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Theorem elqaalem2 25824

Description: Lemma for elqaa 25826. (Contributed by Mario Carneiro, 23-Jul-2014.) (Revised by AV, 3-Oct-2020.)

**Hypotheses**
Ref	Expression
elqaa.1	⊢ (𝜑 → 𝐴 ∈ ℂ)
elqaa.2	⊢ (𝜑 → 𝐹 ∈ ((Poly‘ℚ) ∖ {0_𝑝}))
elqaa.3	⊢ (𝜑 → (𝐹‘𝐴) = 0)
elqaa.4	⊢ 𝐵 = (coeff‘𝐹)
elqaa.5	⊢ 𝑁 = (𝑘 ∈ ℕ₀ ↦ inf({𝑛 ∈ ℕ ∣ ((𝐵‘𝑘) · 𝑛) ∈ ℤ}, ℝ, < ))
elqaa.6	⊢ 𝑅 = (seq0( · , 𝑁)‘(deg‘𝐹))
elqaa.7	⊢ 𝑃 = (𝑥 ∈ V, 𝑦 ∈ V ↦ ((𝑥 · 𝑦) mod (𝑁‘𝐾)))

**Assertion**
Ref	Expression
elqaalem2	⊢ ((𝜑 ∧ 𝐾 ∈ (0...(deg‘𝐹))) → (𝑅 mod (𝑁‘𝐾)) = 0)

Distinct variable groups: 𝑘,𝑛,𝑥,𝑦,𝐴 𝐵,𝑘,𝑛 𝜑,𝑘 𝑘,𝐾,𝑛,𝑥,𝑦 𝑘,𝑁,𝑛,𝑥,𝑦 𝑅,𝑘 Allowed substitution hints: 𝜑(𝑥,𝑦,𝑛) 𝐵(𝑥,𝑦) 𝑃(𝑥,𝑦,𝑘,𝑛) 𝑅(𝑥,𝑦,𝑛) 𝐹(𝑥,𝑦,𝑘,𝑛)

Proof of Theorem elqaalem2 Dummy variables 𝑚 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfznn0 13590	. . 3 ⊢ (𝐾 ∈ (0...(deg‘𝐹)) → 𝐾 ∈ ℕ₀)
2		elqaa.6	. . . . 5 ⊢ 𝑅 = (seq0( · , 𝑁)‘(deg‘𝐹))
3	2	fveq2i 6891	. . . 4 ⊢ ((𝑘 ∈ ℕ ↦ (𝑘 mod (𝑁‘𝐾)))‘𝑅) = ((𝑘 ∈ ℕ ↦ (𝑘 mod (𝑁‘𝐾)))‘(seq0( · , 𝑁)‘(deg‘𝐹)))
4		nnmulcl 12232	. . . . . 6 ⊢ ((𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ) → (𝑖 · 𝑗) ∈ ℕ)
5	4	adantl 482	. . . . 5 ⊢ (((𝜑 ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → (𝑖 · 𝑗) ∈ ℕ)
6		elfznn0 13590	. . . . . 6 ⊢ (𝑖 ∈ (0...(deg‘𝐹)) → 𝑖 ∈ ℕ₀)
7		elqaa.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ)
8		elqaa.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ ((Poly‘ℚ) ∖ {0_𝑝}))
9		elqaa.3	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘𝐴) = 0)
10		elqaa.4	. . . . . . . . 9 ⊢ 𝐵 = (coeff‘𝐹)
11		elqaa.5	. . . . . . . . 9 ⊢ 𝑁 = (𝑘 ∈ ℕ₀ ↦ inf({𝑛 ∈ ℕ ∣ ((𝐵‘𝑘) · 𝑛) ∈ ℤ}, ℝ, < ))
12	7, 8, 9, 10, 11, 2	elqaalem1 25823	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → ((𝑁‘𝑖) ∈ ℕ ∧ ((𝐵‘𝑖) · (𝑁‘𝑖)) ∈ ℤ))
13	12	simpld 495	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ₀) → (𝑁‘𝑖) ∈ ℕ)
14	13	adantlr 713	. . . . . 6 ⊢ (((𝜑 ∧ 𝐾 ∈ ℕ₀) ∧ 𝑖 ∈ ℕ₀) → (𝑁‘𝑖) ∈ ℕ)
15	6, 14	sylan2 593	. . . . 5 ⊢ (((𝜑 ∧ 𝐾 ∈ ℕ₀) ∧ 𝑖 ∈ (0...(deg‘𝐹))) → (𝑁‘𝑖) ∈ ℕ)
16		eldifi 4125	. . . . . . . 8 ⊢ (𝐹 ∈ ((Poly‘ℚ) ∖ {0_𝑝}) → 𝐹 ∈ (Poly‘ℚ))
17		dgrcl 25738	. . . . . . . 8 ⊢ (𝐹 ∈ (Poly‘ℚ) → (deg‘𝐹) ∈ ℕ₀)
