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Theorem blss 12597

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 12581	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐵 ∈ ran (ball‘𝐷) ↔ ∃𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ^* 𝐵 = (𝑦(ball‘𝐷)𝑟)))
2		elbl 12560	. . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) → (𝑃 ∈ (𝑦(ball‘𝐷)𝑟) ↔ (𝑃 ∈ 𝑋 ∧ (𝑦𝐷𝑃) < 𝑟)))
3		simpl1 984	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) → 𝐷 ∈ (∞Met‘𝑋))
4		simpl2 985	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) → 𝑦 ∈ 𝑋)
5		simpr 109	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) → 𝑃 ∈ 𝑋)
6		xmetcl 12521	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑃 ∈ 𝑋) → (𝑦𝐷𝑃) ∈ ℝ^*)
7	3, 4, 5, 6	syl3anc 1216	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^) ∧ 𝑃 ∈ 𝑋) → (𝑦𝐷𝑃) ∈ ℝ^)
8		simpl3 986	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^) ∧ 𝑃 ∈ 𝑋) → 𝑟 ∈ ℝ^)
9		qbtwnxr 10035	. . . . . . . . . . 11 ⊢ (((𝑦𝐷𝑃) ∈ ℝ^* ∧ 𝑟 ∈ ℝ^* ∧ (𝑦𝐷𝑃) < 𝑟) → ∃𝑧 ∈ ℚ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))
10	9	3expia 1183	. . . . . . . . . 10 ⊢ (((𝑦𝐷𝑃) ∈ ℝ^* ∧ 𝑟 ∈ ℝ^*) → ((𝑦𝐷𝑃) < 𝑟 → ∃𝑧 ∈ ℚ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟)))
11	7, 8, 10	syl2anc 408	. . . . . . . . 9 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) → ((𝑦𝐷𝑃) < 𝑟 → ∃𝑧 ∈ ℚ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟)))
12		qre 9417	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℚ → 𝑧 ∈ ℝ)
13		simpll1 1020	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → 𝐷 ∈ (∞Met‘𝑋))
14		simplr 519	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → 𝑃 ∈ 𝑋)
15		simpll2 1021	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → 𝑦 ∈ 𝑋)
16		xmetsym 12537	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑃𝐷𝑦) = (𝑦𝐷𝑃))
17	13, 14, 15, 16	syl3anc 1216	. . . . . . . . . . . . . . 15 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑃𝐷𝑦) = (𝑦𝐷𝑃))
18		simprrl 528	. . . . . . . . . . . . . . 15 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑦𝐷𝑃) < 𝑧)
19	17, 18	eqbrtrd 3950	. . . . . . . . . . . . . 14 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑃𝐷𝑦) < 𝑧)
20		simprl 520	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → 𝑧 ∈ ℝ)
21		xmetcl 12521	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑃𝐷𝑦) ∈ ℝ^*)
22	13, 14, 15, 21	syl3anc 1216	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑃𝐷𝑦) ∈ ℝ^)
23		rexr 7811	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ ℝ → 𝑧 ∈ ℝ^*)
24	23	ad2antrl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → 𝑧 ∈ ℝ^)
25	22, 24, 19	xrltled 9585	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑃𝐷𝑦) ≤ 𝑧)
26		xmetlecl 12536	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑃 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ (𝑃𝐷𝑦) ≤ 𝑧)) → (𝑃𝐷𝑦) ∈ ℝ)
27	13, 14, 15, 20, 25, 26	syl122anc 1225	. . . . . . . . . . . . . . 15 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑃𝐷𝑦) ∈ ℝ)
28		difrp 9480	. . . . . . . . . . . . . . 15 ⊢ (((𝑃𝐷𝑦) ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((𝑃𝐷𝑦) < 𝑧 ↔ (𝑧 − (𝑃𝐷𝑦)) ∈ ℝ⁺))
29	27, 20, 28	syl2anc 408	. . . . . . . . . . . . . 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 146	. . . . . . . . . . . . 13 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑧 − (𝑃𝐷𝑦)) ∈ ℝ⁺)
31	20, 27	resubcld 8143	. . . . . . . . . . . . . . 15 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑧 − (𝑃𝐷𝑦)) ∈ ℝ)
32	22	xrleidd 9587	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑃𝐷𝑦) ≤ (𝑃𝐷𝑦))
