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Theorem blss 15181

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.)

**Assertion**
Ref	Expression
blss	⊢ ((𝐷 ∈ (∞Met‘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 blss Dummy variables 𝑟 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		blrn 15165	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐵 ∈ ran (ball‘𝐷) ↔ ∃𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ^* 𝐵 = (𝑦(ball‘𝐷)𝑟)))
2		elbl 15144	. . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) → (𝑃 ∈ (𝑦(ball‘𝐷)𝑟) ↔ (𝑃 ∈ 𝑋 ∧ (𝑦𝐷𝑃) < 𝑟)))
3		simpl1 1026	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) → 𝐷 ∈ (∞Met‘𝑋))
4		simpl2 1027	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) → 𝑦 ∈ 𝑋)
5		simpr 110	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) → 𝑃 ∈ 𝑋)
6		xmetcl 15105	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑃 ∈ 𝑋) → (𝑦𝐷𝑃) ∈ ℝ^*)
7	3, 4, 5, 6	syl3anc 1273	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^) ∧ 𝑃 ∈ 𝑋) → (𝑦𝐷𝑃) ∈ ℝ^)
8		simpl3 1028	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^) ∧ 𝑃 ∈ 𝑋) → 𝑟 ∈ ℝ^)
9		qbtwnxr 10523	. . . . . . . . . . 11 ⊢ (((𝑦𝐷𝑃) ∈ ℝ^* ∧ 𝑟 ∈ ℝ^* ∧ (𝑦𝐷𝑃) < 𝑟) → ∃𝑧 ∈ ℚ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))
10	9	3expia 1231	. . . . . . . . . 10 ⊢ (((𝑦𝐷𝑃) ∈ ℝ^* ∧ 𝑟 ∈ ℝ^*) → ((𝑦𝐷𝑃) < 𝑟 → ∃𝑧 ∈ ℚ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟)))
11	7, 8, 10	syl2anc 411	. . . . . . . . 9 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) → ((𝑦𝐷𝑃) < 𝑟 → ∃𝑧 ∈ ℚ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟)))
12		qre 9864	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℚ → 𝑧 ∈ ℝ)
13		simpll1 1062	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → 𝐷 ∈ (∞Met‘𝑋))
14		simplr 529	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → 𝑃 ∈ 𝑋)
15		simpll2 1063	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → 𝑦 ∈ 𝑋)
16		xmetsym 15121	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑃𝐷𝑦) = (𝑦𝐷𝑃))
17	13, 14, 15, 16	syl3anc 1273	. . . . . . . . . . . . . . 15 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑃𝐷𝑦) = (𝑦𝐷𝑃))
18		simprrl 541	. . . . . . . . . . . . . . 15 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑦𝐷𝑃) < 𝑧)
19	17, 18	eqbrtrd 4111	. . . . . . . . . . . . . 14 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑃𝐷𝑦) < 𝑧)
20		simprl 531	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → 𝑧 ∈ ℝ)
21		xmetcl 15105	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑃𝐷𝑦) ∈ ℝ^*)
22	13, 14, 15, 21	syl3anc 1273	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑃𝐷𝑦) ∈ ℝ^)
23		rexr 8230	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ ℝ → 𝑧 ∈ ℝ^*)
24	23	ad2antrl 490	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → 𝑧 ∈ ℝ^)
25	22, 24, 19	xrltled 10039	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑃𝐷𝑦) ≤ 𝑧)
26		xmetlecl 15120	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑃 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ (𝑃𝐷𝑦) ≤ 𝑧)) → (𝑃𝐷𝑦) ∈ ℝ)
27	13, 14, 15, 20, 25, 26	syl122anc 1282	. . . . . . . . . . . . . . 15 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑃𝐷𝑦) ∈ ℝ)
28		difrp 9932	. . . . . . . . . . . . . . 15 ⊢ (((𝑃𝐷𝑦) ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((𝑃𝐷𝑦) < 𝑧 ↔ (𝑧 − (𝑃𝐷𝑦)) ∈ ℝ⁺))
29	27, 20, 28	syl2anc 411	. . . . . . . . . . . . . 14 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → ((𝑃𝐷𝑦) < 𝑧 ↔ (𝑧 − (𝑃𝐷𝑦)) ∈ ℝ⁺))
30	19, 29	mpbid 147	. . . . . . . . . . . . 13 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑧 − (𝑃𝐷𝑦)) ∈ ℝ⁺)
31	20, 27	resubcld 8565	. . . . . . . . . . . . . . 15 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑧 − (𝑃𝐷𝑦)) ∈ ℝ)
32	22	xrleidd 10041	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑃𝐷𝑦) ≤ (𝑃𝐷𝑦))
33	20	recnd 8213	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → 𝑧 ∈ ℂ)
34	27	recnd 8213	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑃𝐷𝑦) ∈ ℂ)
