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Theorem dgrcolem1 26293

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 7431	. . . . . . 7 ⊢ (𝑦 = 1 → ((𝐺‘𝑥)↑𝑦) = ((𝐺‘𝑥)↑1))
3	2	mpteq2dv 5254	. . . . . 6 ⊢ (𝑦 = 1 → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦)) = (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1)))
4	3	fveq2d 6904	. . . . 5 ⊢ (𝑦 = 1 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1))))
5		oveq1 7430	. . . . 5 ⊢ (𝑦 = 1 → (𝑦 · 𝑁) = (1 · 𝑁))
6	4, 5	eqeq12d 2741	. . . 4 ⊢ (𝑦 = 1 → ((deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (𝑦 · 𝑁) ↔ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1))) = (1 · 𝑁)))
7	6	imbi2d 339	. . 3 ⊢ (𝑦 = 1 → ((𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (𝑦 · 𝑁)) ↔ (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1))) = (1 · 𝑁))))
8		oveq2 7431	. . . . . . 7 ⊢ (𝑦 = 𝑑 → ((𝐺‘𝑥)↑𝑦) = ((𝐺‘𝑥)↑𝑑))
9	8	mpteq2dv 5254	. . . . . 6 ⊢ (𝑦 = 𝑑 → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦)) = (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)))
10	9	fveq2d 6904	. . . . 5 ⊢ (𝑦 = 𝑑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))))
11		oveq1 7430	. . . . 5 ⊢ (𝑦 = 𝑑 → (𝑦 · 𝑁) = (𝑑 · 𝑁))
12	10, 11	eqeq12d 2741	. . . 4 ⊢ (𝑦 = 𝑑 → ((deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (𝑦 · 𝑁) ↔ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)))
13	12	imbi2d 339	. . 3 ⊢ (𝑦 = 𝑑 → ((𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (𝑦 · 𝑁)) ↔ (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁))))
14		oveq2 7431	. . . . . . 7 ⊢ (𝑦 = (𝑑 + 1) → ((𝐺‘𝑥)↑𝑦) = ((𝐺‘𝑥)↑(𝑑 + 1)))
15	14	mpteq2dv 5254	. . . . . 6 ⊢ (𝑦 = (𝑑 + 1) → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦)) = (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1))))
16	15	fveq2d 6904	. . . . 5 ⊢ (𝑦 = (𝑑 + 1) → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1)))))
17		oveq1 7430	. . . . 5 ⊢ (𝑦 = (𝑑 + 1) → (𝑦 · 𝑁) = ((𝑑 + 1) · 𝑁))
18	16, 17	eqeq12d 2741	. . . 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 339	. . 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 7431	. . . . . . 7 ⊢ (𝑦 = 𝑀 → ((𝐺‘𝑥)↑𝑦) = ((𝐺‘𝑥)↑𝑀))
21	20	mpteq2dv 5254	. . . . . 6 ⊢ (𝑦 = 𝑀 → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦)) = (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀)))
22	21	fveq2d 6904	. . . . 5 ⊢ (𝑦 = 𝑀 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀))))
23		oveq1 7430	. . . . 5 ⊢ (𝑦 = 𝑀 → (𝑦 · 𝑁) = (𝑀 · 𝑁))
24	22, 23	eqeq12d 2741	. . . 4 ⊢ (𝑦 = 𝑀 → ((deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (𝑦 · 𝑁) ↔ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀))) = (𝑀 · 𝑁)))
25	24	imbi2d 339	. . 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 26217	. . . . . . . . . . 11 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ)
28	26, 27	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺:ℂ⟶ℂ)
29	28	ffvelcdmda 7097	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝐺‘𝑥) ∈ ℂ)
30	29	exp1d 14155	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((𝐺‘𝑥)↑1) = (𝐺‘𝑥))
31	30	mpteq2dva 5252	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1)) = (𝑥 ∈ ℂ ↦ (𝐺‘𝑥)))
32	28	feqmptd 6970	. . . . . . 7 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ ℂ ↦ (𝐺‘𝑥)))
33	31, 32	eqtr4d 2768	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1)) = 𝐺)
34	33	fveq2d 6904	. . . . 5 ⊢ (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1))) = (deg‘𝐺))
35		dgrcolem1.1	. . . . 5 ⊢ 𝑁 = (deg‘𝐺)
36	34, 35	eqtr4di 2783	. . . 4 ⊢ (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1))) = 𝑁)
37		dgrcolem1.3	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ)
38	37	nncnd 12275	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℂ)
39	38	mullidd 11278	. . . 4 ⊢ (𝜑 → (1 · 𝑁) = 𝑁)
40	36, 39	eqtr4d 2768	. . 3 ⊢ (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1))) = (1 · 𝑁))
41	29	adantlr 713	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ 𝑥 ∈ ℂ) → (𝐺‘𝑥) ∈ ℂ)
42		nnnn0 12526	. . . . . . . . . . . . . 14 ⊢ (𝑑 ∈ ℕ → 𝑑 ∈ ℕ₀)
43	42	adantl 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝑑 ∈ ℕ₀)
44	43	adantr 479	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ 𝑥 ∈ ℂ) → 𝑑 ∈ ℕ₀)
45	41, 44	expp1d 14161	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ 𝑥 ∈ ℂ) → ((𝐺‘𝑥)↑(𝑑 + 1)) = (((𝐺‘𝑥)↑𝑑) · (𝐺‘𝑥)))
46	45	mpteq2dva 5252	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1))) = (𝑥 ∈ ℂ ↦ (((𝐺‘𝑥)↑𝑑) · (𝐺‘𝑥))))
47		cnex 11235	. . . . . . . . . . . 12 ⊢ ℂ ∈ V
48	47	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ℂ ∈ V)
49		ovexd 7458	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ 𝑥 ∈ ℂ) → ((𝐺‘𝑥)↑𝑑) ∈ V)
50		eqidd 2726	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) = (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)))
51	32	adantr 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝐺 = (𝑥 ∈ ℂ ↦ (𝐺‘𝑥)))
52	48, 49, 41, 50, 51	offval2 7709	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ∘_f · 𝐺) = (𝑥 ∈ ℂ ↦ (((𝐺‘𝑥)↑𝑑) · (𝐺‘𝑥))))
53	46, 52	eqtr4d 2768	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1))) = ((𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ∘_f · 𝐺))
