Theorem vdwnn 16910

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 7356	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑤 → (𝑚 · 𝑑) = (𝑤 · 𝑑))
4	3	oveq2d 7365	. . . . . . . . . 10 ⊢ (𝑚 = 𝑤 → (𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑤 · 𝑑)))
5	4	eleq1d 2813	. . . . . . . . 9 ⊢ (𝑚 = 𝑤 → ((𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑢}) ↔ (𝑎 + (𝑤 · 𝑑)) ∈ (◡𝐹 “ {𝑢})))
6	5	cbvralvw 3207	. . . . . . . 8 ⊢ (∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑢}) ↔ ∀𝑤 ∈ (0...(𝑘 − 1))(𝑎 + (𝑤 · 𝑑)) ∈ (◡𝐹 “ {𝑢}))
7		oveq1 7356	. . . . . . . . . 10 ⊢ (𝑎 = 𝑦 → (𝑎 + (𝑤 · 𝑑)) = (𝑦 + (𝑤 · 𝑑)))
8	7	eleq1d 2813	. . . . . . . . 9 ⊢ (𝑎 = 𝑦 → ((𝑎 + (𝑤 · 𝑑)) ∈ (◡𝐹 “ {𝑢}) ↔ (𝑦 + (𝑤 · 𝑑)) ∈ (◡𝐹 “ {𝑢})))
9	8	ralbidv 3152	. . . . . . . 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 7357	. . . . . . . . . 10 ⊢ (𝑑 = 𝑧 → (𝑤 · 𝑑) = (𝑤 · 𝑧))
12	11	oveq2d 7365	. . . . . . . . 9 ⊢ (𝑑 = 𝑧 → (𝑦 + (𝑤 · 𝑑)) = (𝑦 + (𝑤 · 𝑧)))
13	12	eleq1d 2813	. . . . . . . 8 ⊢ (𝑑 = 𝑧 → ((𝑦 + (𝑤 · 𝑑)) ∈ (◡𝐹 “ {𝑢}) ↔ (𝑦 + (𝑤 · 𝑧)) ∈ (◡𝐹 “ {𝑢})))
14	13	ralbidv 3152	. . . . . . 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 3212	. . . . . 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 7356	. . . . . . . . 9 ⊢ (𝑘 = 𝑥 → (𝑘 − 1) = (𝑥 − 1))
17	16	oveq2d 7365	. . . . . . . 8 ⊢ (𝑘 = 𝑥 → (0...(𝑘 − 1)) = (0...(𝑥 − 1)))
18	17	raleqdv 3289	. . . . . . 7 ⊢ (𝑘 = 𝑥 → (∀𝑤 ∈ (0...(𝑘 − 1))(𝑦 + (𝑤 · 𝑧)) ∈ (◡𝐹 “ {𝑢}) ↔ ∀𝑤 ∈ (0...(𝑥 − 1))(𝑦 + (𝑤 · 𝑧)) ∈ (◡𝐹 “ {𝑢})))
19	18	2rexbidv 3194	. . . . . 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 3405	. . 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 4587	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑢 → {𝑐} = {𝑢})
25	24	imaeq2d 6011	. . . . . . . . . 10 ⊢ (𝑐 = 𝑢 → (◡𝐹 “ {𝑐}) = (◡𝐹 “ {𝑢}))
26	25	eleq2d 2814	. . . . . . . . 9 ⊢ (𝑐 = 𝑢 → ((𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑐}) ↔ (𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑢})))
27	26	ralbidv 3152	. . . . . . . 8 ⊢ (𝑐 = 𝑢 → (∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑐}) ↔ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑢})))
28	27	2rexbidv 3194	. . . . . . 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 3152	. . . . . 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 3208	. . . . 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 4340	. . . . . . 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 3081	. . . . . . 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 3075	. . . . 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 3055	. . . . 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 16909	. 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 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  {crab 3394  ∅c0 4284  {csn 4577  ^◡ccnv 5618   “ cima 5622  ⟶wf 6478  (class class class)co 7349  Fincfn 8872  0cc0 11009  1c1 11010   + caddc 11012   · cmul 11014   − cmin 11347  ℕcn 12128  ...cfz 13410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086  ax-pre-sup 11087

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-oadd 8392  df-er 8625  df-map 8755  df-pm 8756  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-sup 9332  df-inf 9333  df-dju 9797  df-card 9835  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-2 12191  df-n0 12385  df-xnn0 12458  df-z 12472  df-uz 12736  df-rp 12894  df-fz 13411  df-fl 13696  df-hash 14238  df-vdwap 16880  df-vdwmc 16881  df-vdwpc 16882

This theorem is referenced by: (None)