Theorem bdbl 15217

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

Hypothesis

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

Assertion

Ref	Expression
bdbl	|- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( P e. X /\ S e. RR* /\ S <_ R ) ) -> ( P ( ball ` D ) S ) = ( P ( ball ` C ) S ) )

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

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

Proof of Theorem bdbl

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr2 1028	. . . . . 6 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( P e. X /\ S e. RR* /\ S <_ R ) ) -> S e. RR* )
2	1	adantr 276	. . . . 5 |- ( ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( P e. X /\ S e. RR* /\ S <_ R ) ) /\ z e. X ) -> S e. RR* )
3		simpl1 1024	. . . . . . 7 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( P e. X /\ S e. RR* /\ S <_ R ) ) -> C e. ( *Met ` X ) )
4	3	adantr 276	. . . . . 6 |- ( ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( P e. X /\ S e. RR* /\ S <_ R ) ) /\ z e. X ) -> C e. ( *Met ` X ) )
5		simpr1 1027	. . . . . . 7 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( P e. X /\ S e. RR* /\ S <_ R ) ) -> P e. X )
6	5	adantr 276	. . . . . 6 |- ( ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( P e. X /\ S e. RR* /\ S <_ R ) ) /\ z e. X ) -> P e. X )
7		simpr 110	. . . . . 6 |- ( ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( P e. X /\ S e. RR* /\ S <_ R ) ) /\ z e. X ) -> z e. X )
8		xmetcl 15066	. . . . . 6 |- ( ( C e. ( *Met ` X ) /\ P e. X /\ z e. X ) -> ( P C z ) e. RR* )
9	4, 6, 7, 8	syl3anc 1271	. . . . 5 |- ( ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( P e. X /\ S e. RR* /\ S <_ R ) ) /\ z e. X ) -> ( P C z ) e. RR* )
10		simpll2 1061	. . . . 5 |- ( ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( P e. X /\ S e. RR* /\ S <_ R ) ) /\ z e. X ) -> R e. RR* )
11		xrminltinf 11823	. . . . 5 |- ( ( S e. RR* /\ ( P C z ) e. RR* /\ R e. RR* ) -> (inf ( { ( P C z ) , R } , RR* , < ) < S <-> ( ( P C z ) < S \/ R < S ) ) )
12	2, 9, 10, 11	syl3anc 1271	. . . 4 |- ( ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( P e. X /\ S e. RR* /\ S <_ R ) ) /\ z e. X ) -> (inf ( { ( P C z ) , R } , RR* , < ) < S <-> ( ( P C z ) < S \/ R < S ) ) )
13		xmetf 15064	. . . . . . . . 9 |- ( C e. ( *Met ` X ) -> C : ( X X. X ) --> RR* )
14	13	3ad2ant1 1042	. . . . . . . 8 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> C : ( X X. X ) --> RR* )
15	14	adantr 276	. . . . . . 7 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( P e. X /\ S e. RR* /\ S <_ R ) ) -> C : ( X X. X ) --> RR* )
16	15	adantr 276	. . . . . 6 |- ( ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( P e. X /\ S e. RR* /\ S <_ R ) ) /\ z e. X ) -> C : ( X X. X ) --> RR* )
17		stdbdmet.1	. . . . . . 7 |- D = ( x e. X , y e. X |-> inf ( { ( x C y ) , R } , RR* , < ) )
18	17	bdmetval 15214	. . . . . 6 |- ( ( ( C : ( X X. X ) --> RR* /\ R e. RR* ) /\ ( P e. X /\ z e. X ) ) -> ( P D z ) = inf ( { ( P C z ) , R } , RR* , < ) )
19	16, 10, 6, 7, 18	syl22anc 1272	. . . . 5 |- ( ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( P e. X /\ S e. RR* /\ S <_ R ) ) /\ z e. X ) -> ( P D z ) = inf ( { ( P C z ) , R } , RR* , < ) )
20	19	breq1d 4096	. . . 4 |- ( ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( P e. X /\ S e. RR* /\ S <_ R ) ) /\ z e. X ) -> ( ( P D z ) < S <-> inf ( { ( P C z ) , R } , RR* , < ) < S ) )
21		simpr3 1029	. . . . . . . 8 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( P e. X /\ S e. RR* /\ S <_ R ) ) -> S <_ R )
