Theorem vdwnn 16881

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 765	. . 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 767	. . 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 7369	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑤 → (𝑚 · 𝑑) = (𝑤 · 𝑑))
4	3	oveq2d 7378	. . . . . . . . . 10 ⊢ (𝑚 = 𝑤 → (𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑤 · 𝑑)))
5	4	eleq1d 2817	. . . . . . . . 9 ⊢ (𝑚 = 𝑤 → ((𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑢}) ↔ (𝑎 + (𝑤 · 𝑑)) ∈ (◡𝐹 “ {𝑢})))
6	5	cbvralvw 3223	. . . . . . . 8 ⊢ (∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑢}) ↔ ∀𝑤 ∈ (0...(𝑘 − 1))(𝑎 + (𝑤 · 𝑑)) ∈ (◡𝐹 “ {𝑢}))
7		oveq1 7369	. . . . . . . . . 10 ⊢ (𝑎 = 𝑦 → (𝑎 + (𝑤 · 𝑑)) = (𝑦 + (𝑤 · 𝑑)))
8	7	eleq1d 2817	. . . . . . . . 9 ⊢ (𝑎 = 𝑦 → ((𝑎 + (𝑤 · 𝑑)) ∈ (◡𝐹 “ {𝑢}) ↔ (𝑦 + (𝑤 · 𝑑)) ∈ (◡𝐹 “ {𝑢})))
9	8	ralbidv 3170	. . . . . . . 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 282	. . . . . . 7 ⊢ (𝑎 = 𝑦 → (∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑢}) ↔ ∀𝑤 ∈ (0...(𝑘 − 1))(𝑦 + (𝑤 · 𝑑)) ∈ (◡𝐹 “ {𝑢})))
11		oveq2 7370	. . . . . . . . . 10 ⊢ (𝑑 = 𝑧 → (𝑤 · 𝑑) = (𝑤 · 𝑧))
12	11	oveq2d 7378	. . . . . . . . 9 ⊢ (𝑑 = 𝑧 → (𝑦 + (𝑤 · 𝑑)) = (𝑦 + (𝑤 · 𝑧)))
13	12	eleq1d 2817	. . . . . . . 8 ⊢ (𝑑 = 𝑧 → ((𝑦 + (𝑤 · 𝑑)) ∈ (◡𝐹 “ {𝑢}) ↔ (𝑦 + (𝑤 · 𝑧)) ∈ (◡𝐹 “ {𝑢})))
14	13	ralbidv 3170	. . . . . . 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 3226	. . . . . 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 7369	. . . . . . . . 9 ⊢ (𝑘 = 𝑥 → (𝑘 − 1) = (𝑥 − 1))
17	16	oveq2d 7378	. . . . . . . 8 ⊢ (𝑘 = 𝑥 → (0...(𝑘 − 1)) = (0...(𝑥 − 1)))
18	17	raleqdv 3311	. . . . . . 7 ⊢ (𝑘 = 𝑥 → (∀𝑤 ∈ (0...(𝑘 − 1))(𝑦 + (𝑤 · 𝑧)) ∈ (◡𝐹 “ {𝑢}) ↔ ∀𝑤 ∈ (0...(𝑥 − 1))(𝑦 + (𝑤 · 𝑧)) ∈ (◡𝐹 “ {𝑢})))
19	18	2rexbidv 3209	. . . . . 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 282	. . . . 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 317	. . . 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 3415	. . 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 485	. . . . 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 4601	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑢 → {𝑐} = {𝑢})
25	24	imaeq2d 6018	. . . . . . . . . 10 ⊢ (𝑐 = 𝑢 → (◡𝐹 “ {𝑐}) = (◡𝐹 “ {𝑢}))
26	25	eleq2d 2818	. . . . . . . . 9 ⊢ (𝑐 = 𝑢 → ((𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑐}) ↔ (𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑢})))
27	26	ralbidv 3170	. . . . . . . 8 ⊢ (𝑐 = 𝑢 → (∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑐}) ↔ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑢})))
28	27	2rexbidv 3209	. . . . . . 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 3170	. . . . . 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 3224	. . . . 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 327	. . . 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 4350	. . . . . . 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 3099	. . . . . . 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 274	. . . . . 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 3092	. . . . 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 3071	. . . . 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 274	. . . 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 233	. . 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 16880	. 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 402	. 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 230	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 396   ∈ wcel 2106   ≠ wne 2939  ∀wral 3060  ∃wrex 3069  {crab 3405  ∅c0 4287  {csn 4591  ^◡ccnv 5637   “ cima 5641  ⟶wf 6497  (class class class)co 7362  Fincfn 8890  0cc0 11060  1c1 11061   + caddc 11063   · cmul 11065   − cmin 11394  ℕcn 12162  ...cfz 13434

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 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11116  ax-resscn 11117  ax-1cn 11118  ax-icn 11119  ax-addcl 11120  ax-addrcl 11121  ax-mulcl 11122  ax-mulrcl 11123  ax-mulcom 11124  ax-addass 11125  ax-mulass 11126  ax-distr 11127  ax-i2m1 11128  ax-1ne0 11129  ax-1rid 11130  ax-rnegex 11131  ax-rrecex 11132  ax-cnre 11133  ax-pre-lttri 11134  ax-pre-lttrn 11135  ax-pre-ltadd 11136  ax-pre-mulgt0 11137  ax-pre-sup 11138

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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-oadd 8421  df-er 8655  df-map 8774  df-pm 8775  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-sup 9387  df-inf 9388  df-dju 9846  df-card 9884  df-pnf 11200  df-mnf 11201  df-xr 11202  df-ltxr 11203  df-le 11204  df-sub 11396  df-neg 11397  df-nn 12163  df-2 12225  df-n0 12423  df-xnn0 12495  df-z 12509  df-uz 12773  df-rp 12925  df-fz 13435  df-fl 13707  df-hash 14241  df-vdwap 16851  df-vdwmc 16852  df-vdwpc 16853

This theorem is referenced by: (None)