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Theorem blssps 12628
 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 12612 . . 3 (𝐷 ∈ (PsMet‘𝑋) → (𝐵 ∈ ran (ball‘𝐷) ↔ ∃𝑦𝑋𝑟 ∈ ℝ* 𝐵 = (𝑦(ball‘𝐷)𝑟)))
2 elblps 12591 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) → (𝑃 ∈ (𝑦(ball‘𝐷)𝑟) ↔ (𝑃𝑋 ∧ (𝑦𝐷𝑃) < 𝑟)))
3 simpl1 985 . . . . . . . . . . 11 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) → 𝐷 ∈ (PsMet‘𝑋))
4 simpl2 986 . . . . . . . . . . 11 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) → 𝑦𝑋)
5 simpr 109 . . . . . . . . . . 11 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) → 𝑃𝑋)
6 psmetcl 12527 . . . . . . . . . . 11 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑃𝑋) → (𝑦𝐷𝑃) ∈ ℝ*)
73, 4, 5, 6syl3anc 1217 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) → (𝑦𝐷𝑃) ∈ ℝ*)
8 simpl3 987 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) → 𝑟 ∈ ℝ*)
9 qbtwnxr 10064 . . . . . . . . . . 11 (((𝑦𝐷𝑃) ∈ ℝ*𝑟 ∈ ℝ* ∧ (𝑦𝐷𝑃) < 𝑟) → ∃𝑧 ∈ ℚ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))
1093expia 1184 . . . . . . . . . 10 (((𝑦𝐷𝑃) ∈ ℝ*𝑟 ∈ ℝ*) → ((𝑦𝐷𝑃) < 𝑟 → ∃𝑧 ∈ ℚ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟)))
117, 8, 10syl2anc 409 . . . . . . . . 9 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) → ((𝑦𝐷𝑃) < 𝑟 → ∃𝑧 ∈ ℚ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟)))
12 qre 9442 . . . . . . . . . . 11 (𝑧 ∈ ℚ → 𝑧 ∈ ℝ)
13 simpll1 1021 . . . . . . . . . . . . . . . 16 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → 𝐷 ∈ (PsMet‘𝑋))
14 simplr 520 . . . . . . . . . . . . . . . 16 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → 𝑃𝑋)
15 simpll2 1022 . . . . . . . . . . . . . . . 16 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → 𝑦𝑋)
16 psmetsym 12530 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑦𝑋) → (𝑃𝐷𝑦) = (𝑦𝐷𝑃))
1713, 14, 15, 16syl3anc 1217 . . . . . . . . . . . . . . 15 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → (𝑃𝐷𝑦) = (𝑦𝐷𝑃))
18 simprrl 529 . . . . . . . . . . . . . . 15 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → (𝑦𝐷𝑃) < 𝑧)
1917, 18eqbrtrd 3956 . . . . . . . . . . . . . 14 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → (𝑃𝐷𝑦) < 𝑧)
20 simprl 521 . . . . . . . . . . . . . . . 16 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → 𝑧 ∈ ℝ)
21 psmetcl 12527 . . . . . . . . . . . . . . . . . 18 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑦𝑋) → (𝑃𝐷𝑦) ∈ ℝ*)
2213, 14, 15, 21syl3anc 1217 . . . . . . . . . . . . . . . . 17 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → (𝑃𝐷𝑦) ∈ ℝ*)
23 rexr 7833 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ ℝ → 𝑧 ∈ ℝ*)
2423ad2antrl 482 . . . . . . . . . . . . . . . . 17 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → 𝑧 ∈ ℝ*)
2522, 24, 19xrltled 9613 . . . . . . . . . . . . . . . 16 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → (𝑃𝐷𝑦) ≤ 𝑧)
26 psmetlecl 12535 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑃𝑋𝑦𝑋) ∧ (𝑧 ∈ ℝ ∧ (𝑃𝐷𝑦) ≤ 𝑧)) → (𝑃𝐷𝑦) ∈ ℝ)
2713, 14, 15, 20, 25, 26syl122anc 1226 . . . . . . . . . . . . . . 15 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → (𝑃𝐷𝑦) ∈ ℝ)
