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Theorem blin2 24328
Description: Given any two balls and a point in their intersection, there is a ball contained in the intersection with the given center point. (Contributed by Mario Carneiro, 12-Nov-2013.)
Assertion
Ref Expression
blin2 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ (𝐡 ∩ 𝐢)) ∧ (𝐡 ∈ ran (ballβ€˜π·) ∧ 𝐢 ∈ ran (ballβ€˜π·))) β†’ βˆƒπ‘₯ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘₯) βŠ† (𝐡 ∩ 𝐢))
Distinct variable groups:   π‘₯,𝐡   π‘₯,𝐢   π‘₯,𝐷   π‘₯,𝑃   π‘₯,𝑋
Proof of Theorem blin2
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpll 766 . . 3 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ (𝐡 ∩ 𝐢)) ∧ (𝐡 ∈ ran (ballβ€˜π·) ∧ 𝐢 ∈ ran (ballβ€˜π·))) β†’ 𝐷 ∈ (∞Metβ€˜π‘‹))
2 simprl 770 . . 3 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ (𝐡 ∩ 𝐢)) ∧ (𝐡 ∈ ran (ballβ€˜π·) ∧ 𝐢 ∈ ran (ballβ€˜π·))) β†’ 𝐡 ∈ ran (ballβ€˜π·))
3 simplr 768 . . . 4 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ (𝐡 ∩ 𝐢)) ∧ (𝐡 ∈ ran (ballβ€˜π·) ∧ 𝐢 ∈ ran (ballβ€˜π·))) β†’ 𝑃 ∈ (𝐡 ∩ 𝐢))
43elin1d 4194 . . 3 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ (𝐡 ∩ 𝐢)) ∧ (𝐡 ∈ ran (ballβ€˜π·) ∧ 𝐢 ∈ ran (ballβ€˜π·))) β†’ 𝑃 ∈ 𝐡)
5 blss 24324 . . 3 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝐡 ∈ ran (ballβ€˜π·) ∧ 𝑃 ∈ 𝐡) β†’ βˆƒπ‘¦ ∈ ℝ+ (𝑃(ballβ€˜π·)𝑦) βŠ† 𝐡)
61, 2, 4, 5syl3anc 1369 . 2 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ (𝐡 ∩ 𝐢)) ∧ (𝐡 ∈ ran (ballβ€˜π·) ∧ 𝐢 ∈ ran (ballβ€˜π·))) β†’ βˆƒπ‘¦ ∈ ℝ+ (𝑃(ballβ€˜π·)𝑦) βŠ† 𝐡)
7 simprr 772 . . 3 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ (𝐡 ∩ 𝐢)) ∧ (𝐡 ∈ ran (ballβ€˜π·) ∧ 𝐢 ∈ ran (ballβ€˜π·))) β†’ 𝐢 ∈ ran (ballβ€˜π·))
83elin2d 4195 . . 3 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ (𝐡 ∩ 𝐢)) ∧ (𝐡 ∈ ran (ballβ€˜π·) ∧ 𝐢 ∈ ran (ballβ€˜π·))) β†’ 𝑃 ∈ 𝐢)
9 blss 24324 . . 3 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝐢 ∈ ran (ballβ€˜π·) ∧ 𝑃 ∈ 𝐢) β†’ βˆƒπ‘§ ∈ ℝ+ (𝑃(ballβ€˜π·)𝑧) βŠ† 𝐢)
101, 7, 8, 9syl3anc 1369 . 2 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ (𝐡 ∩ 𝐢)) ∧ (𝐡 ∈ ran (ballβ€˜π·) ∧ 𝐢 ∈ ran (ballβ€˜π·))) β†’ βˆƒπ‘§ ∈ ℝ+ (𝑃(ballβ€˜π·)𝑧) βŠ† 𝐢)
