
5		oveq1 7177	. . . . 5 ⊢ (𝑦 = 1 → (𝑦 · 𝑁) = (1 · 𝑁))
6	4, 5	eqeq12d 2754	. . . 4 ⊢ (𝑦 = 1 → ((deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (𝑦 · 𝑁) ↔ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1))) = (1 · 𝑁)))
7	6	imbi2d 344	. . 3 ⊢ (𝑦 = 1 → ((𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (𝑦 · 𝑁)) ↔ (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1))) = (1 · 𝑁))))
8		oveq2 7178	. . . . . . 7 ⊢ (𝑦 = 𝑑 → ((𝐺‘𝑥)↑𝑦) = ((𝐺‘𝑥)↑𝑑))
9	8	mpteq2dv 5126	. . . . . 6 ⊢ (𝑦 = 𝑑 → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦)) = (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)))
10	9	fveq2d 6678	. . . . 5 ⊢ (𝑦 = 𝑑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))))
11		oveq1 7177	. . . . 5 ⊢ (𝑦 = 𝑑 → (𝑦 · 𝑁) = (𝑑 · 𝑁))
12	10, 11	eqeq12d 2754	. . . 4 ⊢ (𝑦 = 𝑑 → ((deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (𝑦 · 𝑁) ↔ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)))
13	12	imbi2d 344	. . 3 ⊢ (𝑦 = 𝑑 → ((𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (𝑦 · 𝑁)) ↔ (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁))))
14		oveq2 7178	. . . . . . 7 ⊢ (𝑦 = (𝑑 + 1) → ((𝐺‘𝑥)↑𝑦) = ((𝐺‘𝑥)↑(𝑑 + 1)))
15	14	mpteq2dv 5126	. . . . . 6 ⊢ (𝑦 = (𝑑 + 1) → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦)) = (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1))))
16	15	fveq2d 6678	. . . . 5 ⊢ (𝑦 = (𝑑 + 1) → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1)))))
17		oveq1 7177	. . . . 5 ⊢ (𝑦 = (𝑑 + 1) → (𝑦 · 𝑁) = ((𝑑 + 1) · 𝑁))
18	16, 17	eqeq12d 2754	. . . 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 344	. . 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 7178	. . . . . . 7 ⊢ (𝑦 = 𝑀 → ((𝐺‘𝑥)↑𝑦) = ((𝐺‘𝑥)↑𝑀))
21	20	mpteq2dv 5126	. . . . . 6 ⊢ (𝑦 = 𝑀 → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦)) = (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀)))
22	21	fveq2d 6678	. . . . 5 ⊢ (𝑦 = 𝑀 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀))))
23		oveq1 7177	. . . . 5 ⊢ (𝑦 = 𝑀 → (𝑦 · 𝑁) = (𝑀 · 𝑁))
24	22, 23	eqeq12d 2754	. . . 4 ⊢ (𝑦 = 𝑀 → ((deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (𝑦 · 𝑁) ↔ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀))) = (𝑀 · 𝑁)))
25	24	imbi2d 344	. . 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 24947	. . . . . . . . . . 11 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ)
28	26, 27	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺:ℂ⟶ℂ)
29	28	ffvelrnda 6861	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝐺‘𝑥) ∈ ℂ)
30	29	exp1d 13597	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((𝐺‘𝑥)↑1) = (𝐺‘𝑥))
31	30	mpteq2dva 5125	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1)) = (𝑥 ∈ ℂ ↦ (𝐺‘𝑥)))
32	28	feqmptd 6737	. . . . . . 7 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ ℂ ↦ (𝐺‘𝑥)))
33	31, 32	eqtr4d 2776	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1)) = 𝐺)
34	33	fveq2d 6678	. . . . 5 ⊢ (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1))) = (deg‘𝐺))
35		dgrcolem1.1	. . . . 5 ⊢ 𝑁 = (deg‘𝐺)
36	34, 35	eqtr4di 2791	. . . 4 ⊢ (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1))) = 𝑁)
37		dgrcolem1.3	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ)
38	37	nncnd 11732	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℂ)
39	38	mulid2d 10737	. . . 4 ⊢ (𝜑 → (1 · 𝑁) = 𝑁)
40	36, 39	eqtr4d 2776	. . 3 ⊢ (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1))) = (1 · 𝑁))
41	29	adantlr 715	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ 𝑥 ∈ ℂ) → (𝐺‘𝑥) ∈ ℂ)
42		nnnn0 11983	. . . . . . . . . . . . . 14 ⊢ (𝑑 ∈ ℕ → 𝑑 ∈ ℕ₀)
43	42	adantl 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝑑 ∈ ℕ₀)
44	43	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ 𝑥 ∈ ℂ) → 𝑑 ∈ ℕ₀)