54	53	fveq2d 6904	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1)))) = (deg‘((𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ∘_f · 𝐺)))
55	54	adantr 479	. . . . . . 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 7430	. . . . . . . . 9 ⊢ ((deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁) → ((deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) + 𝑁) = ((𝑑 · 𝑁) + 𝑁))
57	56	adantl 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → ((deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) + 𝑁) = ((𝑑 · 𝑁) + 𝑁))
58		eqidd 2726	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑦 ∈ ℂ ↦ (𝑦↑𝑑)) = (𝑦 ∈ ℂ ↦ (𝑦↑𝑑)))
59		oveq1 7430	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐺‘𝑥) → (𝑦↑𝑑) = ((𝐺‘𝑥)↑𝑑))
60	41, 51, 58, 59	fmptco 7142	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((𝑦 ∈ ℂ ↦ (𝑦↑𝑑)) ∘ 𝐺) = (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)))
61		ssidd 4002	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ℂ ⊆ ℂ)
62		1cnd 11255	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 1 ∈ ℂ)
63		plypow 26224	. . . . . . . . . . . . 13 ⊢ ((ℂ ⊆ ℂ ∧ 1 ∈ ℂ ∧ 𝑑 ∈ ℕ₀) → (𝑦 ∈ ℂ ↦ (𝑦↑𝑑)) ∈ (Poly‘ℂ))
64	61, 62, 43, 63	syl3anc 1368	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑦 ∈ ℂ ↦ (𝑦↑𝑑)) ∈ (Poly‘ℂ))
65		plyssc 26219	. . . . . . . . . . . . 13 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
66	26	adantr 479	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝐺 ∈ (Poly‘𝑆))
67	65, 66	sselid 3976	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝐺 ∈ (Poly‘ℂ))
68		addcl 11236	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (𝑧 + 𝑤) ∈ ℂ)
69	68	adantl 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ)) → (𝑧 + 𝑤) ∈ ℂ)
70		mulcl 11238	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (𝑧 · 𝑤) ∈ ℂ)
71	70	adantl 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ)) → (𝑧 · 𝑤) ∈ ℂ)
72	64, 67, 69, 71	plyco 26260	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((𝑦 ∈ ℂ ↦ (𝑦↑𝑑)) ∘ 𝐺) ∈ (Poly‘ℂ))
73	60, 72	eqeltrrd 2826	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ∈ (Poly‘ℂ))
74	73	adantr 479	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ∈ (Poly‘ℂ))
75		simpr 483	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁))
76		simpr 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝑑 ∈ ℕ)
77	37	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝑁 ∈ ℕ)
78	76, 77	nnmulcld 12312	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑑 · 𝑁) ∈ ℕ)
79	78	nnne0d 12309	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑑 · 𝑁) ≠ 0)
80	79	adantr 479	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → (𝑑 · 𝑁) ≠ 0)
81	75, 80	eqnetrd 2997	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) ≠ 0)
82		fveq2 6900	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) = 0_𝑝 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (deg‘0_𝑝))
83		dgr0 26282	. . . . . . . . . . . 12 ⊢ (deg‘0_𝑝) = 0
84	82, 83	eqtrdi 2781	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) = 0_𝑝 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = 0)
85	84	necon3i 2962	. . . . . . . . . 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 479	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → 𝐺 ∈ (Poly‘ℂ))
88	37	nnne0d 12309	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ≠ 0)
89		fveq2 6900	. . . . . . . . . . . . . . 15 ⊢ (𝐺 = 0_𝑝 → (deg‘𝐺) = (deg‘0_𝑝))
90	89, 83	eqtrdi 2781	. . . . . . . . . . . . . 14 ⊢ (𝐺 = 0_𝑝 → (deg‘𝐺) = 0)
91	35, 90	eqtrid 2777	. . . . . . . . . . . . 13 ⊢ (𝐺 = 0_𝑝 → 𝑁 = 0)
92	91	necon3i 2962	. . . . . . . . . . . 12 ⊢ (𝑁 ≠ 0 → 𝐺 ≠ 0_𝑝)
93	88, 92	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ≠ 0_𝑝)
94	93	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝐺 ≠ 0_𝑝)
95	94	adantr 479	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → 𝐺 ≠ 0_𝑝)
96		eqid 2725	. . . . . . . . . 10 ⊢ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)))
97	96, 35	dgrmul 26290	. . . . . . . . 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 837	. . . . . . . 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 12267	. . . . . . . . . . 11 ⊢ (𝑑 ∈ ℕ → 𝑑 ∈ ℂ)
100	99	adantl 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝑑 ∈ ℂ)
101	38	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝑁 ∈ ℂ)
102	100, 101	adddirp1d 11286	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((𝑑 + 1) · 𝑁) = ((𝑑 · 𝑁) + 𝑁))
103	102	adantr 479	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → ((𝑑 + 1) · 𝑁) = ((𝑑 · 𝑁) + 𝑁))
104	57, 98, 103	3eqtr4rd 2776	. . . . . . 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 2768	. . . . . 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 411	. . . . 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 412	. . . 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 12277	. 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 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 Vcvv 3461 ⊆ wss 3946 ↦ cmpt 5235 ∘ ccom 5685 ⟶wf 6549 ‘cfv 6553 (class class class)co 7423 ∘_f cof 7687 ℂcc 11152 0cc0 11154 1c1 11155 + caddc 11157 · cmul 11159 ℕcn 12259 ℕ₀cn0 12519 ↑cexp 14076 0_𝑝c0p 25681 Polycply 26203 degcdgr 26206

