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Theorem blssps 22039
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 22023 . . 3 (𝐷 ∈ (PsMet‘𝑋) → (𝐵 ∈ ran (ball‘𝐷) ↔ ∃𝑦𝑋𝑟 ∈ ℝ* 𝐵 = (𝑦(ball‘𝐷)𝑟)))
2 elblps 22002 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) → (𝑃 ∈ (𝑦(ball‘𝐷)𝑟) ↔ (𝑃𝑋 ∧ (𝑦𝐷𝑃) < 𝑟)))
3 simpl1 1057 . . . . . . . . . . 11 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) → 𝐷 ∈ (PsMet‘𝑋))
4 simpl2 1058 . . . . . . . . . . 11 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) → 𝑦𝑋)
5 simpr 476 . . . . . . . . . . 11 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) → 𝑃𝑋)
6 psmetcl 21922 . . . . . . . . . . 11 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑃𝑋) → (𝑦𝐷𝑃) ∈ ℝ*)
73, 4, 5, 6syl3anc 1318 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) → (𝑦𝐷𝑃) ∈ ℝ*)
8 simpl3 1059 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) → 𝑟 ∈ ℝ*)
9 qbtwnxr 11905 . . . . . . . . . . 11 (((𝑦𝐷𝑃) ∈ ℝ*𝑟 ∈ ℝ* ∧ (𝑦𝐷𝑃) < 𝑟) → ∃𝑧 ∈ ℚ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))
1093expia 1259 . . . . . . . . . 10 (((𝑦𝐷𝑃) ∈ ℝ*𝑟 ∈ ℝ*) → ((𝑦𝐷𝑃) < 𝑟 → ∃𝑧 ∈ ℚ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟)))
117, 8, 10syl2anc 691 . . . . . . . . 9 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) → ((𝑦𝐷𝑃) < 𝑟 → ∃𝑧 ∈ ℚ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟)))
12 qre 11669 . . . . . . . . . . 11 (𝑧 ∈ ℚ → 𝑧 ∈ ℝ)
13 simpll1 1093 . . . . . . . . . . . . . . . 16 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → 𝐷 ∈ (PsMet‘𝑋))
14 simplr 788 . . . . . . . . . . . . . . . 16 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → 𝑃𝑋)
15 simpll2 1094 . . . . . . . . . . . . . . . 16 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → 𝑦𝑋)
16 psmetsym 21925 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑦𝑋) → (𝑃𝐷𝑦) = (𝑦𝐷𝑃))
1713, 14, 15, 16syl3anc 1318 . . . . . . . . . . . . . . 15 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → (𝑃𝐷𝑦) = (𝑦𝐷𝑃))
18 simprrl 800 . . . . . . . . . . . . . . 15 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → (𝑦𝐷𝑃) < 𝑧)
1917, 18eqbrtrd 4605 . . . . . . . . . . . . . 14 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → (𝑃𝐷𝑦) < 𝑧)
20 simprl 790 . . . . . . . . . . . . . . . 16 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → 𝑧 ∈ ℝ)
21 psmetcl 21922 . . . . . . . . . . . . . . . . . . 19 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑦𝑋) → (𝑃𝐷𝑦) ∈ ℝ*)
2213, 14, 15, 21syl3anc 1318 . . . . . . . . . . . . . . . . . 18 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → (𝑃𝐷𝑦) ∈ ℝ*)
23 rexr 9964 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ ℝ → 𝑧 ∈ ℝ*)
2423ad2antrl 760 . . . . . . . . . . . . . . . . . 18 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → 𝑧 ∈ ℝ*)
25 xrltle 11858 . . . . . . . . . . . . . . . . . 18 (((𝑃𝐷𝑦) ∈ ℝ*𝑧 ∈ ℝ*) → ((𝑃𝐷𝑦) < 𝑧 → (𝑃𝐷𝑦) ≤ 𝑧))
2622, 24, 25syl2anc 691 . . . . . . . . . . . . . . . . 17 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → ((𝑃𝐷𝑦) < 𝑧 → (𝑃𝐷𝑦) ≤ 𝑧))
2719, 26mpd 15 . . . . . . . . . . . . . . . 16 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → (𝑃𝐷𝑦) ≤ 𝑧)