18	8, 16, 17	3syl 18	. . . . . . 7 ⊢ (𝜑 → (deg‘𝐹) ∈ ℕ₀)
19		nn0uz 12860	. . . . . . 7 ⊢ ℕ₀ = (ℤ_≥‘0)
20	18, 19	eleqtrdi 2843	. . . . . 6 ⊢ (𝜑 → (deg‘𝐹) ∈ (ℤ_≥‘0))
21	20	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ ℕ₀) → (deg‘𝐹) ∈ (ℤ_≥‘0))
22		nnz 12575	. . . . . . . . . 10 ⊢ (𝑖 ∈ ℕ → 𝑖 ∈ ℤ)
23	22	ad2antrl 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → 𝑖 ∈ ℤ)
24	7, 8, 9, 10, 11, 2	elqaalem1 25823	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐾 ∈ ℕ₀) → ((𝑁‘𝐾) ∈ ℕ ∧ ((𝐵‘𝐾) · (𝑁‘𝐾)) ∈ ℤ))
25	24	simpld 495	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐾 ∈ ℕ₀) → (𝑁‘𝐾) ∈ ℕ)
26	25	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → (𝑁‘𝐾) ∈ ℕ)
27	23, 26	zmodcld 13853	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → (𝑖 mod (𝑁‘𝐾)) ∈ ℕ₀)
28	27	nn0zd 12580	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → (𝑖 mod (𝑁‘𝐾)) ∈ ℤ)
29		nnz 12575	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℤ)
30	29	ad2antll 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → 𝑗 ∈ ℤ)
31	30, 26	zmodcld 13853	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → (𝑗 mod (𝑁‘𝐾)) ∈ ℕ₀)
32	31	nn0zd 12580	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → (𝑗 mod (𝑁‘𝐾)) ∈ ℤ)
33	26	nnrpd 13010	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → (𝑁‘𝐾) ∈ ℝ⁺)
34		nnre 12215	. . . . . . . . 9 ⊢ (𝑖 ∈ ℕ → 𝑖 ∈ ℝ)
35	34	ad2antrl 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → 𝑖 ∈ ℝ)
36		modabs2 13866	. . . . . . . 8 ⊢ ((𝑖 ∈ ℝ ∧ (𝑁‘𝐾) ∈ ℝ⁺) → ((𝑖 mod (𝑁‘𝐾)) mod (𝑁‘𝐾)) = (𝑖 mod (𝑁‘𝐾)))
37	35, 33, 36	syl2anc 584	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → ((𝑖 mod (𝑁‘𝐾)) mod (𝑁‘𝐾)) = (𝑖 mod (𝑁‘𝐾)))
38		nnre 12215	. . . . . . . . 9 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℝ)
39	38	ad2antll 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → 𝑗 ∈ ℝ)
40		modabs2 13866	. . . . . . . 8 ⊢ ((𝑗 ∈ ℝ ∧ (𝑁‘𝐾) ∈ ℝ⁺) → ((𝑗 mod (𝑁‘𝐾)) mod (𝑁‘𝐾)) = (𝑗 mod (𝑁‘𝐾)))
41	39, 33, 40	syl2anc 584	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → ((𝑗 mod (𝑁‘𝐾)) mod (𝑁‘𝐾)) = (𝑗 mod (𝑁‘𝐾)))
