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Theorem sblpnf 27507
Description: The infinity ball in the absolute value metric is just the whole space.  S analog of blpnf 18419. (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 3826 . . 3  |-  ( S  e.  { RR ,  CC }  ->  ( S  =  RR  \/  S  =  CC ) )
3 sblpnf.d . . . . 5  |-  D  =  ( ( abs  o.  -  )  |`  ( S  X.  S ) )
4 eqid 2435 . . . . . . 7  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
54remet 18813 . . . . . 6  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  e.  ( Met `  RR )
6 xpeq12 4889 . . . . . . . . 9  |-  ( ( S  =  RR  /\  S  =  RR )  ->  ( S  X.  S
)  =  ( RR 
X.  RR ) )
76anidms 627 . . . . . . . 8  |-  ( S  =  RR  ->  ( S  X.  S )  =  ( RR  X.  RR ) )
87reseq2d 5138 . . . . . . 7  |-  ( S  =  RR  ->  (
( abs  o.  -  )  |`  ( S  X.  S
) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) )
9 fveq2 5720 . . . . . . 7  |-  ( S  =  RR  ->  ( Met `  S )  =  ( Met `  RR ) )
108, 9eleq12d 2503 . . . . . 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 2519 . . . 4  |-  ( S  =  RR  ->  D  e.  ( Met `  S
) )
13 relco 5360 . . . . . . . . 9  |-  Rel  ( abs  o.  -  )
14 resdm 5176 . . . . . . . . 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 12133 . . . . . . . . . . . 12  |-  abs : CC
--> RR
17 ax-resscn 9039 . . . . . . . . . . . 12  |-  RR  C_  CC
18 fss 5591 . . . . . . . . . . . 12  |-  ( ( abs : CC --> RR  /\  RR  C_  CC )  ->  abs : CC --> CC )
1916, 17, 18mp2an 654 . . . . . . . . . . 11  |-  abs : CC
--> CC
20 subf 9299 . . . . . . . . . . 11  |-  -  :
( CC  X.  CC )
--> CC
21 fco 5592 . . . . . . . . . . 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 5588 . . . . . . . . 9  |-  dom  ( abs  o.  -  )  =  ( CC  X.  CC )
2423reseq2i 5135 . . . . . . . 8  |-  ( ( abs  o.  -  )  |` 
dom  ( abs  o.  -  ) )  =  ( ( abs  o.  -  )  |`  ( CC 
X.  CC ) )
2515, 24eqtr3i 2457 . . . . . . 7  |-  ( abs 
o.  -  )  =  ( ( abs  o.  -  )  |`  ( CC 
X.  CC ) )
26 cnmet 18798 . . . . . . 7  |-  ( abs 
o.  -  )  e.  ( Met `  CC )
2725, 26eqeltrri 2506 . . . . . 6  |-  ( ( abs  o.  -  )  |`  ( CC  X.  CC ) )  e.  ( Met `  CC )
28 xpeq12 4889 . . . . . . . . 9  |-  ( ( S  =  CC  /\  S  =  CC )  ->  ( S  X.  S
)  =  ( CC 
X.  CC ) )
2928anidms 627 . . . . . . . 8  |-  ( S  =  CC  ->  ( S  X.  S )  =  ( CC  X.  CC ) )
3029reseq2d 5138 . . . . . . 7  |-  ( S  =  CC  ->  (
( abs  o.  -  )  |`  ( S  X.  S
) )  =  ( ( abs  o.  -  )  |`  ( CC  X.  CC ) ) )
31 fveq2 5720 . . . . . . 7  |-  ( S  =  CC  ->  ( Met `  S )  =  ( Met `  CC ) )
3230, 31eleq12d 2503 . . . . . 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 2519 . . . 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 18419 . 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 1652    e. wcel 1725    C_ wss 3312   {cpr 3807    X. cxp 4868   dom cdm 4870    |` cres 4872    o. ccom 4874   Rel wrel 4875   -->wf 5442   ` cfv 5446  (class class class)co 6073   CCcc 8980   RRcr 8981    +oocpnf 9109    - cmin 9283   abscabs 12031   Metcme 16679   ballcbl 16680
This theorem is referenced by:  dvconstbi  27519
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-xneg 10702  df-xadd 10703  df-xmul 10704  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-psmet 16686  df-xmet 16687  df-met 16688  df-bl 16689
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