Theorem blin2 24748

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 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝐵 ∩ 𝐶))

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.

Step	Hyp	Ref	Expression
1		simpll 779	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → 𝐷 ∈ (∞Met‘𝑋))
2		simprl 783	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → 𝐵 ∈ ran (ball‘𝐷))
3		simplr 781	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → 𝑃 ∈ (𝐵 ∩ 𝐶))
4	3	elin1d 4101	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → 𝑃 ∈ 𝐵)
5		blss 24744	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ ran (ball‘𝐷) ∧ 𝑃 ∈ 𝐵) → ∃𝑦 ∈ ℝ+ (𝑃(ball‘𝐷)𝑦) ⊆ 𝐵)
6	1, 2, 4, 5	syl3anc 1398	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → ∃𝑦 ∈ ℝ+ (𝑃(ball‘𝐷)𝑦) ⊆ 𝐵)
7		simprr 785	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → 𝐶 ∈ ran (ball‘𝐷))
8	3	elin2d 4102	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → 𝑃 ∈ 𝐶)
9		blss 24744	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐶 ∈ ran (ball‘𝐷) ∧ 𝑃 ∈ 𝐶) → ∃𝑧 ∈ ℝ+ (𝑃(ball‘𝐷)𝑧) ⊆ 𝐶)
10	1, 7, 8, 9	syl3anc 1398	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → ∃𝑧 ∈ ℝ+ (𝑃(ball‘𝐷)𝑧) ⊆ 𝐶)
11		reeanv 3234	. . 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 4141	. . . . 5 ⊢ (((𝑃(ball‘𝐷)𝑦) ⊆ 𝐵 ∧ (𝑃(ball‘𝐷)𝑧) ⊆ 𝐶) → ((𝑃(ball‘𝐷)𝑦) ∩ (𝑃(ball‘𝐷)𝑧)) ⊆ (𝐵 ∩ 𝐶))
13		inss1 4133	. . . . . . . . . . 11 ⊢ (𝐵 ∩ 𝐶) ⊆ 𝐵
14		blf 24726	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋)
15		frn 6717	. . . . . . . . . . . . . 14 ⊢ ((ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋 → ran (ball‘𝐷) ⊆ 𝒫 𝑋)
16	1, 14, 15	3syl 19	. . . . . . . . . . . . 13 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → ran (ball‘𝐷) ⊆ 𝒫 𝑋)
17	16, 2	sseldd 3909	. . . . . . . . . . . 12 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → 𝐵 ∈ 𝒫 𝑋)
18	17	elpwid 4565	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → 𝐵 ⊆ 𝑋)
19	13, 18	sstrid 3919	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → (𝐵 ∩ 𝐶) ⊆ 𝑋)
20	19, 3	sseldd 3909	. . . . . . . . 9 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → 𝑃 ∈ 𝑋)
21	1, 20	jca 521	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → (𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋))
22		rpxr 13130	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ+ → 𝑦 ∈ ℝ*)
23		rpxr 13130	. . . . . . . . 9 ⊢ (𝑧 ∈ ℝ+ → 𝑧 ∈ ℝ*)
24	22, 23	anim12i 625	. . . . . . . 8 ⊢ ((𝑦 ∈ ℝ+ ∧ 𝑧 ∈ ℝ+) → (𝑦 ∈ ℝ* ∧ 𝑧 ∈ ℝ*))
25		blin 24740	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑦 ∈ ℝ* ∧ 𝑧 ∈ ℝ*)) → ((𝑃(ball‘𝐷)𝑦) ∩ (𝑃(ball‘𝐷)𝑧)) = (𝑃(ball‘𝐷)if(𝑦 ≤ 𝑧, 𝑦, 𝑧)))
26	21, 24, 25	syl2an 608	. . . . . . 7 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) ∧ (𝑦 ∈ ℝ+ ∧ 𝑧 ∈ ℝ+)) → ((𝑃(ball‘𝐷)𝑦) ∩ (𝑃(ball‘𝐷)𝑧)) = (𝑃(ball‘𝐷)if(𝑦 ≤ 𝑧, 𝑦, 𝑧)))
