Theorem rami 17186

Description: The defining property of a Ramsey number. (Contributed by Mario Carneiro, 22-Apr-2015.)

Hypotheses

Ref	Expression
rami.c	⊢ 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})
rami.m	⊢ (𝜑 → 𝑀 ∈ ℕ0)
rami.r	⊢ (𝜑 → 𝑅 ∈ 𝑉)
rami.f	⊢ (𝜑 → 𝐹:𝑅⟶ℕ0)
rami.x	⊢ (𝜑 → (𝑀 Ramsey 𝐹) ∈ ℕ0)
rami.s	⊢ (𝜑 → 𝑆 ∈ 𝑊)
rami.l	⊢ (𝜑 → (𝑀 Ramsey 𝐹) ≤ (♯‘𝑆))
rami.g	⊢ (𝜑 → 𝐺:(𝑆𝐶𝑀)⟶𝑅)

Assertion

Ref	Expression
rami	⊢ (𝜑 → ∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑆((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝐺 “ {𝑐})))

Distinct variable groups:   𝑥,𝑐,𝐶   𝐺,𝑐,𝑥   𝜑,𝑐,𝑥   𝑆,𝑐,𝑥   𝐹,𝑐,𝑥   𝑎,𝑏,𝑐,𝑖,𝑥,𝑀   𝑅,𝑐,𝑥   𝑉,𝑐,𝑥

Allowed substitution hints:   𝜑(𝑖, 𝑎, 𝑏)   𝐶(𝑖, 𝑎, 𝑏)   𝑅(𝑖, 𝑎, 𝑏)   𝑆(𝑖, 𝑎, 𝑏)   𝐹(𝑖, 𝑎, 𝑏)   𝐺(𝑖, 𝑎, 𝑏)   𝑉(𝑖, 𝑎, 𝑏)   𝑊(𝑥, 𝑖, 𝑎, 𝑏, 𝑐)

