Theorem bdbl 12850

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 989	. . . . . 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 985	. . . . . . 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 988	. . . . . . 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 12699	. . . . . 6 |- ( ( C e. ( *Met ` X ) /\ P e. X /\ z e. X ) -> ( P C z ) e. RR* )
9	4, 6, 7, 8	syl3anc 1217	. . . . 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 1022	. . . . 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 11146	. . . . 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 1217	. . . 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 12697	. . . . . . . . 9 |- ( C e. ( *Met ` X ) -> C : ( X X. X ) --> RR* )
14	13	3ad2ant1 1003	. . . . . . . 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 12847	. . . . . 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 1218	. . . . 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 3971	. . . 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 990	. . . . . . . 8 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( P e. X /\ S e. RR* /\ S <_ R ) ) -> S <_ R )
22		simpl2 986	. . . . . . . . 9 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( P e. X /\ S e. RR* /\ S <_ R ) ) -> R e. RR* )
23		xrlenlt 7921	. . . . . . . . 9 |- ( ( S e. RR* /\ R e. RR* ) -> ( S <_ R <-> -. R < S ) )
24	1, 22, 23	syl2anc 409	. . . . . . . 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 734	. . . . . . 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 718	. . . . . 6 |- ( ( R < S \/ ( P C z ) < S ) <-> ( ( P C z ) < S \/ R < S ) )
29	27, 28	bitrdi 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 2697	. 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 12848	. . . 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 12736	. . 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 1217	. 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 12736	. . 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 1217	. 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 2197	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 698   /\ w3a 963   = wceq 1332   e. wcel 2125   {crab 2436   {cpr 3557   class class class wbr 3961   X. cxp 4577   -->wf 5159   ` cfv 5163  (class class class)co 5814   e. cmpo 5816  infcinf 6915   0cc0 7711   RR*cxr 7890   < clt 7891   <_ cle 7892   *Metcxmet 12327   ballcbl 12329

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-coll 4075  ax-sep 4078  ax-nul 4086  ax-pow 4130  ax-pr 4164  ax-un 4388  ax-setind 4490  ax-iinf 4541  ax-cnex 7802  ax-resscn 7803  ax-1cn 7804  ax-1re 7805  ax-icn 7806  ax-addcl 7807  ax-addrcl 7808  ax-mulcl 7809  ax-mulrcl 7810  ax-addcom 7811  ax-mulcom 7812  ax-addass 7813  ax-mulass 7814  ax-distr 7815  ax-i2m1 7816  ax-0lt1 7817  ax-1rid 7818  ax-0id 7819  ax-rnegex 7820  ax-precex 7821  ax-cnre 7822  ax-pre-ltirr 7823  ax-pre-ltwlin 7824  ax-pre-lttrn 7825  ax-pre-apti 7826  ax-pre-ltadd 7827  ax-pre-mulgt0 7828  ax-pre-mulext 7829  ax-arch 7830  ax-caucvg 7831

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ne 2325  df-nel 2420  df-ral 2437  df-rex 2438  df-reu 2439  df-rmo 2440  df-rab 2441  df-v 2711  df-sbc 2934  df-csb 3028  df-dif 3100  df-un 3102  df-in 3104  df-ss 3111  df-nul 3391  df-if 3502  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-int 3804  df-iun 3847  df-br 3962  df-opab 4022  df-mpt 4023  df-tr 4059  df-id 4248  df-po 4251  df-iso 4252  df-iord 4321  df-on 4323  df-ilim 4324  df-suc 4326  df-iom 4544  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-rn 4590  df-res 4591  df-ima 4592  df-iota 5128  df-fun 5165  df-fn 5166  df-f 5167  df-f1 5168  df-fo 5169  df-f1o 5170  df-fv 5171  df-isom 5172  df-riota 5770  df-ov 5817  df-oprab 5818  df-mpo 5819  df-1st 6078  df-2nd 6079  df-recs 6242  df-frec 6328  df-map 6584  df-sup 6916  df-inf 6917  df-pnf 7893  df-mnf 7894  df-xr 7895  df-ltxr 7896  df-le 7897  df-sub 8027  df-neg 8028  df-reap 8429  df-ap 8436  df-div 8525  df-inn 8813  df-2 8871  df-3 8872  df-4 8873  df-n0 9070  df-z 9147  df-uz 9419  df-rp 9539  df-xneg 9657  df-xadd 9658  df-icc 9777  df-seqfrec 10323  df-exp 10397  df-cj 10719  df-re 10720  df-im 10721  df-rsqrt 10875  df-abs 10876  df-psmet 12334  df-xmet 12335  df-bl 12337

This theorem is referenced by:  bdmopn  12851