22		simpl2 1025	. . . . . . . . 9 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( P e. X /\ S e. RR* /\ S <_ R ) ) -> R e. RR* )
23		xrlenlt 8234	. . . . . . . . 9 |- ( ( S e. RR* /\ R e. RR* ) -> ( S <_ R <-> -. R < S ) )
24	1, 22, 23	syl2anc 411	. . . . . . . 8 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( P e. X /\ S e. RR* /\ S <_ R ) ) -> ( S <_ R <-> -. R < S ) )
25	21, 24	mpbid 147	. . . . . . 7 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( P e. X /\ S e. RR* /\ S <_ R ) ) -> -. R < S )
26		biorf 749	. . . . . . 7 |- ( -. R < S -> ( ( P C z ) < S <-> ( R < S \/ ( P C z ) < S ) ) )
27	25, 26	syl 14	. . . . . 6 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( P e. X /\ S e. RR* /\ S <_ R ) ) -> ( ( P C z ) < S <-> ( R < S \/ ( P C z ) < S ) ) )
28		orcom 733	. . . . . 6 |- ( ( R < S \/ ( P C z ) < S ) <-> ( ( P C z ) < S \/ R < S ) )
29	27, 28	bitrdi 196	. . . . 5 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( P e. X /\ S e. RR* /\ S <_ R ) ) -> ( ( P C z ) < S <-> ( ( P C z ) < S \/ R < S ) ) )
30	29	adantr 276	. . . 4 |- ( ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( P e. X /\ S e. RR* /\ S <_ R ) ) /\ z e. X ) -> ( ( P C z ) < S <-> ( ( P C z ) < S \/ R < S ) ) )
31	12, 20, 30	3bitr4d 220	. . 3 |- ( ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( P e. X /\ S e. RR* /\ S <_ R ) ) /\ z e. X ) -> ( ( P D z ) < S <-> ( P C z ) < S ) )
32	31	rabbidva 2788	. 2 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( P e. X /\ S e. RR* /\ S <_ R ) ) -> { z e. X | ( P D z ) < S } = { z e. X | ( P C z ) < S } )
33	17	bdxmet 15215	. . . 4 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> D e. ( *Met ` X ) )
34	33	adantr 276	. . 3 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( P e. X /\ S e. RR* /\ S <_ R ) ) -> D e. ( *Met ` X ) )
35		blval 15103	. . 3 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ S e. RR* ) -> ( P ( ball ` D ) S ) = { z e. X | ( P D z ) < S } )
36	34, 5, 1, 35	syl3anc 1271	. 2 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( P e. X /\ S e. RR* /\ S <_ R ) ) -> ( P ( ball ` D ) S ) = { z e. X | ( P D z ) < S } )
37		blval 15103	. . 3 |- ( ( C e. ( *Met ` X ) /\ P e. X /\ S e. RR* ) -> ( P ( ball ` C ) S ) = { z e. X | ( P C z ) < S } )
38	3, 5, 1, 37	syl3anc 1271	. 2 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( P e. X /\ S e. RR* /\ S <_ R ) ) -> ( P ( ball ` C ) S ) = { z e. X | ( P C z ) < S } )
39	32, 36, 38	3eqtr4d 2272	1 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( P e. X /\ S e. RR* /\ S <_ R ) ) -> ( P ( ball ` D ) S ) = ( P ( ball ` C ) S ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713   /\ w3a 1002   = wceq 1395   e. wcel 2200   {crab 2512   {cpr 3668   class class class wbr 4086   X. cxp 4721   -->wf 5320   ` cfv 5324  (class class class)co 6013   e. cmpo 6015  infcinf 7173   0cc0 8022   RR*cxr 8203   < clt 8204   <_ cle 8205   *Metcxmet 14540   ballcbl 14542

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138  ax-pre-mulgt0 8139  ax-pre-mulext 8140  ax-arch 8141  ax-caucvg 8142

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-isom 5333  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-map 6814  df-sup 7174  df-inf 7175  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-reap 8745  df-ap 8752  df-div 8843  df-inn 9134  df-2 9192  df-3 9193  df-4 9194  df-n0 9393  df-z 9470  df-uz 9746  df-rp 9879  df-xneg 9997  df-xadd 9998  df-icc 10120  df-seqfrec 10700  df-exp 10791  df-cj 11393  df-re 11394  df-im 11395  df-rsqrt 11549  df-abs 11550  df-psmet 14547  df-xmet 14548  df-bl 14550

This theorem is referenced by:  bdmopn  15218