28 difrp 9507 . . . . . . . . . . . . . . 15 (((𝑃𝐷𝑦) ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((𝑃𝐷𝑦) < 𝑧 ↔ (𝑧 − (𝑃𝐷𝑦)) ∈ ℝ+))
2927, 20, 28syl2anc 409 . . . . . . . . . . . . . 14 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → ((𝑃𝐷𝑦) < 𝑧 ↔ (𝑧 − (𝑃𝐷𝑦)) ∈ ℝ+))
3019, 29mpbid 146 . . . . . . . . . . . . 13 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → (𝑧 − (𝑃𝐷𝑦)) ∈ ℝ+)
3120, 27resubcld 8165 . . . . . . . . . . . . . . 15 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → (𝑧 − (𝑃𝐷𝑦)) ∈ ℝ)
3222xrleidd 9615 . . . . . . . . . . . . . . . 16 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → (𝑃𝐷𝑦) ≤ (𝑃𝐷𝑦))
3320recnd 7816 . . . . . . . . . . . . . . . . 17 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → 𝑧 ∈ ℂ)
3427recnd 7816 . . . . . . . . . . . . . . . . 17 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → (𝑃𝐷𝑦) ∈ ℂ)
3533, 34nncand 8100 . . . . . . . . . . . . . . . 16 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → (𝑧 − (𝑧 − (𝑃𝐷𝑦))) = (𝑃𝐷𝑦))
3632, 35breqtrrd 3962 . . . . . . . . . . . . . . 15 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → (𝑃𝐷𝑦) ≤ (𝑧 − (𝑧 − (𝑃𝐷𝑦))))
37 blss2ps 12607 . . . . . . . . . . . . . . 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 1232 . . . . . . . . . . . . . 14 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → (𝑃(ball‘𝐷)(𝑧 − (𝑃𝐷𝑦))) ⊆ (𝑦(ball‘𝐷)𝑧))
39 simpll3 1023 . . . . . . . . . . . . . . 15 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → 𝑟 ∈ ℝ*)
40 simprrr 530 . . . . . . . . . . . . . . . 16 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → 𝑧 < 𝑟)
4124, 39, 40xrltled 9613 . . . . . . . . . . . . . . 15 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → 𝑧𝑟)
42 ssblps 12626 . . . . . . . . . . . . . . 15 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋) ∧ (𝑧 ∈ ℝ*𝑟 ∈ ℝ*) ∧ 𝑧𝑟) → (𝑦(ball‘𝐷)𝑧) ⊆ (𝑦(ball‘𝐷)𝑟))
4313, 15, 24, 39, 41, 42syl221anc 1228 . . . . . . . . . . . . . 14 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → (𝑦(ball‘𝐷)𝑧) ⊆ (𝑦(ball‘𝐷)𝑟))
4438, 43sstrd 3110 . . . . . . . . . . . . 13 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → (𝑃(ball‘𝐷)(𝑧 − (𝑃𝐷𝑦))) ⊆ (𝑦(ball‘𝐷)𝑟))
45 oveq2 5788 . . . . . . . . . . . . . . 15 (𝑥 = (𝑧 − (𝑃𝐷𝑦)) → (𝑃(ball‘𝐷)𝑥) = (𝑃(ball‘𝐷)(𝑧 − (𝑃𝐷𝑦))))
4645sseq1d 3129 . . . . . . . . . . . . . 14 (𝑥 = (𝑧 − (𝑃𝐷𝑦)) → ((𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟) ↔ (𝑃(ball‘𝐷)(𝑧 − (𝑃𝐷𝑦))) ⊆ (𝑦(ball‘𝐷)𝑟)))
4746rspcev 2792 . . . . . . . . . . . . 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 409 . . . . . . . . . . . 12 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟))
4948expr 373 . . . . . . . . . . 11 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ 𝑧 ∈ ℝ) → (((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟)))
5012, 49sylan2 284 . . . . . . . . . 10 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ 𝑧 ∈ ℚ) → (((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟)))
5150rexlimdva 2552 . . . . . . . . 9 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) → (∃𝑧 ∈ ℚ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟)))
5211, 51syld 45 . . . . . . . 8 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) → ((𝑦𝐷𝑃) < 𝑟 → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟)))
