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Theorem sblpnf 27539
Description: The infinity ball in the absolute value metric is just the whole space.  S analog of blpnf 17954. (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 3660 . . 3  |-  ( S  e.  { RR ,  CC }  ->  ( S  =  RR  \/  S  =  CC ) )
3 sblpnf.d . . . . 5  |-  D  =  ( ( abs  o.  -  )  |`  ( S  X.  S ) )
4 eqid 2283 . . . . . . 7  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
54remet 18296 . . . . . 6  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  e.  ( Met `  RR )
6 xpeq12 4708 . . . . . . . . 9  |-  ( ( S  =  RR  /\  S  =  RR )  ->  ( S  X.  S
)  =  ( RR 
X.  RR ) )
76anidms 626 . . . . . . . 8  |-  ( S  =  RR  ->  ( S  X.  S )  =  ( RR  X.  RR ) )
87reseq2d 4955 . . . . . . 7  |-  ( S  =  RR  ->  (
( abs  o.  -  )  |`  ( S  X.  S
) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) )
9 fveq2 5525 . . . . . . 7  |-  ( S  =  RR  ->  ( Met `  S )  =  ( Met `  RR ) )
108, 9eleq12d 2351 . . . . . 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 2367 . . . 4  |-  ( S  =  RR  ->  D  e.  ( Met `  S
) )
13 relco 5171 . . . . . . . . 9  |-  Rel  ( abs  o.  -  )
14 resdm 4993 . . . . . . . . 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 11821 . . . . . . . . . . . 12  |-  abs : CC
--> RR
17 ax-resscn 8794 . . . . . . . . . . . 12  |-  RR  C_  CC
18 fss 5397 . . . . . . . . . . . 12  |-  ( ( abs : CC --> RR  /\  RR  C_  CC )  ->  abs : CC --> CC )
1916, 17, 18mp2an 653 . . . . . . . . . . 11  |-  abs : CC
--> CC
20 subf 9053 . . . . . . . . . . 11  |-  -  :
( CC  X.  CC )
--> CC
21 fco 5398 . . . . . . . . . . 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 5394 . . . . . . . . 9  |-  dom  ( abs  o.  -  )  =  ( CC  X.  CC )
2423reseq2i 4952 . . . . . . . 8  |-  ( ( abs  o.  -  )  |` 
dom  ( abs  o.  -  ) )  =  ( ( abs  o.  -  )  |`  ( CC 
X.  CC ) )
2515, 24eqtr3i 2305 . . . . . . 7  |-  ( abs 
o.  -  )  =  ( ( abs  o.  -  )  |`  ( CC 
X.  CC ) )
26 cnmet 18281 . . . . . . 7  |-  ( abs 
o.  -  )  e.  ( Met `  CC )
2725, 26eqeltrri 2354 . . . . . 6  |-  ( ( abs  o.  -  )  |`  ( CC  X.  CC ) )  e.  ( Met `  CC )
28 xpeq12 4708 . . . . . . . . 9  |-  ( ( S  =  CC  /\  S  =  CC )  ->  ( S  X.  S
)  =  ( CC 
X.  CC ) )
2928anidms 626 . . . . . . . 8  |-  ( S  =  CC  ->  ( S  X.  S )  =  ( CC  X.  CC ) )
3029reseq2d 4955 . . . . . . 7  |-  ( S  =  CC  ->  (
( abs  o.  -  )  |`  ( S  X.  S
) )  =  ( ( abs  o.  -  )  |`  ( CC  X.  CC ) ) )
31 fveq2 5525 . . . . . . 7  |-  ( S  =  CC  ->  ( Met `  S )  =  ( Met `  CC ) )
3230, 31eleq12d 2351 . . . . . 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 2367 . . . 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 17954 . 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 1623    e. wcel 1684    C_ wss 3152   {cpr 3641    X. cxp 4687   dom cdm 4689    |` cres 4691    o. ccom 4693   Rel wrel 4694   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736    +oocpnf 8864    - cmin 9037   abscabs 11719   Metcme 16370   ballcbl 16371
This theorem is referenced by:  dvconstbi  27551
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-xmet 16373  df-met 16374  df-bl 16375
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