27	26	sseq1d 3939	. . . . . 6 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) ∧ (𝑦 ∈ ℝ+ ∧ 𝑧 ∈ ℝ+)) → (((𝑃(ball‘𝐷)𝑦) ∩ (𝑃(ball‘𝐷)𝑧)) ⊆ (𝐵 ∩ 𝐶) ↔ (𝑃(ball‘𝐷)if(𝑦 ≤ 𝑧, 𝑦, 𝑧)) ⊆ (𝐵 ∩ 𝐶)))
28		ifcl 4527	. . . . . . . 8 ⊢ ((𝑦 ∈ ℝ+ ∧ 𝑧 ∈ ℝ+) → if(𝑦 ≤ 𝑧, 𝑦, 𝑧) ∈ ℝ+)
29		oveq2 7428	. . . . . . . . . . 11 ⊢ (𝑥 = if(𝑦 ≤ 𝑧, 𝑦, 𝑧) → (𝑃(ball‘𝐷)𝑥) = (𝑃(ball‘𝐷)if(𝑦 ≤ 𝑧, 𝑦, 𝑧)))
30	29	sseq1d 3939	. . . . . . . . . 10 ⊢ (𝑥 = if(𝑦 ≤ 𝑧, 𝑦, 𝑧) → ((𝑃(ball‘𝐷)𝑥) ⊆ (𝐵 ∩ 𝐶) ↔ (𝑃(ball‘𝐷)if(𝑦 ≤ 𝑧, 𝑦, 𝑧)) ⊆ (𝐵 ∩ 𝐶)))
31	30	rspcev 3576	. . . . . . . . 9 ⊢ ((if(𝑦 ≤ 𝑧, 𝑦, 𝑧) ∈ ℝ+ ∧ (𝑃(ball‘𝐷)if(𝑦 ≤ 𝑧, 𝑦, 𝑧)) ⊆ (𝐵 ∩ 𝐶)) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝐵 ∩ 𝐶))
32	31	ex 418	. . . . . . . 8 ⊢ (if(𝑦 ≤ 𝑧, 𝑦, 𝑧) ∈ ℝ+ → ((𝑃(ball‘𝐷)if(𝑦 ≤ 𝑧, 𝑦, 𝑧)) ⊆ (𝐵 ∩ 𝐶) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝐵 ∩ 𝐶)))
33	28, 32	syl 18	. . . . . . 7 ⊢ ((𝑦 ∈ ℝ+ ∧ 𝑧 ∈ ℝ+) → ((𝑃(ball‘𝐷)if(𝑦 ≤ 𝑧, 𝑦, 𝑧)) ⊆ (𝐵 ∩ 𝐶) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝐵 ∩ 𝐶)))
34	33	adantl 487	. . . . . 6 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) ∧ (𝑦 ∈ ℝ+ ∧ 𝑧 ∈ ℝ+)) → ((𝑃(ball‘𝐷)if(𝑦 ≤ 𝑧, 𝑦, 𝑧)) ⊆ (𝐵 ∩ 𝐶) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝐵 ∩ 𝐶)))
35	27, 34	sylbid 243	. . . . 5 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) ∧ (𝑦 ∈ ℝ+ ∧ 𝑧 ∈ ℝ+)) → (((𝑃(ball‘𝐷)𝑦) ∩ (𝑃(ball‘𝐷)𝑧)) ⊆ (𝐵 ∩ 𝐶) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝐵 ∩ 𝐶)))
36	12, 35	syl5 35	. . . 4 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (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) ⊆ (𝐵 ∩ 𝐶)))
37	36	rexlimdvva 3219	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (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) ⊆ (𝐵 ∩ 𝐶)))
38	11, 37	biimtrrid 246	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (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) ⊆ (𝐵 ∩ 𝐶)))
39	6, 10, 38	mp2and 712	1 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝐵 ∩ 𝐶))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∃wrex 3086   ⊆ wss 3895   ∩ cin 4089  ifcif 4481  𝒫 cpw 4556   class class class wbr 5102   × cxp 5648  ran crn 5651  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420  ℝ^*cxr 11342   ≤ cle 11344  ℝ⁺crp 13120  ∞Metcxmet 21663  ballcbl 21665

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277  ax-pre-sup 11278

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-er 8717  df-map 8849  df-en 8974  df-dom 8975  df-sdom 8976  df-sup 9434  df-inf 9435  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-div 11974  df-nn 12336  df-2 12405  df-n0 12607  df-z 12694  df-uz 12966  df-q 13076  df-rp 13121  df-xneg 13241  df-xadd 13242  df-xmul 13243  df-psmet 21670  df-xmet 21671  df-bl 21673

This theorem is used by:  blbas  24749