5352expimpd 361 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) → ((𝑃𝑋 ∧ (𝑦𝐷𝑃) < 𝑟) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟)))
542, 53sylbid 149 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) → (𝑃 ∈ (𝑦(ball‘𝐷)𝑟) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟)))
55 eleq2 2204 . . . . . . 7 (𝐵 = (𝑦(ball‘𝐷)𝑟) → (𝑃𝐵𝑃 ∈ (𝑦(ball‘𝐷)𝑟)))
56 sseq2 3124 . . . . . . . 8 (𝐵 = (𝑦(ball‘𝐷)𝑟) → ((𝑃(ball‘𝐷)𝑥) ⊆ 𝐵 ↔ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟)))
5756rexbidv 2439 . . . . . . 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 233 . . . . . 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 156 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) → (𝐵 = (𝑦(ball‘𝐷)𝑟) → (𝑃𝐵 → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐵)))
60593expib 1185 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → ((𝑦𝑋𝑟 ∈ ℝ*) → (𝐵 = (𝑦(ball‘𝐷)𝑟) → (𝑃𝐵 → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐵))))
6160rexlimdvv 2559 . . 3 (𝐷 ∈ (PsMet‘𝑋) → (∃𝑦𝑋𝑟 ∈ ℝ* 𝐵 = (𝑦(ball‘𝐷)𝑟) → (𝑃𝐵 → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐵)))
621, 61sylbid 149 . 2 (𝐷 ∈ (PsMet‘𝑋) → (𝐵 ∈ ran (ball‘𝐷) → (𝑃𝐵 → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐵)))
63623imp 1176 1 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐵 ∈ ran (ball‘𝐷) ∧ 𝑃𝐵) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 963   = wceq 1332   ∈ wcel 1481  ∃wrex 2418   ⊆ wss 3074   class class class wbr 3935  ran crn 4546  ‘cfv 5129  (class class class)co 5780  ℝcr 7641  ℝ*cxr 7821   < clt 7822   ≤ cle 7823   − cmin 7955  ℚcq 9436  ℝ+crp 9468  PsMetcpsmet 12180  ballcbl 12183 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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4052  ax-pow 4104  ax-pr 4137  ax-un 4361  ax-setind 4458  ax-cnex 7733  ax-resscn 7734  ax-1cn 7735  ax-1re 7736  ax-icn 7737  ax-addcl 7738  ax-addrcl 7739  ax-mulcl 7740  ax-mulrcl 7741  ax-addcom 7742  ax-mulcom 7743  ax-addass 7744  ax-mulass 7745  ax-distr 7746  ax-i2m1 7747  ax-0lt1 7748  ax-1rid 7749  ax-0id 7750  ax-rnegex 7751  ax-precex 7752  ax-cnre 7753  ax-pre-ltirr 7754  ax-pre-ltwlin 7755  ax-pre-lttrn 7756  ax-pre-apti 7757  ax-pre-ltadd 7758  ax-pre-mulgt0 7759  ax-pre-mulext 7760  ax-arch 7761 This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3076  df-un 3078  df-in 3080  df-ss 3087  df-if 3478  df-pw 3515  df-sn 3536  df-pr 3537  df-op 3539  df-uni 3743  df-int 3778  df-iun 3821  df-br 3936  df-opab 3996  df-mpt 3997  df-id 4221  df-po 4224  df-iso 4225  df-xp 4551  df-rel 4552  df-cnv 4553  df-co 4554  df-dm 4555  df-rn 4556  df-res 4557  df-ima 4558  df-iota 5094  df-fun 5131  df-fn 5132  df-f 5133  df-fv 5137  df-riota 5736  df-ov 5783  df-oprab 5784  df-mpo 5785  df-1st 6044  df-2nd 6045  df-map 6550  df-pnf 7824  df-mnf 7825  df-xr 7826  df-ltxr 7827  df-le 7828  df-sub 7957  df-neg 7958  df-reap 8359  df-ap 8366  df-div 8455  df-inn 8743  df-2 8801  df-n0 9000  df-z 9077  df-uz 9349  df-q 9437  df-rp 9469  df-xneg 9587  df-xadd 9588  df-psmet 12188  df-bl 12191 This theorem is referenced by:  blssexps  12630
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