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Theorem blss 23794
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.)
Assertion
Ref Expression
blss ((𝐷 ∈ (∞Metβ€˜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 blss
Dummy variables π‘Ÿ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 blrn 23778 . . 3 (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ (𝐡 ∈ ran (ballβ€˜π·) ↔ βˆƒπ‘¦ ∈ 𝑋 βˆƒπ‘Ÿ ∈ ℝ* 𝐡 = (𝑦(ballβ€˜π·)π‘Ÿ)))
2 elbl 23757 . . . . . . 7 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) β†’ (𝑃 ∈ (𝑦(ballβ€˜π·)π‘Ÿ) ↔ (𝑃 ∈ 𝑋 ∧ (𝑦𝐷𝑃) < π‘Ÿ)))
3 simpl1 1192 . . . . . . . . . . 11 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) β†’ 𝐷 ∈ (∞Metβ€˜π‘‹))
4 simpl2 1193 . . . . . . . . . . 11 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) β†’ 𝑦 ∈ 𝑋)
5 simpr 486 . . . . . . . . . . 11 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) β†’ 𝑃 ∈ 𝑋)
6 xmetcl 23700 . . . . . . . . . . 11 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ 𝑃 ∈ 𝑋) β†’ (𝑦𝐷𝑃) ∈ ℝ*)
73, 4, 5, 6syl3anc 1372 . . . . . . . . . 10 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) β†’ (𝑦𝐷𝑃) ∈ ℝ*)
8 simpl3 1194 . . . . . . . . . 10 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) β†’ π‘Ÿ ∈ ℝ*)
9 qbtwnxr 13125 . . . . . . . . . . 11 (((𝑦𝐷𝑃) ∈ ℝ* ∧ π‘Ÿ ∈ ℝ* ∧ (𝑦𝐷𝑃) < π‘Ÿ) β†’ βˆƒπ‘§ ∈ β„š ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))
1093expia 1122 . . . . . . . . . 10 (((𝑦𝐷𝑃) ∈ ℝ* ∧ π‘Ÿ ∈ ℝ*) β†’ ((𝑦𝐷𝑃) < π‘Ÿ β†’ βˆƒπ‘§ ∈ β„š ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ)))
117, 8, 10syl2anc 585 . . . . . . . . 9 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) β†’ ((𝑦𝐷𝑃) < π‘Ÿ β†’ βˆƒπ‘§ ∈ β„š ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ)))
12 qre 12883 . . . . . . . . . . 11 (𝑧 ∈ β„š β†’ 𝑧 ∈ ℝ)
13 simpll1 1213 . . . . . . . . . . . . . . . 16 ((((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ 𝐷 ∈ (∞Metβ€˜π‘‹))
14 simplr 768 . . . . . . . . . . . . . . . 16 ((((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ 𝑃 ∈ 𝑋)
15 simpll2 1214 . . . . . . . . . . . . . . . 16 ((((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ 𝑦 ∈ 𝑋)
16 xmetsym 23716 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ (𝑃𝐷𝑦) = (𝑦𝐷𝑃))
1713, 14, 15, 16syl3anc 1372 . . . . . . . . . . . . . . 15 ((((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ (𝑃𝐷𝑦) = (𝑦𝐷𝑃))
18 simprrl 780 . . . . . . . . . . . . . . 15 ((((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ (𝑦𝐷𝑃) < 𝑧)
1917, 18eqbrtrd 5128 . . . . . . . . . . . . . 14 ((((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ (𝑃𝐷𝑦) < 𝑧)
20 simprl 770 . . . . . . . . . . . . . . . 16 ((((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ 𝑧 ∈ ℝ)
21 xmetcl 23700 . . . . . . . . . . . . . . . . . 18 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ (𝑃𝐷𝑦) ∈ ℝ*)
2213, 14, 15, 21syl3anc 1372 . . . . . . . . . . . . . . . . 17 ((((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ (𝑃𝐷𝑦) ∈ ℝ*)
23 rexr 11206 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ ℝ β†’ 𝑧 ∈ ℝ*)
2423ad2antrl 727 . . . . . . . . . . . . . . . . 17 ((((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ 𝑧 ∈ ℝ*)
2522, 24, 19xrltled 13075 . . . . . . . . . . . . . . . 16 ((((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ (𝑃𝐷𝑦) ≀ 𝑧)
26 xmetlecl 23715 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ (𝑃 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ (𝑃𝐷𝑦) ≀ 𝑧)) β†’ (𝑃𝐷𝑦) ∈ ℝ)
2713, 14, 15, 20, 25, 26syl122anc 1380 . . . . . . . . . . . . . . 15 ((((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ (𝑃𝐷𝑦) ∈ ℝ)
28 difrp 12958 . . . . . . . . . . . . . . 15 (((𝑃𝐷𝑦) ∈ ℝ ∧ 𝑧 ∈ ℝ) β†’ ((𝑃𝐷𝑦) < 𝑧 ↔ (𝑧 βˆ’ (𝑃𝐷𝑦)) ∈ ℝ+))
2927, 20, 28syl2anc 585 . . . . . . . . . . . . . 14 ((((𝐷 ∈ (∞Metβ€˜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 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ (𝑧 βˆ’ (𝑃𝐷𝑦)) ∈ ℝ+)
3120, 27resubcld 11588 . . . . . . . . . . . . . . 15 ((((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ (𝑧 βˆ’ (𝑃𝐷𝑦)) ∈ ℝ)