42	28, 23, 32, 30, 33, 37, 41	modmul12d 13886	. . . . . 6 ⊢ (((𝜑 ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → (((𝑖 mod (𝑁‘𝐾)) · (𝑗 mod (𝑁‘𝐾))) mod (𝑁‘𝐾)) = ((𝑖 · 𝑗) mod (𝑁‘𝐾)))
43		oveq1 7412	. . . . . . . . . 10 ⊢ (𝑘 = 𝑖 → (𝑘 mod (𝑁‘𝐾)) = (𝑖 mod (𝑁‘𝐾)))
44		eqid 2732	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ ↦ (𝑘 mod (𝑁‘𝐾))) = (𝑘 ∈ ℕ ↦ (𝑘 mod (𝑁‘𝐾)))
45		ovex 7438	. . . . . . . . . 10 ⊢ (𝑖 mod (𝑁‘𝐾)) ∈ V
46	43, 44, 45	fvmpt 6995	. . . . . . . . 9 ⊢ (𝑖 ∈ ℕ → ((𝑘 ∈ ℕ ↦ (𝑘 mod (𝑁‘𝐾)))‘𝑖) = (𝑖 mod (𝑁‘𝐾)))
47	46	ad2antrl 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → ((𝑘 ∈ ℕ ↦ (𝑘 mod (𝑁‘𝐾)))‘𝑖) = (𝑖 mod (𝑁‘𝐾)))
48		oveq1 7412	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (𝑘 mod (𝑁‘𝐾)) = (𝑗 mod (𝑁‘𝐾)))
49		ovex 7438	. . . . . . . . . 10 ⊢ (𝑗 mod (𝑁‘𝐾)) ∈ V
50	48, 44, 49	fvmpt 6995	. . . . . . . . 9 ⊢ (𝑗 ∈ ℕ → ((𝑘 ∈ ℕ ↦ (𝑘 mod (𝑁‘𝐾)))‘𝑗) = (𝑗 mod (𝑁‘𝐾)))
51	50	ad2antll 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → ((𝑘 ∈ ℕ ↦ (𝑘 mod (𝑁‘𝐾)))‘𝑗) = (𝑗 mod (𝑁‘𝐾)))
52	47, 51	oveq12d 7423	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → (((𝑘 ∈ ℕ ↦ (𝑘 mod (𝑁‘𝐾)))‘𝑖)𝑃((𝑘 ∈ ℕ ↦ (𝑘 mod (𝑁‘𝐾)))‘𝑗)) = ((𝑖 mod (𝑁‘𝐾))𝑃(𝑗 mod (𝑁‘𝐾))))
53		oveq12 7414	. . . . . . . . . 10 ⊢ ((𝑥 = (𝑖 mod (𝑁‘𝐾)) ∧ 𝑦 = (𝑗 mod (𝑁‘𝐾))) → (𝑥 · 𝑦) = ((𝑖 mod (𝑁‘𝐾)) · (𝑗 mod (𝑁‘𝐾))))
54	53	oveq1d 7420	. . . . . . . . 9 ⊢ ((𝑥 = (𝑖 mod (𝑁‘𝐾)) ∧ 𝑦 = (𝑗 mod (𝑁‘𝐾))) → ((𝑥 · 𝑦) mod (𝑁‘𝐾)) = (((𝑖 mod (𝑁‘𝐾)) · (𝑗 mod (𝑁‘𝐾))) mod (𝑁‘𝐾)))
55		elqaa.7	. . . . . . . . 9 ⊢ 𝑃 = (𝑥 ∈ V, 𝑦 ∈ V ↦ ((𝑥 · 𝑦) mod (𝑁‘𝐾)))
56		ovex 7438	. . . . . . . . 9 ⊢ (((𝑖 mod (𝑁‘𝐾)) · (𝑗 mod (𝑁‘𝐾))) mod (𝑁‘𝐾)) ∈ V
57	54, 55, 56	ovmpoa 7559	. . . . . . . 8 ⊢ (((𝑖 mod (𝑁‘𝐾)) ∈ V ∧ (𝑗 mod (𝑁‘𝐾)) ∈ V) → ((𝑖 mod (𝑁‘𝐾))𝑃(𝑗 mod (𝑁‘𝐾))) = (((𝑖 mod (𝑁‘𝐾)) · (𝑗 mod (𝑁‘𝐾))) mod (𝑁‘𝐾)))
58	45, 49, 57	mp2an 690	. . . . . . 7 ⊢ ((𝑖 mod (𝑁‘𝐾))𝑃(𝑗 mod (𝑁‘𝐾))) = (((𝑖 mod (𝑁‘𝐾)) · (𝑗 mod (𝑁‘𝐾))) mod (𝑁‘𝐾))
59	52, 58	eqtrdi 2788	. . . . . 6 ⊢ (((𝜑 ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → (((𝑘 ∈ ℕ ↦ (𝑘 mod (𝑁‘𝐾)))‘𝑖)𝑃((𝑘 ∈ ℕ ↦ (𝑘 mod (𝑁‘𝐾)))‘𝑗)) = (((𝑖 mod (𝑁‘𝐾)) · (𝑗 mod (𝑁‘𝐾))) mod (𝑁‘𝐾)))
60		oveq1 7412	. . . . . . . 8 ⊢ (𝑘 = (𝑖 · 𝑗) → (𝑘 mod (𝑁‘𝐾)) = ((𝑖 · 𝑗) mod (𝑁‘𝐾)))
61		ovex 7438	. . . . . . . 8 ⊢ ((𝑖 · 𝑗) mod (𝑁‘𝐾)) ∈ V
