Theorem bdmopn 13555

Description: The standard bounded metric corresponding to C generates the same topology as C. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 19-May-2023.)

Hypotheses

Ref	Expression
stdbdmet.1	|- D = ( x e. X , y e. X |-> inf ( { ( x C y ) , R } , RR* , < ) )
stdbdmopn.2	|- J = ( MetOpen ` C )

Assertion

Ref	Expression
bdmopn	|- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> J = ( MetOpen ` D ) )

Distinct variable groups:   x, y, C   x, R, y   x, X, y

Allowed substitution hints:   D( x, y)   J( x, y)

Proof of Theorem bdmopn

Dummy variables r s z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rpxr 9630	. . . . . . . 8 |- ( r e. RR+ -> r e. RR* )
2	1	ad2antll 491	. . . . . . 7 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> r e. RR* )
3		simpl2 1001	. . . . . . 7 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> R e. RR* )
4		xrmincl 11240	. . . . . . 7 |- ( ( r e. RR* /\ R e. RR* ) -> inf ( { r , R } , RR* , < ) e. RR* )
5	2, 3, 4	syl2anc 411	. . . . . 6 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> inf ( { r , R } , RR* , < ) e. RR* )
6		rpre 9629	. . . . . . 7 |- ( r e. RR+ -> r e. RR )
7	6	ad2antll 491	. . . . . 6 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> r e. RR )
8		0xr 7978	. . . . . . . 8 |- 0 e. RR*
9	8	a1i 9	. . . . . . 7 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> 0 e. RR* )
10		rpgt0 9634	. . . . . . . . 9 |- ( r e. RR+ -> 0 < r )
11	10	ad2antll 491	. . . . . . . 8 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> 0 < r )
12		simpl3 1002	. . . . . . . 8 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> 0 < R )
13		xrltmininf 11244	. . . . . . . . 9 |- ( ( 0 e. RR* /\ r e. RR* /\ R e. RR* ) -> ( 0 < inf ( { r , R } , RR* , < ) <-> ( 0 < r /\ 0 < R ) ) )
14	8, 2, 3, 13	mp3an2i 1342	. . . . . . . 8 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> ( 0 < inf ( { r , R } , RR* , < ) <-> ( 0 < r /\ 0 < R ) ) )
15	11, 12, 14	mpbir2and 944	. . . . . . 7 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> 0 < inf ( { r , R } , RR* , < ) )
16	9, 5, 15	xrltled 9768	. . . . . 6 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> 0 <_ inf ( { r , R } , RR* , < ) )
17		xrmin1inf 11241	. . . . . . 7 |- ( ( r e. RR* /\ R e. RR* ) -> inf ( { r , R } , RR* , < ) <_ r )
18	2, 3, 17	syl2anc 411	. . . . . 6 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> inf ( { r , R } , RR* , < ) <_ r )
19		xrrege0 9794	. . . . . 6 |- ( ( (inf ( { r , R } , RR* , < ) e. RR* /\ r e. RR ) /\ ( 0 <_ inf ( { r , R } , RR* , < ) /\ inf ( { r , R } , RR* , < ) <_ r ) ) -> inf ( { r , R } , RR* , < ) e. RR )
20	5, 7, 16, 18, 19	syl22anc 1239	. . . . 5 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> inf ( { r , R } , RR* , < ) e. RR )
21	20, 15	elrpd 9662	. . . 4 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> inf ( { r , R } , RR* , < ) e. RR+ )
22		simprl 529	. . . . . . 7 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> z e. X )
23		xrmin2inf 11242	. . . . . . . 8 |- ( ( r e. RR* /\ R e. RR* ) -> inf ( { r , R } , RR* , < ) <_ R )
24	2, 3, 23	syl2anc 411	. . . . . . 7 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> inf ( { r , R } , RR* , < ) <_ R )
25	22, 5, 24	3jca 1177	. . . . . 6 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> ( z e. X /\ inf ( { r , R } , RR* , < ) e. RR* /\ inf ( { r , R } , RR* , < ) <_ R ) )
26		stdbdmet.1	. . . . . . 7 |- D = ( x e. X , y e. X |-> inf ( { ( x C y ) , R } , RR* , < ) )
27	26	bdbl 13554	. . . . . 6 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ inf ( { r , R } , RR* , < ) e. RR* /\ inf ( { r , R } , RR* , < ) <_ R ) ) -> ( z ( ball ` D )inf ( { r , R } , RR* , < ) ) = ( z ( ball ` C )inf ( { r , R } , RR* , < ) ) )
