Theorem bdbl 12672

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 988	. . . . . 6 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( P e. X /\ S e. RR* /\ S <_ R ) ) -> S e. RR* )
2	1	adantr 274	. . . . 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 984	. . . . . . 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 274	. . . . . 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 987	. . . . . . 7 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( P e. X /\ S e. RR* /\ S <_ R ) ) -> P e. X )
6	5	adantr 274	. . . . . 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 109	. . . . . 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 12521	. . . . . 6 |- ( ( C e. ( *Met ` X ) /\ P e. X /\ z e. X ) -> ( P C z ) e. RR* )
9	4, 6, 7, 8	syl3anc 1216	. . . . 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 1021	. . . . 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 11041	. . . . 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 1216	. . . 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 12519	. . . . . . . . 9 |- ( C e. ( *Met ` X ) -> C : ( X X. X ) --> RR* )
14	13	3ad2ant1 1002	. . . . . . . 8 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> C : ( X X. X ) --> RR* )
15	14	adantr 274	. . . . . . 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 274	. . . . . 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 12669	. . . . . 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 1217	. . . . 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 3939	. . . 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 989	. . . . . . . 8 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( P e. X /\ S e. RR* /\ S <_ R ) ) -> S <_ R )
22		simpl2 985	. . . . . . . . 9 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( P e. X /\ S e. RR* /\ S <_ R ) ) -> R e. RR* )
23		xrlenlt 7829	. . . . . . . . 9 |- ( ( S e. RR* /\ R e. RR* ) -> ( S <_ R <-> -. R < S ) )
24	1, 22, 23	syl2anc 408	. . . . . . . 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 146	. . . . . . 7 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( P e. X /\ S e. RR* /\ S <_ R ) ) -> -. R < S )
26		biorf 733	. . . . . . 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 717	. . . . . 6 |- ( ( R < S \/ ( P C z ) < S ) <-> ( ( P C z ) < S \/ R < S ) )
29	27, 28	syl6bb 195	. . . . 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 274	. . . 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 219	. . 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 2674	. 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 12670	. . . 4 |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> D e. ( *Met ` X ) )
34	33	adantr 274	. . 3 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( P e. X /\ S e. RR* /\ S <_ R ) ) -> D e. ( *Met ` X ) )
35		blval 12558	. . 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 1216	. 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 12558	. . 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 1216	. 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 2182	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 103   <-> wb 104   \/ wo 697   /\ w3a 962   = wceq 1331   e. wcel 1480   {crab 2420   {cpr 3528   class class class wbr 3929   X. cxp 4537   -->wf 5119   ` cfv 5123  (class class class)co 5774   e. cmpo 5776  infcinf 6870   0cc0 7620   RR*cxr 7799   < clt 7800   <_ cle 7801   *Metcxmet 12149   ballcbl 12151

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739  ax-caucvg 7740

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-map 6544  df-sup 6871  df-inf 6872  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-3 8780  df-4 8781  df-n0 8978  df-z 9055  df-uz 9327  df-rp 9442  df-xneg 9559  df-xadd 9560  df-icc 9678  df-seqfrec 10219  df-exp 10293  df-cj 10614  df-re 10615  df-im 10616  df-rsqrt 10770  df-abs 10771  df-psmet 12156  df-xmet 12157  df-bl 12159

This theorem is referenced by:  bdmopn  12673