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Theorem blssps 24748

Description: Any point 𝑃 in a ball 𝐵 can be centered in another ball that is a subset of 𝐵. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 24-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)

**Assertion**
Ref	Expression
blssps	⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐵 ∈ ran (ball‘𝐷) ∧ 𝑃 ∈ 𝐵) → ∃𝑥 ∈ ℝ⁺ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐵)

Distinct variable groups: 𝑥,𝐵 𝑥,𝐷 𝑥,𝑃 𝑥,𝑋

Proof of Theorem blssps Dummy variables 𝑟 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		blrnps 24732	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝐵 ∈ ran (ball‘𝐷) ↔ ∃𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ^* 𝐵 = (𝑦(ball‘𝐷)𝑟)))
2		elblps 24711	. . . . . . 7 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) → (𝑃 ∈ (𝑦(ball‘𝐷)𝑟) ↔ (𝑃 ∈ 𝑋 ∧ (𝑦𝐷𝑃) < 𝑟)))
3		simpl1 1210	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) → 𝐷 ∈ (PsMet‘𝑋))
4		simpl2 1211	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) → 𝑦 ∈ 𝑋)
5		simpr 490	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) → 𝑃 ∈ 𝑋)
6		psmetcl 24631	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑃 ∈ 𝑋) → (𝑦𝐷𝑃) ∈ ℝ^*)
7	3, 4, 5, 6	syl3anc 1398	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^) ∧ 𝑃 ∈ 𝑋) → (𝑦𝐷𝑃) ∈ ℝ^)
8		simpl3 1212	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^) ∧ 𝑃 ∈ 𝑋) → 𝑟 ∈ ℝ^)
9		qbtwnxr 13333	. . . . . . . . . . 11 ⊢ (((𝑦𝐷𝑃) ∈ ℝ^* ∧ 𝑟 ∈ ℝ^* ∧ (𝑦𝐷𝑃) < 𝑟) → ∃𝑧 ∈ ℚ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))
10	9	3expia 1139	. . . . . . . . . 10 ⊢ (((𝑦𝐷𝑃) ∈ ℝ^* ∧ 𝑟 ∈ ℝ^*) → ((𝑦𝐷𝑃) < 𝑟 → ∃𝑧 ∈ ℚ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟)))
11	7, 8, 10	syl2anc 596	. . . . . . . . 9 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) → ((𝑦𝐷𝑃) < 𝑟 → ∃𝑧 ∈ ℚ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟)))
12		qre 13083	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℚ → 𝑧 ∈ ℝ)
13		simpll1 1231	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → 𝐷 ∈ (PsMet‘𝑋))
14		simplr 781	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → 𝑃 ∈ 𝑋)
15		simpll2 1232	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → 𝑦 ∈ 𝑋)
16		psmetsym 24634	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑃𝐷𝑦) = (𝑦𝐷𝑃))
17	13, 14, 15, 16	syl3anc 1398	. . . . . . . . . . . . . . 15 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑃𝐷𝑦) = (𝑦𝐷𝑃))
18		simprrl 793	. . . . . . . . . . . . . . 15 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑦𝐷𝑃) < 𝑧)
19	17, 18	eqbrtrd 5126	. . . . . . . . . . . . . 14 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑃𝐷𝑦) < 𝑧)
20		simprl 783	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → 𝑧 ∈ ℝ)
21		psmetcl 24631	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑃𝐷𝑦) ∈ ℝ^*)
22	13, 14, 15, 21	syl3anc 1398	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑃𝐷𝑦) ∈ ℝ^)
23		rexr 11358	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ ℝ → 𝑧 ∈ ℝ^*)
24	23	ad2antrl 741	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → 𝑧 ∈ ℝ^)
25	22, 24, 19	xrltled 13282	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑃𝐷𝑦) ≤ 𝑧)
26		psmetlecl 24639	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑃 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ (𝑃𝐷𝑦) ≤ 𝑧)) → (𝑃𝐷𝑦) ∈ ℝ)
27	13, 14, 15, 20, 25, 26	syl122anc 1406	. . . . . . . . . . . . . . 15 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑃𝐷𝑦) ∈ ℝ)
28		difrp 13163	. . . . . . . . . . . . . . 15 ⊢ (((𝑃𝐷𝑦) ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((𝑃𝐷𝑦) < 𝑧 ↔ (𝑧 − (𝑃𝐷𝑦)) ∈ ℝ⁺))
29	27, 20, 28	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → ((𝑃𝐷𝑦) < 𝑧 ↔ (𝑧 − (𝑃𝐷𝑦)) ∈ ℝ⁺))
30	19, 29	mpbid 235	. . . . . . . . . . . . 13 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑧 − (𝑃𝐷𝑦)) ∈ ℝ⁺)
31	20, 27	resubcld 11747	. . . . . . . . . . . . . . 15 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑧 − (𝑃𝐷𝑦)) ∈ ℝ)
32	22	xrleidd 13284	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑃𝐷𝑦) ≤ (𝑃𝐷𝑦))
33	20	recnd 11340	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → 𝑧 ∈ ℂ)
34	27	recnd 11340	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑃𝐷𝑦) ∈ ℂ)
35	33, 34	nncand 11677	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑧 − (𝑧 − (𝑃𝐷𝑦))) = (𝑃𝐷𝑦))
36	32, 35	breqtrrd 5132	. . . . . . . . . . . . . . 15 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑃𝐷𝑦) ≤ (𝑧 − (𝑧 − (𝑃𝐷𝑦))))
37		blss2ps 24727	. . . . . . . . . . . . . . 15 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ ((𝑧 − (𝑃𝐷𝑦)) ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (𝑃𝐷𝑦) ≤ (𝑧 − (𝑧 − (𝑃𝐷𝑦))))) → (𝑃(ball‘𝐷)(𝑧 − (𝑃𝐷𝑦))) ⊆ (𝑦(ball‘𝐷)𝑧))
