Theorem vdwnn 17053

Description: Van der Waerden's theorem, infinitary version. For any finite coloring 𝐹 of the positive integers, there is a color 𝑐 that contains arbitrarily long arithmetic progressions. (Contributed by Mario Carneiro, 13-Sep-2014.)

Assertion

Ref	Expression
vdwnn	⊢ ((𝑅 ∈ Fin ∧ 𝐹:ℕ⟶𝑅) → ∃𝑐 ∈ 𝑅 ∀𝑘 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑐}))

Distinct variable groups:   𝑎,𝑐,𝑑,𝑘,𝑚,𝐹   𝑅,𝑐

Allowed substitution hints:   𝑅(𝑘,𝑚,𝑎,𝑑)

Proof of Theorem vdwnn

Dummy variables 𝑢 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll 778	. . 3 ⊢ (((𝑅 ∈ Fin ∧ 𝐹:ℕ⟶𝑅) ∧ ¬ ∃𝑐 ∈ 𝑅 ∀𝑘 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑐})) → 𝑅 ∈ Fin)
2		simplr 780	. . 3 ⊢ (((𝑅 ∈ Fin ∧ 𝐹:ℕ⟶𝑅) ∧ ¬ ∃𝑐 ∈ 𝑅 ∀𝑘 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑐})) → 𝐹:ℕ⟶𝑅)
3		oveq1 7417	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑤 → (𝑚 · 𝑑) = (𝑤 · 𝑑))
4	3	oveq2d 7426	. . . . . . . . . 10 ⊢ (𝑚 = 𝑤 → (𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑤 · 𝑑)))
5	4	eleq1d 2848	. . . . . . . . 9 ⊢ (𝑚 = 𝑤 → ((𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑢}) ↔ (𝑎 + (𝑤 · 𝑑)) ∈ (◡𝐹 “ {𝑢})))
6	5	cbvralvw 3243	. . . . . . . 8 ⊢ (∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑢}) ↔ ∀𝑤 ∈ (0...(𝑘 − 1))(𝑎 + (𝑤 · 𝑑)) ∈ (◡𝐹 “ {𝑢}))
7		oveq1 7417	. . . . . . . . . 10 ⊢ (𝑎 = 𝑦 → (𝑎 + (𝑤 · 𝑑)) = (𝑦 + (𝑤 · 𝑑)))
8	7	eleq1d 2848	. . . . . . . . 9 ⊢ (𝑎 = 𝑦 → ((𝑎 + (𝑤 · 𝑑)) ∈ (◡𝐹 “ {𝑢}) ↔ (𝑦 + (𝑤 · 𝑑)) ∈ (◡𝐹 “ {𝑢})))
9	8	ralbidv 3188	. . . . . . . 8 ⊢ (𝑎 = 𝑦 → (∀𝑤 ∈ (0...(𝑘 − 1))(𝑎 + (𝑤 · 𝑑)) ∈ (◡𝐹 “ {𝑢}) ↔ ∀𝑤 ∈ (0...(𝑘 − 1))(𝑦 + (𝑤 · 𝑑)) ∈ (◡𝐹 “ {𝑢})))
10	6, 9	bitrid 286	. . . . . . 7 ⊢ (𝑎 = 𝑦 → (∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑢}) ↔ ∀𝑤 ∈ (0...(𝑘 − 1))(𝑦 + (𝑤 · 𝑑)) ∈ (◡𝐹 “ {𝑢})))
11		oveq2 7418	. . . . . . . . . 10 ⊢ (𝑑 = 𝑧 → (𝑤 · 𝑑) = (𝑤 · 𝑧))
12	11	oveq2d 7426	. . . . . . . . 9 ⊢ (𝑑 = 𝑧 → (𝑦 + (𝑤 · 𝑑)) = (𝑦 + (𝑤 · 𝑧)))
13	12	eleq1d 2848	. . . . . . . 8 ⊢ (𝑑 = 𝑧 → ((𝑦 + (𝑤 · 𝑑)) ∈ (◡𝐹 “ {𝑢}) ↔ (𝑦 + (𝑤 · 𝑧)) ∈ (◡𝐹 “ {𝑢})))
14	13	ralbidv 3188	. . . . . . 7 ⊢ (𝑑 = 𝑧 → (∀𝑤 ∈ (0...(𝑘 − 1))(𝑦 + (𝑤 · 𝑑)) ∈ (◡𝐹 “ {𝑢}) ↔ ∀𝑤 ∈ (0...(𝑘 − 1))(𝑦 + (𝑤 · 𝑧)) ∈ (◡𝐹 “ {𝑢})))
15	10, 14	cbvrex2vw 3248	. . . . . 6 ⊢ (∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑢}) ↔ ∃𝑦 ∈ ℕ ∃𝑧 ∈ ℕ ∀𝑤 ∈ (0...(𝑘 − 1))(𝑦 + (𝑤 · 𝑧)) ∈ (◡𝐹 “ {𝑢}))
16		oveq1 7417	. . . . . . . . 9 ⊢ (𝑘 = 𝑥 → (𝑘 − 1) = (𝑥 − 1))