45	41, 44	expp1d 13603	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ 𝑥 ∈ ℂ) → ((𝐺‘𝑥)↑(𝑑 + 1)) = (((𝐺‘𝑥)↑𝑑) · (𝐺‘𝑥)))
46	45	mpteq2dva 5125	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1))) = (𝑥 ∈ ℂ ↦ (((𝐺‘𝑥)↑𝑑) · (𝐺‘𝑥))))
47		cnex 10696	. . . . . . . . . . . 12 ⊢ ℂ ∈ V
48	47	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ℂ ∈ V)
49		ovexd 7205	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ 𝑥 ∈ ℂ) → ((𝐺‘𝑥)↑𝑑) ∈ V)
50		eqidd 2739	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) = (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)))
51	32	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝐺 = (𝑥 ∈ ℂ ↦ (𝐺‘𝑥)))
52	48, 49, 41, 50, 51	offval2 7444	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ∘_f · 𝐺) = (𝑥 ∈ ℂ ↦ (((𝐺‘𝑥)↑𝑑) · (𝐺‘𝑥))))
53	46, 52	eqtr4d 2776	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1))) = ((𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ∘_f · 𝐺))
54	53	fveq2d 6678	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1)))) = (deg‘((𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ∘_f · 𝐺)))
55	54	adantr 484	. . . . . . 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 7177	. . . . . . . . 9 ⊢ ((deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁) → ((deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) + 𝑁) = ((𝑑 · 𝑁) + 𝑁))
57	56	adantl 485	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → ((deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) + 𝑁) = ((𝑑 · 𝑁) + 𝑁))
58		eqidd 2739	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑦 ∈ ℂ ↦ (𝑦↑𝑑)) = (𝑦 ∈ ℂ ↦ (𝑦↑𝑑)))
59		oveq1 7177	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐺‘𝑥) → (𝑦↑𝑑) = ((𝐺‘𝑥)↑𝑑))
60	41, 51, 58, 59	fmptco 6901	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((𝑦 ∈ ℂ ↦ (𝑦↑𝑑)) ∘ 𝐺) = (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)))
61		ssidd 3900	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ℂ ⊆ ℂ)
62		1cnd 10714	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 1 ∈ ℂ)
63		plypow 24954	. . . . . . . . . . . . 13 ⊢ ((ℂ ⊆ ℂ ∧ 1 ∈ ℂ ∧ 𝑑 ∈ ℕ₀) → (𝑦 ∈ ℂ ↦ (𝑦↑𝑑)) ∈ (Poly‘ℂ))
64	61, 62, 43, 63	syl3anc 1372	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑦 ∈ ℂ ↦ (𝑦↑𝑑)) ∈ (Poly‘ℂ))
65		plyssc 24949	. . . . . . . . . . . . 13 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
66	26	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝐺 ∈ (Poly‘𝑆))
67	65, 66	sseldi 3875	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝐺 ∈ (Poly‘ℂ))
68		addcl 10697	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (𝑧 + 𝑤) ∈ ℂ)
69	68	adantl 485	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ)) → (𝑧 + 𝑤) ∈ ℂ)
70		mulcl 10699	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (𝑧 · 𝑤) ∈ ℂ)
71	70	adantl 485	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ)) → (𝑧 · 𝑤) ∈ ℂ)
72	64, 67, 69, 71	plyco 24990	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((𝑦 ∈ ℂ ↦ (𝑦↑𝑑)) ∘ 𝐺) ∈ (Poly‘ℂ))
73	60, 72	eqeltrrd 2834	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ∈ (Poly‘ℂ))
74	73	adantr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ∈ (Poly‘ℂ))
75		simpr 488	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁))
76		simpr 488	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝑑 ∈ ℕ)
77	37	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝑁 ∈ ℕ)
78	76, 77	nnmulcld 11769	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑑 · 𝑁) ∈ ℕ)
79	78	nnne0d 11766	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑑 · 𝑁) ≠ 0)
80	79	adantr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → (𝑑 · 𝑁) ≠ 0)
81	75, 80	eqnetrd 3001	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) ≠ 0)