28 psmetlecl 21930 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑃𝑋𝑦𝑋) ∧ (𝑧 ∈ ℝ ∧ (𝑃𝐷𝑦) ≤ 𝑧)) → (𝑃𝐷𝑦) ∈ ℝ)
2913, 14, 15, 20, 27, 28syl122anc 1327 . . . . . . . . . . . . . . 15 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → (𝑃𝐷𝑦) ∈ ℝ)
30 difrp 11744 . . . . . . . . . . . . . . 15 (((𝑃𝐷𝑦) ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((𝑃𝐷𝑦) < 𝑧 ↔ (𝑧 − (𝑃𝐷𝑦)) ∈ ℝ+))
3129, 20, 30syl2anc 691 . . . . . . . . . . . . . 14 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → ((𝑃𝐷𝑦) < 𝑧 ↔ (𝑧 − (𝑃𝐷𝑦)) ∈ ℝ+))
3219, 31mpbid 221 . . . . . . . . . . . . 13 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → (𝑧 − (𝑃𝐷𝑦)) ∈ ℝ+)
3320, 29resubcld 10337 . . . . . . . . . . . . . . 15 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → (𝑧 − (𝑃𝐷𝑦)) ∈ ℝ)
34 xrleid 11859 . . . . . . . . . . . . . . . . 17 ((𝑃𝐷𝑦) ∈ ℝ* → (𝑃𝐷𝑦) ≤ (𝑃𝐷𝑦))
3522, 34syl 17 . . . . . . . . . . . . . . . 16 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → (𝑃𝐷𝑦) ≤ (𝑃𝐷𝑦))
3620recnd 9947 . . . . . . . . . . . . . . . . 17 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → 𝑧 ∈ ℂ)
3729recnd 9947 . . . . . . . . . . . . . . . . 17 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → (𝑃𝐷𝑦) ∈ ℂ)
3836, 37nncand 10276 . . . . . . . . . . . . . . . 16 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → (𝑧 − (𝑧 − (𝑃𝐷𝑦))) = (𝑃𝐷𝑦))
3935, 38breqtrrd 4611 . . . . . . . . . . . . . . 15 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → (𝑃𝐷𝑦) ≤ (𝑧 − (𝑧 − (𝑃𝐷𝑦))))
40 blss2ps 22018 . . . . . . . . . . . . . . 15 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑦𝑋) ∧ ((𝑧 − (𝑃𝐷𝑦)) ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (𝑃𝐷𝑦) ≤ (𝑧 − (𝑧 − (𝑃𝐷𝑦))))) → (𝑃(ball‘𝐷)(𝑧 − (𝑃𝐷𝑦))) ⊆ (𝑦(ball‘𝐷)𝑧))
4113, 14, 15, 33, 20, 39, 40syl33anc 1333 . . . . . . . . . . . . . 14 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → (𝑃(ball‘𝐷)(𝑧 − (𝑃𝐷𝑦))) ⊆ (𝑦(ball‘𝐷)𝑧))
42 simpll3 1095 . . . . . . . . . . . . . . 15 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → 𝑟 ∈ ℝ*)
43 simprrr 801 . . . . . . . . . . . . . . . 16 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → 𝑧 < 𝑟)
44 xrltle 11858 . . . . . . . . . . . . . . . . 17 ((𝑧 ∈ ℝ*𝑟 ∈ ℝ*) → (𝑧 < 𝑟𝑧𝑟))
4524, 42, 44syl2anc 691 . . . . . . . . . . . . . . . 16 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → (𝑧 < 𝑟𝑧𝑟))
4643, 45mpd 15 . . . . . . . . . . . . . . 15 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → 𝑧𝑟)
47 ssblps 22037 . . . . . . . . . . . . . . 15 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋) ∧ (𝑧 ∈ ℝ*𝑟 ∈ ℝ*) ∧ 𝑧𝑟) → (𝑦(ball‘𝐷)𝑧) ⊆ (𝑦(ball‘𝐷)𝑟))
4813, 15, 24, 42, 46, 47syl221anc 1329 . . . . . . . . . . . . . 14 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → (𝑦(ball‘𝐷)𝑧) ⊆ (𝑦(ball‘𝐷)𝑟))
4941, 48sstrd 3578 . . . . . . . . . . . . 13 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → (𝑃(ball‘𝐷)(𝑧 − (𝑃𝐷𝑦))) ⊆ (𝑦(ball‘𝐷)𝑟))
50 oveq2 6557 . . . . . . . . . . . . . . 15 (𝑥 = (𝑧 − (𝑃𝐷𝑦)) → (𝑃(ball‘𝐷)𝑥) = (𝑃(ball‘𝐷)(𝑧 − (𝑃𝐷𝑦))))
