Theorem blin2 13226

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 524	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → 𝐷 ∈ (∞Met‘𝑋))
2		simprl 526	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → 𝐵 ∈ ran (ball‘𝐷))
3		simplr 525	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → 𝑃 ∈ (𝐵 ∩ 𝐶))
4	3	elin1d 3316	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → 𝑃 ∈ 𝐵)
5		blss 13222	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ ran (ball‘𝐷) ∧ 𝑃 ∈ 𝐵) → ∃𝑦 ∈ ℝ+ (𝑃(ball‘𝐷)𝑦) ⊆ 𝐵)
6	1, 2, 4, 5	syl3anc 1233	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → ∃𝑦 ∈ ℝ+ (𝑃(ball‘𝐷)𝑦) ⊆ 𝐵)
7		simprr 527	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → 𝐶 ∈ ran (ball‘𝐷))
8	3	elin2d 3317	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → 𝑃 ∈ 𝐶)
9		blss 13222	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐶 ∈ ran (ball‘𝐷) ∧ 𝑃 ∈ 𝐶) → ∃𝑧 ∈ ℝ+ (𝑃(ball‘𝐷)𝑧) ⊆ 𝐶)
10	1, 7, 8, 9	syl3anc 1233	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → ∃𝑧 ∈ ℝ+ (𝑃(ball‘𝐷)𝑧) ⊆ 𝐶)
11		reeanv 2639	. . 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 3355	. . . . 5 ⊢ (((𝑃(ball‘𝐷)𝑦) ⊆ 𝐵 ∧ (𝑃(ball‘𝐷)𝑧) ⊆ 𝐶) → ((𝑃(ball‘𝐷)𝑦) ∩ (𝑃(ball‘𝐷)𝑧)) ⊆ (𝐵 ∩ 𝐶))
13		inss1 3347	. . . . . . . . . . 11 ⊢ (𝐵 ∩ 𝐶) ⊆ 𝐵
14		blf 13204	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋)
15		frn 5356	. . . . . . . . . . . . . 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 3148	. . . . . . . . . . . 12 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → 𝐵 ∈ 𝒫 𝑋)
18	17	elpwid 3577	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → 𝐵 ⊆ 𝑋)
19	13, 18	sstrid 3158	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → (𝐵 ∩ 𝐶) ⊆ 𝑋)
20	19, 3	sseldd 3148	. . . . . . . . 9 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → 𝑃 ∈ 𝑋)
21	1, 20	jca 304	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → (𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋))
22		rpxr 9618	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ+ → 𝑦 ∈ ℝ*)
23		rpxr 9618	. . . . . . . . 9 ⊢ (𝑧 ∈ ℝ+ → 𝑧 ∈ ℝ*)
24	22, 23	anim12i 336	. . . . . . . 8 ⊢ ((𝑦 ∈ ℝ+ ∧ 𝑧 ∈ ℝ+) → (𝑦 ∈ ℝ* ∧ 𝑧 ∈ ℝ*))
25		blininf 13218	. . . . . . . 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 287	. . . . . . 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 3176	. . . . . 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 11237	. . . . . . . 8 ⊢ ((𝑦 ∈ ℝ+ ∧ 𝑧 ∈ ℝ+) → inf({𝑦, 𝑧}, ℝ*, < ) ∈ ℝ+)
29		oveq2 5861	. . . . . . . . . . 11 ⊢ (𝑥 = inf({𝑦, 𝑧}, ℝ*, < ) → (𝑃(ball‘𝐷)𝑥) = (𝑃(ball‘𝐷)inf({𝑦, 𝑧}, ℝ*, < )))
30	29	sseq1d 3176	. . . . . . . . . 10 ⊢ (𝑥 = inf({𝑦, 𝑧}, ℝ*, < ) → ((𝑃(ball‘𝐷)𝑥) ⊆ (𝐵 ∩ 𝐶) ↔ (𝑃(ball‘𝐷)inf({𝑦, 𝑧}, ℝ*, < )) ⊆ (𝐵 ∩ 𝐶)))
31	30	rspcev 2834	. . . . . . . . 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 114	. . . . . . . 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 275	. . . . . 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 149	. . . . 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 2595	. . 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	syl5bir 152	. 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 431	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 103   = wceq 1348   ∈ wcel 2141  ∃wrex 2449   ∩ cin 3120   ⊆ wss 3121  𝒫 cpw 3566  {cpr 3584   × cxp 4609  ran crn 4612  ⟶wf 5194  ‘cfv 5198  (class class class)co 5853  infcinf 6960  ℝ^*cxr 7953   < clt 7954  ℝ⁺crp 9610  ∞Metcxmet 12774  ballcbl 12776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-map 6628  df-sup 6961  df-inf 6962  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-xneg 9729  df-xadd 9730  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-psmet 12781  df-xmet 12782  df-bl 12784

This theorem is referenced by:  blbas  13227