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Theorem blssps 24464

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 24448	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝐵 ∈ ran (ball‘𝐷) ↔ ∃𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ^* 𝐵 = (𝑦(ball‘𝐷)𝑟)))
2		elblps 24427	. . . . . . 7 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) → (𝑃 ∈ (𝑦(ball‘𝐷)𝑟) ↔ (𝑃 ∈ 𝑋 ∧ (𝑦𝐷𝑃) < 𝑟)))
3		simpl1 1204	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) → 𝐷 ∈ (PsMet‘𝑋))
4		simpl2 1205	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) → 𝑦 ∈ 𝑋)
5		simpr 488	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) → 𝑃 ∈ 𝑋)
6		psmetcl 24347	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑃 ∈ 𝑋) → (𝑦𝐷𝑃) ∈ ℝ^*)
7	3, 4, 5, 6	syl3anc 1389	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^) ∧ 𝑃 ∈ 𝑋) → (𝑦𝐷𝑃) ∈ ℝ^)
8		simpl3 1206	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^) ∧ 𝑃 ∈ 𝑋) → 𝑟 ∈ ℝ^)
9		qbtwnxr 13200	. . . . . . . . . . 11 ⊢ (((𝑦𝐷𝑃) ∈ ℝ^* ∧ 𝑟 ∈ ℝ^* ∧ (𝑦𝐷𝑃) < 𝑟) → ∃𝑧 ∈ ℚ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))
10	9	3expia 1133	. . . . . . . . . 10 ⊢ (((𝑦𝐷𝑃) ∈ ℝ^* ∧ 𝑟 ∈ ℝ^*) → ((𝑦𝐷𝑃) < 𝑟 → ∃𝑧 ∈ ℚ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟)))
11	7, 8, 10	syl2anc 593	. . . . . . . . 9 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) → ((𝑦𝐷𝑃) < 𝑟 → ∃𝑧 ∈ ℚ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟)))
12		qre 12951	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℚ → 𝑧 ∈ ℝ)
13		simpll1 1225	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → 𝐷 ∈ (PsMet‘𝑋))
14		simplr 778	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → 𝑃 ∈ 𝑋)
15		simpll2 1226	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → 𝑦 ∈ 𝑋)
16		psmetsym 24350	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑃𝐷𝑦) = (𝑦𝐷𝑃))
17	13, 14, 15, 16	syl3anc 1389	. . . . . . . . . . . . . . 15 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑃𝐷𝑦) = (𝑦𝐷𝑃))
18		simprrl 790	. . . . . . . . . . . . . . 15 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑦𝐷𝑃) < 𝑧)
19	17, 18	eqbrtrd 5121	. . . . . . . . . . . . . 14 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑃𝐷𝑦) < 𝑧)
20		simprl 780	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → 𝑧 ∈ ℝ)
21		psmetcl 24347	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑃𝐷𝑦) ∈ ℝ^*)
22	13, 14, 15, 21	syl3anc 1389	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑃𝐷𝑦) ∈ ℝ^)
23		rexr 11225	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ ℝ → 𝑧 ∈ ℝ^*)
24	23	ad2antrl 738	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → 𝑧 ∈ ℝ^)
25	22, 24, 19	xrltled 13149	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑃𝐷𝑦) ≤ 𝑧)
26		psmetlecl 24355	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑃 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ (𝑃𝐷𝑦) ≤ 𝑧)) → (𝑃𝐷𝑦) ∈ ℝ)
27	13, 14, 15, 20, 25, 26	syl122anc 1397	. . . . . . . . . . . . . . 15 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑃𝐷𝑦) ∈ ℝ)
28		difrp 13030	. . . . . . . . . . . . . . 15 ⊢ (((𝑃𝐷𝑦) ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((𝑃𝐷𝑦) < 𝑧 ↔ (𝑧 − (𝑃𝐷𝑦)) ∈ ℝ⁺))
29	27, 20, 28	syl2anc 593	. . . . . . . . . . . . . 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 234	. . . . . . . . . . . . 13 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑧 − (𝑃𝐷𝑦)) ∈ ℝ⁺)
31	20, 27	resubcld 11612	. . . . . . . . . . . . . . 15 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑧 − (𝑃𝐷𝑦)) ∈ ℝ)
32	22	xrleidd 13151	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑃𝐷𝑦) ≤ (𝑃𝐷𝑦))
33	20	recnd 11207	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → 𝑧 ∈ ℂ)
34	27	recnd 11207	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑃𝐷𝑦) ∈ ℂ)
35	33, 34	nncand 11544	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑧 − (𝑧 − (𝑃𝐷𝑦))) = (𝑃𝐷𝑦))
36	32, 35	breqtrrd 5127	. . . . . . . . . . . . . . 15 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → (𝑃𝐷𝑦) ≤ (𝑧 − (𝑧 − (𝑃𝐷𝑦))))
37		blss2ps 24443	. . . . . . . . . . . . . . 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 1403	. . . . . . . . . . . . . 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 1227	. . . . . . . . . . . . . . 15 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → 𝑟 ∈ ℝ^)
40		simprrr 791	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → 𝑧 < 𝑟)
41	24, 39, 40	xrltled 13149	. . . . . . . . . . . . . . 15 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < 𝑟))) → 𝑧 ≤ 𝑟)
42		ssblps 24462	. . . . . . . . . . . . . . 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 1399	. . . . . . . . . . . . . 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 3946	. . . . . . . . . . . . 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 7400	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑧 − (𝑃𝐷𝑦)) → (𝑃(ball‘𝐷)𝑥) = (𝑃(ball‘𝐷)(𝑧 − (𝑃𝐷𝑦))))
46	45	sseq1d 3967	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑧 − (𝑃𝐷𝑦)) → ((𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟) ↔ (𝑃(ball‘𝐷)(𝑧 − (𝑃𝐷𝑦))) ⊆ (𝑦(ball‘𝐷)𝑟)))
47	46	rspcev 3581	. . . . . . . . . . . . 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 593	. . . . . . . . . . . 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 460	. . . . . . . . . . 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 602	. . . . . . . . . 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 3162	. . . . . . . . 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 47	. . . . . . . 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 457	. . . . . . 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 242	. . . . . 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 2850	. . . . . . 7 ⊢ (𝐵 = (𝑦(ball‘𝐷)𝑟) → (𝑃 ∈ 𝐵 ↔ 𝑃 ∈ (𝑦(ball‘𝐷)𝑟)))
56		sseq2 3962	. . . . . . . 8 ⊢ (𝐵 = (𝑦(ball‘𝐷)𝑟) → ((𝑃(ball‘𝐷)𝑥) ⊆ 𝐵 ↔ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟)))
57	56	rexbidv 3185	. . . . . . 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 346	. . . . . 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 249	. . . . 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 1134	. . . 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 3217	. . 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 242	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝐵 ∈ ran (ball‘𝐷) → (𝑃 ∈ 𝐵 → ∃𝑥 ∈ ℝ⁺ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐵)))
63	62	3imp 1122	1 ⊢ ((𝐷 ∈ (PsMet‘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 208 ∧ wa 399 ∧ w3a 1097 = wceq 1559 ∈ wcel 2141 ∃wrex 3085 ⊆ wss 3904 class class class wbr 5099 ran crn 5646 ‘cfv 6517 (class class class)co 7392 ℝcr 11069 ℝ^*cxr 11212 < clt 11213 ≤ cle 11214 − cmin 11411 ℚcq 12946 ℝ⁺crp 12990 PsMetcpsmet 21388 ballcbl 21391

