Theorem bdmopn 14043

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 9663	. . . . . . . 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 11276	. . . . . . 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 9662	. . . . . . 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 8006	. . . . . . . 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 9667	. . . . . . . . 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 11280	. . . . . . . . 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 9801	. . . . . 6 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( z e. X /\ r e. RR+ ) ) -> 0 <_ inf ( { r , R } , RR* , < ) )
17		xrmin1inf 11277	. . . . . . 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 9827	. . . . . 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 9695	. . . 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 11278	. . . . . . . 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 14042	. . . . . 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 2183	. . . 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 4008	. . . . . 6 |- ( s = inf ( { r , R } , RR* , < ) -> ( s <_ r <-> inf ( { r , R } , RR* , < ) <_ r ) )
31		oveq2 5885	. . . . . . 7 |- ( s = inf ( { r , R } , RR* , < ) -> ( z ( ball ` C ) s ) = ( z ( ball ` C )inf ( { r , R } , RR* , < ) ) )
32		oveq2 5885	. . . . . . 7 |- ( s = inf ( { r , R } , RR* , < ) -> ( z ( ball ` D ) s ) = ( z ( ball ` D )inf ( { r , R } , RR* , < ) ) )
33	31, 32	eqeq12d 2192	. . . . . 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 2843	. . . 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 2559	. 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 14040	. . 3 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> D e. ( *Met ` X ) )
40		stdbdmopn.2	. . . 4 |- J = ( MetOpen ` C )
41		eqid 2177	. . . 4 |- ( MetOpen ` D ) = ( MetOpen ` D )
42	40, 41	metequiv2 14035	. . 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 2148   A.wral 2455   E.wrex 2456   {cpr 3595   class class class wbr 4005   ` cfv 5218  (class class class)co 5877   e. cmpo 5879  infcinf 6984   RRcr 7812   0cc0 7813   RR*cxr 7993   < clt 7994   <_ cle 7995   RR+crp 9655   *Metcxmet 13479   ballcbl 13481   MetOpencmopn 13484

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931  ax-arch 7932  ax-caucvg 7933

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 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-isom 5227  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-frec 6394  df-map 6652  df-sup 6985  df-inf 6986  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-n0 9179  df-z 9256  df-uz 9531  df-q 9622  df-rp 9656  df-xneg 9774  df-xadd 9775  df-icc 9897  df-seqfrec 10448  df-exp 10522  df-cj 10853  df-re 10854  df-im 10855  df-rsqrt 11009  df-abs 11010  df-topgen 12714  df-psmet 13486  df-xmet 13487  df-bl 13489  df-mopn 13490  df-top 13537  df-bases 13582

This theorem is referenced by:  mopnex  14044