62	60, 44, 61	fvmpt 6995	. . . . . . 7 ⊢ ((𝑖 · 𝑗) ∈ ℕ → ((𝑘 ∈ ℕ ↦ (𝑘 mod (𝑁‘𝐾)))‘(𝑖 · 𝑗)) = ((𝑖 · 𝑗) mod (𝑁‘𝐾)))
63	5, 62	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → ((𝑘 ∈ ℕ ↦ (𝑘 mod (𝑁‘𝐾)))‘(𝑖 · 𝑗)) = ((𝑖 · 𝑗) mod (𝑁‘𝐾)))
64	42, 59, 63	3eqtr4rd 2783	. . . . 5 ⊢ (((𝜑 ∧ 𝐾 ∈ ℕ₀) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → ((𝑘 ∈ ℕ ↦ (𝑘 mod (𝑁‘𝐾)))‘(𝑖 · 𝑗)) = (((𝑘 ∈ ℕ ↦ (𝑘 mod (𝑁‘𝐾)))‘𝑖)𝑃((𝑘 ∈ ℕ ↦ (𝑘 mod (𝑁‘𝐾)))‘𝑗)))
65		oveq1 7412	. . . . . . . . 9 ⊢ (𝑘 = (𝑁‘𝑖) → (𝑘 mod (𝑁‘𝐾)) = ((𝑁‘𝑖) mod (𝑁‘𝐾)))
66		ovex 7438	. . . . . . . . 9 ⊢ ((𝑁‘𝑖) mod (𝑁‘𝐾)) ∈ V
67	65, 44, 66	fvmpt 6995	. . . . . . . 8 ⊢ ((𝑁‘𝑖) ∈ ℕ → ((𝑘 ∈ ℕ ↦ (𝑘 mod (𝑁‘𝐾)))‘(𝑁‘𝑖)) = ((𝑁‘𝑖) mod (𝑁‘𝐾)))
68	14, 67	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 ∈ ℕ₀) ∧ 𝑖 ∈ ℕ₀) → ((𝑘 ∈ ℕ ↦ (𝑘 mod (𝑁‘𝐾)))‘(𝑁‘𝑖)) = ((𝑁‘𝑖) mod (𝑁‘𝐾)))
69		fveq2 6888	. . . . . . . . . 10 ⊢ (𝑘 = 𝑖 → (𝑁‘𝑘) = (𝑁‘𝑖))
70	69	oveq1d 7420	. . . . . . . . 9 ⊢ (𝑘 = 𝑖 → ((𝑁‘𝑘) mod (𝑁‘𝐾)) = ((𝑁‘𝑖) mod (𝑁‘𝐾)))
71		eqid 2732	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ₀ ↦ ((𝑁‘𝑘) mod (𝑁‘𝐾))) = (𝑘 ∈ ℕ₀ ↦ ((𝑁‘𝑘) mod (𝑁‘𝐾)))
72	70, 71, 66	fvmpt 6995	. . . . . . . 8 ⊢ (𝑖 ∈ ℕ₀ → ((𝑘 ∈ ℕ₀ ↦ ((𝑁‘𝑘) mod (𝑁‘𝐾)))‘𝑖) = ((𝑁‘𝑖) mod (𝑁‘𝐾)))
73	72	adantl 482	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 ∈ ℕ₀) ∧ 𝑖 ∈ ℕ₀) → ((𝑘 ∈ ℕ₀ ↦ ((𝑁‘𝑘) mod (𝑁‘𝐾)))‘𝑖) = ((𝑁‘𝑖) mod (𝑁‘𝐾)))
74	68, 73	eqtr4d 2775	. . . . . 6 ⊢ (((𝜑 ∧ 𝐾 ∈ ℕ₀) ∧ 𝑖 ∈ ℕ₀) → ((𝑘 ∈ ℕ ↦ (𝑘 mod (𝑁‘𝐾)))‘(𝑁‘𝑖)) = ((𝑘 ∈ ℕ₀ ↦ ((𝑁‘𝑘) mod (𝑁‘𝐾)))‘𝑖))
75	6, 74	sylan2 593	. . . . 5 ⊢ (((𝜑 ∧ 𝐾 ∈ ℕ₀) ∧ 𝑖 ∈ (0...(deg‘𝐹))) → ((𝑘 ∈ ℕ ↦ (𝑘 mod (𝑁‘𝐾)))‘(𝑁‘𝑖)) = ((𝑘 ∈ ℕ₀ ↦ ((𝑁‘𝑘) mod (𝑁‘𝐾)))‘𝑖))
76	5, 15, 21, 64, 75	seqhomo 14011	. . . 4 ⊢ ((𝜑 ∧ 𝐾 ∈ ℕ₀) → ((𝑘 ∈ ℕ ↦ (𝑘 mod (𝑁‘𝐾)))‘(seq0( · , 𝑁)‘(deg‘𝐹))) = (seq0(𝑃, (𝑘 ∈ ℕ₀ ↦ ((𝑁‘𝑘) mod (𝑁‘𝐾))))‘(deg‘𝐹)))
77	3, 76	eqtrid 2784	. . 3 ⊢ ((𝜑 ∧ 𝐾 ∈ ℕ₀) → ((𝑘 ∈ ℕ ↦ (𝑘 mod (𝑁‘𝐾)))‘𝑅) = (seq0(𝑃, (𝑘 ∈ ℕ₀ ↦ ((𝑁‘𝑘) mod (𝑁‘𝐾))))‘(deg‘𝐹)))
78	1, 77	sylan2 593	. 2 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(deg‘𝐹))) → ((𝑘 ∈ ℕ ↦ (𝑘 mod (𝑁‘𝐾)))‘𝑅) = (seq0(𝑃, (𝑘 ∈ ℕ₀ ↦ ((𝑁‘𝑘) mod (𝑁‘𝐾))))‘(deg‘𝐹)))
79		0zd 12566	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℤ)
