Theorem rami 16948

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 5874	. . . . . 6 ⊢ (𝑓 = 𝐺 → ◡𝑓 = ◡𝐺)
2	1	imaeq1d 6059	. . . . 5 ⊢ (𝑓 = 𝐺 → (◡𝑓 “ {𝑐}) = (◡𝐺 “ {𝑐}))
3	2	sseq2d 4015	. . . 4 ⊢ (𝑓 = 𝐺 → ((𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐}) ↔ (𝑥𝐶𝑀) ⊆ (◡𝐺 “ {𝑐})))
4	3	anbi2d 630	. . 3 ⊢ (𝑓 = 𝐺 → (((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐})) ↔ ((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝐺 “ {𝑐}))))
5	4	2rexbidv 3220	. 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 2733	. . . . . . . 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 16944	. . . . . . 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 16943	. . . . . . 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 282	. . . . . 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 1372	. . . . 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 231	. . . 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 5152	. . . . . . . 8 ⊢ (𝑛 = (𝑀 Ramsey 𝐹) → (𝑛 ≤ (♯‘𝑠) ↔ (𝑀 Ramsey 𝐹) ≤ (♯‘𝑠)))
19	18	imbi1d 342	. . . . . . 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 1924	. . . . . 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 3684	. . . . 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 498	. . . 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 17	. . 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 6892	. . . . . 6 ⊢ (𝑠 = 𝑆 → (♯‘𝑠) = (♯‘𝑆))
26	25	breq2d 5161	. . . . 5 ⊢ (𝑠 = 𝑆 → ((𝑀 Ramsey 𝐹) ≤ (♯‘𝑠) ↔ (𝑀 Ramsey 𝐹) ≤ (♯‘𝑆)))
27		oveq1 7416	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑠𝐶𝑀) = (𝑆𝐶𝑀))
28	27	oveq2d 7425	. . . . . 6 ⊢ (𝑠 = 𝑆 → (𝑅 ↑m (𝑠𝐶𝑀)) = (𝑅 ↑m (𝑆𝐶𝑀)))
29		pweq 4617	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → 𝒫 𝑠 = 𝒫 𝑆)
30	29	rexeqdv 3327	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐})) ↔ ∃𝑥 ∈ 𝒫 𝑆((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐}))))
31	30	rexbidv 3179	. . . . . 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 3343	. . . . 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 345	. . . 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 3587	. . 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 66	. 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 7442	. . . 4 ⊢ (𝑆𝐶𝑀) ∈ V
38		elmapg 8833	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑆𝐶𝑀) ∈ V) → (𝐺 ∈ (𝑅 ↑m (𝑆𝐶𝑀)) ↔ 𝐺:(𝑆𝐶𝑀)⟶𝑅))
39	9, 37, 38	sylancl 587	. . 3 ⊢ (𝜑 → (𝐺 ∈ (𝑅 ↑m (𝑆𝐶𝑀)) ↔ 𝐺:(𝑆𝐶𝑀)⟶𝑅))
40	36, 39	mpbird 257	. 2 ⊢ (𝜑 → 𝐺 ∈ (𝑅 ↑m (𝑆𝐶𝑀)))
41	5, 35, 40	rspcdva 3614	1 ⊢ (𝜑 → ∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑆((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝐺 “ {𝑐})))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088  ∀wal 1540   = wceq 1542   ∈ wcel 2107   ≠ wne 2941  ∀wral 3062  ∃wrex 3071  {crab 3433  Vcvv 3475   ⊆ wss 3949  ∅c0 4323  𝒫 cpw 4603  {csn 4629   class class class wbr 5149  ^◡ccnv 5676   “ cima 5680  ⟶wf 6540  ‘cfv 6544  (class class class)co 7409   ∈ cmpo 7411   ↑_m cmap 8820   ≤ cle 11249  ℕ₀cn0 12472  ♯chash 14290   Ramsey cram 16932

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-sup 9437  df-inf 9438  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-n0 12473  df-z 12559  df-uz 12823  df-ram 16934

This theorem is referenced by:  ramlb  16952  ramub1lem2  16960