MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  blssps Structured version   Visualization version   GIF version
Theorem blssps 24281
Description: Any point 𝑃 in a ball 𝐡 can be centered in another ball that is a subset of 𝐡. (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 ∈ (PsMetβ€˜π‘‹) ∧ 𝐡 ∈ ran (ballβ€˜π·) ∧ 𝑃 ∈ 𝐡) β†’ βˆƒπ‘₯ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘₯) βŠ† 𝐡)
Distinct variable groups:   π‘₯,𝐡   π‘₯,𝐷   π‘₯,𝑃   π‘₯,𝑋
Proof of Theorem blssps
Dummy variables π‘Ÿ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 blrnps 24265 . . 3 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ (𝐡 ∈ ran (ballβ€˜π·) ↔ βˆƒπ‘¦ ∈ 𝑋 βˆƒπ‘Ÿ ∈ ℝ* 𝐡 = (𝑦(ballβ€˜π·)π‘Ÿ)))
2 elblps 24244 . . . . . . 7 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) β†’ (𝑃 ∈ (𝑦(ballβ€˜π·)π‘Ÿ) ↔ (𝑃 ∈ 𝑋 ∧ (𝑦𝐷𝑃) < π‘Ÿ)))
3 simpl1 1188 . . . . . . . . . . 11 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) β†’ 𝐷 ∈ (PsMetβ€˜π‘‹))
4 simpl2 1189 . . . . . . . . . . 11 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) β†’ 𝑦 ∈ 𝑋)
5 simpr 484 . . . . . . . . . . 11 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) β†’ 𝑃 ∈ 𝑋)
6 psmetcl 24164 . . . . . . . . . . 11 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ 𝑃 ∈ 𝑋) β†’ (𝑦𝐷𝑃) ∈ ℝ*)
73, 4, 5, 6syl3anc 1368 . . . . . . . . . 10 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) β†’ (𝑦𝐷𝑃) ∈ ℝ*)
8 simpl3 1190 . . . . . . . . . 10 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) β†’ π‘Ÿ ∈ ℝ*)
9 qbtwnxr 13182 . . . . . . . . . . 11 (((𝑦𝐷𝑃) ∈ ℝ* ∧ π‘Ÿ ∈ ℝ* ∧ (𝑦𝐷𝑃) < π‘Ÿ) β†’ βˆƒπ‘§ ∈ β„š ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))
1093expia 1118 . . . . . . . . . 10 (((𝑦𝐷𝑃) ∈ ℝ* ∧ π‘Ÿ ∈ ℝ*) β†’ ((𝑦𝐷𝑃) < π‘Ÿ β†’ βˆƒπ‘§ ∈ β„š ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ)))
117, 8, 10syl2anc 583 . . . . . . . . 9 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) β†’ ((𝑦𝐷𝑃) < π‘Ÿ β†’ βˆƒπ‘§ ∈ β„š ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ)))
12 qre 12938 . . . . . . . . . . 11 (𝑧 ∈ β„š β†’ 𝑧 ∈ ℝ)
13 simpll1 1209 . . . . . . . . . . . . . . . 16 ((((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ 𝐷 ∈ (PsMetβ€˜π‘‹))
14 simplr 766 . . . . . . . . . . . . . . . 16 ((((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ 𝑃 ∈ 𝑋)
15 simpll2 1210 . . . . . . . . . . . . . . . 16 ((((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ 𝑦 ∈ 𝑋)
16 psmetsym 24167 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ (𝑃𝐷𝑦) = (𝑦𝐷𝑃))
1713, 14, 15, 16syl3anc 1368 . . . . . . . . . . . . . . 15 ((((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ (𝑃𝐷𝑦) = (𝑦𝐷𝑃))
18 simprrl 778 . . . . . . . . . . . . . . 15 ((((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ (𝑦𝐷𝑃) < 𝑧)