80	4	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → (𝑖 · 𝑗) ∈ ℕ)
81	19, 79, 13, 80	seqf 13985	. . . . . . 7 ⊢ (𝜑 → seq0( · , 𝑁):ℕ₀⟶ℕ)
82	81, 18	ffvelcdmd 7084	. . . . . 6 ⊢ (𝜑 → (seq0( · , 𝑁)‘(deg‘𝐹)) ∈ ℕ)
83	2, 82	eqeltrid 2837	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℕ)
84	83	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝐾 ∈ ℕ₀) → 𝑅 ∈ ℕ)
85		oveq1 7412	. . . . 5 ⊢ (𝑘 = 𝑅 → (𝑘 mod (𝑁‘𝐾)) = (𝑅 mod (𝑁‘𝐾)))
86		ovex 7438	. . . . 5 ⊢ (𝑅 mod (𝑁‘𝐾)) ∈ V
87	85, 44, 86	fvmpt 6995	. . . 4 ⊢ (𝑅 ∈ ℕ → ((𝑘 ∈ ℕ ↦ (𝑘 mod (𝑁‘𝐾)))‘𝑅) = (𝑅 mod (𝑁‘𝐾)))
88	84, 87	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝐾 ∈ ℕ₀) → ((𝑘 ∈ ℕ ↦ (𝑘 mod (𝑁‘𝐾)))‘𝑅) = (𝑅 mod (𝑁‘𝐾)))
89	1, 88	sylan2 593	. 2 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(deg‘𝐹))) → ((𝑘 ∈ ℕ ↦ (𝑘 mod (𝑁‘𝐾)))‘𝑅) = (𝑅 mod (𝑁‘𝐾)))
90		oveq12 7414	. . . . . . 7 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → (𝑥 · 𝑦) = (𝑖 · 𝑗))
91	90	oveq1d 7420	. . . . . 6 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → ((𝑥 · 𝑦) mod (𝑁‘𝐾)) = ((𝑖 · 𝑗) mod (𝑁‘𝐾)))
92	91, 55, 61	ovmpoa 7559	. . . . 5 ⊢ ((𝑖 ∈ V ∧ 𝑗 ∈ V) → (𝑖𝑃𝑗) = ((𝑖 · 𝑗) mod (𝑁‘𝐾)))
93	92	el2v 3482	. . . 4 ⊢ (𝑖𝑃𝑗) = ((𝑖 · 𝑗) mod (𝑁‘𝐾))
94		nn0mulcl 12504	. . . . . 6 ⊢ ((𝑖 ∈ ℕ₀ ∧ 𝑗 ∈ ℕ₀) → (𝑖 · 𝑗) ∈ ℕ₀)
95	94	nn0zd 12580	. . . . 5 ⊢ ((𝑖 ∈ ℕ₀ ∧ 𝑗 ∈ ℕ₀) → (𝑖 · 𝑗) ∈ ℤ)
96	1, 25	sylan2 593	. . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(deg‘𝐹))) → (𝑁‘𝐾) ∈ ℕ)
97		zmodcl 13852	. . . . 5 ⊢ (((𝑖 · 𝑗) ∈ ℤ ∧ (𝑁‘𝐾) ∈ ℕ) → ((𝑖 · 𝑗) mod (𝑁‘𝐾)) ∈ ℕ₀)
98	95, 96, 97	syl2anr 597	. . . 4 ⊢ (((𝜑 ∧ 𝐾 ∈ (0...(deg‘𝐹))) ∧ (𝑖 ∈ ℕ₀ ∧ 𝑗 ∈ ℕ₀)) → ((𝑖 · 𝑗) mod (𝑁‘𝐾)) ∈ ℕ₀)
99	93, 98	eqeltrid 2837	. . 3 ⊢ (((𝜑 ∧ 𝐾 ∈ (0...(deg‘𝐹))) ∧ (𝑖 ∈ ℕ₀ ∧ 𝑗 ∈ ℕ₀)) → (𝑖𝑃𝑗) ∈ ℕ₀)
100		fveq2 6888	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑚 → (𝐵‘𝑘) = (𝐵‘𝑚))
101	100	oveq1d 7420	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑚 → ((𝐵‘𝑘) · 𝑛) = ((𝐵‘𝑚) · 𝑛))
102	101	eleq1d 2818	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑚 → (((𝐵‘𝑘) · 𝑛) ∈ ℤ ↔ ((𝐵‘𝑚) · 𝑛) ∈ ℤ))
103	102	rabbidv 3440	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑚 → {𝑛 ∈ ℕ ∣ ((𝐵‘𝑘) · 𝑛) ∈ ℤ} = {𝑛 ∈ ℕ ∣ ((𝐵‘𝑚) · 𝑛) ∈ ℤ})
