Theorem blin2 15071

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 527	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → 𝐷 ∈ (∞Met‘𝑋))
2		simprl 529	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → 𝐵 ∈ ran (ball‘𝐷))
3		simplr 528	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → 𝑃 ∈ (𝐵 ∩ 𝐶))
4	3	elin1d 3373	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → 𝑃 ∈ 𝐵)
5		blss 15067	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ ran (ball‘𝐷) ∧ 𝑃 ∈ 𝐵) → ∃𝑦 ∈ ℝ+ (𝑃(ball‘𝐷)𝑦) ⊆ 𝐵)
6	1, 2, 4, 5	syl3anc 1252	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → ∃𝑦 ∈ ℝ+ (𝑃(ball‘𝐷)𝑦) ⊆ 𝐵)
7		simprr 531	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → 𝐶 ∈ ran (ball‘𝐷))
8	3	elin2d 3374	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → 𝑃 ∈ 𝐶)
9		blss 15067	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐶 ∈ ran (ball‘𝐷) ∧ 𝑃 ∈ 𝐶) → ∃𝑧 ∈ ℝ+ (𝑃(ball‘𝐷)𝑧) ⊆ 𝐶)
10	1, 7, 8, 9	syl3anc 1252	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → ∃𝑧 ∈ ℝ+ (𝑃(ball‘𝐷)𝑧) ⊆ 𝐶)
11		reeanv 2681	. . 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 3412	. . . . 5 ⊢ (((𝑃(ball‘𝐷)𝑦) ⊆ 𝐵 ∧ (𝑃(ball‘𝐷)𝑧) ⊆ 𝐶) → ((𝑃(ball‘𝐷)𝑦) ∩ (𝑃(ball‘𝐷)𝑧)) ⊆ (𝐵 ∩ 𝐶))
13		inss1 3404	. . . . . . . . . . 11 ⊢ (𝐵 ∩ 𝐶) ⊆ 𝐵
14		blf 15049	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋)
15		frn 5458	. . . . . . . . . . . . . 14 ⊢ ((ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋 → ran (ball‘𝐷) ⊆ 𝒫 𝑋)
16	1, 14, 15	3syl 17	. . . . . . . . . . . . 13 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → ran (ball‘𝐷) ⊆ 𝒫 𝑋)
17	16, 2	sseldd 3205	. . . . . . . . . . . 12 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → 𝐵 ∈ 𝒫 𝑋)
18	17	elpwid 3640	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → 𝐵 ⊆ 𝑋)
19	13, 18	sstrid 3215	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → (𝐵 ∩ 𝐶) ⊆ 𝑋)
20	19, 3	sseldd 3205	. . . . . . . . 9 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → 𝑃 ∈ 𝑋)
21	1, 20	jca 306	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → (𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋))
22		rpxr 9825	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ+ → 𝑦 ∈ ℝ*)
23		rpxr 9825	. . . . . . . . 9 ⊢ (𝑧 ∈ ℝ+ → 𝑧 ∈ ℝ*)
24	22, 23	anim12i 338	. . . . . . . 8 ⊢ ((𝑦 ∈ ℝ+ ∧ 𝑧 ∈ ℝ+) → (𝑦 ∈ ℝ* ∧ 𝑧 ∈ ℝ*))
25		blininf 15063	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑦 ∈ ℝ* ∧ 𝑧 ∈ ℝ*)) → ((𝑃(ball‘𝐷)𝑦) ∩ (𝑃(ball‘𝐷)𝑧)) = (𝑃(ball‘𝐷)inf({𝑦, 𝑧}, ℝ*, < )))
26	21, 24, 25	syl2an 289	. . . . . . 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)inf({𝑦, 𝑧}, ℝ*, < )))
27	26	sseq1d 3233	. . . . . 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)inf({𝑦, 𝑧}, ℝ*, < )) ⊆ (𝐵 ∩ 𝐶)))
28		xrminrpcl 11751	. . . . . . . 8 ⊢ ((𝑦 ∈ ℝ+ ∧ 𝑧 ∈ ℝ+) → inf({𝑦, 𝑧}, ℝ*, < ) ∈ ℝ+)
29		oveq2 5982	. . . . . . . . . . 11 ⊢ (𝑥 = inf({𝑦, 𝑧}, ℝ*, < ) → (𝑃(ball‘𝐷)𝑥) = (𝑃(ball‘𝐷)inf({𝑦, 𝑧}, ℝ*, < )))
30	29	sseq1d 3233	. . . . . . . . . 10 ⊢ (𝑥 = inf({𝑦, 𝑧}, ℝ*, < ) → ((𝑃(ball‘𝐷)𝑥) ⊆ (𝐵 ∩ 𝐶) ↔ (𝑃(ball‘𝐷)inf({𝑦, 𝑧}, ℝ*, < )) ⊆ (𝐵 ∩ 𝐶)))
31	30	rspcev 2887	. . . . . . . . 9 ⊢ ((inf({𝑦, 𝑧}, ℝ*, < ) ∈ ℝ+ ∧ (𝑃(ball‘𝐷)inf({𝑦, 𝑧}, ℝ*, < )) ⊆ (𝐵 ∩ 𝐶)) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝐵 ∩ 𝐶))
32	31	ex 115	. . . . . . . 8 ⊢ (inf({𝑦, 𝑧}, ℝ*, < ) ∈ ℝ+ → ((𝑃(ball‘𝐷)inf({𝑦, 𝑧}, ℝ*, < )) ⊆ (𝐵 ∩ 𝐶) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝐵 ∩ 𝐶)))
33	28, 32	syl 14	. . . . . . 7 ⊢ ((𝑦 ∈ ℝ+ ∧ 𝑧 ∈ ℝ+) → ((𝑃(ball‘𝐷)inf({𝑦, 𝑧}, ℝ*, < )) ⊆ (𝐵 ∩ 𝐶) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝐵 ∩ 𝐶)))
34	33	adantl 277	. . . . . 6 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) ∧ (𝑦 ∈ ℝ+ ∧ 𝑧 ∈ ℝ+)) → ((𝑃(ball‘𝐷)inf({𝑦, 𝑧}, ℝ*, < )) ⊆ (𝐵 ∩ 𝐶) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝐵 ∩ 𝐶)))
35	27, 34	sylbid 150	. . . . 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 32	. . . 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 2636	. . 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 153	. 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 433	1 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝐵 ∩ 𝐶))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1375   ∈ wcel 2180  ∃wrex 2489   ∩ cin 3176   ⊆ wss 3177  𝒫 cpw 3629  {cpr 3647   × cxp 4694  ran crn 4697  ⟶wf 5290  ‘cfv 5294  (class class class)co 5974  infcinf 7118  ℝ^*cxr 8148   < clt 8149  ℝ⁺crp 9817  ∞Metcxmet 14465  ballcbl 14467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-coll 4178  ax-sep 4181  ax-nul 4189  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-iinf 4657  ax-cnex 8058  ax-resscn 8059  ax-1cn 8060  ax-1re 8061  ax-icn 8062  ax-addcl 8063  ax-addrcl 8064  ax-mulcl 8065  ax-mulrcl 8066  ax-addcom 8067  ax-mulcom 8068  ax-addass 8069  ax-mulass 8070  ax-distr 8071  ax-i2m1 8072  ax-0lt1 8073  ax-1rid 8074  ax-0id 8075  ax-rnegex 8076  ax-precex 8077  ax-cnre 8078  ax-pre-ltirr 8079  ax-pre-ltwlin 8080  ax-pre-lttrn 8081  ax-pre-apti 8082  ax-pre-ltadd 8083  ax-pre-mulgt0 8084  ax-pre-mulext 8085  ax-arch 8086  ax-caucvg 8087

