Theorem bdmopn 14401

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 9679	. . . . . . . 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 1003	. . . . . . 7 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> R e. RR* )
4		xrmincl 11292	. . . . . . 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 9678	. . . . . . 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 8022	. . . . . . . 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 9683	. . . . . . . . 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 1004	. . . . . . . 8 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> 0 < R )
13		xrltmininf 11296	. . . . . . . . 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 1353	. . . . . . . 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 946	. . . . . . 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 9817	. . . . . 6 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> 0 <_ inf ( { r , R } , RR* , < ) )
17		xrmin1inf 11293	. . . . . . 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 9843	. . . . . 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 1250	. . . . 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 9711	. . . 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 11294	. . . . . . . 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 1179	. . . . . 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 14400	. . . . . 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 2195	. . . 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 4021	. . . . . 6 |- ( s = inf ( { r , R } , RR* , < ) -> ( s <_ r <-> inf ( { r , R } , RR* , < ) <_ r ) )
31		oveq2 5899	. . . . . . 7 |- ( s = inf ( { r , R } , RR* , < ) -> ( z ( ball ` C ) s ) = ( z ( ball ` C )inf ( { r , R } , RR* , < ) ) )
32		oveq2 5899	. . . . . . 7 |- ( s = inf ( { r , R } , RR* , < ) -> ( z ( ball ` D ) s ) = ( z ( ball ` D )inf ( { r , R } , RR* , < ) ) )
33	31, 32	eqeq12d 2204	. . . . . 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 2856	. . . 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 1247	. . 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 2572	. 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 999	. . 3 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> C e. ( *Met ` X ) )
39	26	bdxmet 14398	. . 3 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> D e. ( *Met ` X ) )
40		stdbdmopn.2	. . . 4 |- J = ( MetOpen ` C )
41		eqid 2189	. . . 4 |- ( MetOpen ` D ) = ( MetOpen ` D )
42	40, 41	metequiv2 14393	. . 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 980   = wceq 1364   e. wcel 2160   A.wral 2468   E.wrex 2469   {cpr 3608   class class class wbr 4018   ` cfv 5231  (class class class)co 5891   e. cmpo 5893  infcinf 7000   RRcr 7828   0cc0 7829   RR*cxr 8009   < clt 8010   <_ cle 8011   RR+crp 9671   *Metcxmet 13810   ballcbl 13812   MetOpencmopn 13815

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602  ax-cnex 7920  ax-resscn 7921  ax-1cn 7922  ax-1re 7923  ax-icn 7924  ax-addcl 7925  ax-addrcl 7926  ax-mulcl 7927  ax-mulrcl 7928  ax-addcom 7929  ax-mulcom 7930  ax-addass 7931  ax-mulass 7932  ax-distr 7933  ax-i2m1 7934  ax-0lt1 7935  ax-1rid 7936  ax-0id 7937  ax-rnegex 7938  ax-precex 7939  ax-cnre 7940  ax-pre-ltirr 7941  ax-pre-ltwlin 7942  ax-pre-lttrn 7943  ax-pre-apti 7944  ax-pre-ltadd 7945  ax-pre-mulgt0 7946  ax-pre-mulext 7947  ax-arch 7948  ax-caucvg 7949

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-po 4311  df-iso 4312  df-iord 4381  df-on 4383  df-ilim 4384  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-isom 5240  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-recs 6324  df-frec 6410  df-map 6668  df-sup 7001  df-inf 7002  df-pnf 8012  df-mnf 8013  df-xr 8014  df-ltxr 8015  df-le 8016  df-sub 8148  df-neg 8149  df-reap 8550  df-ap 8557  df-div 8648  df-inn 8938  df-2 8996  df-3 8997  df-4 8998  df-n0 9195  df-z 9272  df-uz 9547  df-q 9638  df-rp 9672  df-xneg 9790  df-xadd 9791  df-icc 9913  df-seqfrec 10464  df-exp 10538  df-cj 10869  df-re 10870  df-im 10871  df-rsqrt 11025  df-abs 11026  df-topgen 12731  df-psmet 13817  df-xmet 13818  df-bl 13820  df-mopn 13821  df-top 13895  df-bases 13940

This theorem is referenced by:  mopnex  14402