104	103	infeq1d 9468	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑚 → inf({𝑛 ∈ ℕ ∣ ((𝐵‘𝑘) · 𝑛) ∈ ℤ}, ℝ, < ) = inf({𝑛 ∈ ℕ ∣ ((𝐵‘𝑚) · 𝑛) ∈ ℤ}, ℝ, < ))
105	104	cbvmptv 5260	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ₀ ↦ inf({𝑛 ∈ ℕ ∣ ((𝐵‘𝑘) · 𝑛) ∈ ℤ}, ℝ, < )) = (𝑚 ∈ ℕ₀ ↦ inf({𝑛 ∈ ℕ ∣ ((𝐵‘𝑚) · 𝑛) ∈ ℤ}, ℝ, < ))
106	11, 105	eqtri 2760	. . . . . . . . . . 11 ⊢ 𝑁 = (𝑚 ∈ ℕ₀ ↦ inf({𝑛 ∈ ℕ ∣ ((𝐵‘𝑚) · 𝑛) ∈ ℤ}, ℝ, < ))
107	7, 8, 9, 10, 106, 2	elqaalem1 25823	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((𝑁‘𝑘) ∈ ℕ ∧ ((𝐵‘𝑘) · (𝑁‘𝑘)) ∈ ℤ))
108	107	simpld 495	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝑁‘𝑘) ∈ ℕ)
109	108	adantlr 713	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐾 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → (𝑁‘𝑘) ∈ ℕ)
110	109	nnzd 12581	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → (𝑁‘𝑘) ∈ ℤ)
111	25	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → (𝑁‘𝐾) ∈ ℕ)
112	110, 111	zmodcld 13853	. . . . . 6 ⊢ (((𝜑 ∧ 𝐾 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → ((𝑁‘𝑘) mod (𝑁‘𝐾)) ∈ ℕ₀)
113	112	fmpttd 7111	. . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ ℕ₀) → (𝑘 ∈ ℕ₀ ↦ ((𝑁‘𝑘) mod (𝑁‘𝐾))):ℕ₀⟶ℕ₀)
114	1, 113	sylan2 593	. . . 4 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(deg‘𝐹))) → (𝑘 ∈ ℕ₀ ↦ ((𝑁‘𝑘) mod (𝑁‘𝐾))):ℕ₀⟶ℕ₀)
115		ffvelcdm 7080	. . . 4 ⊢ (((𝑘 ∈ ℕ₀ ↦ ((𝑁‘𝑘) mod (𝑁‘𝐾))):ℕ₀⟶ℕ₀ ∧ 𝑖 ∈ ℕ₀) → ((𝑘 ∈ ℕ₀ ↦ ((𝑁‘𝑘) mod (𝑁‘𝐾)))‘𝑖) ∈ ℕ₀)
116	114, 6, 115	syl2an 596	. . 3 ⊢ (((𝜑 ∧ 𝐾 ∈ (0...(deg‘𝐹))) ∧ 𝑖 ∈ (0...(deg‘𝐹))) → ((𝑘 ∈ ℕ₀ ↦ ((𝑁‘𝑘) mod (𝑁‘𝐾)))‘𝑖) ∈ ℕ₀)
117		c0ex 11204	. . . . 5 ⊢ 0 ∈ V
118		vex 3478	. . . . 5 ⊢ 𝑖 ∈ V
119		oveq12 7414	. . . . . . 7 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑖) → (𝑥 · 𝑦) = (0 · 𝑖))
120	119	oveq1d 7420	. . . . . 6 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑖) → ((𝑥 · 𝑦) mod (𝑁‘𝐾)) = ((0 · 𝑖) mod (𝑁‘𝐾)))
121		ovex 7438	. . . . . 6 ⊢ ((0 · 𝑖) mod (𝑁‘𝐾)) ∈ V
122	120, 55, 121	ovmpoa 7559	. . . . 5 ⊢ ((0 ∈ V ∧ 𝑖 ∈ V) → (0𝑃𝑖) = ((0 · 𝑖) mod (𝑁‘𝐾)))
123	117, 118, 122	mp2an 690	. . . 4 ⊢ (0𝑃𝑖) = ((0 · 𝑖) mod (𝑁‘𝐾))
124		nn0cn 12478	. . . . . . 7 ⊢ (𝑖 ∈ ℕ₀ → 𝑖 ∈ ℂ)