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-inf2 9680 ax-cnex 11210 ax-resscn 11211 ax-1cn 11212 ax-icn 11213 ax-addcl 11214 ax-addrcl 11215 ax-mulcl 11216 ax-mulrcl 11217 ax-mulcom 11218 ax-addass 11219 ax-mulass 11220 ax-distr 11221 ax-i2m1 11222 ax-1ne0 11223 ax-1rid 11224 ax-rnegex 11225 ax-rrecex 11226 ax-cnre 11227 ax-pre-lttri 11228 ax-pre-lttrn 11229 ax-pre-ltadd 11230 ax-pre-mulgt0 11231 ax-pre-sup 11232

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5579 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-se 5637 df-we 5638 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-pred 6311 df-ord 6378 df-on 6379 df-lim 6380 df-suc 6381 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-of 7689 df-om 7876 df-1st 8002 df-2nd 8003 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-map 8856 df-pm 8857 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-sup 9481 df-inf 9482 df-oi 9549 df-card 9978 df-pnf 11296 df-mnf 11297 df-xr 11298 df-ltxr 11299 df-le 11300 df-sub 11492 df-neg 11493 df-div 11918 df-nn 12260 df-2 12322 df-3 12323 df-n0 12520 df-z 12606 df-uz 12870 df-rp 13024 df-fz 13534 df-fzo 13677 df-fl 13807 df-seq 14017 df-exp 14077 df-hash 14343 df-cj 15099 df-re 15100 df-im 15101 df-sqrt 15235 df-abs 15236 df-clim 15485 df-rlim 15486 df-sum 15686 df-0p 25682 df-ply 26207 df-coe 26209 df-dgr 26210

This theorem is referenced by: dgrcolem2 26294

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