Theorem bdmopn 15605

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 10062	. . . . . . . 8 |- ( r e. RR+ -> r e. RR* )
2	1	ad2antll 495	. . . . . . 7 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> r e. RR* )
3		simpl2 1032	. . . . . . 7 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> R e. RR* )
4		xrmincl 12032	. . . . . . 7 |- ( ( r e. RR* /\ R e. RR* ) -> inf ( { r , R } , RR* , < ) e. RR* )
5	2, 3, 4	syl2anc 415	. . . . . 6 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> inf ( { r , R } , RR* , < ) e. RR* )
6		rpre 10061	. . . . . . 7 |- ( r e. RR+ -> r e. RR )
7	6	ad2antll 495	. . . . . 6 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> r e. RR )
8		0xr 8372	. . . . . . . 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 10066	. . . . . . . . 9 |- ( r e. RR+ -> 0 < r )
11	10	ad2antll 495	. . . . . . . 8 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> 0 < r )
12		simpl3 1033	. . . . . . . 8 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> 0 < R )
13		xrltmininf 12036	. . . . . . . . 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 1383	. . . . . . . 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 957	. . . . . . 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 10201	. . . . . 6 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> 0 <_ inf ( { r , R } , RR* , < ) )
17		xrmin1inf 12033	. . . . . . 7 |- ( ( r e. RR* /\ R e. RR* ) -> inf ( { r , R } , RR* , < ) <_ r )
18	2, 3, 17	syl2anc 415	. . . . . 6 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> inf ( { r , R } , RR* , < ) <_ r )
19		xrrege0 10227	. . . . . 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 1279	. . . . 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 10094	. . . 4 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> inf ( { r , R } , RR* , < ) e. RR+ )
22		simprl 535	. . . . . . 7 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> z e. X )
23		xrmin2inf 12034	. . . . . . . 8 |- ( ( r e. RR* /\ R e. RR* ) -> inf ( { r , R } , RR* , < ) <_ R )
24	2, 3, 23	syl2anc 415	. . . . . . 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 1208	. . . . . 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 15604	. . . . . 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 2244	. . . 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 4133	. . . . . 6 |- ( s = inf ( { r , R } , RR* , < ) -> ( s <_ r <-> inf ( { r , R } , RR* , < ) <_ r ) )
31		oveq2 6093	. . . . . . 7 |- ( s = inf ( { r , R } , RR* , < ) -> ( z ( ball ` C ) s ) = ( z ( ball ` C )inf ( { r , R } , RR* , < ) ) )
32		oveq2 6093	. . . . . . 7 |- ( s = inf ( { r , R } , RR* , < ) -> ( z ( ball ` D ) s ) = ( z ( ball ` D )inf ( { r , R } , RR* , < ) ) )
33	31, 32	eqeq12d 2253	. . . . . 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 477	. . . . 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 2929	. . . 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 1276	. . 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 2632	. 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 1028	. . 3 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> C e. ( *Met ` X ) )
39	26	bdxmet 15602	. . 3 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> D e. ( *Met ` X ) )
40		stdbdmopn.2	. . . 4 |- J = ( MetOpen ` C )
41		eqid 2238	. . . 4 |- ( MetOpen ` D ) = ( MetOpen ` D )
42	40, 41	metequiv2 15597	. . 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 415	. 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 ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   {cpr 3710   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   e. cmpo 6087  infcinf 7323   RRcr 8178   0cc0 8179   RR*cxr 8359   < clt 8360   <_ cle 8361   RR+crp 10054   *Metcxmet 14873   ballcbl 14875   MetOpencmopn 14878

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-map 6924  df-sup 7324  df-inf 7325  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-n0 9564  df-z 9645  df-uz 9922  df-q 10020  df-rp 10055  df-xneg 10174  df-xadd 10175  df-icc 10297  df-seqfrec 10885  df-exp 10976  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-topgen 13614  df-psmet 14880  df-xmet 14881  df-bl 14883  df-mopn 14884  df-top 15099  df-bases 15144

This theorem is used by:  mopnex  15606