Theorem bdbl 14671

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 1006	. . . . . 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 1002	. . . . . . 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 1005	. . . . . . 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 14520	. . . . . 6 |- ( ( C e. ( *Met ` X ) /\ P e. X /\ z e. X ) -> ( P C z ) e. RR* )
9	4, 6, 7, 8	syl3anc 1249	. . . . 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 1039	. . . . 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 11415	. . . . 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 1249	. . . 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 14518	. . . . . . . . 9 |- ( C e. ( *Met ` X ) -> C : ( X X. X ) --> RR* )
14	13	3ad2ant1 1020	. . . . . . . 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 14668	. . . . . 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 1250	. . . . 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 4039	. . . 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 1007	. . . . . . . 8 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( P e. X /\ S e. RR* /\ S <_ R ) ) -> S <_ R )
22		simpl2 1003	. . . . . . . . 9 |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( P e. X /\ S e. RR* /\ S <_ R ) ) -> R e. RR* )
23		xrlenlt 8084	. . . . . . . . 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 745	. . . . . . 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 729	. . . . . 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 2748	. 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 14669	. . . 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 14557	. . 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 1249	. 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 14557	. . 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 1249	. 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 2236	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 709   /\ w3a 980   = wceq 1364   e. wcel 2164   {crab 2476   {cpr 3619   class class class wbr 4029   X. cxp 4657   -->wf 5250   ` cfv 5254  (class class class)co 5918   e. cmpo 5920  infcinf 7042   0cc0 7872   RR*cxr 8053   < clt 8054   <_ cle 8055   *Metcxmet 14032   ballcbl 14034

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-isom 5263  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-map 6704  df-sup 7043  df-inf 7044  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-rp 9720  df-xneg 9838  df-xadd 9839  df-icc 9961  df-seqfrec 10519  df-exp 10610  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143  df-psmet 14039  df-xmet 14040  df-bl 14042

This theorem is referenced by:  bdmopn  14672