82		fveq2 6674	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) = 0_𝑝 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (deg‘0_𝑝))
83		dgr0 25011	. . . . . . . . . . . 12 ⊢ (deg‘0_𝑝) = 0
84	82, 83	eqtrdi 2789	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) = 0_𝑝 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = 0)
85	84	necon3i 2966	. . . . . . . . . 10 ⊢ ((deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) ≠ 0 → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ≠ 0_𝑝)
86	81, 85	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ≠ 0_𝑝)
87	67	adantr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → 𝐺 ∈ (Poly‘ℂ))
88	37	nnne0d 11766	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ≠ 0)
89		fveq2 6674	. . . . . . . . . . . . . . 15 ⊢ (𝐺 = 0_𝑝 → (deg‘𝐺) = (deg‘0_𝑝))
90	89, 83	eqtrdi 2789	. . . . . . . . . . . . . 14 ⊢ (𝐺 = 0_𝑝 → (deg‘𝐺) = 0)
91	35, 90	syl5eq 2785	. . . . . . . . . . . . 13 ⊢ (𝐺 = 0_𝑝 → 𝑁 = 0)
92	91	necon3i 2966	. . . . . . . . . . . 12 ⊢ (𝑁 ≠ 0 → 𝐺 ≠ 0_𝑝)
93	88, 92	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ≠ 0_𝑝)
94	93	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝐺 ≠ 0_𝑝)
95	94	adantr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → 𝐺 ≠ 0_𝑝)
96		eqid 2738	. . . . . . . . . 10 ⊢ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)))
97	96, 35	dgrmul 25019	. . . . . . . . 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 838	. . . . . . . 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 11724	. . . . . . . . . . 11 ⊢ (𝑑 ∈ ℕ → 𝑑 ∈ ℂ)
100	99	adantl 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝑑 ∈ ℂ)
101	38	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝑁 ∈ ℂ)
102	100, 101	adddirp1d 10745	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((𝑑 + 1) · 𝑁) = ((𝑑 · 𝑁) + 𝑁))
103	102	adantr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → ((𝑑 + 1) · 𝑁) = ((𝑑 · 𝑁) + 𝑁))
104	57, 98, 103	3eqtr4rd 2784	. . . . . . 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 2776	. . . . . 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 416	. . . . 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 417	. . . 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 29	. . 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 11734	. 2 ⊢ (𝑀 ∈ ℕ → (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀))) = (𝑀 · 𝑁)))
110	1, 109	mpcom 38	1 ⊢ (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀))) = (𝑀 · 𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ≠ wne 2934 Vcvv 3398 ⊆ wss 3843 ↦ cmpt 5110 ∘ ccom 5529 ⟶wf 6335 ‘cfv 6339 (class class class)co 7170 ∘_f cof 7423 ℂcc 10613 0cc0 10615 1c1 10616 + caddc 10618 · cmul 10620 ℕcn 11716 ℕ₀cn0 11976 ↑cexp 13521 0_𝑝c0p 24421 Polycply 24933 degcdgr 24936

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-inf2 9177 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-pre-sup 10693

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-se 5484 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-isom 6348 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-of 7425 df-om 7600 df-1st 7714 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-er 8320 df-map 8439 df-pm 8440 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-sup 8979 df-inf 8980 df-oi 9047 df-card 9441 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-div 11376 df-nn 11717 df-2 11779 df-3 11780 df-n0 11977 df-z 12063 df-uz 12325 df-rp 12473 df-fz 12982 df-fzo 13125 df-fl 13253 df-seq 13461 df-exp 13522 df-hash 13783 df-cj 14548 df-re 14549 df-im 14550 df-sqrt 14684 df-abs 14685 df-clim 14935 df-rlim 14936 df-sum 15136 df-0p 24422 df-ply 24937 df-coe 24939 df-dgr 24940

This theorem is referenced by: dgrcolem2 25023

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