5150sseq1d 3595 . . . . . . . . . . . . . 14 (𝑥 = (𝑧 − (𝑃𝐷𝑦)) → ((𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟) ↔ (𝑃(ball‘𝐷)(𝑧 − (𝑃𝐷𝑦))) ⊆ (𝑦(ball‘𝐷)𝑟)))
5251rspcev 3282 . . . . . . . . . . . . 13 (((𝑧 − (𝑃𝐷𝑦)) ∈ ℝ+ ∧ (𝑃(ball‘𝐷)(𝑧 − (𝑃𝐷𝑦))) ⊆ (𝑦(ball‘𝐷)𝑟)) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟))
5332, 49, 52syl2anc 691 . . . . . . . . . . . 12 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟))) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟))
5453expr 641 . . . . . . . . . . 11 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ 𝑧 ∈ ℝ) → (((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟)))
5512, 54sylan2 490 . . . . . . . . . 10 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) ∧ 𝑧 ∈ ℚ) → (((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟)))
5655rexlimdva 3013 . . . . . . . . 9 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) → (∃𝑧 ∈ ℚ ((𝑦𝐷𝑃) < 𝑧𝑧 < 𝑟) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟)))
5711, 56syld 46 . . . . . . . 8 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) ∧ 𝑃𝑋) → ((𝑦𝐷𝑃) < 𝑟 → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟)))
5857expimpd 627 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) → ((𝑃𝑋 ∧ (𝑦𝐷𝑃) < 𝑟) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟)))
592, 58sylbid 229 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) → (𝑃 ∈ (𝑦(ball‘𝐷)𝑟) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟)))
60 eleq2 2677 . . . . . . 7 (𝐵 = (𝑦(ball‘𝐷)𝑟) → (𝑃𝐵𝑃 ∈ (𝑦(ball‘𝐷)𝑟)))
61 sseq2 3590 . . . . . . . 8 (𝐵 = (𝑦(ball‘𝐷)𝑟) → ((𝑃(ball‘𝐷)𝑥) ⊆ 𝐵 ↔ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟)))
6261rexbidv 3034 . . . . . . 7 (𝐵 = (𝑦(ball‘𝐷)𝑟) → (∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐵 ↔ ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑦(ball‘𝐷)𝑟)))
6360, 62imbi12d 333 . . . . . 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))))
6459, 63syl5ibrcom 236 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) → (𝐵 = (𝑦(ball‘𝐷)𝑟) → (𝑃𝐵 → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐵)))
65643expib 1260 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → ((𝑦𝑋𝑟 ∈ ℝ*) → (𝐵 = (𝑦(ball‘𝐷)𝑟) → (𝑃𝐵 → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐵))))
6665rexlimdvv 3019 . . 3 (𝐷 ∈ (PsMet‘𝑋) → (∃𝑦𝑋𝑟 ∈ ℝ* 𝐵 = (𝑦(ball‘𝐷)𝑟) → (𝑃𝐵 → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐵)))
671, 66sylbid 229 . 2 (𝐷 ∈ (PsMet‘𝑋) → (𝐵 ∈ ran (ball‘𝐷) → (𝑃𝐵 → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐵)))
68673imp 1249 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 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wrex 2897  wss 3540   class class class wbr 4583  ran crn 5039  cfv 5804  (class class class)co 6549  cr 9814  *cxr 9952   < clt 9953  cle 9954  cmin 10145  cq 11664  +crp 11708  PsMetcpsmet 19551  ballcbl 19554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-sup 8231  df-inf 8232  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-psmet 19559  df-bl 19562
This theorem is referenced by:  blssexps  22041
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