1917, 18eqbrtrd 5163 . . . . . . . . . . . . . 14 ((((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ (𝑃𝐷𝑦) < 𝑧)
20 simprl 768 . . . . . . . . . . . . . . . 16 ((((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ 𝑧 ∈ ℝ)
21 psmetcl 24164 . . . . . . . . . . . . . . . . . 18 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ (𝑃𝐷𝑦) ∈ ℝ*)
2213, 14, 15, 21syl3anc 1368 . . . . . . . . . . . . . . . . 17 ((((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ (𝑃𝐷𝑦) ∈ ℝ*)
23 rexr 11261 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ ℝ β†’ 𝑧 ∈ ℝ*)
2423ad2antrl 725 . . . . . . . . . . . . . . . . 17 ((((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ 𝑧 ∈ ℝ*)
2522, 24, 19xrltled 13132 . . . . . . . . . . . . . . . 16 ((((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ (𝑃𝐷𝑦) ≀ 𝑧)
26 psmetlecl 24172 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝑃 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ (𝑃𝐷𝑦) ≀ 𝑧)) β†’ (𝑃𝐷𝑦) ∈ ℝ)
2713, 14, 15, 20, 25, 26syl122anc 1376 . . . . . . . . . . . . . . 15 ((((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ (𝑃𝐷𝑦) ∈ ℝ)
28 difrp 13015 . . . . . . . . . . . . . . 15 (((𝑃𝐷𝑦) ∈ ℝ ∧ 𝑧 ∈ ℝ) β†’ ((𝑃𝐷𝑦) < 𝑧 ↔ (𝑧 βˆ’ (𝑃𝐷𝑦)) ∈ ℝ+))
2927, 20, 28syl2anc 583 . . . . . . . . . . . . . 14 ((((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ ((𝑃𝐷𝑦) < 𝑧 ↔ (𝑧 βˆ’ (𝑃𝐷𝑦)) ∈ ℝ+))
3019, 29mpbid 231 . . . . . . . . . . . . 13 ((((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ (𝑧 βˆ’ (𝑃𝐷𝑦)) ∈ ℝ+)
3120, 27resubcld 11643 . . . . . . . . . . . . . . 15 ((((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ (𝑧 βˆ’ (𝑃𝐷𝑦)) ∈ ℝ)
3222xrleidd 13134 . . . . . . . . . . . . . . . 16 ((((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ (𝑃𝐷𝑦) ≀ (𝑃𝐷𝑦))
3320recnd 11243 . . . . . . . . . . . . . . . . 17 ((((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ 𝑧 ∈ β„‚)
3427recnd 11243 . . . . . . . . . . . . . . . . 17 ((((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ (𝑃𝐷𝑦) ∈ β„‚)
3533, 34nncand 11577 . . . . . . . . . . . . . . . 16 ((((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ (𝑧 βˆ’ (𝑧 βˆ’ (𝑃𝐷𝑦))) = (𝑃𝐷𝑦))
3632, 35breqtrrd 5169 . . . . . . . . . . . . . . 15 ((((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ (𝑃𝐷𝑦) ≀ (𝑧 βˆ’ (𝑧 βˆ’ (𝑃𝐷𝑦))))
37 blss2ps 24260 . . . . . . . . . . . . . . 15 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ ((𝑧 βˆ’ (𝑃𝐷𝑦)) ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (𝑃𝐷𝑦) ≀ (𝑧 βˆ’ (𝑧 βˆ’ (𝑃𝐷𝑦))))) β†’ (𝑃(ballβ€˜π·)(𝑧 βˆ’ (𝑃𝐷𝑦))) βŠ† (𝑦(ballβ€˜π·)𝑧))
3813, 14, 15, 31, 20, 36, 37syl33anc 1382 . . . . . . . . . . . . . 14 ((((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ (𝑃(ballβ€˜π·)(𝑧 βˆ’ (𝑃𝐷𝑦))) βŠ† (𝑦(ballβ€˜π·)𝑧))
