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Theorem blin2 24396
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 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝐵𝐶))
Distinct variable groups:   𝑥,𝐵   𝑥,𝐶   𝑥,𝐷   𝑥,𝑃   𝑥,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 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → 𝐷 ∈ (∞Met‘𝑋))
2 simprl 769 . . 3 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → 𝐵 ∈ ran (ball‘𝐷))
3 simplr 767 . . . 4 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → 𝑃 ∈ (𝐵𝐶))
43elin1d 4196 . . 3 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → 𝑃𝐵)
5 blss 24392 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ ran (ball‘𝐷) ∧ 𝑃𝐵) → ∃𝑦 ∈ ℝ+ (𝑃(ball‘𝐷)𝑦) ⊆ 𝐵)
61, 2, 4, 5syl3anc 1368 . 2 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → ∃𝑦 ∈ ℝ+ (𝑃(ball‘𝐷)𝑦) ⊆ 𝐵)
7 simprr 771 . . 3 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → 𝐶 ∈ ran (ball‘𝐷))
83elin2d 4197 . . 3 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → 𝑃𝐶)
9 blss 24392 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐶 ∈ ran (ball‘𝐷) ∧ 𝑃𝐶) → ∃𝑧 ∈ ℝ+ (𝑃(ball‘𝐷)𝑧) ⊆ 𝐶)
101, 7, 8, 9syl3anc 1368 . 2 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → ∃𝑧 ∈ ℝ+ (𝑃(ball‘𝐷)𝑧) ⊆ 𝐶)
11 reeanv 3216 . . 3 (∃𝑦 ∈ ℝ+𝑧 ∈ ℝ+ ((𝑃(ball‘𝐷)𝑦) ⊆ 𝐵 ∧ (𝑃(ball‘𝐷)𝑧) ⊆ 𝐶) ↔ (∃𝑦 ∈ ℝ+ (𝑃(ball‘𝐷)𝑦) ⊆ 𝐵 ∧ ∃𝑧 ∈ ℝ+ (𝑃(ball‘𝐷)𝑧) ⊆ 𝐶))
12 ss2in 4235 . . . . 5 (((𝑃(ball‘𝐷)𝑦) ⊆ 𝐵 ∧ (𝑃(ball‘𝐷)𝑧) ⊆ 𝐶) → ((𝑃(ball‘𝐷)𝑦) ∩ (𝑃(ball‘𝐷)𝑧)) ⊆ (𝐵𝐶))
13 inss1 4227 . . . . . . . . . . 11 (𝐵𝐶) ⊆ 𝐵
14 blf 24374 . . . . . . . . . . . . . 14 (𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋)
15 frn 6730 . . . . . . . . . . . . . 14 ((ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋 → ran (ball‘𝐷) ⊆ 𝒫 𝑋)
161, 14, 153syl 18 . . . . . . . . . . . . 13 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → ran (ball‘𝐷) ⊆ 𝒫 𝑋)
1716, 2sseldd 3977 . . . . . . . . . . . 12 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → 𝐵 ∈ 𝒫 𝑋)
1817elpwid 4613 . . . . . . . . . . 11 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → 𝐵𝑋)
1913, 18sstrid 3988 . . . . . . . . . 10 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → (𝐵𝐶) ⊆ 𝑋)
2019, 3sseldd 3977 . . . . . . . . 9 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → 𝑃𝑋)
211, 20jca 510 . . . . . . . 8 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → (𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋))
22 rpxr 13023 . . . . . . . . 9 (𝑦 ∈ ℝ+𝑦 ∈ ℝ*)
23 rpxr 13023 . . . . . . . . 9 (𝑧 ∈ ℝ+𝑧 ∈ ℝ*)
2422, 23anim12i 611 . . . . . . . 8 ((𝑦 ∈ ℝ+𝑧 ∈ ℝ+) → (𝑦 ∈ ℝ*𝑧 ∈ ℝ*))
25 blin 24388 . . . . . . . 8 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋) ∧ (𝑦 ∈ ℝ*𝑧 ∈ ℝ*)) → ((𝑃(ball‘𝐷)𝑦) ∩ (𝑃(ball‘𝐷)𝑧)) = (𝑃(ball‘𝐷)if(𝑦𝑧, 𝑦, 𝑧)))
2621, 24, 25syl2an 594 . . . . . . 7 ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) ∧ (𝑦 ∈ ℝ+𝑧 ∈ ℝ+)) → ((𝑃(ball‘𝐷)𝑦) ∩ (𝑃(ball‘𝐷)𝑧)) = (𝑃(ball‘𝐷)if(𝑦𝑧, 𝑦, 𝑧)))
2726sseq1d 4008 . . . . . 6 ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) ∧ (𝑦 ∈ ℝ+𝑧 ∈ ℝ+)) → (((𝑃(ball‘𝐷)𝑦) ∩ (𝑃(ball‘𝐷)𝑧)) ⊆ (𝐵𝐶) ↔ (𝑃(ball‘𝐷)if(𝑦𝑧, 𝑦, 𝑧)) ⊆ (𝐵𝐶)))