17	16	oveq2d 7426	. . . . . . . 8 ⊢ (𝑘 = 𝑥 → (0...(𝑘 − 1)) = (0...(𝑥 − 1)))
18	17	raleqdv 3323	. . . . . . 7 ⊢ (𝑘 = 𝑥 → (∀𝑤 ∈ (0...(𝑘 − 1))(𝑦 + (𝑤 · 𝑧)) ∈ (◡𝐹 “ {𝑢}) ↔ ∀𝑤 ∈ (0...(𝑥 − 1))(𝑦 + (𝑤 · 𝑧)) ∈ (◡𝐹 “ {𝑢})))
19	18	2rexbidv 3230	. . . . . 6 ⊢ (𝑘 = 𝑥 → (∃𝑦 ∈ ℕ ∃𝑧 ∈ ℕ ∀𝑤 ∈ (0...(𝑘 − 1))(𝑦 + (𝑤 · 𝑧)) ∈ (◡𝐹 “ {𝑢}) ↔ ∃𝑦 ∈ ℕ ∃𝑧 ∈ ℕ ∀𝑤 ∈ (0...(𝑥 − 1))(𝑦 + (𝑤 · 𝑧)) ∈ (◡𝐹 “ {𝑢})))
20	15, 19	bitrid 286	. . . . 5 ⊢ (𝑘 = 𝑥 → (∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑢}) ↔ ∃𝑦 ∈ ℕ ∃𝑧 ∈ ℕ ∀𝑤 ∈ (0...(𝑥 − 1))(𝑦 + (𝑤 · 𝑧)) ∈ (◡𝐹 “ {𝑢})))
21	20	notbid 321	. . . 4 ⊢ (𝑘 = 𝑥 → (¬ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑢}) ↔ ¬ ∃𝑦 ∈ ℕ ∃𝑧 ∈ ℕ ∀𝑤 ∈ (0...(𝑥 − 1))(𝑦 + (𝑤 · 𝑧)) ∈ (◡𝐹 “ {𝑢})))
22	21	cbvrabv 3426	. . 3 ⊢ {𝑘 ∈ ℕ ∣ ¬ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑢})} = {𝑥 ∈ ℕ ∣ ¬ ∃𝑦 ∈ ℕ ∃𝑧 ∈ ℕ ∀𝑤 ∈ (0...(𝑥 − 1))(𝑦 + (𝑤 · 𝑧)) ∈ (◡𝐹 “ {𝑢})}
23		simpr 489	. . . . 5 ⊢ (((𝑅 ∈ Fin ∧ 𝐹:ℕ⟶𝑅) ∧ ¬ ∃𝑐 ∈ 𝑅 ∀𝑘 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑐})) → ¬ ∃𝑐 ∈ 𝑅 ∀𝑘 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑐}))
24		sneq 4599	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑢 → {𝑐} = {𝑢})
25	24	imaeq2d 6062	. . . . . . . . . 10 ⊢ (𝑐 = 𝑢 → (◡𝐹 “ {𝑐}) = (◡𝐹 “ {𝑢}))
26	25	eleq2d 2849	. . . . . . . . 9 ⊢ (𝑐 = 𝑢 → ((𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑐}) ↔ (𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑢})))
27	26	ralbidv 3188	. . . . . . . 8 ⊢ (𝑐 = 𝑢 → (∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑐}) ↔ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑢})))
28	27	2rexbidv 3230	. . . . . . 7 ⊢ (𝑐 = 𝑢 → (∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑐}) ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑢})))
29	28	ralbidv 3188	. . . . . 6 ⊢ (𝑐 = 𝑢 → (∀𝑘 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑐}) ↔ ∀𝑘 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑢})))
30	29	cbvrexvw 3244	. . . . 5 ⊢ (∃𝑐 ∈ 𝑅 ∀𝑘 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑐}) ↔ ∃𝑢 ∈ 𝑅 ∀𝑘 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑢}))
31	23, 30	sylnib 331	. . . 4 ⊢ (((𝑅 ∈ Fin ∧ 𝐹:ℕ⟶𝑅) ∧ ¬ ∃𝑐 ∈ 𝑅 ∀𝑘 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑐})) → ¬ ∃𝑢 ∈ 𝑅 ∀𝑘 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑢}))
32		rabn0 4346	. . . . . . 7 ⊢ ({𝑘 ∈ ℕ ∣ ¬ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑢})} ≠ ∅ ↔ ∃𝑘 ∈ ℕ ¬ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑢}))
