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Theorem blss 24369

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 24353	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐵 ∈ ran (ball‘𝐷) ↔ ∃𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ^* 𝐵 = (𝑦(ball‘𝐷)𝑟)))
2		elbl 24332	. . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) → (𝑃 ∈ (𝑦(ball‘𝐷)𝑟) ↔ (𝑃 ∈ 𝑋 ∧ (𝑦𝐷𝑃) < 𝑟)))
3		simpl1 1192	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) → 𝐷 ∈ (∞Met‘𝑋))
4		simpl2 1193	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) → 𝑦 ∈ 𝑋)
5		simpr 484	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) → 𝑃 ∈ 𝑋)
6		xmetcl 24275	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑃 ∈ 𝑋) → (𝑦𝐷𝑃) ∈ ℝ^*)
7	3, 4, 5, 6	syl3anc 1373	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^) ∧ 𝑃 ∈ 𝑋) → (𝑦𝐷𝑃) ∈ ℝ^)
8		simpl3 1194	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^) ∧ 𝑃 ∈ 𝑋) → 𝑟 ∈ ℝ^)
9		qbtwnxr 13115	. . . . . . . . . . 11 ⊢ (((𝑦𝐷𝑃) ∈ ℝ^* ∧ 𝑟 ∈ ℝ^* ∧ (𝑦𝐷𝑃) < 𝑟) → ∃𝑧 ∈ ℚ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))
10	9	3expia 1121	. . . . . . . . . 10 ⊢ (((𝑦𝐷𝑃) ∈ ℝ^* ∧ 𝑟 ∈ ℝ^*) → ((𝑦𝐷𝑃) < 𝑟 → ∃𝑧 ∈ ℚ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟)))
11	7, 8, 10	syl2anc 584	. . . . . . . . 9 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) → ((𝑦𝐷𝑃) < 𝑟 → ∃𝑧 ∈ ℚ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟)))
12		qre 12866	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℚ → 𝑧 ∈ ℝ)
13		simpll1 1213	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → 𝐷 ∈ (∞Met‘𝑋))
14		simplr 768	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → 𝑃 ∈ 𝑋)
15		simpll2 1214	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → 𝑦 ∈ 𝑋)
16		xmetsym 24291	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑃𝐷𝑦) = (𝑦𝐷𝑃))
17	13, 14, 15, 16	syl3anc 1373	. . . . . . . . . . . . . . 15 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑃𝐷𝑦) = (𝑦𝐷𝑃))
18		simprrl 780	. . . . . . . . . . . . . . 15 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑦𝐷𝑃) < 𝑧)
19	17, 18	eqbrtrd 5120	. . . . . . . . . . . . . 14 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑃𝐷𝑦) < 𝑧)
20		simprl 770	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → 𝑧 ∈ ℝ)
21		xmetcl 24275	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑃𝐷𝑦) ∈ ℝ^*)
22	13, 14, 15, 21	syl3anc 1373	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑃𝐷𝑦) ∈ ℝ^)
23		rexr 11178	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ ℝ → 𝑧 ∈ ℝ^*)
24	23	ad2antrl 728	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → 𝑧 ∈ ℝ^)
25	22, 24, 19	xrltled 13064	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑃𝐷𝑦) ≤ 𝑧)
26		xmetlecl 24290	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑃 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ (𝑃𝐷𝑦) ≤ 𝑧)) → (𝑃𝐷𝑦) ∈ ℝ)
27	13, 14, 15, 20, 25, 26	syl122anc 1381	. . . . . . . . . . . . . . 15 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑃𝐷𝑦) ∈ ℝ)
28		difrp 12945	. . . . . . . . . . . . . . 15 ⊢ (((𝑃𝐷𝑦) ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((𝑃𝐷𝑦) < 𝑧 ↔ (𝑧 − (𝑃𝐷𝑦)) ∈ ℝ⁺))
29	27, 20, 28	syl2anc 584	. . . . . . . . . . . . . 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 232	. . . . . . . . . . . . 13 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑧 − (𝑃𝐷𝑦)) ∈ ℝ⁺)
31	20, 27	resubcld 11565	. . . . . . . . . . . . . . 15 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑧 − (𝑃𝐷𝑦)) ∈ ℝ)