3222xrleidd 13077 . . . . . . . . . . . . . . . 16 ((((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ (𝑃𝐷𝑦) ≀ (𝑃𝐷𝑦))
3320recnd 11188 . . . . . . . . . . . . . . . . 17 ((((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ 𝑧 ∈ β„‚)
3427recnd 11188 . . . . . . . . . . . . . . . . 17 ((((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ (𝑃𝐷𝑦) ∈ β„‚)
3533, 34nncand 11522 . . . . . . . . . . . . . . . 16 ((((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ (𝑧 βˆ’ (𝑧 βˆ’ (𝑃𝐷𝑦))) = (𝑃𝐷𝑦))
3632, 35breqtrrd 5134 . . . . . . . . . . . . . . 15 ((((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ (𝑃𝐷𝑦) ≀ (𝑧 βˆ’ (𝑧 βˆ’ (𝑃𝐷𝑦))))
37 blss2 23773 . . . . . . . . . . . . . . 15 (((𝐷 ∈ (∞Metβ€˜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 1386 . . . . . . . . . . . . . 14 ((((𝐷 ∈ (∞Metβ€˜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 1215 . . . . . . . . . . . . . . 15 ((((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ π‘Ÿ ∈ ℝ*)
40 simprrr 781 . . . . . . . . . . . . . . . 16 ((((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ 𝑧 < π‘Ÿ)
4124, 39, 40xrltled 13075 . . . . . . . . . . . . . . 15 ((((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) ∧ 𝑃 ∈ 𝑋) ∧ (𝑧 ∈ ℝ ∧ ((𝑦𝐷𝑃) < 𝑧 ∧ 𝑧 < π‘Ÿ))) β†’ 𝑧 ≀ π‘Ÿ)
42 ssbl 23792 . . . . . . . . . . . . . . 15 (((𝐷 ∈ (∞Metβ€˜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 1382 . . . . . . . . . . . . . 14 ((((𝐷 ∈ (∞Metβ€˜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 3955 . . . . . . . . . . . . 13 ((((𝐷 ∈ (∞Metβ€˜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 7366 . . . . . . . . . . . . . . 15 (π‘₯ = (𝑧 βˆ’ (𝑃𝐷𝑦)) β†’ (𝑃(ballβ€˜π·)π‘₯) = (𝑃(ballβ€˜π·)(𝑧 βˆ’ (𝑃𝐷𝑦))))
4645sseq1d 3976 . . . . . . . . . . . . . 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 3580 . . . . . . . . . . . . 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 585 . . . . . . . . . . . 12 ((((𝐷 ∈ (∞Metβ€˜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 458 . . . . . . . . . . 11 ((((𝐷 ∈ (∞Metβ€˜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 594 . . . . . . . . . 10 ((((𝐷 ∈ (∞Metβ€˜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 ∈ (∞Metβ€˜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 ∈ (∞Metβ€˜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 455 . . . . . . 7 ((𝐷 ∈ (∞Metβ€˜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 ∈ (∞Metβ€˜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 2823 . . . . . . 7 (𝐡 = (𝑦(ballβ€˜π·)π‘Ÿ) β†’ (𝑃 ∈ 𝐡 ↔ 𝑃 ∈ (𝑦(ballβ€˜π·)π‘Ÿ)))
56 sseq2 3971 . . . . . . . 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 345 . . . . . 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 247 . . . . 5 ((𝐷 ∈ (∞Metβ€˜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 1123 . . . 4 (𝐷 ∈ (∞Metβ€˜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 3201 . . 3 (𝐷 ∈ (∞Metβ€˜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 ∈ (∞Metβ€˜π‘‹) β†’ (𝐡 ∈ ran (ballβ€˜π·) β†’ (𝑃 ∈ 𝐡 β†’ βˆƒπ‘₯ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘₯) βŠ† 𝐡)))
63623imp 1112 1 ((𝐷 ∈ (∞Metβ€˜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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3070   βŠ† wss 3911   class class class wbr 5106  ran crn 5635  β€˜cfv 6497  (class class class)co 7358  β„cr 11055  β„*cxr 11193   < clt 11194   ≀ cle 11195   βˆ’ cmin 11390  β„šcq 12878  β„+crp 12920  βˆžMetcxmet 20797  ballcbl 20799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133  ax-pre-sup 11134
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-er 8651  df-map 8770  df-en 8887  df-dom 8888  df-sdom 8889  df-sup 9383  df-inf 9384  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-div 11818  df-nn 12159  df-2 12221  df-n0 12419  df-z 12505  df-uz 12769  df-q 12879  df-rp 12921  df-xneg 13038  df-xadd 13039  df-xmul 13040  df-psmet 20804  df-xmet 20805  df-bl 20807
This theorem is referenced by:  blssex  23796  blin2  23798  metss  23880  metcnp3  23912
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