39 simpll3 1211 . . . . . . . . . . . . . . 15 ((((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ π‘Ÿ ∈ ℝ*)
40 simprrr 779 . . . . . . . . . . . . . . . 16 ((((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ 𝑧 < π‘Ÿ)
4124, 39, 40xrltled 13132 . . . . . . . . . . . . . . 15 ((((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ 𝑧 ≀ π‘Ÿ)
42 ssblps 24279 . . . . . . . . . . . . . . 15 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑧 ∈ ℝ* ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑧 ≀ π‘Ÿ) β†’ (𝑦(ballβ€˜π·)𝑧) βŠ† (𝑦(ballβ€˜π·)π‘Ÿ))
4313, 15, 24, 39, 41, 42syl221anc 1378 . . . . . . . . . . . . . 14 ((((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ (𝑦(ballβ€˜π·)𝑧) βŠ† (𝑦(ballβ€˜π·)π‘Ÿ))
4438, 43sstrd 3987 . . . . . . . . . . . . 13 ((((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ (𝑃(ballβ€˜π·)(𝑧 βˆ’ (𝑃𝐷𝑦))) βŠ† (𝑦(ballβ€˜π·)π‘Ÿ))
45 oveq2 7412 . . . . . . . . . . . . . . 15 (π‘₯ = (𝑧 βˆ’ (𝑃𝐷𝑦)) β†’ (𝑃(ballβ€˜π·)π‘₯) = (𝑃(ballβ€˜π·)(𝑧 βˆ’ (𝑃𝐷𝑦))))
4645sseq1d 4008 . . . . . . . . . . . . . 14 (π‘₯ = (𝑧 βˆ’ (𝑃𝐷𝑦)) β†’ ((𝑃(ballβ€˜π·)π‘₯) βŠ† (𝑦(ballβ€˜π·)π‘Ÿ) ↔ (𝑃(ballβ€˜π·)(𝑧 βˆ’ (𝑃𝐷𝑦))) βŠ† (𝑦(ballβ€˜π·)π‘Ÿ)))
4746rspcev 3606 . . . . . . . . . . . . 13 (((𝑧 βˆ’ (𝑃𝐷𝑦)) ∈ ℝ+ ∧ (𝑃(ballβ€˜π·)(𝑧 βˆ’ (𝑃𝐷𝑦))) βŠ† (𝑦(ballβ€˜π·)π‘Ÿ)) β†’ βˆƒπ‘₯ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘₯) βŠ† (𝑦(ballβ€˜π·)π‘Ÿ))
4830, 44, 47syl2anc 583 . . . . . . . . . . . 12 ((((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ βˆƒπ‘₯ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘₯) βŠ† (𝑦(ballβ€˜π·)π‘Ÿ))
4948expr 456 . . . . . . . . . . 11 ((((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ 𝑧 ∈ ℝ) β†’ (((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ) β†’ βˆƒπ‘₯ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘₯) βŠ† (𝑦(ballβ€˜π·)π‘Ÿ)))
5012, 49sylan2 592 . . . . . . . . . 10 ((((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ 𝑧 ∈ β„š) β†’ (((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ) β†’ βˆƒπ‘₯ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘₯) βŠ† (𝑦(ballβ€˜π·)π‘Ÿ)))
5150rexlimdva 3149 . . . . . . . . 9 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) β†’ (βˆƒπ‘§ ∈ β„š ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ) β†’ βˆƒπ‘₯ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘₯) βŠ† (𝑦(ballβ€˜π·)π‘Ÿ)))
5211, 51syld 47 . . . . . . . 8 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) β†’ ((𝑦𝐷𝑃) < π‘Ÿ β†’ βˆƒπ‘₯ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘₯) βŠ† (𝑦(ballβ€˜π·)π‘Ÿ)))
5352expimpd 453 . . . . . . 7 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) β†’ ((𝑃 ∈ 𝑋 ∧ (𝑦𝐷𝑃) < π‘Ÿ) β†’ βˆƒπ‘₯ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘₯) βŠ† (𝑦(ballβ€˜π·)π‘Ÿ)))