11 reeanv 3222 . . 3 (βˆƒπ‘¦ ∈ ℝ+ βˆƒπ‘§ ∈ ℝ+ ((𝑃(ballβ€˜π·)𝑦) βŠ† 𝐡 ∧ (𝑃(ballβ€˜π·)𝑧) βŠ† 𝐢) ↔ (βˆƒπ‘¦ ∈ ℝ+ (𝑃(ballβ€˜π·)𝑦) βŠ† 𝐡 ∧ βˆƒπ‘§ ∈ ℝ+ (𝑃(ballβ€˜π·)𝑧) βŠ† 𝐢))
12 ss2in 4232 . . . . 5 (((𝑃(ballβ€˜π·)𝑦) βŠ† 𝐡 ∧ (𝑃(ballβ€˜π·)𝑧) βŠ† 𝐢) β†’ ((𝑃(ballβ€˜π·)𝑦) ∩ (𝑃(ballβ€˜π·)𝑧)) βŠ† (𝐡 ∩ 𝐢))
13 inss1 4224 . . . . . . . . . . 11 (𝐡 ∩ 𝐢) βŠ† 𝐡
14 blf 24306 . . . . . . . . . . . . . 14 (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ (ballβ€˜π·):(𝑋 Γ— ℝ*)βŸΆπ’« 𝑋)
15 frn 6723 . . . . . . . . . . . . . 14 ((ballβ€˜π·):(𝑋 Γ— ℝ*)βŸΆπ’« 𝑋 β†’ ran (ballβ€˜π·) βŠ† 𝒫 𝑋)
161, 14, 153syl 18 . . . . . . . . . . . . 13 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ (𝐡 ∩ 𝐢)) ∧ (𝐡 ∈ ran (ballβ€˜π·) ∧ 𝐢 ∈ ran (ballβ€˜π·))) β†’ ran (ballβ€˜π·) βŠ† 𝒫 𝑋)
1716, 2sseldd 3979 . . . . . . . . . . . 12 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ (𝐡 ∩ 𝐢)) ∧ (𝐡 ∈ ran (ballβ€˜π·) ∧ 𝐢 ∈ ran (ballβ€˜π·))) β†’ 𝐡 ∈ 𝒫 𝑋)
1817elpwid 4607 . . . . . . . . . . 11 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ (𝐡 ∩ 𝐢)) ∧ (𝐡 ∈ ran (ballβ€˜π·) ∧ 𝐢 ∈ ran (ballβ€˜π·))) β†’ 𝐡 βŠ† 𝑋)
1913, 18sstrid 3989 . . . . . . . . . 10 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ (𝐡 ∩ 𝐢)) ∧ (𝐡 ∈ ran (ballβ€˜π·) ∧ 𝐢 ∈ ran (ballβ€˜π·))) β†’ (𝐡 ∩ 𝐢) βŠ† 𝑋)
2019, 3sseldd 3979 . . . . . . . . 9 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ (𝐡 ∩ 𝐢)) ∧ (𝐡 ∈ ran (ballβ€˜π·) ∧ 𝐢 ∈ ran (ballβ€˜π·))) β†’ 𝑃 ∈ 𝑋)
211, 20jca 511 . . . . . . . 8 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ (𝐡 ∩ 𝐢)) ∧ (𝐡 ∈ ran (ballβ€˜π·) ∧ 𝐢 ∈ ran (ballβ€˜π·))) β†’ (𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋))
22 rpxr 13009 . . . . . . . . 9 (𝑦 ∈ ℝ+ β†’ 𝑦 ∈ ℝ*)
23 rpxr 13009 . . . . . . . . 9 (𝑧 ∈ ℝ+ β†’ 𝑧 ∈ ℝ*)
2422, 23anim12i 612 . . . . . . . 8 ((𝑦 ∈ ℝ+ ∧ 𝑧 ∈ ℝ+) β†’ (𝑦 ∈ ℝ* ∧ 𝑧 ∈ ℝ*))
25 blin 24320 . . . . . . . 8 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) ∧ (𝑦 ∈ ℝ* ∧ 𝑧 ∈ ℝ*)) β†’ ((𝑃(ballβ€˜π·)𝑦) ∩ (𝑃(ballβ€˜π·)𝑧)) = (𝑃(ballβ€˜π·)if(𝑦 ≀ 𝑧, 𝑦, 𝑧)))
2621, 24, 25syl2an 595 . . . . . . 7 ((((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ (𝐡 ∩ 𝐢)) ∧ (𝐡 ∈ ran (ballβ€˜π·) ∧ 𝐢 ∈ ran (ballβ€˜π·))) ∧ (𝑦 ∈ ℝ+ ∧ 𝑧 ∈ ℝ+)) β†’ ((𝑃(ballβ€˜π·)𝑦) ∩ (𝑃(ballβ€˜π·)𝑧)) = (𝑃(ballβ€˜π·)if(𝑦 ≀ 𝑧, 𝑦, 𝑧)))