Proof of Theorem rami

Dummy variables 𝑓 𝑛 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnveq 5851	. . . . . 6 ⊢ (𝑓 = 𝐺 → ◡𝑓 = ◡𝐺)
2	1	imaeq1d 6051	. . . . 5 ⊢ (𝑓 = 𝐺 → (◡𝑓 “ {𝑐}) = (◡𝐺 “ {𝑐}))
3	2	sseq2d 3963	. . . 4 ⊢ (𝑓 = 𝐺 → ((𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐}) ↔ (𝑥𝐶𝑀) ⊆ (◡𝐺 “ {𝑐})))
4	3	anbi2d 642	. . 3 ⊢ (𝑓 = 𝐺 → (((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐})) ↔ ((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝐺 “ {𝑐}))))
5	4	2rexbidv 3228	. 2 ⊢ (𝑓 = 𝐺 → (∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑆((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐})) ↔ ∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑆((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝐺 “ {𝑐}))))
6		rami.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑊)
7		rami.x	. . . . 5 ⊢ (𝜑 → (𝑀 Ramsey 𝐹) ∈ ℕ0)
8		rami.m	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℕ0)
9		rami.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ 𝑉)
10		rami.f	. . . . . 6 ⊢ (𝜑 → 𝐹:𝑅⟶ℕ0)
11		rami.c	. . . . . . . 8 ⊢ 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})
12		eqid 2761	. . . . . . . 8 ⊢ {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅 ↑m (𝑠𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐})))} = {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅 ↑m (𝑠𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐})))}
13	11, 12	ramtcl2 17182	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) → ((𝑀 Ramsey 𝐹) ∈ ℕ0 ↔ {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅 ↑m (𝑠𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐})))} ≠ ∅))
14	11, 12	ramtcl 17181	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) → ((𝑀 Ramsey 𝐹) ∈ {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅 ↑m (𝑠𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐})))} ↔ {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅 ↑m (𝑠𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐})))} ≠ ∅))
15	13, 14	bitr4d 285	. . . . . 6 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) → ((𝑀 Ramsey 𝐹) ∈ ℕ0 ↔ (𝑀 Ramsey 𝐹) ∈ {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅 ↑m (𝑠𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐})))}))
16	8, 9, 10, 15	syl3anc 1398	. . . . 5 ⊢ (𝜑 → ((𝑀 Ramsey 𝐹) ∈ ℕ0 ↔ (𝑀 Ramsey 𝐹) ∈ {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅 ↑m (𝑠𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐})))}))
17	7, 16	mpbid 235	. . . 4 ⊢ (𝜑 → (𝑀 Ramsey 𝐹) ∈ {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅 ↑m (𝑠𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐})))})
18		breq1 5106	. . . . . . . 8 ⊢ (𝑛 = (𝑀 Ramsey 𝐹) → (𝑛 ≤ (♯‘𝑠) ↔ (𝑀 Ramsey 𝐹) ≤ (♯‘𝑠)))
19	18	imbi1d 344	. . . . . . 7 ⊢ (𝑛 = (𝑀 Ramsey 𝐹) → ((𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅 ↑m (𝑠𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐}))) ↔ ((𝑀 Ramsey 𝐹) ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅 ↑m (𝑠𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐})))))
20	19	albidv 1953	. . . . . 6 ⊢ (𝑛 = (𝑀 Ramsey 𝐹) → (∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅 ↑m (𝑠𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐}))) ↔ ∀𝑠((𝑀 Ramsey 𝐹) ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅 ↑m (𝑠𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐})))))
21	20	elrab 3645	. . . . 5 ⊢ ((𝑀 Ramsey 𝐹) ∈ {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅 ↑m (𝑠𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐})))} ↔ ((𝑀 Ramsey 𝐹) ∈ ℕ0 ∧ ∀𝑠((𝑀 Ramsey 𝐹) ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅 ↑m (𝑠𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐})))))
22	21	simprbi 503	. . . 4 ⊢ ((𝑀 Ramsey 𝐹) ∈ {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅 ↑m (𝑠𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐})))} → ∀𝑠((𝑀 Ramsey 𝐹) ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅 ↑m (𝑠𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐}))))
23	17, 22	syl 18	. . 3 ⊢ (𝜑 → ∀𝑠((𝑀 Ramsey 𝐹) ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅 ↑m (𝑠𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐}))))
24		rami.l	. . 3 ⊢ (𝜑 → (𝑀 Ramsey 𝐹) ≤ (♯‘𝑆))
25		fveq2 6883	. . . . . 6 ⊢ (𝑠 = 𝑆 → (♯‘𝑠) = (♯‘𝑆))
26	25	breq2d 5115	. . . . 5 ⊢ (𝑠 = 𝑆 → ((𝑀 Ramsey 𝐹) ≤ (♯‘𝑠) ↔ (𝑀 Ramsey 𝐹) ≤ (♯‘𝑆)))
27		oveq1 7425	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑠𝐶𝑀) = (𝑆𝐶𝑀))
28	27	oveq2d 7434	. . . . . 6 ⊢ (𝑠 = 𝑆 → (𝑅 ↑m (𝑠𝐶𝑀)) = (𝑅 ↑m (𝑆𝐶𝑀)))
29		pweq 4571	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → 𝒫 𝑠 = 𝒫 𝑆)
30	29	rexeqdv 3321	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐})) ↔ ∃𝑥 ∈ 𝒫 𝑆((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐}))))
31	30	rexbidv 3187	. . . . . 6 ⊢ (𝑠 = 𝑆 → (∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐})) ↔ ∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑆((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐}))))
32	28, 31	raleqbidv 3335	. . . . 5 ⊢ (𝑠 = 𝑆 → (∀𝑓 ∈ (𝑅 ↑m (𝑠𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐})) ↔ ∀𝑓 ∈ (𝑅 ↑m (𝑆𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑆((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐}))))
33	26, 32	imbi12d 347	. . . 4 ⊢ (𝑠 = 𝑆 → (((𝑀 Ramsey 𝐹) ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅 ↑m (𝑠𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐}))) ↔ ((𝑀 Ramsey 𝐹) ≤ (♯‘𝑆) → ∀𝑓 ∈ (𝑅 ↑m (𝑆𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑆((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐})))))
34	33	spcgv 3551	. . 3 ⊢ (𝑆 ∈ 𝑊 → (∀𝑠((𝑀 Ramsey 𝐹) ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅 ↑m (𝑠𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐}))) → ((𝑀 Ramsey 𝐹) ≤ (♯‘𝑆) → ∀𝑓 ∈ (𝑅 ↑m (𝑆𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑆((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐})))))
35	6, 23, 24, 34	syl3c 67	. 2 ⊢ (𝜑 → ∀𝑓 ∈ (𝑅 ↑m (𝑆𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑆((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐})))
36		rami.g	. . 3 ⊢ (𝜑 → 𝐺:(𝑆𝐶𝑀)⟶𝑅)
37		ovex 7451	. . . 4 ⊢ (𝑆𝐶𝑀) ∈ V
38		elmapg 8852	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑆𝐶𝑀) ∈ V) → (𝐺 ∈ (𝑅 ↑m (𝑆𝐶𝑀)) ↔ 𝐺:(𝑆𝐶𝑀)⟶𝑅))
39	9, 37, 38	sylancl 598	. . 3 ⊢ (𝜑 → (𝐺 ∈ (𝑅 ↑m (𝑆𝐶𝑀)) ↔ 𝐺:(𝑆𝐶𝑀)⟶𝑅))
40	36, 39	mpbird 260	. 2 ⊢ (𝜑 → 𝐺 ∈ (𝑅 ↑m (𝑆𝐶𝑀)))
41	5, 35, 40	rspcdva 3578	1 ⊢ (𝜑 → ∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑆((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝐺 “ {𝑐})))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077  ∃wrex 3087  {crab 3413  Vcvv 3451   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557  {csn 4584   class class class wbr 5103  ^◡ccnv 5650   “ cima 5654  ⟶wf 6533  ‘cfv 6537  (class class class)co 7418   ∈ cmpo 7420   ↑_m cmap 8840   ≤ cle 11337  ℕ₀cn0 12599  ♯chash 14467   Ramsey cram 17170

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-cnex 11249  ax-resscn 11250  ax-1cn 11251  ax-icn 11252  ax-addcl 11253  ax-addrcl 11254  ax-mulcl 11255  ax-mulrcl 11256  ax-mulcom 11257  ax-addass 11258  ax-mulass 11259  ax-distr 11260  ax-i2m1 11261  ax-1ne0 11262  ax-1rid 11263  ax-rnegex 11264  ax-rrecex 11265  ax-cnre 11266  ax-pre-lttri 11267  ax-pre-lttrn 11268  ax-pre-ltadd 11269  ax-pre-mulgt0 11270

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-1st 7999  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-er 8710  df-map 8842  df-en 8967  df-dom 8968  df-sdom 8969  df-sup 9427  df-inf 9428  df-pnf 11338  df-mnf 11339  df-xr 11340  df-ltxr 11341  df-le 11342  df-sub 11536  df-neg 11537  df-nn 12329  df-n0 12600  df-z 12687  df-uz 12959  df-ram 17172

This theorem is used by:  ramlb  17190  ramub1lem2  17198