35	33, 34	nncand 8500	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑧 − (𝑧 − (𝑃𝐷𝑦))) = (𝑃𝐷𝑦))
36	32, 35	breqtrrd 4117	. . . . . . . . . . . . . . 15 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑃𝐷𝑦) ≤ (𝑧 − (𝑧 − (𝑃𝐷𝑦))))
37		blss2 15160	. . . . . . . . . . . . . . 15 ⊢ (((𝐷 ∈ (∞Met‘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 1288	. . . . . . . . . . . . . 14 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑃(ball‘𝐷)(𝑧 − (𝑃𝐷𝑦))) ⊆ (𝑦(ball‘𝐷)𝑧))
39		simpll3 1064	. . . . . . . . . . . . . . 15 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → 𝑟 ∈ ℝ^)
40		simprrr 542	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → 𝑧 < 𝑟)
41	24, 39, 40	xrltled 10039	. . . . . . . . . . . . . . 15 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → 𝑧 ≤ 𝑟)
42		ssbl 15179	. . . . . . . . . . . . . . 15 ⊢ (((𝐷 ∈ (∞Met‘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 1284	. . . . . . . . . . . . . 14 ⊢ ((((𝐷 ∈ (∞Met‘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 3236	. . . . . . . . . . . . 13 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑃(ball‘𝐷)(𝑧 − (𝑃𝐷𝑦))) ⊆ (𝑦(ball‘𝐷)𝑟))
45		oveq2 6031	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑧 − (𝑃𝐷𝑦)) → (𝑃(ball‘𝐷)𝑥) = (𝑃(ball‘𝐷)(𝑧 − (𝑃𝐷𝑦))))
46	45	sseq1d 3255	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑧 − (𝑃𝐷𝑦)) → ((𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟) ↔ (𝑃(ball‘𝐷)(𝑧 − (𝑃𝐷𝑦))) ⊆ (𝑦(ball‘𝐷)𝑟)))
47	46	rspcev 2909	. . . . . . . . . . . . 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 411	. . . . . . . . . . . 12 ⊢ ((((𝐷 ∈ (∞Met‘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 375	. . . . . . . . . . 11 ⊢ ((((𝐷 ∈ (∞Met‘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 286	. . . . . . . . . 10 ⊢ ((((𝐷 ∈ (∞Met‘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 2649	. . . . . . . . 9 ⊢ (((𝐷 ∈ (∞Met‘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 45	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) → ((𝑦𝐷𝑃) < 𝑟 → ∃𝑥 ∈ ℝ⁺ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟)))
53	52	expimpd 363	. . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) → ((𝑃 ∈ 𝑋 ∧ (𝑦𝐷𝑃) < 𝑟) → ∃𝑥 ∈ ℝ⁺ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟)))
54	2, 53	sylbid 150	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) → (𝑃 ∈ (𝑦(ball‘𝐷)𝑟) → ∃𝑥 ∈ ℝ⁺ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟)))
55		eleq2 2294	. . . . . . 7 ⊢ (𝐵 = (𝑦(ball‘𝐷)𝑟) → (𝑃 ∈ 𝐵 ↔ 𝑃 ∈ (𝑦(ball‘𝐷)𝑟)))
56		sseq2 3250	. . . . . . . 8 ⊢ (𝐵 = (𝑦(ball‘𝐷)𝑟) → ((𝑃(ball‘𝐷)𝑥) ⊆ 𝐵 ↔ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟)))
57	56	rexbidv 2532	. . . . . . 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 234	. . . . . 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 157	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) → (𝐵 = (𝑦(ball‘𝐷)𝑟) → (𝑃 ∈ 𝐵 → ∃𝑥 ∈ ℝ⁺ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐵)))
60	59	3expib 1232	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ((𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) → (𝐵 = (𝑦(ball‘𝐷)𝑟) → (𝑃 ∈ 𝐵 → ∃𝑥 ∈ ℝ⁺ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐵))))
61	60	rexlimdvv 2656	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (∃𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ^* 𝐵 = (𝑦(ball‘𝐷)𝑟) → (𝑃 ∈ 𝐵 → ∃𝑥 ∈ ℝ⁺ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐵)))
62	1, 61	sylbid 150	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐵 ∈ ran (ball‘𝐷) → (𝑃 ∈ 𝐵 → ∃𝑥 ∈ ℝ⁺ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐵)))
63	62	3imp 1219	1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ ran (ball‘𝐷) ∧ 𝑃 ∈ 𝐵) → ∃𝑥 ∈ ℝ⁺ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐵)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 1004 = wceq 1397 ∈ wcel 2201 ∃wrex 2510 ⊆ wss 3199 class class class wbr 4089 ran crn 4728 ‘cfv 5328 (class class class)co 6023 ℝcr 8036 ℝ^*cxr 8218 < clt 8219 ≤ cle 8220 − cmin 8355 ℚcq 9858 ℝ⁺crp 9893 ∞Metcxmet 14574 ballcbl 14576

