Theorem vdwnn 16924

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 766	. . 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 768	. . 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 7363	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑤 → (𝑚 · 𝑑) = (𝑤 · 𝑑))
4	3	oveq2d 7372	. . . . . . . . . 10 ⊢ (𝑚 = 𝑤 → (𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑤 · 𝑑)))
5	4	eleq1d 2819	. . . . . . . . 9 ⊢ (𝑚 = 𝑤 → ((𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑢}) ↔ (𝑎 + (𝑤 · 𝑑)) ∈ (◡𝐹 “ {𝑢})))
6	5	cbvralvw 3212	. . . . . . . 8 ⊢ (∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑢}) ↔ ∀𝑤 ∈ (0...(𝑘 − 1))(𝑎 + (𝑤 · 𝑑)) ∈ (◡𝐹 “ {𝑢}))
7		oveq1 7363	. . . . . . . . . 10 ⊢ (𝑎 = 𝑦 → (𝑎 + (𝑤 · 𝑑)) = (𝑦 + (𝑤 · 𝑑)))
8	7	eleq1d 2819	. . . . . . . . 9 ⊢ (𝑎 = 𝑦 → ((𝑎 + (𝑤 · 𝑑)) ∈ (◡𝐹 “ {𝑢}) ↔ (𝑦 + (𝑤 · 𝑑)) ∈ (◡𝐹 “ {𝑢})))
9	8	ralbidv 3157	. . . . . . . 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 283	. . . . . . 7 ⊢ (𝑎 = 𝑦 → (∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑢}) ↔ ∀𝑤 ∈ (0...(𝑘 − 1))(𝑦 + (𝑤 · 𝑑)) ∈ (◡𝐹 “ {𝑢})))
11		oveq2 7364	. . . . . . . . . 10 ⊢ (𝑑 = 𝑧 → (𝑤 · 𝑑) = (𝑤 · 𝑧))
12	11	oveq2d 7372	. . . . . . . . 9 ⊢ (𝑑 = 𝑧 → (𝑦 + (𝑤 · 𝑑)) = (𝑦 + (𝑤 · 𝑧)))
13	12	eleq1d 2819	. . . . . . . 8 ⊢ (𝑑 = 𝑧 → ((𝑦 + (𝑤 · 𝑑)) ∈ (◡𝐹 “ {𝑢}) ↔ (𝑦 + (𝑤 · 𝑧)) ∈ (◡𝐹 “ {𝑢})))
14	13	ralbidv 3157	. . . . . . 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 3217	. . . . . 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 7363	. . . . . . . . 9 ⊢ (𝑘 = 𝑥 → (𝑘 − 1) = (𝑥 − 1))
17	16	oveq2d 7372	. . . . . . . 8 ⊢ (𝑘 = 𝑥 → (0...(𝑘 − 1)) = (0...(𝑥 − 1)))
18	17	raleqdv 3294	. . . . . . 7 ⊢ (𝑘 = 𝑥 → (∀𝑤 ∈ (0...(𝑘 − 1))(𝑦 + (𝑤 · 𝑧)) ∈ (◡𝐹 “ {𝑢}) ↔ ∀𝑤 ∈ (0...(𝑥 − 1))(𝑦 + (𝑤 · 𝑧)) ∈ (◡𝐹 “ {𝑢})))
19	18	2rexbidv 3199	. . . . . 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 283	. . . . 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 318	. . . 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 3407	. . 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 484	. . . . 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 4588	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑢 → {𝑐} = {𝑢})
25	24	imaeq2d 6017	. . . . . . . . . 10 ⊢ (𝑐 = 𝑢 → (◡𝐹 “ {𝑐}) = (◡𝐹 “ {𝑢}))
26	25	eleq2d 2820	. . . . . . . . 9 ⊢ (𝑐 = 𝑢 → ((𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑐}) ↔ (𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑢})))
27	26	ralbidv 3157	. . . . . . . 8 ⊢ (𝑐 = 𝑢 → (∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑐}) ↔ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑢})))
28	27	2rexbidv 3199	. . . . . . 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 3157	. . . . . 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 3213	. . . . 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 328	. . . 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 4339	. . . . . . 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 3086	. . . . . . 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 275	. . . . . 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 3080	. . . . 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 3060	. . . . 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 275	. . . 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 234	. . 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 16923	. 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 401	. 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 231	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 395   ∈ wcel 2113   ≠ wne 2930  ∀wral 3049  ∃wrex 3058  {crab 3397  ∅c0 4283  {csn 4578  ^◡ccnv 5621   “ cima 5625  ⟶wf 6486  (class class class)co 7356  Fincfn 8881  0cc0 11024  1c1 11025   + caddc 11027   · cmul 11029   − cmin 11362  ℕcn 12143  ...cfz 13421

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-cnex 11080  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-addrcl 11085  ax-mulcl 11086  ax-mulrcl 11087  ax-mulcom 11088  ax-addass 11089  ax-mulass 11090  ax-distr 11091  ax-i2m1 11092  ax-1ne0 11093  ax-1rid 11094  ax-rnegex 11095  ax-rrecex 11096  ax-cnre 11097  ax-pre-lttri 11098  ax-pre-lttrn 11099  ax-pre-ltadd 11100  ax-pre-mulgt0 11101  ax-pre-sup 11102

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-oadd 8399  df-er 8633  df-map 8763  df-pm 8764  df-en 8882  df-dom 8883  df-sdom 8884  df-fin 8885  df-sup 9343  df-inf 9344  df-dju 9811  df-card 9849  df-pnf 11166  df-mnf 11167  df-xr 11168  df-ltxr 11169  df-le 11170  df-sub 11364  df-neg 11365  df-nn 12144  df-2 12206  df-n0 12400  df-xnn0 12473  df-z 12487  df-uz 12750  df-rp 12904  df-fz 13422  df-fl 13710  df-hash 14252  df-vdwap 16894  df-vdwmc 16895  df-vdwpc 16896

This theorem is referenced by: (None)