Theorem metdsge 22652

Description: The distance from the point 𝐴 to the set 𝑆 is greater than 𝑅 iff the 𝑅-ball around 𝐴 misses 𝑆. (Contributed by Mario Carneiro, 4-Sep-2015.) (Proof shortened by AV, 30-Sep-2020.)

Hypothesis

Ref	Expression
metdscn.f	⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))

Assertion

Ref	Expression
metdsge	⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ 𝑅 ∈ ℝ*) → (𝑅 ≤ (𝐹‘𝐴) ↔ (𝑆 ∩ (𝐴(ball‘𝐷)𝑅)) = ∅))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐷,𝑦   𝑥,𝑆,𝑦   𝑥,𝑋,𝑦

Allowed substitution hints:   𝑅(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem metdsge

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl3 1066	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ 𝑅 ∈ ℝ*) → 𝐴 ∈ 𝑋)
2		metdscn.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))
3	2	metdsval 22650	. . . 4 ⊢ (𝐴 ∈ 𝑋 → (𝐹‘𝐴) = inf(ran (𝑦 ∈ 𝑆 ↦ (𝐴𝐷𝑦)), ℝ*, < ))
4	1, 3	syl 17	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ 𝑅 ∈ ℝ*) → (𝐹‘𝐴) = inf(ran (𝑦 ∈ 𝑆 ↦ (𝐴𝐷𝑦)), ℝ*, < ))
5	4	breq2d 4665	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ 𝑅 ∈ ℝ*) → (𝑅 ≤ (𝐹‘𝐴) ↔ 𝑅 ≤ inf(ran (𝑦 ∈ 𝑆 ↦ (𝐴𝐷𝑦)), ℝ*, < )))
6		simpll1 1100	. . . . . 6 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ 𝑅 ∈ ℝ*) ∧ 𝑤 ∈ 𝑆) → 𝐷 ∈ (∞Met‘𝑋))
7	1	adantr 481	. . . . . 6 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ 𝑅 ∈ ℝ*) ∧ 𝑤 ∈ 𝑆) → 𝐴 ∈ 𝑋)
8		simpl2 1065	. . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ 𝑅 ∈ ℝ*) → 𝑆 ⊆ 𝑋)
9	8	sselda 3603	. . . . . 6 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ 𝑅 ∈ ℝ*) ∧ 𝑤 ∈ 𝑆) → 𝑤 ∈ 𝑋)
10		xmetcl 22136	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) → (𝐴𝐷𝑤) ∈ ℝ*)
11	6, 7, 9, 10	syl3anc 1326	. . . . 5 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ 𝑅 ∈ ℝ*) ∧ 𝑤 ∈ 𝑆) → (𝐴𝐷𝑤) ∈ ℝ*)
12		oveq2 6658	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝐴𝐷𝑦) = (𝐴𝐷𝑤))
13	12	cbvmptv 4750	. . . . 5 ⊢ (𝑦 ∈ 𝑆 ↦ (𝐴𝐷𝑦)) = (𝑤 ∈ 𝑆 ↦ (𝐴𝐷𝑤))
14	11, 13	fmptd 6385	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ 𝑅 ∈ ℝ*) → (𝑦 ∈ 𝑆 ↦ (𝐴𝐷𝑦)):𝑆⟶ℝ*)
15		frn 6053	. . . 4 ⊢ ((𝑦 ∈ 𝑆 ↦ (𝐴𝐷𝑦)):𝑆⟶ℝ* → ran (𝑦 ∈ 𝑆 ↦ (𝐴𝐷𝑦)) ⊆ ℝ*)
16	14, 15	syl 17	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ 𝑅 ∈ ℝ*) → ran (𝑦 ∈ 𝑆 ↦ (𝐴𝐷𝑦)) ⊆ ℝ*)
17		simpr 477	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ 𝑅 ∈ ℝ*) → 𝑅 ∈ ℝ*)
18		infxrgelb 12165	. . 3 ⊢ ((ran (𝑦 ∈ 𝑆 ↦ (𝐴𝐷𝑦)) ⊆ ℝ* ∧ 𝑅 ∈ ℝ*) → (𝑅 ≤ inf(ran (𝑦 ∈ 𝑆 ↦ (𝐴𝐷𝑦)), ℝ*, < ) ↔ ∀𝑧 ∈ ran (𝑦 ∈ 𝑆 ↦ (𝐴𝐷𝑦))𝑅 ≤ 𝑧))
