MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  blin2 Structured version   Visualization version   GIF version

Theorem blin2 24439
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 767 . . 3 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → 𝐷 ∈ (∞Met‘𝑋))
2 simprl 771 . . 3 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → 𝐵 ∈ ran (ball‘𝐷))
3 simplr 769 . . . 4 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → 𝑃 ∈ (𝐵𝐶))
43elin1d 4204 . . 3 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → 𝑃𝐵)
5 blss 24435 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ ran (ball‘𝐷) ∧ 𝑃𝐵) → ∃𝑦 ∈ ℝ+ (𝑃(ball‘𝐷)𝑦) ⊆ 𝐵)
61, 2, 4, 5syl3anc 1373 . 2 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → ∃𝑦 ∈ ℝ+ (𝑃(ball‘𝐷)𝑦) ⊆ 𝐵)
7 simprr 773 . . 3 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → 𝐶 ∈ ran (ball‘𝐷))
83elin2d 4205 . . 3 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → 𝑃𝐶)
9 blss 24435 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐶 ∈ ran (ball‘𝐷) ∧ 𝑃𝐶) → ∃𝑧 ∈ ℝ+ (𝑃(ball‘𝐷)𝑧) ⊆ 𝐶)
101, 7, 8, 9syl3anc 1373 . 2 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → ∃𝑧 ∈ ℝ+ (𝑃(ball‘𝐷)𝑧) ⊆ 𝐶)
11 reeanv 3229 . . 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 4245 . . . . 5 (((𝑃(ball‘𝐷)𝑦) ⊆ 𝐵 ∧ (𝑃(ball‘𝐷)𝑧) ⊆ 𝐶) → ((𝑃(ball‘𝐷)𝑦) ∩ (𝑃(ball‘𝐷)𝑧)) ⊆ (𝐵𝐶))
13 inss1 4237 . . . . . . . . . . 11 (𝐵𝐶) ⊆ 𝐵
14 blf 24417 . . . . . . . . . . . . . 14 (𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋)
15 frn 6743 . . . . . . . . . . . . . 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 3984 . . . . . . . . . . . 12 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → 𝐵 ∈ 𝒫 𝑋)
1817elpwid 4609 . . . . . . . . . . 11 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → 𝐵𝑋)
1913, 18sstrid 3995 . . . . . . . . . 10 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → (𝐵𝐶) ⊆ 𝑋)
2019, 3sseldd 3984 . . . . . . . . 9 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → 𝑃𝑋)
211, 20jca 511 . . . . . . . 8 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → (𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋))
22 rpxr 13044 . . . . . . . . 9 (𝑦 ∈ ℝ+𝑦 ∈ ℝ*)
23 rpxr 13044 . . . . . . . . 9 (𝑧 ∈ ℝ+𝑧 ∈ ℝ*)
2422, 23anim12i 613 . . . . . . . 8 ((𝑦 ∈ ℝ+𝑧 ∈ ℝ+) → (𝑦 ∈ ℝ*𝑧 ∈ ℝ*))
25 blin 24431 . . . . . . . 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 596 . . . . . . 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 4015 . . . . . 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 4571 . . . . . . . 8 ((𝑦 ∈ ℝ+𝑧 ∈ ℝ+) → if(𝑦𝑧, 𝑦, 𝑧) ∈ ℝ+)
29 oveq2 7439 . . . . . . . . . . 11 (𝑥 = if(𝑦𝑧, 𝑦, 𝑧) → (𝑃(ball‘𝐷)𝑥) = (𝑃(ball‘𝐷)if(𝑦𝑧, 𝑦, 𝑧)))
3029sseq1d 4015 . . . . . . . . . 10 (𝑥 = if(𝑦𝑧, 𝑦, 𝑧) → ((𝑃(ball‘𝐷)𝑥) ⊆ (𝐵𝐶) ↔ (𝑃(ball‘𝐷)if(𝑦𝑧, 𝑦, 𝑧)) ⊆ (𝐵𝐶)))
3130rspcev 3622 . . . . . . . . 9 ((if(𝑦𝑧, 𝑦, 𝑧) ∈ ℝ+ ∧ (𝑃(ball‘𝐷)if(𝑦𝑧, 𝑦, 𝑧)) ⊆ (𝐵𝐶)) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝐵𝐶))
3231ex 412 . . . . . . . 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 481 . . . . . 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 240 . . . . 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 3213 . . 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 243 . 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 699 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 395   = wceq 1540  wcel 2108  wrex 3070  cin 3950  wss 3951  ifcif 4525  𝒫 cpw 4600   class class class wbr 5143   × cxp 5683  ran crn 5686  wf 6557  cfv 6561  (class class class)co 7431  *cxr 11294  cle 11296  +crp 13034  ∞Metcxmet 21349  ballcbl 21351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-sup 9482  df-inf 9483  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-n0 12527  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-psmet 21356  df-xmet 21357  df-bl 21359
This theorem is referenced by:  blbas  24440
  Copyright terms: Public domain W3C validator