32	22	xrleidd 13066	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑃𝐷𝑦) ≤ (𝑃𝐷𝑦))
33	20	recnd 11160	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → 𝑧 ∈ ℂ)
34	27	recnd 11160	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑃𝐷𝑦) ∈ ℂ)
35	33, 34	nncand 11497	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑧 − (𝑧 − (𝑃𝐷𝑦))) = (𝑃𝐷𝑦))
36	32, 35	breqtrrd 5126	. . . . . . . . . . . . . . 15 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑃𝐷𝑦) ≤ (𝑧 − (𝑧 − (𝑃𝐷𝑦))))
37		blss2 24348	. . . . . . . . . . . . . . 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 1387	. . . . . . . . . . . . . 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 1215	. . . . . . . . . . . . . . 15 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → 𝑟 ∈ ℝ^)
40		simprrr 781	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → 𝑧 < 𝑟)
41	24, 39, 40	xrltled 13064	. . . . . . . . . . . . . . 15 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → 𝑧 ≤ 𝑟)
42		ssbl 24367	. . . . . . . . . . . . . . 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 1383	. . . . . . . . . . . . . 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 3944	. . . . . . . . . . . . 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 7366	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑧 − (𝑃𝐷𝑦)) → (𝑃(ball‘𝐷)𝑥) = (𝑃(ball‘𝐷)(𝑧 − (𝑃𝐷𝑦))))
46	45	sseq1d 3965	. . . . . . . . . . . . . 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 584	. . . . . . . . . . . 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 456	. . . . . . . . . . 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 593	. . . . . . . . . 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 3137	. . . . . . . . 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 47	. . . . . . . 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 453	. . . . . . 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 240	. . . . . 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 2825	. . . . . . 7 ⊢ (𝐵 = (𝑦(ball‘𝐷)𝑟) → (𝑃 ∈ 𝐵 ↔ 𝑃 ∈ (𝑦(ball‘𝐷)𝑟)))
56		sseq2 3960	. . . . . . . 8 ⊢ (𝐵 = (𝑦(ball‘𝐷)𝑟) → ((𝑃(ball‘𝐷)𝑥) ⊆ 𝐵 ↔ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟)))
57	56	rexbidv 3160	. . . . . . 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 344	. . . . . 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 247	. . . . 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 1122	. . . 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 3192	. . 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 240	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐵 ∈ ran (ball‘𝐷) → (𝑃 ∈ 𝐵 → ∃𝑥 ∈ ℝ⁺ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐵)))
63	62	3imp 1110	1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ ran (ball‘𝐷) ∧ 𝑃 ∈ 𝐵) → ∃𝑥 ∈ ℝ⁺ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ∃wrex 3060 ⊆ wss 3901 class class class wbr 5098 ran crn 5625 ‘cfv 6492 (class class class)co 7358 ℝcr 11025 ℝ^*cxr 11165 < clt 11166 ≤ cle 11167 − cmin 11364 ℚcq 12861 ℝ⁺crp 12905 ∞Metcxmet 21294 ballcbl 21296

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-er 8635 df-map 8765 df-en 8884 df-dom 8885 df-sdom 8886 df-sup 9345 df-inf 9346 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-n0 12402 df-z 12489 df-uz 12752 df-q 12862 df-rp 12906 df-xneg 13026 df-xadd 13027 df-xmul 13028 df-psmet 21301 df-xmet 21302 df-bl 21304

This theorem is referenced by: blssex 24371 blin2 24373 metss 24452 metcnp3 24484

W3C validator