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Theorem sblpnf 27642
Description: The infinity ball in the absolute value metric is just the whole space.  S analog of blpnf 17970. (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 3673 . . 3  |-  ( S  e.  { RR ,  CC }  ->  ( S  =  RR  \/  S  =  CC ) )
3 sblpnf.d . . . . 5  |-  D  =  ( ( abs  o.  -  )  |`  ( S  X.  S ) )
4 eqid 2296 . . . . . . 7  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
54remet 18312 . . . . . 6  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  e.  ( Met `  RR )
6 xpeq12 4724 . . . . . . . . 9  |-  ( ( S  =  RR  /\  S  =  RR )  ->  ( S  X.  S
)  =  ( RR 
X.  RR ) )
76anidms 626 . . . . . . . 8  |-  ( S  =  RR  ->  ( S  X.  S )  =  ( RR  X.  RR ) )
87reseq2d 4971 . . . . . . 7  |-  ( S  =  RR  ->  (
( abs  o.  -  )  |`  ( S  X.  S
) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) )
9 fveq2 5541 . . . . . . 7  |-  ( S  =  RR  ->  ( Met `  S )  =  ( Met `  RR ) )
108, 9eleq12d 2364 . . . . . 6  |-  ( S  =  RR  ->  (
( ( abs  o.  -  )  |`  ( S  X.  S ) )  e.  ( Met `  S
)  <->  ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) )  e.  ( Met `  RR ) ) )
115, 10mpbiri 224 . . . . 5  |-  ( S  =  RR  ->  (
( abs  o.  -  )  |`  ( S  X.  S
) )  e.  ( Met `  S ) )
123, 11syl5eqel 2380 . . . 4  |-  ( S  =  RR  ->  D  e.  ( Met `  S
) )
13 relco 5187 . . . . . . . . 9  |-  Rel  ( abs  o.  -  )
14 resdm 5009 . . . . . . . . 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 11837 . . . . . . . . . . . 12  |-  abs : CC
--> RR
17 ax-resscn 8810 . . . . . . . . . . . 12  |-  RR  C_  CC
18 fss 5413 . . . . . . . . . . . 12  |-  ( ( abs : CC --> RR  /\  RR  C_  CC )  ->  abs : CC --> CC )
1916, 17, 18mp2an 653 . . . . . . . . . . 11  |-  abs : CC
--> CC
20 subf 9069 . . . . . . . . . . 11  |-  -  :
( CC  X.  CC )
--> CC
21 fco 5414 . . . . . . . . . . 11  |-  ( ( abs : CC --> CC  /\  -  : ( CC  X.  CC ) --> CC )  -> 
( abs  o.  -  ) : ( CC  X.  CC ) --> CC )
2219, 20, 21mp2an 653 . . . . . . . . . 10  |-  ( abs 
o.  -  ) :
( CC  X.  CC )
--> CC
2322fdmi 5410 . . . . . . . . 9  |-  dom  ( abs  o.  -  )  =  ( CC  X.  CC )
2423reseq2i 4968 . . . . . . . 8  |-  ( ( abs  o.  -  )  |` 
dom  ( abs  o.  -  ) )  =  ( ( abs  o.  -  )  |`  ( CC 
X.  CC ) )
2515, 24eqtr3i 2318 . . . . . . 7  |-  ( abs 
o.  -  )  =  ( ( abs  o.  -  )  |`  ( CC 
X.  CC ) )
26 cnmet 18297 . . . . . . 7  |-  ( abs 
o.  -  )  e.  ( Met `  CC )
2725, 26eqeltrri 2367 . . . . . 6  |-  ( ( abs  o.  -  )  |`  ( CC  X.  CC ) )  e.  ( Met `  CC )
28 xpeq12 4724 . . . . . . . . 9  |-  ( ( S  =  CC  /\  S  =  CC )  ->  ( S  X.  S
)  =  ( CC 
X.  CC ) )
2928anidms 626 . . . . . . . 8  |-  ( S  =  CC  ->  ( S  X.  S )  =  ( CC  X.  CC ) )
3029reseq2d 4971 . . . . . . 7  |-  ( S  =  CC  ->  (
( abs  o.  -  )  |`  ( S  X.  S
) )  =  ( ( abs  o.  -  )  |`  ( CC  X.  CC ) ) )
31 fveq2 5541 . . . . . . 7  |-  ( S  =  CC  ->  ( Met `  S )  =  ( Met `  CC ) )
3230, 31eleq12d 2364 . . . . . 6  |-  ( S  =  CC  ->  (
( ( abs  o.  -  )  |`  ( S  X.  S ) )  e.  ( Met `  S
)  <->  ( ( abs 
o.  -  )  |`  ( CC  X.  CC ) )  e.  ( Met `  CC ) ) )
3327, 32mpbiri 224 . . . . 5  |-  ( S  =  CC  ->  (
( abs  o.  -  )  |`  ( S  X.  S
) )  e.  ( Met `  S ) )
343, 33syl5eqel 2380 . . . 4  |-  ( S  =  CC  ->  D  e.  ( Met `  S
) )
3512, 34jaoi 368 . . 3  |-  ( ( S  =  RR  \/  S  =  CC )  ->  D  e.  ( Met `  S ) )
361, 2, 353syl 18 . 2  |-  ( ph  ->  D  e.  ( Met `  S ) )
37 blpnf 17970 . 2  |-  ( ( D  e.  ( Met `  S )  /\  P  e.  S )  ->  ( P ( ball `  D
)  +oo )  =  S )
3836, 37sylan 457 1  |-  ( (
ph  /\  P  e.  S )  ->  ( P ( ball `  D
)  +oo )  =  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696    C_ wss 3165   {cpr 3654    X. cxp 4703   dom cdm 4705    |` cres 4707    o. ccom 4709   Rel wrel 4710   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752    +oocpnf 8880    - cmin 9053   abscabs 11735   Metcme 16386   ballcbl 16387
This theorem is referenced by:  dvconstbi  27654
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-xmet 16389  df-met 16390  df-bl 16391
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