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Theorem blssps 23131
 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) ∧ 𝑃𝐵) → ∃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.
StepHypRef Expression
1 blrnps 23115 . . 3 (𝐷 ∈ (PsMet‘𝑋) → (𝐵 ∈ ran (ball‘𝐷) ↔ ∃𝑦𝑋𝑟 ∈ ℝ* 𝐵 = (𝑦(ball‘𝐷)𝑟)))
2 elblps 23094 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) → (𝑃 ∈ (𝑦(ball‘𝐷)𝑟) ↔ (𝑃𝑋 ∧ (𝑦𝐷𝑃) < 𝑟)))
3 simpl1 1188 . . . . . . . . . . 11 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) → 𝐷 ∈ (PsMet‘𝑋))
4 simpl2 1189 . . . . . . . . . . 11 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) → 𝑦𝑋)
5 simpr 488 . . . . . . . . . . 11 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) → 𝑃𝑋)
6 psmetcl 23014 . . . . . . . . . . 11 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑃𝑋) → (𝑦𝐷𝑃) ∈ ℝ*)
73, 4, 5, 6syl3anc 1368 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) → (𝑦𝐷𝑃) ∈ ℝ*)
8 simpl3 1190 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) → 𝑟 ∈ ℝ*)
9 qbtwnxr 12639 . . . . . . . . . . 11 (((𝑦𝐷𝑃) ∈ ℝ*𝑟 ∈ ℝ* ∧ (𝑦𝐷𝑃) < 𝑟) → ∃𝑧 ∈ ℚ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))
1093expia 1118 . . . . . . . . . 10 (((𝑦𝐷𝑃) ∈ ℝ*𝑟 ∈ ℝ*) → ((𝑦𝐷𝑃) < 𝑟 → ∃𝑧 ∈ ℚ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟)))
117, 8, 10syl2anc 587 . . . . . . . . 9 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) → ((𝑦𝐷𝑃) < 𝑟 → ∃𝑧 ∈ ℚ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟)))
12 qre 12398 . . . . . . . . . . 11 (𝑧 ∈ ℚ → 𝑧 ∈ ℝ)
13 simpll1 1209 . . . . . . . . . . . . . . . 16 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → 𝐷 ∈ (PsMet‘𝑋))
14 simplr 768 . . . . . . . . . . . . . . . 16 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → 𝑃𝑋)
15 simpll2 1210 . . . . . . . . . . . . . . . 16 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → 𝑦𝑋)
16 psmetsym 23017 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑦𝑋) → (𝑃𝐷𝑦) = (𝑦𝐷𝑃))
1713, 14, 15, 16syl3anc 1368 . . . . . . . . . . . . . . 15 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → (𝑃𝐷𝑦) = (𝑦𝐷𝑃))
18 simprrl 780 . . . . . . . . . . . . . . 15 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → (𝑦𝐷𝑃) < 𝑧)
1917, 18eqbrtrd 5057 . . . . . . . . . . . . . 14 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → (𝑃𝐷𝑦) < 𝑧)
20 simprl 770 . . . . . . . . . . . . . . . 16 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → 𝑧 ∈ ℝ)
21 psmetcl 23014 . . . . . . . . . . . . . . . . . 18 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑦𝑋) → (𝑃𝐷𝑦) ∈ ℝ*)
2213, 14, 15, 21syl3anc 1368 . . . . . . . . . . . . . . . . 17 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → (𝑃𝐷𝑦) ∈ ℝ*)
23 rexr 10730 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ ℝ → 𝑧 ∈ ℝ*)
2423ad2antrl 727 . . . . . . . . . . . . . . . . 17 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → 𝑧 ∈ ℝ*)
2522, 24, 19xrltled 12589 . . . . . . . . . . . . . . . 16 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → (𝑃𝐷𝑦) ≤ 𝑧)
26 psmetlecl 23022 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑃𝑋𝑦𝑋) ∧ (𝑧 ∈ ℝ ∧ (𝑃𝐷𝑦) ≤ 𝑧)) → (𝑃𝐷𝑦) ∈ ℝ)
2713, 14, 15, 20, 25, 26syl122anc 1376 . . . . . . . . . . . . . . 15 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → (𝑃𝐷𝑦) ∈ ℝ)