38	13, 14, 15, 31, 20, 36, 37	syl33anc 1412	. . . . . . . . . . . . . 14 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑃(ball‘𝐷)(𝑧 − (𝑃𝐷𝑦))) ⊆ (𝑦(ball‘𝐷)𝑧))
39		simpll3 1233	. . . . . . . . . . . . . . 15 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → 𝑟 ∈ ℝ^)
40		simprrr 794	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → 𝑧 < 𝑟)
41	24, 39, 40	xrltled 13282	. . . . . . . . . . . . . . 15 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → 𝑧 ≤ 𝑟)
42		ssblps 24746	. . . . . . . . . . . . . . 15 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑧 ∈ ℝ^* ∧ 𝑟 ∈ ℝ^*) ∧ 𝑧 ≤ 𝑟) → (𝑦(ball‘𝐷)𝑧) ⊆ (𝑦(ball‘𝐷)𝑟))
43	13, 15, 24, 39, 41, 42	syl221anc 1408	. . . . . . . . . . . . . 14 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑦(ball‘𝐷)𝑧) ⊆ (𝑦(ball‘𝐷)𝑟))
44	38, 43	sstrd 3918	. . . . . . . . . . . . 13 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑃(ball‘𝐷)(𝑧 − (𝑃𝐷𝑦))) ⊆ (𝑦(ball‘𝐷)𝑟))
45		oveq2 7428	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑧 − (𝑃𝐷𝑦)) → (𝑃(ball‘𝐷)𝑥) = (𝑃(ball‘𝐷)(𝑧 − (𝑃𝐷𝑦))))
46	45	sseq1d 3939	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑧 − (𝑃𝐷𝑦)) → ((𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟) ↔ (𝑃(ball‘𝐷)(𝑧 − (𝑃𝐷𝑦))) ⊆ (𝑦(ball‘𝐷)𝑟)))
47	46	rspcev 3576	. . . . . . . . . . . . 13 ⊢ (((𝑧 − (𝑃𝐷𝑦)) ∈ ℝ⁺ ∧ (𝑃(ball‘𝐷)(𝑧 − (𝑃𝐷𝑦))) ⊆ (𝑦(ball‘𝐷)𝑟)) → ∃𝑥 ∈ ℝ⁺ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟))
48	30, 44, 47	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → ∃𝑥 ∈ ℝ⁺ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟))
49	48	expr 462	. . . . . . . . . . 11 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ 𝑧 ∈ ℝ) → (((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟) → ∃𝑥 ∈ ℝ⁺ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟)))
50	12, 49	sylan2 605	. . . . . . . . . 10 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ 𝑧 ∈ ℚ) → (((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟) → ∃𝑥 ∈ ℝ⁺ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟)))
51	50	rexlimdva 3163	. . . . . . . . 9 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) → (∃𝑧 ∈ ℚ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟) → ∃𝑥 ∈ ℝ⁺ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟)))
52	11, 51	syld 48	. . . . . . . 8 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) → ((𝑦𝐷𝑃) < 𝑟 → ∃𝑥 ∈ ℝ⁺ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟)))
53	52	expimpd 459	. . . . . . 7 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) → ((𝑃 ∈ 𝑋 ∧ (𝑦𝐷𝑃) < 𝑟) → ∃𝑥 ∈ ℝ⁺ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟)))
54	2, 53	sylbid 243	. . . . . 6 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) → (𝑃 ∈ (𝑦(ball‘𝐷)𝑟) → ∃𝑥 ∈ ℝ⁺ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟)))
55		eleq2 2849	. . . . . . 7 ⊢ (𝐵 = (𝑦(ball‘𝐷)𝑟) → (𝑃 ∈ 𝐵 ↔ 𝑃 ∈ (𝑦(ball‘𝐷)𝑟)))
56		sseq2 3934	. . . . . . . 8 ⊢ (𝐵 = (𝑦(ball‘𝐷)𝑟) → ((𝑃(ball‘𝐷)𝑥) ⊆ 𝐵 ↔ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟)))
57	56	rexbidv 3186	. . . . . . 7 ⊢ (𝐵 = (𝑦(ball‘𝐷)𝑟) → (∃𝑥 ∈ ℝ⁺ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐵 ↔ ∃𝑥 ∈ ℝ⁺ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟)))
58	55, 57	imbi12d 347	. . . . . 6 ⊢ (𝐵 = (𝑦(ball‘𝐷)𝑟) → ((𝑃 ∈ 𝐵 → ∃𝑥 ∈ ℝ⁺ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐵) ↔ (𝑃 ∈ (𝑦(ball‘𝐷)𝑟) → ∃𝑥 ∈ ℝ⁺ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟))))
59	54, 58	syl5ibrcom 250	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) → (𝐵 = (𝑦(ball‘𝐷)𝑟) → (𝑃 ∈ 𝐵 → ∃𝑥 ∈ ℝ⁺ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐵)))
60	59	3expib 1140	. . . 4 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ((𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) → (𝐵 = (𝑦(ball‘𝐷)𝑟) → (𝑃 ∈ 𝐵 → ∃𝑥 ∈ ℝ⁺ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐵))))
61	60	rexlimdvv 3218	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (∃𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ^* 𝐵 = (𝑦(ball‘𝐷)𝑟) → (𝑃 ∈ 𝐵 → ∃𝑥 ∈ ℝ⁺ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐵)))
62	1, 61	sylbid 243	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝐵 ∈ ran (ball‘𝐷) → (𝑃 ∈ 𝐵 → ∃𝑥 ∈ ℝ⁺ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐵)))
63	62	3imp 1128	1 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐵 ∈ ran (ball‘𝐷) ∧ 𝑃 ∈ 𝐵) → ∃𝑥 ∈ ℝ⁺ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐵)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ∃wrex 3086 ⊆ wss 3895 class class class wbr 5102 ran crn 5651 ‘cfv 6538 (class class class)co 7420 ℝcr 11202 ℝ^*cxr 11345 < clt 11346 ≤ cle 11347 − cmin 11544 ℚcq 13078 ℝ⁺crp 13123 PsMetcpsmet 21667 ballcbl 21670