33	20	recnd 7794	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → 𝑧 ∈ ℂ)
34	27	recnd 7794	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑃𝐷𝑦) ∈ ℂ)
35	33, 34	nncand 8078	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑧 − (𝑧 − (𝑃𝐷𝑦))) = (𝑃𝐷𝑦))
36	32, 35	breqtrrd 3956	. . . . . . . . . . . . . . 15 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑃𝐷𝑦) ≤ (𝑧 − (𝑧 − (𝑃𝐷𝑦))))
37		blss2 12576	. . . . . . . . . . . . . . 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 1231	. . . . . . . . . . . . . 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 1022	. . . . . . . . . . . . . . 15 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → 𝑟 ∈ ℝ^)
40		simprrr 529	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → 𝑧 < 𝑟)
41	24, 39, 40	xrltled 9585	. . . . . . . . . . . . . . 15 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → 𝑧 ≤ 𝑟)
42		ssbl 12595	. . . . . . . . . . . . . . 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 1227	. . . . . . . . . . . . . 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 3107	. . . . . . . . . . . . 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 5782	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑧 − (𝑃𝐷𝑦)) → (𝑃(ball‘𝐷)𝑥) = (𝑃(ball‘𝐷)(𝑧 − (𝑃𝐷𝑦))))
46	45	sseq1d 3126	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑧 − (𝑃𝐷𝑦)) → ((𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟) ↔ (𝑃(ball‘𝐷)(𝑧 − (𝑃𝐷𝑦))) ⊆ (𝑦(ball‘𝐷)𝑟)))
47	46	rspcev 2789	. . . . . . . . . . . . 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 408	. . . . . . . . . . . 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 372	. . . . . . . . . . 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 284	. . . . . . . . . 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 2549	. . . . . . . . 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 360	. . . . . . 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 149	. . . . . 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 2203	. . . . . . 7 ⊢ (𝐵 = (𝑦(ball‘𝐷)𝑟) → (𝑃 ∈ 𝐵 ↔ 𝑃 ∈ (𝑦(ball‘𝐷)𝑟)))
56		sseq2 3121	. . . . . . . 8 ⊢ (𝐵 = (𝑦(ball‘𝐷)𝑟) → ((𝑃(ball‘𝐷)𝑥) ⊆ 𝐵 ↔ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟)))
57	56	rexbidv 2438	. . . . . . 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 233	. . . . . 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 156	. . . . 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 1184	. . . 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 2556	. . 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 149	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐵 ∈ ran (ball‘𝐷) → (𝑃 ∈ 𝐵 → ∃𝑥 ∈ ℝ⁺ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐵)))
63	62	3imp 1175	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 103 ↔ wb 104 ∧ w3a 962 = wceq 1331 ∈ wcel 1480 ∃wrex 2417 ⊆ wss 3071 class class class wbr 3929 ran crn 4540 ‘cfv 5123 (class class class)co 5774 ℝcr 7619 ℝ^*cxr 7799 < clt 7800 ≤ cle 7801 − cmin 7933 ℚcq 9411 ℝ⁺crp 9441 ∞Metcxmet 12149 ballcbl 12151

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 ax-arch 7739

This theorem depends on definitions: df-bi 116 df-stab 816 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-po 4218 df-iso 4219 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-map 6544 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-n0 8978 df-z 9055 df-uz 9327 df-q 9412 df-rp 9442 df-xneg 9559 df-xadd 9560 df-psmet 12156 df-xmet 12157 df-bl 12159

This theorem is referenced by: blssex 12599 blin2 12601 metss 12663 metcnp3 12680

W3C validator