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Theorem blin2 23934
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 765 . . 3 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ (𝐡 ∩ 𝐢)) ∧ (𝐡 ∈ ran (ballβ€˜π·) ∧ 𝐢 ∈ ran (ballβ€˜π·))) β†’ 𝐷 ∈ (∞Metβ€˜π‘‹))
2 simprl 769 . . 3 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ (𝐡 ∩ 𝐢)) ∧ (𝐡 ∈ ran (ballβ€˜π·) ∧ 𝐢 ∈ ran (ballβ€˜π·))) β†’ 𝐡 ∈ ran (ballβ€˜π·))
3 simplr 767 . . . 4 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ (𝐡 ∩ 𝐢)) ∧ (𝐡 ∈ ran (ballβ€˜π·) ∧ 𝐢 ∈ ran (ballβ€˜π·))) β†’ 𝑃 ∈ (𝐡 ∩ 𝐢))
43elin1d 4198 . . 3 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ (𝐡 ∩ 𝐢)) ∧ (𝐡 ∈ ran (ballβ€˜π·) ∧ 𝐢 ∈ ran (ballβ€˜π·))) β†’ 𝑃 ∈ 𝐡)
5 blss 23930 . . 3 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝐡 ∈ ran (ballβ€˜π·) ∧ 𝑃 ∈ 𝐡) β†’ βˆƒπ‘¦ ∈ ℝ+ (𝑃(ballβ€˜π·)𝑦) βŠ† 𝐡)
61, 2, 4, 5syl3anc 1371 . 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 771 . . 3 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ (𝐡 ∩ 𝐢)) ∧ (𝐡 ∈ ran (ballβ€˜π·) ∧ 𝐢 ∈ ran (ballβ€˜π·))) β†’ 𝐢 ∈ ran (ballβ€˜π·))
83elin2d 4199 . . 3 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ (𝐡 ∩ 𝐢)) ∧ (𝐡 ∈ ran (ballβ€˜π·) ∧ 𝐢 ∈ ran (ballβ€˜π·))) β†’ 𝑃 ∈ 𝐢)
9 blss 23930 . . 3 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝐢 ∈ ran (ballβ€˜π·) ∧ 𝑃 ∈ 𝐢) β†’ βˆƒπ‘§ ∈ ℝ+ (𝑃(ballβ€˜π·)𝑧) βŠ† 𝐢)
101, 7, 8, 9syl3anc 1371 . 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 3226 . . 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 4236 . . . . 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 4228 . . . . . . . . . . 11 (𝐡 ∩ 𝐢) βŠ† 𝐡
14 blf 23912 . . . . . . . . . . . . . 14 (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ (ballβ€˜π·):(𝑋 Γ— ℝ*)βŸΆπ’« 𝑋)
15 frn 6724 . . . . . . . . . . . . . 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 3983 . . . . . . . . . . . 12 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ (𝐡 ∩ 𝐢)) ∧ (𝐡 ∈ ran (ballβ€˜π·) ∧ 𝐢 ∈ ran (ballβ€˜π·))) β†’ 𝐡 ∈ 𝒫 𝑋)
1817elpwid 4611 . . . . . . . . . . 11 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ (𝐡 ∩ 𝐢)) ∧ (𝐡 ∈ ran (ballβ€˜π·) ∧ 𝐢 ∈ ran (ballβ€˜π·))) β†’ 𝐡 βŠ† 𝑋)
1913, 18sstrid 3993 . . . . . . . . . 10 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ (𝐡 ∩ 𝐢)) ∧ (𝐡 ∈ ran (ballβ€˜π·) ∧ 𝐢 ∈ ran (ballβ€˜π·))) β†’ (𝐡 ∩ 𝐢) βŠ† 𝑋)
2019, 3sseldd 3983 . . . . . . . . 9 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ (𝐡 ∩ 𝐢)) ∧ (𝐡 ∈ ran (ballβ€˜π·) ∧ 𝐢 ∈ ran (ballβ€˜π·))) β†’ 𝑃 ∈ 𝑋)
211, 20jca 512 . . . . . . . 8 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ (𝐡 ∩ 𝐢)) ∧ (𝐡 ∈ ran (ballβ€˜π·) ∧ 𝐢 ∈ ran (ballβ€˜π·))) β†’ (𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋))
22 rpxr 12982 . . . . . . . . 9 (𝑦 ∈ ℝ+ β†’ 𝑦 ∈ ℝ*)
23 rpxr 12982 . . . . . . . . 9 (𝑧 ∈ ℝ+ β†’ 𝑧 ∈ ℝ*)
2422, 23anim12i 613 . . . . . . . 8 ((𝑦 ∈ ℝ+ ∧ 𝑧 ∈ ℝ+) β†’ (𝑦 ∈ ℝ* ∧ 𝑧 ∈ ℝ*))
25 blin 23926 . . . . . . . 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 596 . . . . . . 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 4013 . . . . . 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 4573 . . . . . . . 8 ((𝑦 ∈ ℝ+ ∧ 𝑧 ∈ ℝ+) β†’ if(𝑦 ≀ 𝑧, 𝑦, 𝑧) ∈ ℝ+)
29 oveq2 7416 . . . . . . . . . . 11 (π‘₯ = if(𝑦 ≀ 𝑧, 𝑦, 𝑧) β†’ (𝑃(ballβ€˜π·)π‘₯) = (𝑃(ballβ€˜π·)if(𝑦 ≀ 𝑧, 𝑦, 𝑧)))
3029sseq1d 4013 . . . . . . . . . 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 3612 . . . . . . . . 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 413 . . . . . . . 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 482 . . . . . 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 3211 . . 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 697 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 396   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3070   ∩ cin 3947   βŠ† wss 3948  ifcif 4528  π’« cpw 4602   class class class wbr 5148   Γ— cxp 5674  ran crn 5677  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7408  β„*cxr 11246   ≀ cle 11248  β„+crp 12973  βˆžMetcxmet 20928  ballcbl 20930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  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 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-sup 9436  df-inf 9437  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-div 11871  df-nn 12212  df-2 12274  df-n0 12472  df-z 12558  df-uz 12822  df-q 12932  df-rp 12974  df-xneg 13091  df-xadd 13092  df-xmul 13093  df-psmet 20935  df-xmet 20936  df-bl 20938
This theorem is referenced by:  blbas  23935
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