Theorem equivcau 22906

Description: If the metric 𝐷 is "strongly finer" than 𝐶 (meaning that there is a positive real constant 𝑅 such that 𝐶(𝑥, 𝑦) ≤ 𝑅 · 𝐷(𝑥, 𝑦)), all the 𝐷-Cauchy sequences are also 𝐶-Cauchy. (Using this theorem twice in each direction states that if two metrics are strongly equivalent, then they have the same Cauchy sequences.) (Contributed by Mario Carneiro, 14-Sep-2015.)

Hypotheses

Ref	Expression
equivcau.1	⊢ (𝜑 → 𝐶 ∈ (Met‘𝑋))
equivcau.2	⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
equivcau.3	⊢ (𝜑 → 𝑅 ∈ ℝ+)
equivcau.4	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐶𝑦) ≤ (𝑅 · (𝑥𝐷𝑦)))

Assertion

Ref	Expression
equivcau	⊢ (𝜑 → (Cau‘𝐷) ⊆ (Cau‘𝐶))

Distinct variable groups:   𝑥,𝑦,𝐶   𝑥,𝐷,𝑦   𝜑,𝑥,𝑦   𝑥,𝑅,𝑦   𝑥,𝑋,𝑦

Proof of Theorem equivcau

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

Step	Hyp	Ref	Expression
1		simpr 476	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋 ↑pm ℂ)) ∧ 𝑟 ∈ ℝ+) → 𝑟 ∈ ℝ+)
2		equivcau.3	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ ℝ+)
3	2	ad2antrr 758	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋 ↑pm ℂ)) ∧ 𝑟 ∈ ℝ+) → 𝑅 ∈ ℝ+)
4	1, 3	rpdivcld 11765	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋 ↑pm ℂ)) ∧ 𝑟 ∈ ℝ+) → (𝑟 / 𝑅) ∈ ℝ+)
5		oveq2 6557	. . . . . . . . 9 ⊢ (𝑠 = (𝑟 / 𝑅) → ((𝑓‘𝑘)(ball‘𝐷)𝑠) = ((𝑓‘𝑘)(ball‘𝐷)(𝑟 / 𝑅)))
6	5	feq3d 5945	. . . . . . . 8 ⊢ (𝑠 = (𝑟 / 𝑅) → ((𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐷)𝑠) ↔ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐷)(𝑟 / 𝑅))))
7	6	rexbidv 3034	. . . . . . 7 ⊢ (𝑠 = (𝑟 / 𝑅) → (∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐷)𝑠) ↔ ∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐷)(𝑟 / 𝑅))))
8	7	rspcv 3278	. . . . . 6 ⊢ ((𝑟 / 𝑅) ∈ ℝ+ → (∀𝑠 ∈ ℝ+ ∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐷)𝑠) → ∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐷)(𝑟 / 𝑅))))
9	4, 8	syl 17	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋 ↑pm ℂ)) ∧ 𝑟 ∈ ℝ+) → (∀𝑠 ∈ ℝ+ ∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐷)𝑠) → ∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐷)(𝑟 / 𝑅))))
10		simprr 792	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝑋 ↑pm ℂ)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑘 ∈ ℤ ∧ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐷)(𝑟 / 𝑅)))) → (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐷)(𝑟 / 𝑅)))
11		elpmi 7762	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (𝑋 ↑pm ℂ) → (𝑓:dom 𝑓⟶𝑋 ∧ dom 𝑓 ⊆ ℂ))
12	11	simpld 474	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (𝑋 ↑pm ℂ) → 𝑓:dom 𝑓⟶𝑋)
13	12	ad3antlr 763	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝑋 ↑pm ℂ)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑘 ∈ ℤ ∧ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐷)(𝑟 / 𝑅)))) → 𝑓:dom 𝑓⟶𝑋)
14		resss 5342	. . . . . . . . . . . 12 ⊢ (𝑓 ↾ (ℤ≥‘𝑘)) ⊆ 𝑓
15		dmss 5245	. . . . . . . . . . . 12 ⊢ ((𝑓 ↾ (ℤ≥‘𝑘)) ⊆ 𝑓 → dom (𝑓 ↾ (ℤ≥‘𝑘)) ⊆ dom 𝑓)
16	14, 15	ax-mp 5	. . . . . . . . . . 11 ⊢ dom (𝑓 ↾ (ℤ≥‘𝑘)) ⊆ dom 𝑓
17		uzid 11578	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℤ → 𝑘 ∈ (ℤ≥‘𝑘))