33		rexnal 3117	. . . . . . 7 ⊢ (∃𝑘 ∈ ℕ ¬ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑢}) ↔ ¬ ∀𝑘 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑢}))
34	32, 33	bitri 278	. . . . . 6 ⊢ ({𝑘 ∈ ℕ ∣ ¬ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑢})} ≠ ∅ ↔ ¬ ∀𝑘 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑢}))
35	34	ralbii 3111	. . . . 5 ⊢ (∀𝑢 ∈ 𝑅 {𝑘 ∈ ℕ ∣ ¬ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑢})} ≠ ∅ ↔ ∀𝑢 ∈ 𝑅 ¬ ∀𝑘 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑢}))
36		ralnex 3091	. . . . 5 ⊢ (∀𝑢 ∈ 𝑅 ¬ ∀𝑘 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑢}) ↔ ¬ ∃𝑢 ∈ 𝑅 ∀𝑘 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑢}))
37	35, 36	bitri 278	. . . 4 ⊢ (∀𝑢 ∈ 𝑅 {𝑘 ∈ ℕ ∣ ¬ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑢})} ≠ ∅ ↔ ¬ ∃𝑢 ∈ 𝑅 ∀𝑘 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑢}))
38	31, 37	sylibr 237	. . 3 ⊢ (((𝑅 ∈ Fin ∧ 𝐹:ℕ⟶𝑅) ∧ ¬ ∃𝑐 ∈ 𝑅 ∀𝑘 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑐})) → ∀𝑢 ∈ 𝑅 {𝑘 ∈ ℕ ∣ ¬ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑢})} ≠ ∅)
39	1, 2, 22, 38	vdwnnlem3 17052	. 2 ⊢ ¬ ((𝑅 ∈ Fin ∧ 𝐹:ℕ⟶𝑅) ∧ ¬ ∃𝑐 ∈ 𝑅 ∀𝑘 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑐}))
40		iman 406	. 2 ⊢ (((𝑅 ∈ Fin ∧ 𝐹:ℕ⟶𝑅) → ∃𝑐 ∈ 𝑅 ∀𝑘 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑐})) ↔ ¬ ((𝑅 ∈ Fin ∧ 𝐹:ℕ⟶𝑅) ∧ ¬ ∃𝑐 ∈ 𝑅 ∀𝑘 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑐})))
41	39, 40	mpbir 234	1 ⊢ ((𝑅 ∈ Fin ∧ 𝐹:ℕ⟶𝑅) → ∃𝑐 ∈ 𝑅 ∀𝑘 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑐}))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  ∃wrex 3089  {crab 3416  ∅c0 4286  {csn 4589  ^◡ccnv 5660   “ cima 5664  ⟶wf 6532  (class class class)co 7410  Fincfn 8939  0cc0 11095  1c1 11096   + caddc 11098   · cmul 11100   − cmin 11436  ℕcn 12228  ...cfz 13530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172  ax-pre-sup 11173

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-oadd 8453  df-er 8690  df-map 8822  df-pm 8823  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9398  df-inf 9399  df-dju 9883  df-card 9921  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-nn 12229  df-2 12298  df-n0 12500  df-xnn0 12573  df-z 12587  df-uz 12858  df-rp 13012  df-fz 13531  df-fl 13821  df-hash 14363  df-vdwap 17023  df-vdwmc 17024  df-vdwpc 17025

This theorem is referenced by: (None)