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-cnex 11259 ax-resscn 11260 ax-1cn 11261 ax-icn 11262 ax-addcl 11263 ax-addrcl 11264 ax-mulcl 11265 ax-mulrcl 11266 ax-mulcom 11267 ax-addass 11268 ax-mulass 11269 ax-distr 11270 ax-i2m1 11271 ax-1ne0 11272 ax-1rid 11273 ax-rnegex 11274 ax-rrecex 11275 ax-cnre 11276 ax-pre-lttri 11277 ax-pre-lttrn 11278 ax-pre-ltadd 11279 ax-pre-mulgt0 11280 ax-pre-sup 11281

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8302 df-wrecs 8333 df-recs 8382 df-rdg 8421 df-er 8720 df-map 8852 df-en 8977 df-dom 8978 df-sdom 8979 df-sup 9437 df-inf 9438 df-pnf 11348 df-mnf 11349 df-xr 11350 df-ltxr 11351 df-le 11352 df-sub 11546 df-neg 11547 df-div 11977 df-nn 12339 df-2 12408 df-n0 12610 df-z 12697 df-uz 12969 df-q 13079 df-rp 13124 df-xneg 13244 df-xadd 13245 df-xmul 13246 df-psmet 21675 df-bl 21678

This theorem is used by: blssexps 24750

W3C validator