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Theorem blssps 24343
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 24327 . . 3 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ (𝐡 ∈ ran (ballβ€˜π·) ↔ βˆƒπ‘¦ ∈ 𝑋 βˆƒπ‘Ÿ ∈ ℝ* 𝐡 = (𝑦(ballβ€˜π·)π‘Ÿ)))
2 elblps 24306 . . . . . . 7 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) β†’ (𝑃 ∈ (𝑦(ballβ€˜π·)π‘Ÿ) ↔ (𝑃 ∈ 𝑋 ∧ (𝑦𝐷𝑃) < π‘Ÿ)))
3 simpl1 1189 . . . . . . . . . . 11 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) β†’ 𝐷 ∈ (PsMetβ€˜π‘‹))
4 simpl2 1190 . . . . . . . . . . 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 24226 . . . . . . . . . . 11 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ 𝑃 ∈ 𝑋) β†’ (𝑦𝐷𝑃) ∈ ℝ*)
73, 4, 5, 6syl3anc 1369 . . . . . . . . . 10 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) β†’ (𝑦𝐷𝑃) ∈ ℝ*)
8 simpl3 1191 . . . . . . . . . 10 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) β†’ π‘Ÿ ∈ ℝ*)
9 qbtwnxr 13212 . . . . . . . . . . 11 (((𝑦𝐷𝑃) ∈ ℝ* ∧ π‘Ÿ ∈ ℝ* ∧ (𝑦𝐷𝑃) < π‘Ÿ) β†’ βˆƒπ‘§ ∈ β„š ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))
1093expia 1119 . . . . . . . . . 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 12968 . . . . . . . . . . 11 (𝑧 ∈ β„š β†’ 𝑧 ∈ ℝ)
13 simpll1 1210 . . . . . . . . . . . . . . . 16 ((((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ 𝐷 ∈ (PsMetβ€˜π‘‹))
14 simplr 768 . . . . . . . . . . . . . . . 16 ((((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ 𝑃 ∈ 𝑋)
15 simpll2 1211 . . . . . . . . . . . . . . . 16 ((((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ 𝑦 ∈ 𝑋)
16 psmetsym 24229 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ (𝑃𝐷𝑦) = (𝑦𝐷𝑃))
1713, 14, 15, 16syl3anc 1369 . . . . . . . . . . . . . . 15 ((((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ (𝑃𝐷𝑦) = (𝑦𝐷𝑃))
18 simprrl 780 . . . . . . . . . . . . . . 15 ((((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ (𝑦𝐷𝑃) < 𝑧)
1917, 18eqbrtrd 5170 . . . . . . . . . . . . . 14 ((((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ (𝑃𝐷𝑦) < 𝑧)
20 simprl 770 . . . . . . . . . . . . . . . 16 ((((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ 𝑧 ∈ ℝ)
21 psmetcl 24226 . . . . . . . . . . . . . . . . . 18 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ (𝑃𝐷𝑦) ∈ ℝ*)
2213, 14, 15, 21syl3anc 1369 . . . . . . . . . . . . . . . . 17 ((((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ (𝑃𝐷𝑦) ∈ ℝ*)
23 rexr 11291 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ ℝ β†’ 𝑧 ∈ ℝ*)
2423ad2antrl 727 . . . . . . . . . . . . . . . . 17 ((((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ 𝑧 ∈ ℝ*)
2522, 24, 19xrltled 13162 . . . . . . . . . . . . . . . 16 ((((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ (𝑃𝐷𝑦) ≀ 𝑧)
26 psmetlecl 24234 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝑃 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ (𝑃𝐷𝑦) ≀ 𝑧)) β†’ (𝑃𝐷𝑦) ∈ ℝ)
2713, 14, 15, 20, 25, 26syl122anc 1377 . . . . . . . . . . . . . . 15 ((((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ (𝑃𝐷𝑦) ∈ ℝ)
28 difrp 13045 . . . . . . . . . . . . . . 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 11673 . . . . . . . . . . . . . . 15 ((((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ (𝑧 βˆ’ (𝑃𝐷𝑦)) ∈ ℝ)
3222xrleidd 13164 . . . . . . . . . . . . . . . 16 ((((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ (𝑃𝐷𝑦) ≀ (𝑃𝐷𝑦))
3320recnd 11273 . . . . . . . . . . . . . . . . 17 ((((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ 𝑧 ∈ β„‚)
3427recnd 11273 . . . . . . . . . . . . . . . . 17 ((((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ (𝑃𝐷𝑦) ∈ β„‚)
3533, 34nncand 11607 . . . . . . . . . . . . . . . 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 5176 . . . . . . . . . . . . . . 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 24322 . . . . . . . . . . . . . . 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 1383 . . . . . . . . . . . . . 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 1212 . . . . . . . . . . . . . . 15 ((((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ π‘Ÿ ∈ ℝ*)
40 simprrr 781 . . . . . . . . . . . . . . . 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 13162 . . . . . . . . . . . . . . 15 ((((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ 𝑧 ≀ π‘Ÿ)
42 ssblps 24341 . . . . . . . . . . . . . . 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 1379 . . . . . . . . . . . . . 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 3990 . . . . . . . . . . . . 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 7428 . . . . . . . . . . . . . . 15 (π‘₯ = (𝑧 βˆ’ (𝑃𝐷𝑦)) β†’ (𝑃(ballβ€˜π·)π‘₯) = (𝑃(ballβ€˜π·)(𝑧 βˆ’ (𝑃𝐷𝑦))))
4645sseq1d 4011 . . . . . . . . . . . . . 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 3609 . . . . . . . . . . . . 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 3152 . . . . . . . . 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 2818 . . . . . . 7 (𝐡 = (𝑦(ballβ€˜π·)π‘Ÿ) β†’ (𝑃 ∈ 𝐡 ↔ 𝑃 ∈ (𝑦(ballβ€˜π·)π‘Ÿ)))
56 sseq2 4006 . . . . . . . 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 3175 . . . . . . 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 1120 . . . 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 3207 . . 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 1109 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 1085   = wceq 1534   ∈ wcel 2099  βˆƒwrex 3067   βŠ† wss 3947   class class class wbr 5148  ran crn 5679  β€˜cfv 6548  (class class class)co 7420  β„cr 11138  β„*cxr 11278   < clt 11279   ≀ cle 11280   βˆ’ cmin 11475  β„šcq 12963  β„+crp 13007  PsMetcpsmet 21263  ballcbl 21266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-cnex 11195  ax-resscn 11196  ax-1cn 11197  ax-icn 11198  ax-addcl 11199  ax-addrcl 11200  ax-mulcl 11201  ax-mulrcl 11202  ax-mulcom 11203  ax-addass 11204  ax-mulass 11205  ax-distr 11206  ax-i2m1 11207  ax-1ne0 11208  ax-1rid 11209  ax-rnegex 11210  ax-rrecex 11211  ax-cnre 11212  ax-pre-lttri 11213  ax-pre-lttrn 11214  ax-pre-ltadd 11215  ax-pre-mulgt0 11216  ax-pre-sup 11217
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-1st 7993  df-2nd 7994  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-er 8725  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-sup 9466  df-inf 9467  df-pnf 11281  df-mnf 11282  df-xr 11283  df-ltxr 11284  df-le 11285  df-sub 11477  df-neg 11478  df-div 11903  df-nn 12244  df-2 12306  df-n0 12504  df-z 12590  df-uz 12854  df-q 12964  df-rp 13008  df-xneg 13125  df-xadd 13126  df-xmul 13127  df-psmet 21271  df-bl 21274
This theorem is referenced by:  blssexps  24345
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