Theorem rami 17017

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 5880	. . . . . 6 ⊢ (𝑓 = 𝐺 → ◡𝑓 = ◡𝐺)
2	1	imaeq1d 6068	. . . . 5 ⊢ (𝑓 = 𝐺 → (◡𝑓 “ {𝑐}) = (◡𝐺 “ {𝑐}))
3	2	sseq2d 4012	. . . 4 ⊢ (𝑓 = 𝐺 → ((𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐}) ↔ (𝑥𝐶𝑀) ⊆ (◡𝐺 “ {𝑐})))
4	3	anbi2d 628	. . 3 ⊢ (𝑓 = 𝐺 → (((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐})) ↔ ((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝐺 “ {𝑐}))))
5	4	2rexbidv 3210	. 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 2726	. . . . . . . 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 17013	. . . . . . 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 17012	. . . . . . 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 281	. . . . . 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 1368	. . . . 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 5156	. . . . . . . 8 ⊢ (𝑛 = (𝑀 Ramsey 𝐹) → (𝑛 ≤ (♯‘𝑠) ↔ (𝑀 Ramsey 𝐹) ≤ (♯‘𝑠)))
19	18	imbi1d 340	. . . . . . 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 1916	. . . . . 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 3681	. . . . 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 495	. . . 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 6901	. . . . . 6 ⊢ (𝑠 = 𝑆 → (♯‘𝑠) = (♯‘𝑆))
26	25	breq2d 5165	. . . . 5 ⊢ (𝑠 = 𝑆 → ((𝑀 Ramsey 𝐹) ≤ (♯‘𝑠) ↔ (𝑀 Ramsey 𝐹) ≤ (♯‘𝑆)))
27		oveq1 7431	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑠𝐶𝑀) = (𝑆𝐶𝑀))
28	27	oveq2d 7440	. . . . . 6 ⊢ (𝑠 = 𝑆 → (𝑅 ↑m (𝑠𝐶𝑀)) = (𝑅 ↑m (𝑆𝐶𝑀)))
29		pweq 4621	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → 𝒫 𝑠 = 𝒫 𝑆)
30	29	rexeqdv 3316	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐})) ↔ ∃𝑥 ∈ 𝒫 𝑆((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐}))))
31	30	rexbidv 3169	. . . . . 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 3330	. . . . 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 343	. . . 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 3582	. . 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 7457	. . . 4 ⊢ (𝑆𝐶𝑀) ∈ V
38		elmapg 8868	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑆𝐶𝑀) ∈ V) → (𝐺 ∈ (𝑅 ↑m (𝑆𝐶𝑀)) ↔ 𝐺:(𝑆𝐶𝑀)⟶𝑅))
39	9, 37, 38	sylancl 584	. . 3 ⊢ (𝜑 → (𝐺 ∈ (𝑅 ↑m (𝑆𝐶𝑀)) ↔ 𝐺:(𝑆𝐶𝑀)⟶𝑅))
40	36, 39	mpbird 256	. 2 ⊢ (𝜑 → 𝐺 ∈ (𝑅 ↑m (𝑆𝐶𝑀)))
41	5, 35, 40	rspcdva 3609	1 ⊢ (𝜑 → ∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑆((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝐺 “ {𝑐})))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084  ∀wal 1532   = wceq 1534   ∈ wcel 2099   ≠ wne 2930  ∀wral 3051  ∃wrex 3060  {crab 3419  Vcvv 3462   ⊆ wss 3947  ∅c0 4325  𝒫 cpw 4607  {csn 4633   class class class wbr 5153  ^◡ccnv 5681   “ cima 5685  ⟶wf 6550  ‘cfv 6554  (class class class)co 7424   ∈ cmpo 7426   ↑_m cmap 8855   ≤ cle 11299  ℕ₀cn0 12524  ♯chash 14347   Ramsey cram 17001

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-er 8734  df-map 8857  df-en 8975  df-dom 8976  df-sdom 8977  df-sup 9485  df-inf 9486  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-nn 12265  df-n0 12525  df-z 12611  df-uz 12875  df-ram 17003

This theorem is referenced by:  ramlb  17021  ramub1lem2  17029