125	124	mul02d 11408	. . . . . 6 ⊢ (𝑖 ∈ ℕ₀ → (0 · 𝑖) = 0)
126	125	oveq1d 7420	. . . . 5 ⊢ (𝑖 ∈ ℕ₀ → ((0 · 𝑖) mod (𝑁‘𝐾)) = (0 mod (𝑁‘𝐾)))
127	96	nnrpd 13010	. . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(deg‘𝐹))) → (𝑁‘𝐾) ∈ ℝ⁺)
128		0mod 13863	. . . . . 6 ⊢ ((𝑁‘𝐾) ∈ ℝ⁺ → (0 mod (𝑁‘𝐾)) = 0)
129	127, 128	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(deg‘𝐹))) → (0 mod (𝑁‘𝐾)) = 0)
130	126, 129	sylan9eqr 2794	. . . 4 ⊢ (((𝜑 ∧ 𝐾 ∈ (0...(deg‘𝐹))) ∧ 𝑖 ∈ ℕ₀) → ((0 · 𝑖) mod (𝑁‘𝐾)) = 0)
131	123, 130	eqtrid 2784	. . 3 ⊢ (((𝜑 ∧ 𝐾 ∈ (0...(deg‘𝐹))) ∧ 𝑖 ∈ ℕ₀) → (0𝑃𝑖) = 0)
132		oveq12 7414	. . . . . . 7 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 0) → (𝑥 · 𝑦) = (𝑖 · 0))
133	132	oveq1d 7420	. . . . . 6 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 0) → ((𝑥 · 𝑦) mod (𝑁‘𝐾)) = ((𝑖 · 0) mod (𝑁‘𝐾)))
134		ovex 7438	. . . . . 6 ⊢ ((𝑖 · 0) mod (𝑁‘𝐾)) ∈ V
135	133, 55, 134	ovmpoa 7559	. . . . 5 ⊢ ((𝑖 ∈ V ∧ 0 ∈ V) → (𝑖𝑃0) = ((𝑖 · 0) mod (𝑁‘𝐾)))
136	118, 117, 135	mp2an 690	. . . 4 ⊢ (𝑖𝑃0) = ((𝑖 · 0) mod (𝑁‘𝐾))
137	124	mul01d 11409	. . . . . 6 ⊢ (𝑖 ∈ ℕ₀ → (𝑖 · 0) = 0)
138	137	oveq1d 7420	. . . . 5 ⊢ (𝑖 ∈ ℕ₀ → ((𝑖 · 0) mod (𝑁‘𝐾)) = (0 mod (𝑁‘𝐾)))
139	138, 129	sylan9eqr 2794	. . . 4 ⊢ (((𝜑 ∧ 𝐾 ∈ (0...(deg‘𝐹))) ∧ 𝑖 ∈ ℕ₀) → ((𝑖 · 0) mod (𝑁‘𝐾)) = 0)
140	136, 139	eqtrid 2784	. . 3 ⊢ (((𝜑 ∧ 𝐾 ∈ (0...(deg‘𝐹))) ∧ 𝑖 ∈ ℕ₀) → (𝑖𝑃0) = 0)
141		simpr 485	. . 3 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(deg‘𝐹))) → 𝐾 ∈ (0...(deg‘𝐹)))
142	18	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(deg‘𝐹))) → (deg‘𝐹) ∈ ℕ₀)
143	1	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(deg‘𝐹))) → 𝐾 ∈ ℕ₀)
144		fveq2 6888	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑁‘𝑘) = (𝑁‘𝐾))
145	144	oveq1d 7420	. . . . . 6 ⊢ (𝑘 = 𝐾 → ((𝑁‘𝑘) mod (𝑁‘𝐾)) = ((𝑁‘𝐾) mod (𝑁‘𝐾)))
146		ovex 7438	. . . . . 6 ⊢ ((𝑁‘𝐾) mod (𝑁‘𝐾)) ∈ V
147	145, 71, 146	fvmpt 6995	. . . . 5 ⊢ (𝐾 ∈ ℕ₀ → ((𝑘 ∈ ℕ₀ ↦ ((𝑁‘𝑘) mod (𝑁‘𝐾)))‘𝐾) = ((𝑁‘𝐾) mod (𝑁‘𝐾)))
148	143, 147	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(deg‘𝐹))) → ((𝑘 ∈ ℕ₀ ↦ ((𝑁‘𝑘) mod (𝑁‘𝐾)))‘𝐾) = ((𝑁‘𝐾) mod (𝑁‘𝐾)))
149	96	nncnd 12224	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(deg‘𝐹))) → (𝑁‘𝐾) ∈ ℂ)
