Theorem bdmopn 15247

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 9896	. . . . . . . 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 1027	. . . . . . 7 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> R e. RR* )
4		xrmincl 11844	. . . . . . 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 9895	. . . . . . 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 8226	. . . . . . . 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 9900	. . . . . . . . 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 1028	. . . . . . . 8 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> 0 < R )
13		xrltmininf 11848	. . . . . . . . 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 1378	. . . . . . . 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 952	. . . . . . 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 10034	. . . . . 6 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> 0 <_ inf ( { r , R } , RR* , < ) )
17		xrmin1inf 11845	. . . . . . 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 10060	. . . . . 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 1274	. . . . 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 9928	. . . 4 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> inf ( { r , R } , RR* , < ) e. RR+ )
22		simprl 531	. . . . . . 7 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> z e. X )
23		xrmin2inf 11846	. . . . . . . 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 1203	. . . . . 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 15246	. . . . . 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 2237	. . . 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 4091	. . . . . 6 |- ( s = inf ( { r , R } , RR* , < ) -> ( s <_ r <-> inf ( { r , R } , RR* , < ) <_ r ) )
31		oveq2 6026	. . . . . . 7 |- ( s = inf ( { r , R } , RR* , < ) -> ( z ( ball ` C ) s ) = ( z ( ball ` C )inf ( { r , R } , RR* , < ) ) )
32		oveq2 6026	. . . . . . 7 |- ( s = inf ( { r , R } , RR* , < ) -> ( z ( ball ` D ) s ) = ( z ( ball ` D )inf ( { r , R } , RR* , < ) ) )
33	31, 32	eqeq12d 2246	. . . . . 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 2910	. . . 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 1271	. . 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 2614	. 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 1023	. . 3 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> C e. ( *Met ` X ) )
39	26	bdxmet 15244	. . 3 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> D e. ( *Met ` X ) )
40		stdbdmopn.2	. . . 4 |- J = ( MetOpen ` C )
41		eqid 2231	. . . 4 |- ( MetOpen ` D ) = ( MetOpen ` D )
42	40, 41	metequiv2 15239	. . 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 1004   = wceq 1397   e. wcel 2202   A.wral 2510   E.wrex 2511   {cpr 3670   class class class wbr 4088   ` cfv 5326  (class class class)co 6018   e. cmpo 6020  infcinf 7182   RRcr 8031   0cc0 8032   RR*cxr 8213   < clt 8214   <_ cle 8215   RR+crp 9888   *Metcxmet 14569   ballcbl 14571   MetOpencmopn 14574

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150  ax-arch 8151  ax-caucvg 8152

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-map 6819  df-sup 7183  df-inf 7184  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-n0 9403  df-z 9480  df-uz 9756  df-q 9854  df-rp 9889  df-xneg 10007  df-xadd 10008  df-icc 10130  df-seqfrec 10711  df-exp 10802  df-cj 11420  df-re 11421  df-im 11422  df-rsqrt 11576  df-abs 11577  df-topgen 13361  df-psmet 14576  df-xmet 14577  df-bl 14579  df-mopn 14580  df-top 14741  df-bases 14786

This theorem is referenced by:  mopnex  15248