28 difrp 12473 . . . . . . . . . . . . . . 15 (((𝑃𝐷𝑦) ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((𝑃𝐷𝑦) < 𝑧 ↔ (𝑧 − (𝑃𝐷𝑦)) ∈ ℝ+))
2927, 20, 28syl2anc 587 . . . . . . . . . . . . . 14 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → ((𝑃𝐷𝑦) < 𝑧 ↔ (𝑧 − (𝑃𝐷𝑦)) ∈ ℝ+))
3019, 29mpbid 235 . . . . . . . . . . . . 13 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → (𝑧 − (𝑃𝐷𝑦)) ∈ ℝ+)
3120, 27resubcld 11111 . . . . . . . . . . . . . . 15 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → (𝑧 − (𝑃𝐷𝑦)) ∈ ℝ)
3222xrleidd 12591 . . . . . . . . . . . . . . . 16 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → (𝑃𝐷𝑦) ≤ (𝑃𝐷𝑦))
3320recnd 10712 . . . . . . . . . . . . . . . . 17 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → 𝑧 ∈ ℂ)
3427recnd 10712 . . . . . . . . . . . . . . . . 17 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → (𝑃𝐷𝑦) ∈ ℂ)
3533, 34nncand 11045 . . . . . . . . . . . . . . . 16 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → (𝑧 − (𝑧 − (𝑃𝐷𝑦))) = (𝑃𝐷𝑦))
3632, 35breqtrrd 5063 . . . . . . . . . . . . . . 15 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → (𝑃𝐷𝑦) ≤ (𝑧 − (𝑧 − (𝑃𝐷𝑦))))
37 blss2ps 23110 . . . . . . . . . . . . . . 15 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑦𝑋) ∧ ((𝑧 − (𝑃𝐷𝑦)) ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (𝑃𝐷𝑦) ≤ (𝑧 − (𝑧 − (𝑃𝐷𝑦))))) → (𝑃(ball‘𝐷)(𝑧 − (𝑃𝐷𝑦))) ⊆ (𝑦(ball‘𝐷)𝑧))
3813, 14, 15, 31, 20, 36, 37syl33anc 1382 . . . . . . . . . . . . . 14 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → (𝑃(ball‘𝐷)(𝑧 − (𝑃𝐷𝑦))) ⊆ (𝑦(ball‘𝐷)𝑧))
39 simpll3 1211 . . . . . . . . . . . . . . 15 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → 𝑟 ∈ ℝ*)
40 simprrr 781 . . . . . . . . . . . . . . . 16 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → 𝑧 < 𝑟)
4124, 39, 40xrltled 12589 . . . . . . . . . . . . . . 15 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → 𝑧𝑟)
42 ssblps 23129 . . . . . . . . . . . . . . 15 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋) ∧ (𝑧 ∈ ℝ*𝑟 ∈ ℝ*) ∧ 𝑧𝑟) → (𝑦(ball‘𝐷)𝑧) ⊆ (𝑦(ball‘𝐷)𝑟))
4313, 15, 24, 39, 41, 42syl221anc 1378 . . . . . . . . . . . . . 14 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → (𝑦(ball‘𝐷)𝑧) ⊆ (𝑦(ball‘𝐷)𝑟))
4438, 43sstrd 3904 . . . . . . . . . . . . 13 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → (𝑃(ball‘𝐷)(𝑧 − (𝑃𝐷𝑦))) ⊆ (𝑦(ball‘𝐷)𝑟))
45 oveq2 7163 . . . . . . . . . . . . . . 15 (𝑥 = (𝑧 − (𝑃𝐷𝑦)) → (𝑃(ball‘𝐷)𝑥) = (𝑃(ball‘𝐷)(𝑧 − (𝑃𝐷𝑦))))
4645sseq1d 3925 . . . . . . . . . . . . . 14 (𝑥 = (𝑧 − (𝑃𝐷𝑦)) → ((𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟) ↔ (𝑃(ball‘𝐷)(𝑧 − (𝑃𝐷𝑦))) ⊆ (𝑦(ball‘𝐷)𝑟)))
4746rspcev 3543 . . . . . . . . . . . . 13 (((𝑧 − (𝑃𝐷𝑦)) ∈ ℝ+ ∧ (𝑃(ball‘𝐷)(𝑧 − (𝑃𝐷𝑦))) ⊆ (𝑦(ball‘𝐷)𝑟)) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟))
4830, 44, 47syl2anc 587 . . . . . . . . . . . 12 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟))
4948expr 460 . . . . . . . . . . 11 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ 𝑧 ∈ ℝ) → (((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟)))
5012, 49sylan2 595 . . . . . . . . . 10 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ 𝑧 ∈ ℚ) → (((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟)))
5150rexlimdva 3208 . . . . . . . . 9 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) → (∃𝑧 ∈ ℚ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟)))
