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Theorem blin2 24259
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 764 . . 3 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ (𝐡 ∩ 𝐢)) ∧ (𝐡 ∈ ran (ballβ€˜π·) ∧ 𝐢 ∈ ran (ballβ€˜π·))) β†’ 𝐷 ∈ (∞Metβ€˜π‘‹))
2 simprl 768 . . 3 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ (𝐡 ∩ 𝐢)) ∧ (𝐡 ∈ ran (ballβ€˜π·) ∧ 𝐢 ∈ ran (ballβ€˜π·))) β†’ 𝐡 ∈ ran (ballβ€˜π·))
3 simplr 766 . . . 4 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ (𝐡 ∩ 𝐢)) ∧ (𝐡 ∈ ran (ballβ€˜π·) ∧ 𝐢 ∈ ran (ballβ€˜π·))) β†’ 𝑃 ∈ (𝐡 ∩ 𝐢))
43elin1d 4191 . . 3 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ (𝐡 ∩ 𝐢)) ∧ (𝐡 ∈ ran (ballβ€˜π·) ∧ 𝐢 ∈ ran (ballβ€˜π·))) β†’ 𝑃 ∈ 𝐡)
5 blss 24255 . . 3 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝐡 ∈ ran (ballβ€˜π·) ∧ 𝑃 ∈ 𝐡) β†’ βˆƒπ‘¦ ∈ ℝ+ (𝑃(ballβ€˜π·)𝑦) βŠ† 𝐡)
61, 2, 4, 5syl3anc 1368 . 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 770 . . 3 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ (𝐡 ∩ 𝐢)) ∧ (𝐡 ∈ ran (ballβ€˜π·) ∧ 𝐢 ∈ ran (ballβ€˜π·))) β†’ 𝐢 ∈ ran (ballβ€˜π·))
83elin2d 4192 . . 3 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ (𝐡 ∩ 𝐢)) ∧ (𝐡 ∈ ran (ballβ€˜π·) ∧ 𝐢 ∈ ran (ballβ€˜π·))) β†’ 𝑃 ∈ 𝐢)
9 blss 24255 . . 3 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝐢 ∈ ran (ballβ€˜π·) ∧ 𝑃 ∈ 𝐢) β†’ βˆƒπ‘§ ∈ ℝ+ (𝑃(ballβ€˜π·)𝑧) βŠ† 𝐢)
101, 7, 8, 9syl3anc 1368 . 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 3218 . . 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 4229 . . . . 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 4221 . . . . . . . . . . 11 (𝐡 ∩ 𝐢) βŠ† 𝐡
14 blf 24237 . . . . . . . . . . . . . 14 (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ (ballβ€˜π·):(𝑋 Γ— ℝ*)βŸΆπ’« 𝑋)
15 frn 6715 . . . . . . . . . . . . . 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 3976 . . . . . . . . . . . 12 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ (𝐡 ∩ 𝐢)) ∧ (𝐡 ∈ ran (ballβ€˜π·) ∧ 𝐢 ∈ ran (ballβ€˜π·))) β†’ 𝐡 ∈ 𝒫 𝑋)
1817elpwid 4604 . . . . . . . . . . 11 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ (𝐡 ∩ 𝐢)) ∧ (𝐡 ∈ ran (ballβ€˜π·) ∧ 𝐢 ∈ ran (ballβ€˜π·))) β†’ 𝐡 βŠ† 𝑋)
1913, 18sstrid 3986 . . . . . . . . . 10 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ (𝐡 ∩ 𝐢)) ∧ (𝐡 ∈ ran (ballβ€˜π·) ∧ 𝐢 ∈ ran (ballβ€˜π·))) β†’ (𝐡 ∩ 𝐢) βŠ† 𝑋)
2019, 3sseldd 3976 . . . . . . . . 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 12981 . . . . . . . . 9 (𝑦 ∈ ℝ+ β†’ 𝑦 ∈ ℝ*)
23 rpxr 12981 . . . . . . . . 9 (𝑧 ∈ ℝ+ β†’ 𝑧 ∈ ℝ*)
2422, 23anim12i 612 . . . . . . . 8 ((𝑦 ∈ ℝ+ ∧ 𝑧 ∈ ℝ+) β†’ (𝑦 ∈ ℝ* ∧ 𝑧 ∈ ℝ*))
25 blin 24251 . . . . . . . 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 4006 . . . . . 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 4566 . . . . . . . 8 ((𝑦 ∈ ℝ+ ∧ 𝑧 ∈ ℝ+) β†’ if(𝑦 ≀ 𝑧, 𝑦, 𝑧) ∈ ℝ+)
29 oveq2 7410 . . . . . . . . . . 11 (π‘₯ = if(𝑦 ≀ 𝑧, 𝑦, 𝑧) β†’ (𝑃(ballβ€˜π·)π‘₯) = (𝑃(ballβ€˜π·)if(𝑦 ≀ 𝑧, 𝑦, 𝑧)))
3029sseq1d 4006 . . . . . . . . . 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 3604 . . . . . . . . 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 3203 . . 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 696 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 1533   ∈ wcel 2098  βˆƒwrex 3062   ∩ cin 3940   βŠ† wss 3941  ifcif 4521  π’« cpw 4595   class class class wbr 5139   Γ— cxp 5665  ran crn 5668  βŸΆwf 6530  β€˜cfv 6534  (class class class)co 7402  β„*cxr 11245   ≀ cle 11247  β„+crp 12972  βˆžMetcxmet 21215  ballcbl 21217
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 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184  ax-pre-sup 11185
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 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7358  df-ov 7405  df-oprab 7406  df-mpo 7407  df-om 7850  df-1st 7969  df-2nd 7970  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-er 8700  df-map 8819  df-en 8937  df-dom 8938  df-sdom 8939  df-sup 9434  df-inf 9435  df-pnf 11248  df-mnf 11249  df-xr 11250  df-ltxr 11251  df-le 11252  df-sub 11444  df-neg 11445  df-div 11870  df-nn 12211  df-2 12273  df-n0 12471  df-z 12557  df-uz 12821  df-q 12931  df-rp 12973  df-xneg 13090  df-xadd 13091  df-xmul 13092  df-psmet 21222  df-xmet 21223  df-bl 21225
This theorem is referenced by:  blbas  24260
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