28	25, 27	syldan 282	. . . . 5 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> ( z ( ball ` D )inf ( { r , R } , RR* , < ) ) = ( z ( ball ` C )inf ( { r , R } , RR* , < ) ) )
29	28	eqcomd 2181	. . . 4 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> ( z ( ball ` C )inf ( { r , R } , RR* , < ) ) = ( z ( ball ` D )inf ( { r , R } , RR* , < ) ) )
30		breq1 4001	. . . . . 6 |- ( s = inf ( { r , R } , RR* , < ) -> ( s <_ r <-> inf ( { r , R } , RR* , < ) <_ r ) )
31		oveq2 5873	. . . . . . 7 |- ( s = inf ( { r , R } , RR* , < ) -> ( z ( ball ` C ) s ) = ( z ( ball ` C )inf ( { r , R } , RR* , < ) ) )
32		oveq2 5873	. . . . . . 7 |- ( s = inf ( { r , R } , RR* , < ) -> ( z ( ball ` D ) s ) = ( z ( ball ` D )inf ( { r , R } , RR* , < ) ) )
33	31, 32	eqeq12d 2190	. . . . . 6 |- ( s = inf ( { r , R } , RR* , < ) -> ( ( z ( ball ` C ) s ) = ( z ( ball ` D ) s ) <-> ( z ( ball ` C )inf ( { r , R } , RR* , < ) ) = ( z ( ball ` D )inf ( { r , R } , RR* , < ) ) ) )
34	30, 33	anbi12d 473	. . . . 5 |- ( s = inf ( { r , R } , RR* , < ) -> ( ( s <_ r /\ ( z ( ball ` C ) s ) = ( z ( ball ` D ) s ) ) <-> (inf ( { r , R } , RR* , < ) <_ r /\ ( z ( ball ` C )inf ( { r , R } , RR* , < ) ) = ( z ( ball ` D )inf ( { r , R } , RR* , < ) ) ) ) )
35	34	rspcev 2839	. . . 4 |- ( (inf ( { r , R } , RR* , < ) e. RR+ /\ (inf ( { r , R } , RR* , < ) <_ r /\ ( z ( ball ` C )inf ( { r , R } , RR* , < ) ) = ( z ( ball ` D )inf ( { r , R } , RR* , < ) ) ) ) -> E. s e. RR+ ( s <_ r /\ ( z ( ball ` C ) s ) = ( z ( ball ` D ) s ) ) )
36	21, 18, 29, 35	syl12anc 1236	. . 3 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> E. s e. RR+ ( s <_ r /\ ( z ( ball ` C ) s ) = ( z ( ball ` D ) s ) ) )
37	36	ralrimivva 2557	. 2 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> A. z e. X A. r e. RR+ E. s e. RR+ ( s <_ r /\ ( z ( ball ` C ) s ) = ( z ( ball ` D ) s ) ) )
38		simp1 997	. . 3 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> C e. ( *Met ` X ) )
39	26	bdxmet 13552	. . 3 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> D e. ( *Met ` X ) )
40		stdbdmopn.2	. . . 4 |- J = ( MetOpen ` C )
41		eqid 2175	. . . 4 |- ( MetOpen ` D ) = ( MetOpen ` D )
42	40, 41	metequiv2 13547	. . 3 |- ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` X ) ) -> ( A. z e. X A. r e. RR+ E. s e. RR+ ( s <_ r /\ ( z ( ball ` C ) s ) = ( z ( ball ` D ) s ) ) -> J = ( MetOpen ` D ) ) )
43	38, 39, 42	syl2anc 411	. 2 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> ( A. z e. X A. r e. RR+ E. s e. RR+ ( s <_ r /\ ( z ( ball ` C ) s ) = ( z ( ball ` D ) s ) ) -> J = ( MetOpen ` D ) ) )
44	37, 43	mpd 13	1 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> J = ( MetOpen ` D ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 978   = wceq 1353   e. wcel 2146   A.wral 2453   E.wrex 2454   {cpr 3590   class class class wbr 3998   ` cfv 5208  (class class class)co 5865   e. cmpo 5867  infcinf 6972   RRcr 7785   0cc0 7786   RR*cxr 7965   < clt 7966   <_ cle 7967   RR+crp 9622   *Metcxmet 13031   ballcbl 13033   MetOpencmopn 13036

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904  ax-arch 7905  ax-caucvg 7906

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-ilim 4363  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-isom 5217  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-frec 6382  df-map 6640  df-sup 6973  df-inf 6974  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513  df-div 8602  df-inn 8891  df-2 8949  df-3 8950  df-4 8951  df-n0 9148  df-z 9225  df-uz 9500  df-q 9591  df-rp 9623  df-xneg 9741  df-xadd 9742  df-icc 9864  df-seqfrec 10414  df-exp 10488  df-cj 10817  df-re 10818  df-im 10819  df-rsqrt 10973  df-abs 10974  df-topgen 12629  df-psmet 13038  df-xmet 13039  df-bl 13041  df-mopn 13042  df-top 13047  df-bases 13092

This theorem is referenced by:  mopnex  13556