5211, 51syld 47 . . . . . . . 8 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) → ((𝑦𝐷𝑃) < 𝑟 → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟)))
5352expimpd 457 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) → ((𝑃𝑋 ∧ (𝑦𝐷𝑃) < 𝑟) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟)))
542, 53sylbid 243 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) → (𝑃 ∈ (𝑦(ball‘𝐷)𝑟) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟)))
55 eleq2 2840 . . . . . . 7 (𝐵 = (𝑦(ball‘𝐷)𝑟) → (𝑃𝐵𝑃 ∈ (𝑦(ball‘𝐷)𝑟)))
56 sseq2 3920 . . . . . . . 8 (𝐵 = (𝑦(ball‘𝐷)𝑟) → ((𝑃(ball‘𝐷)𝑥) ⊆ 𝐵 ↔ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟)))
5756rexbidv 3221 . . . . . . 7 (𝐵 = (𝑦(ball‘𝐷)𝑟) → (∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐵 ↔ ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟)))
5855, 57imbi12d 348 . . . . . 6 (𝐵 = (𝑦(ball‘𝐷)𝑟) → ((𝑃𝐵 → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐵) ↔ (𝑃 ∈ (𝑦(ball‘𝐷)𝑟) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟))))
5954, 58syl5ibrcom 250 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) → (𝐵 = (𝑦(ball‘𝐷)𝑟) → (𝑃𝐵 → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐵)))
60593expib 1119 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → ((𝑦𝑋𝑟 ∈ ℝ*) → (𝐵 = (𝑦(ball‘𝐷)𝑟) → (𝑃𝐵 → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐵))))
6160rexlimdvv 3217 . . 3 (𝐷 ∈ (PsMet‘𝑋) → (∃𝑦𝑋𝑟 ∈ ℝ* 𝐵 = (𝑦(ball‘𝐷)𝑟) → (𝑃𝐵 → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐵)))
621, 61sylbid 243 . 2 (𝐷 ∈ (PsMet‘𝑋) → (𝐵 ∈ ran (ball‘𝐷) → (𝑃𝐵 → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐵)))
63623imp 1108 1 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐵 ∈ ran (ball‘𝐷) ∧ 𝑃𝐵) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∃wrex 3071   ⊆ wss 3860   class class class wbr 5035  ran crn 5528  ‘cfv 6339  (class class class)co 7155  ℝcr 10579  ℝ*cxr 10717   < clt 10718   ≤ cle 10719   − cmin 10913  ℚcq 12393  ℝ+crp 12435  PsMetcpsmet 20155  ballcbl 20158 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464  ax-cnex 10636  ax-resscn 10637  ax-1cn 10638  ax-icn 10639  ax-addcl 10640  ax-addrcl 10641  ax-mulcl 10642  ax-mulrcl 10643  ax-mulcom 10644  ax-addass 10645  ax-mulass 10646  ax-distr 10647  ax-i2m1 10648  ax-1ne0 10649  ax-1rid 10650  ax-rnegex 10651  ax-rrecex 10652  ax-cnre 10653  ax-pre-lttri 10654  ax-pre-lttrn 10655  ax-pre-ltadd 10656  ax-pre-mulgt0 10657  ax-pre-sup 10658 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-tr 5142  df-id 5433  df-eprel 5438  df-po 5446  df-so 5447  df-fr 5486  df-we 5488  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-pred 6130  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7585  df-1st 7698  df-2nd 7699  df-wrecs 7962  df-recs 8023  df-rdg 8061  df-er 8304  df-map 8423  df-en 8533  df-dom 8534  df-sdom 8535  df-sup 8944  df-inf 8945  df-pnf 10720  df-mnf 10721  df-xr 10722  df-ltxr 10723  df-le 10724  df-sub 10915  df-neg 10916  df-div 11341  df-nn 11680  df-2 11742  df-n0 11940  df-z 12026  df-uz 12288  df-q 12394  df-rp 12436  df-xneg 12553  df-xadd 12554  df-xmul 12555  df-psmet 20163  df-bl 20166 This theorem is referenced by:  blssexps  23133
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