19	16, 17, 18	syl2anc 693	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ 𝑅 ∈ ℝ*) → (𝑅 ≤ inf(ran (𝑦 ∈ 𝑆 ↦ (𝐴𝐷𝑦)), ℝ*, < ) ↔ ∀𝑧 ∈ ran (𝑦 ∈ 𝑆 ↦ (𝐴𝐷𝑦))𝑅 ≤ 𝑧))
20	17	adantr 481	. . . . . . 7 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ 𝑅 ∈ ℝ*) ∧ 𝑤 ∈ 𝑆) → 𝑅 ∈ ℝ*)
21		elbl2 22195	. . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ*) ∧ (𝐴 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝑤 ∈ (𝐴(ball‘𝐷)𝑅) ↔ (𝐴𝐷𝑤) < 𝑅))
22	6, 20, 7, 9, 21	syl22anc 1327	. . . . . 6 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ 𝑅 ∈ ℝ*) ∧ 𝑤 ∈ 𝑆) → (𝑤 ∈ (𝐴(ball‘𝐷)𝑅) ↔ (𝐴𝐷𝑤) < 𝑅))
23		xrltnle 10105	. . . . . . 7 ⊢ (((𝐴𝐷𝑤) ∈ ℝ* ∧ 𝑅 ∈ ℝ*) → ((𝐴𝐷𝑤) < 𝑅 ↔ ¬ 𝑅 ≤ (𝐴𝐷𝑤)))
24	11, 20, 23	syl2anc 693	. . . . . 6 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ 𝑅 ∈ ℝ*) ∧ 𝑤 ∈ 𝑆) → ((𝐴𝐷𝑤) < 𝑅 ↔ ¬ 𝑅 ≤ (𝐴𝐷𝑤)))
25	22, 24	bitrd 268	. . . . 5 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ 𝑅 ∈ ℝ*) ∧ 𝑤 ∈ 𝑆) → (𝑤 ∈ (𝐴(ball‘𝐷)𝑅) ↔ ¬ 𝑅 ≤ (𝐴𝐷𝑤)))
26	25	con2bid 344	. . . 4 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ 𝑅 ∈ ℝ*) ∧ 𝑤 ∈ 𝑆) → (𝑅 ≤ (𝐴𝐷𝑤) ↔ ¬ 𝑤 ∈ (𝐴(ball‘𝐷)𝑅)))
27	26	ralbidva 2985	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ 𝑅 ∈ ℝ*) → (∀𝑤 ∈ 𝑆 𝑅 ≤ (𝐴𝐷𝑤) ↔ ∀𝑤 ∈ 𝑆 ¬ 𝑤 ∈ (𝐴(ball‘𝐷)𝑅)))
28		ovex 6678	. . . . 5 ⊢ (𝐴𝐷𝑤) ∈ V
29	28	rgenw 2924	. . . 4 ⊢ ∀𝑤 ∈ 𝑆 (𝐴𝐷𝑤) ∈ V
30		breq2 4657	. . . . 5 ⊢ (𝑧 = (𝐴𝐷𝑤) → (𝑅 ≤ 𝑧 ↔ 𝑅 ≤ (𝐴𝐷𝑤)))
31	13, 30	ralrnmpt 6368	. . . 4 ⊢ (∀𝑤 ∈ 𝑆 (𝐴𝐷𝑤) ∈ V → (∀𝑧 ∈ ran (𝑦 ∈ 𝑆 ↦ (𝐴𝐷𝑦))𝑅 ≤ 𝑧 ↔ ∀𝑤 ∈ 𝑆 𝑅 ≤ (𝐴𝐷𝑤)))
32	29, 31	ax-mp 5	. . 3 ⊢ (∀𝑧 ∈ ran (𝑦 ∈ 𝑆 ↦ (𝐴𝐷𝑦))𝑅 ≤ 𝑧 ↔ ∀𝑤 ∈ 𝑆 𝑅 ≤ (𝐴𝐷𝑤))
33		disj 4017	. . 3 ⊢ ((𝑆 ∩ (𝐴(ball‘𝐷)𝑅)) = ∅ ↔ ∀𝑤 ∈ 𝑆 ¬ 𝑤 ∈ (𝐴(ball‘𝐷)𝑅))
34	27, 32, 33	3bitr4g 303	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ 𝑅 ∈ ℝ*) → (∀𝑧 ∈ ran (𝑦 ∈ 𝑆 ↦ (𝐴𝐷𝑦))𝑅 ≤ 𝑧 ↔ (𝑆 ∩ (𝐴(ball‘𝐷)𝑅)) = ∅))
35	5, 19, 34	3bitrd 294	1 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ 𝑅 ∈ ℝ*) → (𝑅 ≤ (𝐹‘𝐴) ↔ (𝑆 ∩ (𝐴(ball‘𝐷)𝑅)) = ∅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915   class class class wbr 4653   ↦ cmpt 4729  ran crn 5115  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  infcinf 8347  ℝ^*cxr 10073   < clt 10074   ≤ cle 10075  ∞Metcxmt 19731  ballcbl 19733

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-psmet 19738  df-xmet 19739  df-bl 19741

This theorem is referenced by:  metds0  22653  metdstri  22654  metdseq0  22657  lebnumlem3  22762