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Theorem equivcau 24824

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 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋 ↑_pm ℂ)) ∧ 𝑟 ∈ ℝ⁺) → 𝑟 ∈ ℝ⁺)
2		equivcau.3	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
3	2	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋 ↑_pm ℂ)) ∧ 𝑟 ∈ ℝ⁺) → 𝑅 ∈ ℝ⁺)
4	1, 3	rpdivcld 13035	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋 ↑_pm ℂ)) ∧ 𝑟 ∈ ℝ⁺) → (𝑟 / 𝑅) ∈ ℝ⁺)
5		oveq2 7419	. . . . . . . . 9 ⊢ (𝑠 = (𝑟 / 𝑅) → ((𝑓‘𝑘)(ball‘𝐷)𝑠) = ((𝑓‘𝑘)(ball‘𝐷)(𝑟 / 𝑅)))
6	5	feq3d 6704	. . . . . . . 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 3178	. . . . . . 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 3608	. . . . . 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 771	. . . . . . . 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 8842	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (𝑋 ↑_pm ℂ) → (𝑓:dom 𝑓⟶𝑋 ∧ dom 𝑓 ⊆ ℂ))
12	11	simpld 495	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (𝑋 ↑_pm ℂ) → 𝑓:dom 𝑓⟶𝑋)
13	12	ad3antlr 729	. . . . . . . . . 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 6006	. . . . . . . . . . . 12 ⊢ (𝑓 ↾ (ℤ_≥‘𝑘)) ⊆ 𝑓
15		dmss 5902	. . . . . . . . . . . 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 12839	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℤ → 𝑘 ∈ (ℤ_≥‘𝑘))
18	17	ad2antrl 726	. . . . . . . . . . . 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 6726	. . . . . . . . . . . . 13 ⊢ ((𝑓 ↾ (ℤ_≥‘𝑘)):(ℤ_≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐷)(𝑟 / 𝑅)) → dom (𝑓 ↾ (ℤ_≥‘𝑘)) = (ℤ_≥‘𝑘))
20	19	ad2antll 727	. . . . . . . . . . . 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 2836	. . . . . . . . . . 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 3980	. . . . . . . . . 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 7087	. . . . . . . . 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 2732	. . . . . . . . . . . . 13 ⊢ (MetOpen‘𝐶) = (MetOpen‘𝐶)
25		eqid 2732	. . . . . . . . . . . . 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 24027	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → (𝑥(ball‘𝐷)(𝑟 / 𝑅)) ⊆ (𝑥(ball‘𝐶)𝑟))
30	29	expr 457	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑟 ∈ ℝ⁺ → (𝑥(ball‘𝐷)(𝑟 / 𝑅)) ⊆ (𝑥(ball‘𝐶)𝑟)))
31	30	ralrimiva 3146	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 (𝑟 ∈ ℝ⁺ → (𝑥(ball‘𝐷)(𝑟 / 𝑅)) ⊆ (𝑥(ball‘𝐶)𝑟)))
32	31	ad3antrrr 728	. . . . . . . . 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 767	. . . . . . . . 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 7418	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑓‘𝑘) → (𝑥(ball‘𝐷)(𝑟 / 𝑅)) = ((𝑓‘𝑘)(ball‘𝐷)(𝑟 / 𝑅)))
35		oveq1 7418	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑓‘𝑘) → (𝑥(ball‘𝐶)𝑟) = ((𝑓‘𝑘)(ball‘𝐶)𝑟))
36	34, 35	sseq12d 4015	. . . . . . . . . . 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 340	. . . . . . . . . 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 3608	. . . . . . . . 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 6735	. . . . . . 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 457	. . . . . 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 3168	. . . . 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 3154	. . 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 4073	. 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 23847	. . 3 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
47		caufval 24799	. . 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 23847	. . 3 ⊢ (𝐶 ∈ (Met‘𝑋) → 𝐶 ∈ (∞Met‘𝑋))
50		caufval 24799	. . 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 4029	1 ⊢ (𝜑 → (Cau‘𝐷) ⊆ (Cau‘𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ∃wrex 3070 {crab 3432 ⊆ wss 3948 class class class wbr 5148 dom cdm 5676 ↾ cres 5678 ⟶wf 6539 ‘cfv 6543 (class class class)co 7411 ↑_pm cpm 8823 ℂcc 11110 · cmul 11117 ≤ cle 11251 / cdiv 11873 ℤcz 12560 ℤ_≥cuz 12824 ℝ⁺crp 12976 ∞Metcxmet 20935 Metcmet 20936 ballcbl 20937 MetOpencmopn 20940 Cauccau 24777

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-1st 7977 df-2nd 7978 df-er 8705 df-map 8824 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-z 12561 df-uz 12825 df-rp 12977 df-xadd 13095 df-psmet 20942 df-xmet 20943 df-met 20944 df-bl 20945 df-cau 24780

This theorem is referenced by: (None)

W3C validator