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Theorem equivcau 24817

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 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋 ↑_pm ℂ)) ∧ 𝑟 ∈ ℝ⁺) → 𝑟 ∈ ℝ⁺)
2		equivcau.3	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
3	2	ad2antrr 725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋 ↑_pm ℂ)) ∧ 𝑟 ∈ ℝ⁺) → 𝑅 ∈ ℝ⁺)
4	1, 3	rpdivcld 13033	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋 ↑_pm ℂ)) ∧ 𝑟 ∈ ℝ⁺) → (𝑟 / 𝑅) ∈ ℝ⁺)
5		oveq2 7417	. . . . . . . . 9 ⊢ (𝑠 = (𝑟 / 𝑅) → ((𝑓‘𝑘)(ball‘𝐷)𝑠) = ((𝑓‘𝑘)(ball‘𝐷)(𝑟 / 𝑅)))
6	5	feq3d 6705	. . . . . . . 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 3179	. . . . . . 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 3609	. . . . . 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 772	. . . . . . . 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 8840	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (𝑋 ↑_pm ℂ) → (𝑓:dom 𝑓⟶𝑋 ∧ dom 𝑓 ⊆ ℂ))
12	11	simpld 496	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (𝑋 ↑_pm ℂ) → 𝑓:dom 𝑓⟶𝑋)
13	12	ad3antlr 730	. . . . . . . . . 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 6007	. . . . . . . . . . . 12 ⊢ (𝑓 ↾ (ℤ_≥‘𝑘)) ⊆ 𝑓
15		dmss 5903	. . . . . . . . . . . 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 12837	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℤ → 𝑘 ∈ (ℤ_≥‘𝑘))
18	17	ad2antrl 727	. . . . . . . . . . . 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 6727	. . . . . . . . . . . . 13 ⊢ ((𝑓 ↾ (ℤ_≥‘𝑘)):(ℤ_≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐷)(𝑟 / 𝑅)) → dom (𝑓 ↾ (ℤ_≥‘𝑘)) = (ℤ_≥‘𝑘))
20	19	ad2antll 728	. . . . . . . . . . . 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 2837	. . . . . . . . . . 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	sselid 3981	. . . . . . . . . 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	ffvelcdmd 7088	. . . . . . . . 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 2733	. . . . . . . . . . . . 13 ⊢ (MetOpen‘𝐶) = (MetOpen‘𝐶)
25		eqid 2733	. . . . . . . . . . . . 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 24020	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → (𝑥(ball‘𝐷)(𝑟 / 𝑅)) ⊆ (𝑥(ball‘𝐶)𝑟))
30	29	expr 458	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑟 ∈ ℝ⁺ → (𝑥(ball‘𝐷)(𝑟 / 𝑅)) ⊆ (𝑥(ball‘𝐶)𝑟)))
31	30	ralrimiva 3147	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 (𝑟 ∈ ℝ⁺ → (𝑥(ball‘𝐷)(𝑟 / 𝑅)) ⊆ (𝑥(ball‘𝐶)𝑟)))
32	31	ad3antrrr 729	. . . . . . . . 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 768	. . . . . . . . 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 7416	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑓‘𝑘) → (𝑥(ball‘𝐷)(𝑟 / 𝑅)) = ((𝑓‘𝑘)(ball‘𝐷)(𝑟 / 𝑅)))
35		oveq1 7416	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑓‘𝑘) → (𝑥(ball‘𝐶)𝑟) = ((𝑓‘𝑘)(ball‘𝐶)𝑟))
36	34, 35	sseq12d 4016	. . . . . . . . . . 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 341	. . . . . . . . . 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 3609	. . . . . . . . 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 66	. . . . . . . 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 6736	. . . . . . 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 458	. . . . . 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 3169	. . . . 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 47	. . . 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 3155	. . 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 4074	. 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 23840	. . 3 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
47		caufval 24792	. . 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 23840	. . 3 ⊢ (𝐶 ∈ (Met‘𝑋) → 𝐶 ∈ (∞Met‘𝑋))
50		caufval 24792	. . 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 4030	1 ⊢ (𝜑 → (Cau‘𝐷) ⊆ (Cau‘𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 {crab 3433 ⊆ wss 3949 class class class wbr 5149 dom cdm 5677 ↾ cres 5679 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 ↑_pm cpm 8821 ℂcc 11108 · cmul 11115 ≤ cle 11249 / cdiv 11871 ℤcz 12558 ℤ_≥cuz 12822 ℝ⁺crp 12974 ∞Metcxmet 20929 Metcmet 20930 ballcbl 20931 MetOpencmopn 20934 Cauccau 24770

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-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-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-id 5575 df-po 5589 df-so 5590 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-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-1st 7975 df-2nd 7976 df-er 8703 df-map 8822 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-z 12559 df-uz 12823 df-rp 12975 df-xadd 13093 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-cau 24773

This theorem is referenced by: (None)

W3C validator