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 e. X |-> inf ( ran ( y e. S |-> ( x D y ) ) , RR* , < ) )

Assertion

Ref	Expression
metdsge	|- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ R e. RR* ) -> ( R <_ ( F ` A ) <-> ( S i^i ( 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 e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ R e. RR* ) -> A e. X )
2		metdscn.f	. . . . 5 |- F = ( x e. X |-> inf ( ran ( y e. S |-> ( x D y ) ) , RR* , < ) )
3	2	metdsval 22650	. . . 4 |- ( A e. X -> ( F ` A ) = inf ( ran ( y e. S |-> ( A D y ) ) , RR* , < ) )
4	1, 3	syl 17	. . 3 |- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ R e. RR* ) -> ( F ` A ) = inf ( ran ( y e. S |-> ( A D y ) ) , RR* , < ) )
5	4	breq2d 4665	. 2 |- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ R e. RR* ) -> ( R <_ ( F ` A ) <-> R <_ inf ( ran ( y e. S |-> ( A D y ) ) , RR* , < ) ) )
6		simpll1 1100	. . . . . 6 |- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ R e. RR* ) /\ w e. S ) -> D e. ( *Met ` X ) )
7	1	adantr 481	. . . . . 6 |- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ R e. RR* ) /\ w e. S ) -> A e. X )
8		simpl2 1065	. . . . . . 7 |- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ R e. RR* ) -> S C_ X )
9	8	sselda 3603	. . . . . 6 |- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ R e. RR* ) /\ w e. S ) -> w e. X )
10		xmetcl 22136	. . . . . 6 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ w e. X ) -> ( A D w ) e. RR* )
11	6, 7, 9, 10	syl3anc 1326	. . . . 5 |- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ R e. RR* ) /\ w e. S ) -> ( A D w ) e. RR* )
12		oveq2 6658	. . . . . 6 |- ( y = w -> ( A D y ) = ( A D w ) )
13	12	cbvmptv 4750	. . . . 5 |- ( y e. S |-> ( A D y ) ) = ( w e. S |-> ( A D w ) )
14	11, 13	fmptd 6385	. . . 4 |- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ R e. RR* ) -> ( y e. S |-> ( A D y ) ) : S --> RR* )
15		frn 6053	. . . 4 |- ( ( y e. S |-> ( A D y ) ) : S --> RR* -> ran ( y e. S |-> ( A D y ) ) C_ RR* )
16	14, 15	syl 17	. . 3 |- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ R e. RR* ) -> ran ( y e. S |-> ( A D y ) ) C_ RR* )
17		simpr 477	. . 3 |- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ R e. RR* ) -> R e. RR* )
18		infxrgelb 12165	. . 3 |- ( ( ran ( y e. S |-> ( A D y ) ) C_ RR* /\ R e. RR* ) -> ( R <_ inf ( ran ( y e. S |-> ( A D y ) ) , RR* , < ) <-> A. z e. ran ( y e. S |-> ( A D y ) ) R <_ z ) )
19	16, 17, 18	syl2anc 693	. 2 |- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ R e. RR* ) -> ( R <_ inf ( ran ( y e. S |-> ( A D y ) ) , RR* , < ) <-> A. z e. ran ( y e. S |-> ( A D y ) ) R <_ z ) )
20	17	adantr 481	. . . . . . 7 |- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ R e. RR* ) /\ w e. S ) -> R e. RR* )
21		elbl2 22195	. . . . . . 7 |- ( ( ( D e. ( *Met ` X ) /\ R e. RR* ) /\ ( A e. X /\ w e. X ) ) -> ( w e. ( A ( ball ` D ) R ) <-> ( A D w ) < R ) )
22	6, 20, 7, 9, 21	syl22anc 1327	. . . . . 6 |- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ R e. RR* ) /\ w e. S ) -> ( w e. ( A ( ball ` D ) R ) <-> ( A D w ) < R ) )
23		xrltnle 10105	. . . . . . 7 |- ( ( ( A D w ) e. RR* /\ R e. RR* ) -> ( ( A D w ) < R <-> -. R <_ ( A D w ) ) )
24	11, 20, 23	syl2anc 693	. . . . . 6 |- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ R e. RR* ) /\ w e. S ) -> ( ( A D w ) < R <-> -. R <_ ( A D w ) ) )
25	22, 24	bitrd 268	. . . . 5 |- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ R e. RR* ) /\ w e. S ) -> ( w e. ( A ( ball ` D ) R ) <-> -. R <_ ( A D w ) ) )
26	25	con2bid 344	. . . 4 |- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ R e. RR* ) /\ w e. S ) -> ( R <_ ( A D w ) <-> -. w e. ( A ( ball ` D ) R ) ) )
27	26	ralbidva 2985	. . 3 |- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ R e. RR* ) -> ( A. w e. S R <_ ( A D w ) <-> A. w e. S -. w e. ( A ( ball ` D ) R ) ) )
28		ovex 6678	. . . . 5 |- ( A D w ) e. _V
29	28	rgenw 2924	. . . 4 |- A. w e. S ( A D w ) e. _V
30		breq2 4657	. . . . 5 |- ( z = ( A D w ) -> ( R <_ z <-> R <_ ( A D w ) ) )
31	13, 30	ralrnmpt 6368	. . . 4 |- ( A. w e. S ( A D w ) e. _V -> ( A. z e. ran ( y e. S |-> ( A D y ) ) R <_ z <-> A. w e. S R <_ ( A D w ) ) )
32	29, 31	ax-mp 5	. . 3 |- ( A. z e. ran ( y e. S |-> ( A D y ) ) R <_ z <-> A. w e. S R <_ ( A D w ) )
33		disj 4017	. . 3 |- ( ( S i^i ( A ( ball ` D ) R ) ) = (/) <-> A. w e. S -. w e. ( A ( ball ` D ) R ) )
34	27, 32, 33	3bitr4g 303	. 2 |- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ R e. RR* ) -> ( A. z e. ran ( y e. S |-> ( A D y ) ) R <_ z <-> ( S i^i ( A ( ball ` D ) R ) ) = (/) ) )
35	5, 19, 34	3bitrd 294	1 |- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ R e. RR* ) -> ( R <_ ( F ` A ) <-> ( S i^i ( A ( ball ` D ) R ) ) = (/) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   i^i cin 3573   C_ 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   RR*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