542, 53sylbid 239 . . . . . 6 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) β†’ (𝑃 ∈ (𝑦(ballβ€˜π·)π‘Ÿ) β†’ βˆƒπ‘₯ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘₯) βŠ† (𝑦(ballβ€˜π·)π‘Ÿ)))
55 eleq2 2816 . . . . . . 7 (𝐡 = (𝑦(ballβ€˜π·)π‘Ÿ) β†’ (𝑃 ∈ 𝐡 ↔ 𝑃 ∈ (𝑦(ballβ€˜π·)π‘Ÿ)))
56 sseq2 4003 . . . . . . . 8 (𝐡 = (𝑦(ballβ€˜π·)π‘Ÿ) β†’ ((𝑃(ballβ€˜π·)π‘₯) βŠ† 𝐡 ↔ (𝑃(ballβ€˜π·)π‘₯) βŠ† (𝑦(ballβ€˜π·)π‘Ÿ)))
5756rexbidv 3172 . . . . . . 7 (𝐡 = (𝑦(ballβ€˜π·)π‘Ÿ) β†’ (βˆƒπ‘₯ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘₯) βŠ† 𝐡 ↔ βˆƒπ‘₯ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘₯) βŠ† (𝑦(ballβ€˜π·)π‘Ÿ)))
5855, 57imbi12d 344 . . . . . 6 (𝐡 = (𝑦(ballβ€˜π·)π‘Ÿ) β†’ ((𝑃 ∈ 𝐡 β†’ βˆƒπ‘₯ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘₯) βŠ† 𝐡) ↔ (𝑃 ∈ (𝑦(ballβ€˜π·)π‘Ÿ) β†’ βˆƒπ‘₯ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘₯) βŠ† (𝑦(ballβ€˜π·)π‘Ÿ))))
5954, 58syl5ibrcom 246 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) β†’ (𝐡 = (𝑦(ballβ€˜π·)π‘Ÿ) β†’ (𝑃 ∈ 𝐡 β†’ βˆƒπ‘₯ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘₯) βŠ† 𝐡)))
60593expib 1119 . . . 4 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ ((𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) β†’ (𝐡 = (𝑦(ballβ€˜π·)π‘Ÿ) β†’ (𝑃 ∈ 𝐡 β†’ βˆƒπ‘₯ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘₯) βŠ† 𝐡))))
6160rexlimdvv 3204 . . 3 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ (βˆƒπ‘¦ ∈ 𝑋 βˆƒπ‘Ÿ ∈ ℝ* 𝐡 = (𝑦(ballβ€˜π·)π‘Ÿ) β†’ (𝑃 ∈ 𝐡 β†’ βˆƒπ‘₯ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘₯) βŠ† 𝐡)))
621, 61sylbid 239 . 2 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ (𝐡 ∈ ran (ballβ€˜π·) β†’ (𝑃 ∈ 𝐡 β†’ βˆƒπ‘₯ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘₯) βŠ† 𝐡)))
63623imp 1108 1 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝐡 ∈ ran (ballβ€˜π·) ∧ 𝑃 ∈ 𝐡) β†’ βˆƒπ‘₯ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘₯) βŠ† 𝐡)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3064   βŠ† wss 3943   class class class wbr 5141  ran crn 5670  β€˜cfv 6536  (class class class)co 7404  β„cr 11108  β„*cxr 11248   < clt 11249   ≀ cle 11250   βˆ’ cmin 11445  β„šcq 12933  β„+crp 12977  PsMetcpsmet 21220  ballcbl 21223
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186  ax-pre-sup 11187
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-sup 9436  df-inf 9437  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11447  df-neg 11448  df-div 11873  df-nn 12214  df-2 12276  df-n0 12474  df-z 12560  df-uz 12824  df-q 12934  df-rp 12978  df-xneg 13095  df-xadd 13096  df-xmul 13097  df-psmet 21228  df-bl 21231
This theorem is referenced by:  blssexps  24283
  Copyright terms: Public domain W3C validator