2726sseq1d 4009 . . . . . 6 ((((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ (𝐡 ∩ 𝐢)) ∧ (𝐡 ∈ ran (ballβ€˜π·) ∧ 𝐢 ∈ ran (ballβ€˜π·))) ∧ (𝑦 ∈ ℝ+ ∧ 𝑧 ∈ ℝ+)) β†’ (((𝑃(ballβ€˜π·)𝑦) ∩ (𝑃(ballβ€˜π·)𝑧)) βŠ† (𝐡 ∩ 𝐢) ↔ (𝑃(ballβ€˜π·)if(𝑦 ≀ 𝑧, 𝑦, 𝑧)) βŠ† (𝐡 ∩ 𝐢)))
28 ifcl 4569 . . . . . . . 8 ((𝑦 ∈ ℝ+ ∧ 𝑧 ∈ ℝ+) β†’ if(𝑦 ≀ 𝑧, 𝑦, 𝑧) ∈ ℝ+)
29 oveq2 7422 . . . . . . . . . . 11 (π‘₯ = if(𝑦 ≀ 𝑧, 𝑦, 𝑧) β†’ (𝑃(ballβ€˜π·)π‘₯) = (𝑃(ballβ€˜π·)if(𝑦 ≀ 𝑧, 𝑦, 𝑧)))
3029sseq1d 4009 . . . . . . . . . 10 (π‘₯ = if(𝑦 ≀ 𝑧, 𝑦, 𝑧) β†’ ((𝑃(ballβ€˜π·)π‘₯) βŠ† (𝐡 ∩ 𝐢) ↔ (𝑃(ballβ€˜π·)if(𝑦 ≀ 𝑧, 𝑦, 𝑧)) βŠ† (𝐡 ∩ 𝐢)))
3130rspcev 3608 . . . . . . . . 9 ((if(𝑦 ≀ 𝑧, 𝑦, 𝑧) ∈ ℝ+ ∧ (𝑃(ballβ€˜π·)if(𝑦 ≀ 𝑧, 𝑦, 𝑧)) βŠ† (𝐡 ∩ 𝐢)) β†’ βˆƒπ‘₯ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘₯) βŠ† (𝐡 ∩ 𝐢))
3231ex 412 . . . . . . . 8 (if(𝑦 ≀ 𝑧, 𝑦, 𝑧) ∈ ℝ+ β†’ ((𝑃(ballβ€˜π·)if(𝑦 ≀ 𝑧, 𝑦, 𝑧)) βŠ† (𝐡 ∩ 𝐢) β†’ βˆƒπ‘₯ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘₯) βŠ† (𝐡 ∩ 𝐢)))
3328, 32syl 17 . . . . . . 7 ((𝑦 ∈ ℝ+ ∧ 𝑧 ∈ ℝ+) β†’ ((𝑃(ballβ€˜π·)if(𝑦 ≀ 𝑧, 𝑦, 𝑧)) βŠ† (𝐡 ∩ 𝐢) β†’ βˆƒπ‘₯ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘₯) βŠ† (𝐡 ∩ 𝐢)))
3433adantl 481 . . . . . 6 ((((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ (𝐡 ∩ 𝐢)) ∧ (𝐡 ∈ ran (ballβ€˜π·) ∧ 𝐢 ∈ ran (ballβ€˜π·))) ∧ (𝑦 ∈ ℝ+ ∧ 𝑧 ∈ ℝ+)) β†’ ((𝑃(ballβ€˜π·)if(𝑦 ≀ 𝑧, 𝑦, 𝑧)) βŠ† (𝐡 ∩ 𝐢) β†’ βˆƒπ‘₯ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘₯) βŠ† (𝐡 ∩ 𝐢)))
3527, 34sylbid 239 . . . . 5 ((((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ (𝐡 ∩ 𝐢)) ∧ (𝐡 ∈ ran (ballβ€˜π·) ∧ 𝐢 ∈ ran (ballβ€˜π·))) ∧ (𝑦 ∈ ℝ+ ∧ 𝑧 ∈ ℝ+)) β†’ (((𝑃(ballβ€˜π·)𝑦) ∩ (𝑃(ballβ€˜π·)𝑧)) βŠ† (𝐡 ∩ 𝐢) β†’ βˆƒπ‘₯ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘₯) βŠ† (𝐡 ∩ 𝐢)))
3612, 35syl5 34 . . . 4 ((((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ (𝐡 ∩ 𝐢)) ∧ (𝐡 ∈ ran (ballβ€˜π·) ∧ 𝐢 ∈ ran (ballβ€˜π·))) ∧ (𝑦 ∈ ℝ+ ∧ 𝑧 ∈ ℝ+)) β†’ (((𝑃(ballβ€˜π·)𝑦) βŠ† 𝐡 ∧ (𝑃(ballβ€˜π·)𝑧) βŠ† 𝐢) β†’ βˆƒπ‘₯ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘₯) βŠ† (𝐡 ∩ 𝐢)))