This theorem depends on definitions:  df-bi 117  df-stab 835  df-dc 839  df-3or 984  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-nel 2476  df-ral 2493  df-rex 2494  df-reu 2495  df-rmo 2496  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-nul 3472  df-if 3583  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-iun 3946  df-br 4063  df-opab 4125  df-mpt 4126  df-tr 4162  df-id 4361  df-po 4364  df-iso 4365  df-iord 4434  df-on 4436  df-ilim 4437  df-suc 4439  df-iom 4660  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-isom 5303  df-riota 5927  df-ov 5977  df-oprab 5978  df-mpo 5979  df-1st 6256  df-2nd 6257  df-recs 6421  df-frec 6507  df-map 6767  df-sup 7119  df-inf 7120  df-pnf 8151  df-mnf 8152  df-xr 8153  df-ltxr 8154  df-le 8155  df-sub 8287  df-neg 8288  df-reap 8690  df-ap 8697  df-div 8788  df-inn 9079  df-2 9137  df-3 9138  df-4 9139  df-n0 9338  df-z 9415  df-uz 9691  df-q 9783  df-rp 9818  df-xneg 9936  df-xadd 9937  df-seqfrec 10637  df-exp 10728  df-cj 11319  df-re 11320  df-im 11321  df-rsqrt 11475  df-abs 11476  df-psmet 14472  df-xmet 14473  df-bl 14475

This theorem is referenced by:  blbas  15072