150	96	nnne0d 12258	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(deg‘𝐹))) → (𝑁‘𝐾) ≠ 0)
151	149, 150	dividd 11984	. . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(deg‘𝐹))) → ((𝑁‘𝐾) / (𝑁‘𝐾)) = 1)
152		1z 12588	. . . . . 6 ⊢ 1 ∈ ℤ
153	151, 152	eqeltrdi 2841	. . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(deg‘𝐹))) → ((𝑁‘𝐾) / (𝑁‘𝐾)) ∈ ℤ)
154	96	nnred 12223	. . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(deg‘𝐹))) → (𝑁‘𝐾) ∈ ℝ)
155		mod0 13837	. . . . . 6 ⊢ (((𝑁‘𝐾) ∈ ℝ ∧ (𝑁‘𝐾) ∈ ℝ⁺) → (((𝑁‘𝐾) mod (𝑁‘𝐾)) = 0 ↔ ((𝑁‘𝐾) / (𝑁‘𝐾)) ∈ ℤ))
156	154, 127, 155	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(deg‘𝐹))) → (((𝑁‘𝐾) mod (𝑁‘𝐾)) = 0 ↔ ((𝑁‘𝐾) / (𝑁‘𝐾)) ∈ ℤ))
157	153, 156	mpbird 256	. . . 4 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(deg‘𝐹))) → ((𝑁‘𝐾) mod (𝑁‘𝐾)) = 0)
158	148, 157	eqtrd 2772	. . 3 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(deg‘𝐹))) → ((𝑘 ∈ ℕ₀ ↦ ((𝑁‘𝑘) mod (𝑁‘𝐾)))‘𝐾) = 0)
159	99, 116, 131, 140, 141, 142, 158	seqz 14012	. 2 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(deg‘𝐹))) → (seq0(𝑃, (𝑘 ∈ ℕ₀ ↦ ((𝑁‘𝑘) mod (𝑁‘𝐾))))‘(deg‘𝐹)) = 0)
160	78, 89, 159	3eqtr3d 2780	1 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(deg‘𝐹))) → (𝑅 mod (𝑁‘𝐾)) = 0)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 {crab 3432 Vcvv 3474 ∖ cdif 3944 {csn 4627 ↦ cmpt 5230 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ∈ cmpo 7407 infcinf 9432 ℂcc 11104 ℝcr 11105 0cc0 11106 1c1 11107 · cmul 11111 < clt 11244 / cdiv 11867 ℕcn 12208 ℕ₀cn0 12468 ℤcz 12554 ℤ_≥cuz 12818 ℚcq 12928 ℝ⁺crp 12970 ...cfz 13480 mod cmo 13830 seqcseq 13962 0_𝑝c0p 25177 Polycply 25689 coeffccoe 25691 degcdgr 25692

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 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 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 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-q 12929 df-rp 12971 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-rlim 15429 df-sum 15629 df-0p 25178 df-ply 25693 df-coe 25695 df-dgr 25696

This theorem is referenced by: elqaalem3 25825

W3C validator