18	17	ad2antrl 760	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝑋 ↑pm ℂ)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑘 ∈ ℤ ∧ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐷)(𝑟 / 𝑅)))) → 𝑘 ∈ (ℤ≥‘𝑘))
19		fdm 5964	. . . . . . . . . . . . 13 ⊢ ((𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐷)(𝑟 / 𝑅)) → dom (𝑓 ↾ (ℤ≥‘𝑘)) = (ℤ≥‘𝑘))
20	19	ad2antll 761	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝑋 ↑pm ℂ)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑘 ∈ ℤ ∧ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐷)(𝑟 / 𝑅)))) → dom (𝑓 ↾ (ℤ≥‘𝑘)) = (ℤ≥‘𝑘))
21	18, 20	eleqtrrd 2691	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝑋 ↑pm ℂ)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑘 ∈ ℤ ∧ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐷)(𝑟 / 𝑅)))) → 𝑘 ∈ dom (𝑓 ↾ (ℤ≥‘𝑘)))
22	16, 21	sseldi 3566	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝑋 ↑pm ℂ)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑘 ∈ ℤ ∧ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐷)(𝑟 / 𝑅)))) → 𝑘 ∈ dom 𝑓)
23	13, 22	ffvelrnd 6268	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝑋 ↑pm ℂ)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑘 ∈ ℤ ∧ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐷)(𝑟 / 𝑅)))) → (𝑓‘𝑘) ∈ 𝑋)
24		eqid 2610	. . . . . . . . . . . . 13 ⊢ (MetOpen‘𝐶) = (MetOpen‘𝐶)
25		eqid 2610	. . . . . . . . . . . . 13 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷)
26		equivcau.1	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ (Met‘𝑋))
27		equivcau.2	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
28		equivcau.4	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐶𝑦) ≤ (𝑅 · (𝑥𝐷𝑦)))
29	24, 25, 26, 27, 2, 28	metss2lem 22126	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+)) → (𝑥(ball‘𝐷)(𝑟 / 𝑅)) ⊆ (𝑥(ball‘𝐶)𝑟))
30	29	expr 641	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑟 ∈ ℝ+ → (𝑥(ball‘𝐷)(𝑟 / 𝑅)) ⊆ (𝑥(ball‘𝐶)𝑟)))
31	30	ralrimiva 2949	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 (𝑟 ∈ ℝ+ → (𝑥(ball‘𝐷)(𝑟 / 𝑅)) ⊆ (𝑥(ball‘𝐶)𝑟)))
32	31	ad3antrrr 762	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝑋 ↑pm ℂ)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑘 ∈ ℤ ∧ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐷)(𝑟 / 𝑅)))) → ∀𝑥 ∈ 𝑋 (𝑟 ∈ ℝ+ → (𝑥(ball‘𝐷)(𝑟 / 𝑅)) ⊆ (𝑥(ball‘𝐶)𝑟)))
33		simplr 788	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝑋 ↑pm ℂ)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑘 ∈ ℤ ∧ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐷)(𝑟 / 𝑅)))) → 𝑟 ∈ ℝ+)
34		oveq1 6556	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑓‘𝑘) → (𝑥(ball‘𝐷)(𝑟 / 𝑅)) = ((𝑓‘𝑘)(ball‘𝐷)(𝑟 / 𝑅)))
35		oveq1 6556	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑓‘𝑘) → (𝑥(ball‘𝐶)𝑟) = ((𝑓‘𝑘)(ball‘𝐶)𝑟))
36	34, 35	sseq12d 3597	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑓‘𝑘) → ((𝑥(ball‘𝐷)(𝑟 / 𝑅)) ⊆ (𝑥(ball‘𝐶)𝑟) ↔ ((𝑓‘𝑘)(ball‘𝐷)(𝑟 / 𝑅)) ⊆ ((𝑓‘𝑘)(ball‘𝐶)𝑟)))
37	36	imbi2d 329	. . . . . . . . . 10 ⊢ (𝑥 = (𝑓‘𝑘) → ((𝑟 ∈ ℝ+ → (𝑥(ball‘𝐷)(𝑟 / 𝑅)) ⊆ (𝑥(ball‘𝐶)𝑟)) ↔ (𝑟 ∈ ℝ+ → ((𝑓‘𝑘)(ball‘𝐷)(𝑟 / 𝑅)) ⊆ ((𝑓‘𝑘)(ball‘𝐶)𝑟))))