28 ifcl 4575 . . . . . . . 8 ((𝑦 ∈ ℝ+𝑧 ∈ ℝ+) → if(𝑦𝑧, 𝑦, 𝑧) ∈ ℝ+)
29 oveq2 7427 . . . . . . . . . . 11 (𝑥 = if(𝑦𝑧, 𝑦, 𝑧) → (𝑃(ball‘𝐷)𝑥) = (𝑃(ball‘𝐷)if(𝑦𝑧, 𝑦, 𝑧)))
3029sseq1d 4008 . . . . . . . . . 10 (𝑥 = if(𝑦𝑧, 𝑦, 𝑧) → ((𝑃(ball‘𝐷)𝑥) ⊆ (𝐵𝐶) ↔ (𝑃(ball‘𝐷)if(𝑦𝑧, 𝑦, 𝑧)) ⊆ (𝐵𝐶)))
3130rspcev 3606 . . . . . . . . 9 ((if(𝑦𝑧, 𝑦, 𝑧) ∈ ℝ+ ∧ (𝑃(ball‘𝐷)if(𝑦𝑧, 𝑦, 𝑧)) ⊆ (𝐵𝐶)) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝐵𝐶))
3231ex 411 . . . . . . . 8 (if(𝑦𝑧, 𝑦, 𝑧) ∈ ℝ+ → ((𝑃(ball‘𝐷)if(𝑦𝑧, 𝑦, 𝑧)) ⊆ (𝐵𝐶) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝐵𝐶)))
3328, 32syl 17 . . . . . . 7 ((𝑦 ∈ ℝ+𝑧 ∈ ℝ+) → ((𝑃(ball‘𝐷)if(𝑦𝑧, 𝑦, 𝑧)) ⊆ (𝐵𝐶) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝐵𝐶)))
3433adantl 480 . . . . . 6 ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) ∧ (𝑦 ∈ ℝ+𝑧 ∈ ℝ+)) → ((𝑃(ball‘𝐷)if(𝑦𝑧, 𝑦, 𝑧)) ⊆ (𝐵𝐶) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝐵𝐶)))
3527, 34sylbid 239 . . . . 5 ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) ∧ (𝑦 ∈ ℝ+𝑧 ∈ ℝ+)) → (((𝑃(ball‘𝐷)𝑦) ∩ (𝑃(ball‘𝐷)𝑧)) ⊆ (𝐵𝐶) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝐵𝐶)))
3612, 35syl5 34 . . . 4 ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) ∧ (𝑦 ∈ ℝ+𝑧 ∈ ℝ+)) → (((𝑃(ball‘𝐷)𝑦) ⊆ 𝐵 ∧ (𝑃(ball‘𝐷)𝑧) ⊆ 𝐶) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝐵𝐶)))
3736rexlimdvva 3201 . . 3 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → (∃𝑦 ∈ ℝ+𝑧 ∈ ℝ+ ((𝑃(ball‘𝐷)𝑦) ⊆ 𝐵 ∧ (𝑃(ball‘𝐷)𝑧) ⊆ 𝐶) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝐵𝐶)))
3811, 37biimtrrid 242 . 2 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → ((∃𝑦 ∈ ℝ+ (𝑃(ball‘𝐷)𝑦) ⊆ 𝐵 ∧ ∃𝑧 ∈ ℝ+ (𝑃(ball‘𝐷)𝑧) ⊆ 𝐶) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝐵𝐶)))
396, 10, 38mp2and 697 1 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  wrex 3059  cin 3943  wss 3944  ifcif 4530  𝒫 cpw 4604   class class class wbr 5149   × cxp 5676  ran crn 5679  wf 6545  cfv 6549  (class class class)co 7419  *cxr 11284  cle 11286  +crp 13014  ∞Metcxmet 21298  ballcbl 21300
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 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11201  ax-resscn 11202  ax-1cn 11203  ax-icn 11204  ax-addcl 11205  ax-addrcl 11206  ax-mulcl 11207  ax-mulrcl 11208  ax-mulcom 11209  ax-addass 11210  ax-mulass 11211  ax-distr 11212  ax-i2m1 11213  ax-1ne0 11214  ax-1rid 11215  ax-rnegex 11216  ax-rrecex 11217  ax-cnre 11218  ax-pre-lttri 11219  ax-pre-lttrn 11220  ax-pre-ltadd 11221  ax-pre-mulgt0 11222  ax-pre-sup 11223
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  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 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  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 9472  df-inf 9473  df-pnf 11287  df-mnf 11288  df-xr 11289  df-ltxr 11290  df-le 11291  df-sub 11483  df-neg 11484  df-div 11909  df-nn 12251  df-2 12313  df-n0 12511  df-z 12597  df-uz 12861  df-q 12971  df-rp 13015  df-xneg 13132  df-xadd 13133  df-xmul 13134  df-psmet 21305  df-xmet 21306  df-bl 21308
This theorem is referenced by:  blbas  24397
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