3736rexlimdvva 3207 . . 3 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ (𝐡 ∩ 𝐢)) ∧ (𝐡 ∈ ran (ballβ€˜π·) ∧ 𝐢 ∈ ran (ballβ€˜π·))) β†’ (βˆƒπ‘¦ ∈ ℝ+ βˆƒπ‘§ ∈ ℝ+ ((𝑃(ballβ€˜π·)𝑦) βŠ† 𝐡 ∧ (𝑃(ballβ€˜π·)𝑧) βŠ† 𝐢) β†’ βˆƒπ‘₯ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘₯) βŠ† (𝐡 ∩ 𝐢)))
3811, 37biimtrrid 242 . 2 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ (𝐡 ∩ 𝐢)) ∧ (𝐡 ∈ ran (ballβ€˜π·) ∧ 𝐢 ∈ ran (ballβ€˜π·))) β†’ ((βˆƒπ‘¦ ∈ ℝ+ (𝑃(ballβ€˜π·)𝑦) βŠ† 𝐡 ∧ βˆƒπ‘§ ∈ ℝ+ (𝑃(ballβ€˜π·)𝑧) βŠ† 𝐢) β†’ βˆƒπ‘₯ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘₯) βŠ† (𝐡 ∩ 𝐢)))
396, 10, 38mp2and 698 1 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ (𝐡 ∩ 𝐢)) ∧ (𝐡 ∈ ran (ballβ€˜π·) ∧ 𝐢 ∈ ran (ballβ€˜π·))) β†’ βˆƒπ‘₯ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘₯) βŠ† (𝐡 ∩ 𝐢))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  βˆƒwrex 3066   ∩ cin 3944   βŠ† wss 3945  ifcif 4524  π’« cpw 4598   class class class wbr 5142   Γ— cxp 5670  ran crn 5673  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7414  β„*cxr 11271   ≀ cle 11273  β„+crp 13000  βˆžMetcxmet 21257  ballcbl 21259
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 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11188  ax-resscn 11189  ax-1cn 11190  ax-icn 11191  ax-addcl 11192  ax-addrcl 11193  ax-mulcl 11194  ax-mulrcl 11195  ax-mulcom 11196  ax-addass 11197  ax-mulass 11198  ax-distr 11199  ax-i2m1 11200  ax-1ne0 11201  ax-1rid 11202  ax-rnegex 11203  ax-rrecex 11204  ax-cnre 11205  ax-pre-lttri 11206  ax-pre-lttrn 11207  ax-pre-ltadd 11208  ax-pre-mulgt0 11209  ax-pre-sup 11210
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 2937  df-nel 3043  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-er 8718  df-map 8840  df-en 8958  df-dom 8959  df-sdom 8960  df-sup 9459  df-inf 9460  df-pnf 11274  df-mnf 11275  df-xr 11276  df-ltxr 11277  df-le 11278  df-sub 11470  df-neg 11471  df-div 11896  df-nn 12237  df-2 12299  df-n0 12497  df-z 12583  df-uz 12847  df-q 12957  df-rp 13001  df-xneg 13118  df-xadd 13119  df-xmul 13120  df-psmet 21264  df-xmet 21265  df-bl 21267
This theorem is referenced by:  blbas  24329
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