38	37	rspcv 3278	. . . . . . . . 9 ⊢ ((𝑓‘𝑘) ∈ 𝑋 → (∀𝑥 ∈ 𝑋 (𝑟 ∈ ℝ+ → (𝑥(ball‘𝐷)(𝑟 / 𝑅)) ⊆ (𝑥(ball‘𝐶)𝑟)) → (𝑟 ∈ ℝ+ → ((𝑓‘𝑘)(ball‘𝐷)(𝑟 / 𝑅)) ⊆ ((𝑓‘𝑘)(ball‘𝐶)𝑟))))
39	23, 32, 33, 38	syl3c 64	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝑋 ↑pm ℂ)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑘 ∈ ℤ ∧ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐷)(𝑟 / 𝑅)))) → ((𝑓‘𝑘)(ball‘𝐷)(𝑟 / 𝑅)) ⊆ ((𝑓‘𝑘)(ball‘𝐶)𝑟))
40	10, 39	fssd 5970	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝑋 ↑pm ℂ)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑘 ∈ ℤ ∧ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐷)(𝑟 / 𝑅)))) → (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐶)𝑟))
41	40	expr 641	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑓 ∈ (𝑋 ↑pm ℂ)) ∧ 𝑟 ∈ ℝ+) ∧ 𝑘 ∈ ℤ) → ((𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐷)(𝑟 / 𝑅)) → (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐶)𝑟)))
42	41	reximdva 3000	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋 ↑pm ℂ)) ∧ 𝑟 ∈ ℝ+) → (∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐷)(𝑟 / 𝑅)) → ∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐶)𝑟)))
43	9, 42	syld 46	. . . 4 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋 ↑pm ℂ)) ∧ 𝑟 ∈ ℝ+) → (∀𝑠 ∈ ℝ+ ∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐷)𝑠) → ∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐶)𝑟)))
44	43	ralrimdva 2952	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋 ↑pm ℂ)) → (∀𝑠 ∈ ℝ+ ∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐷)𝑠) → ∀𝑟 ∈ ℝ+ ∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐶)𝑟)))
45	44	ss2rabdv 3646	. 2 ⊢ (𝜑 → {𝑓 ∈ (𝑋 ↑pm ℂ) ∣ ∀𝑠 ∈ ℝ+ ∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐷)𝑠)} ⊆ {𝑓 ∈ (𝑋 ↑pm ℂ) ∣ ∀𝑟 ∈ ℝ+ ∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐶)𝑟)})
46		metxmet 21949	. . 3 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
47		caufval 22881	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (Cau‘𝐷) = {𝑓 ∈ (𝑋 ↑pm ℂ) ∣ ∀𝑠 ∈ ℝ+ ∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐷)𝑠)})
48	27, 46, 47	3syl 18	. 2 ⊢ (𝜑 → (Cau‘𝐷) = {𝑓 ∈ (𝑋 ↑pm ℂ) ∣ ∀𝑠 ∈ ℝ+ ∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐷)𝑠)})
49		metxmet 21949	. . 3 ⊢ (𝐶 ∈ (Met‘𝑋) → 𝐶 ∈ (∞Met‘𝑋))
50		caufval 22881	. . 3 ⊢ (𝐶 ∈ (∞Met‘𝑋) → (Cau‘𝐶) = {𝑓 ∈ (𝑋 ↑pm ℂ) ∣ ∀𝑟 ∈ ℝ+ ∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐶)𝑟)})
51	26, 49, 50	3syl 18	. 2 ⊢ (𝜑 → (Cau‘𝐶) = {𝑓 ∈ (𝑋 ↑pm ℂ) ∣ ∀𝑟 ∈ ℝ+ ∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐶)𝑟)})
52	45, 48, 51	3sstr4d 3611	1 ⊢ (𝜑 → (Cau‘𝐷) ⊆ (Cau‘𝐶))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {crab 2900   ⊆ wss 3540   class class class wbr 4583  dom cdm 5038   ↾ cres 5040  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑_pm cpm 7745  ℂcc 9813   · cmul 9820   ≤ cle 9954   / cdiv 10563  ℤcz 11254  ℤ_≥cuz 11563  ℝ⁺crp 11708  ∞Metcxmt 19552  Metcme 19553  ballcbl 19554  MetOpencmopn 19557  Caucca 22859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-z 11255  df-uz 11564  df-rp 11709  df-xadd 11823  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-cau 22862

This theorem is referenced by: (None)