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2203 ax-14 2204 ax-ext 2212 ax-sep 4208 ax-pow 4266 ax-pr 4301 ax-un 4532 ax-setind 4637 ax-cnex 8128 ax-resscn 8129 ax-1cn 8130 ax-1re 8131 ax-icn 8132 ax-addcl 8133 ax-addrcl 8134 ax-mulcl 8135 ax-mulrcl 8136 ax-addcom 8137 ax-mulcom 8138 ax-addass 8139 ax-mulass 8140 ax-distr 8141 ax-i2m1 8142 ax-0lt1 8143 ax-1rid 8144 ax-0id 8145 ax-rnegex 8146 ax-precex 8147 ax-cnre 8148 ax-pre-ltirr 8149 ax-pre-ltwlin 8150 ax-pre-lttrn 8151 ax-pre-apti 8152 ax-pre-ltadd 8153 ax-pre-mulgt0 8154 ax-pre-mulext 8155 ax-arch 8156

This theorem depends on definitions: df-bi 117 df-stab 838 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1810 df-eu 2081 df-mo 2082 df-clab 2217 df-cleq 2223 df-clel 2226 df-nfc 2362 df-ne 2402 df-nel 2497 df-ral 2514 df-rex 2515 df-reu 2516 df-rmo 2517 df-rab 2518 df-v 2803 df-sbc 3031 df-csb 3127 df-dif 3201 df-un 3203 df-in 3205 df-ss 3212 df-if 3605 df-pw 3655 df-sn 3676 df-pr 3677 df-op 3679 df-uni 3895 df-int 3930 df-iun 3973 df-br 4090 df-opab 4152 df-mpt 4153 df-id 4392 df-po 4395 df-iso 4396 df-xp 4733 df-rel 4734 df-cnv 4735 df-co 4736 df-dm 4737 df-rn 4738 df-res 4739 df-ima 4740 df-iota 5288 df-fun 5330 df-fn 5331 df-f 5332 df-fv 5336 df-riota 5976 df-ov 6026 df-oprab 6027 df-mpo 6028 df-1st 6308 df-2nd 6309 df-map 6824 df-pnf 8221 df-mnf 8222 df-xr 8223 df-ltxr 8224 df-le 8225 df-sub 8357 df-neg 8358 df-reap 8760 df-ap 8767 df-div 8858 df-inn 9149 df-2 9207 df-n0 9408 df-z 9485 df-uz 9761 df-q 9859 df-rp 9894 df-xneg 10012 df-xadd 10013 df-psmet 14581 df-xmet 14582 df-bl 14584

This theorem is referenced by: blssex 15183 blin2 15185 metss 15247 metcnp3 15264

W3C validator