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714 ax-cnex 11126 ax-resscn 11127 ax-1cn 11128 ax-icn 11129 ax-addcl 11130 ax-addrcl 11131 ax-mulcl 11132 ax-mulrcl 11133 ax-mulcom 11134 ax-addass 11135 ax-mulass 11136 ax-distr 11137 ax-i2m1 11138 ax-1ne0 11139 ax-1rid 11140 ax-rnegex 11141 ax-rrecex 11142 ax-cnre 11143 ax-pre-lttri 11144 ax-pre-lttrn 11145 ax-pre-ltadd 11146 ax-pre-mulgt0 11147 ax-pre-sup 11148

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-iun 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6284 df-ord 6345 df-on 6346 df-lim 6347 df-suc 6348 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-riota 7349 df-ov 7395 df-oprab 7396 df-mpo 7397 df-om 7843 df-1st 7966 df-2nd 7967 df-frecs 8257 df-wrecs 8288 df-recs 8337 df-rdg 8376 df-er 8673 df-map 8805 df-en 8924 df-dom 8925 df-sdom 8926 df-sup 9385 df-inf 9386 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11413 df-neg 11414 df-div 11842 df-nn 12208 df-2 12277 df-n0 12479 df-z 12566 df-uz 12837 df-q 12947 df-rp 12991 df-xneg 13111 df-xadd 13112 df-xmul 13113 df-psmet 21396 df-bl 21399

This theorem is referenced by: blssexps 24466

W3C validator