Users' Mathboxes Mathbox for Steve Rodriguez < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sblpnf Unicode version

Theorem sblpnf 27209
Description: The infinity ball in the absolute value metric is just the whole space.  S analog of blpnf 18329. (Contributed by Steve Rodriguez, 8-Nov-2015.)
Hypotheses
Ref Expression
sblpnf.s  |-  ( ph  ->  S  e.  { RR ,  CC } )
sblpnf.d  |-  D  =  ( ( abs  o.  -  )  |`  ( S  X.  S ) )
Assertion
Ref Expression
sblpnf  |-  ( (
ph  /\  P  e.  S )  ->  ( P ( ball `  D
)  +oo )  =  S )

Proof of Theorem sblpnf
StepHypRef Expression
1 sblpnf.s . . 3  |-  ( ph  ->  S  e.  { RR ,  CC } )
2 elpri 3778 . . 3  |-  ( S  e.  { RR ,  CC }  ->  ( S  =  RR  \/  S  =  CC ) )
3 sblpnf.d . . . . 5  |-  D  =  ( ( abs  o.  -  )  |`  ( S  X.  S ) )
4 eqid 2388 . . . . . . 7  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
54remet 18693 . . . . . 6  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  e.  ( Met `  RR )
6 xpeq12 4838 . . . . . . . . 9  |-  ( ( S  =  RR  /\  S  =  RR )  ->  ( S  X.  S
)  =  ( RR 
X.  RR ) )
76anidms 627 . . . . . . . 8  |-  ( S  =  RR  ->  ( S  X.  S )  =  ( RR  X.  RR ) )
87reseq2d 5087 . . . . . . 7  |-  ( S  =  RR  ->  (
( abs  o.  -  )  |`  ( S  X.  S
) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) )
9 fveq2 5669 . . . . . . 7  |-  ( S  =  RR  ->  ( Met `  S )  =  ( Met `  RR ) )
108, 9eleq12d 2456 . . . . . 6  |-  ( S  =  RR  ->  (
( ( abs  o.  -  )  |`  ( S  X.  S ) )  e.  ( Met `  S
)  <->  ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) )  e.  ( Met `  RR ) ) )
115, 10mpbiri 225 . . . . 5  |-  ( S  =  RR  ->  (
( abs  o.  -  )  |`  ( S  X.  S
) )  e.  ( Met `  S ) )
123, 11syl5eqel 2472 . . . 4  |-  ( S  =  RR  ->  D  e.  ( Met `  S
) )
13 relco 5309 . . . . . . . . 9  |-  Rel  ( abs  o.  -  )
14 resdm 5125 . . . . . . . . 9  |-  ( Rel  ( abs  o.  -  )  ->  ( ( abs 
o.  -  )  |`  dom  ( abs  o.  -  ) )  =  ( abs  o.  -  ) )
1513, 14ax-mp 8 . . . . . . . 8  |-  ( ( abs  o.  -  )  |` 
dom  ( abs  o.  -  ) )  =  ( abs  o.  -  )
16 absf 12069 . . . . . . . . . . . 12  |-  abs : CC
--> RR
17 ax-resscn 8981 . . . . . . . . . . . 12  |-  RR  C_  CC
18 fss 5540 . . . . . . . . . . . 12  |-  ( ( abs : CC --> RR  /\  RR  C_  CC )  ->  abs : CC --> CC )
1916, 17, 18mp2an 654 . . . . . . . . . . 11  |-  abs : CC
--> CC
20 subf 9240 . . . . . . . . . . 11  |-  -  :
( CC  X.  CC )
--> CC
21 fco 5541 . . . . . . . . . . 11  |-  ( ( abs : CC --> CC  /\  -  : ( CC  X.  CC ) --> CC )  -> 
( abs  o.  -  ) : ( CC  X.  CC ) --> CC )
2219, 20, 21mp2an 654 . . . . . . . . . 10  |-  ( abs 
o.  -  ) :
( CC  X.  CC )
--> CC
2322fdmi 5537 . . . . . . . . 9  |-  dom  ( abs  o.  -  )  =  ( CC  X.  CC )
2423reseq2i 5084 . . . . . . . 8  |-  ( ( abs  o.  -  )  |` 
dom  ( abs  o.  -  ) )  =  ( ( abs  o.  -  )  |`  ( CC 
X.  CC ) )
2515, 24eqtr3i 2410 . . . . . . 7  |-  ( abs 
o.  -  )  =  ( ( abs  o.  -  )  |`  ( CC 
X.  CC ) )
26 cnmet 18678 . . . . . . 7  |-  ( abs 
o.  -  )  e.  ( Met `  CC )
2725, 26eqeltrri 2459 . . . . . 6  |-  ( ( abs  o.  -  )  |`  ( CC  X.  CC ) )  e.  ( Met `  CC )
28 xpeq12 4838 . . . . . . . . 9  |-  ( ( S  =  CC  /\  S  =  CC )  ->  ( S  X.  S
)  =  ( CC 
X.  CC ) )
2928anidms 627 . . . . . . . 8  |-  ( S  =  CC  ->  ( S  X.  S )  =  ( CC  X.  CC ) )
3029reseq2d 5087 . . . . . . 7  |-  ( S  =  CC  ->  (
( abs  o.  -  )  |`  ( S  X.  S
) )  =  ( ( abs  o.  -  )  |`  ( CC  X.  CC ) ) )
31 fveq2 5669 . . . . . . 7  |-  ( S  =  CC  ->  ( Met `  S )  =  ( Met `  CC ) )
3230, 31eleq12d 2456 . . . . . 6  |-  ( S  =  CC  ->  (
( ( abs  o.  -  )  |`  ( S  X.  S ) )  e.  ( Met `  S
)  <->  ( ( abs 
o.  -  )  |`  ( CC  X.  CC ) )  e.  ( Met `  CC ) ) )
3327, 32mpbiri 225 . . . . 5  |-  ( S  =  CC  ->  (
( abs  o.  -  )  |`  ( S  X.  S
) )  e.  ( Met `  S ) )
343, 33syl5eqel 2472 . . . 4  |-  ( S  =  CC  ->  D  e.  ( Met `  S
) )
3512, 34jaoi 369 . . 3  |-  ( ( S  =  RR  \/  S  =  CC )  ->  D  e.  ( Met `  S ) )
361, 2, 353syl 19 . 2  |-  ( ph  ->  D  e.  ( Met `  S ) )
37 blpnf 18329 . 2  |-  ( ( D  e.  ( Met `  S )  /\  P  e.  S )  ->  ( P ( ball `  D
)  +oo )  =  S )
3836, 37sylan 458 1  |-  ( (
ph  /\  P  e.  S )  ->  ( P ( ball `  D
)  +oo )  =  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1717    C_ wss 3264   {cpr 3759    X. cxp 4817   dom cdm 4819    |` cres 4821    o. ccom 4823   Rel wrel 4824   -->wf 5391   ` cfv 5395  (class class class)co 6021   CCcc 8922   RRcr 8923    +oocpnf 9051    - cmin 9224   abscabs 11967   Metcme 16614   ballcbl 16615
This theorem is referenced by:  dvconstbi  27221
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-er 6842  df-map 6957  df-en 7047  df-dom 7048  df-sdom 7049  df-sup 7382  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-n0 10155  df-z 10216  df-uz 10422  df-rp 10546  df-xneg 10643  df-xadd 10644  df-xmul 10645  df-seq 11252  df-exp 11311  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-xmet 16620  df-met 16621  df-bl 16622
  Copyright terms: Public domain W3C validator