Theorem bdbl 15226

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 1030	. . . . . 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 1026	. . . . . . 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 1029	. . . . . . 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 15075	. . . . . 6 |- ( ( C e. ( *Met ` X ) /\ P e. X /\ z e. X ) -> ( P C z ) e. RR* )
9	4, 6, 7, 8	syl3anc 1273	. . . . 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 1063	. . . . 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 11832	. . . . 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 1273	. . . 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 15073	. . . . . . . . 9 |- ( C e. ( *Met ` X ) -> C : ( X X. X ) --> RR* )
14	13	3ad2ant1 1044	. . . . . . . 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 15223	. . . . . 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 1274	. . . . 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 4098	. . . 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 1031	. . . . . . . 8 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( P e. X /\ S e. RR* /\ S <_ R ) ) -> S <_ R )
22		simpl2 1027	. . . . . . . . 9 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( P e. X /\ S e. RR* /\ S <_ R ) ) -> R e. RR* )
23		xrlenlt 8243	. . . . . . . . 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 751	. . . . . . 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 735	. . . . . 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 2790	. 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 15224	. . . 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 15112	. . 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 1273	. 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 15112	. . 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 1273	. 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 2274	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 715   /\ w3a 1004   = wceq 1397   e. wcel 2202   {crab 2514   {cpr 3670   class class class wbr 4088   X. cxp 4723   -->wf 5322   ` cfv 5326  (class class class)co 6017   e. cmpo 6019  infcinf 7181   0cc0 8031   RR*cxr 8212   < clt 8213   <_ cle 8214   *Metcxmet 14549   ballcbl 14551

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 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150  ax-caucvg 8151

This theorem depends on definitions:  df-bi 117  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 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-map 6818  df-sup 7182  df-inf 7183  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-n0 9402  df-z 9479  df-uz 9755  df-rp 9888  df-xneg 10006  df-xadd 10007  df-icc 10129  df-seqfrec 10709  df-exp 10800  df-cj 11402  df-re 11403  df-im 11404  df-rsqrt 11558  df-abs 11559  df-psmet 14556  df-xmet 14557  df-bl 14559

This theorem is referenced by:  bdmopn  15227