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Theorem sblpnf 36002
Description: The infinity ball in the absolute value metric is just the whole space.  S analog of blpnf 21082. (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 3989 . . 3  |-  ( S  e.  { RR ,  CC }  ->  ( S  =  RR  \/  S  =  CC ) )
3 sblpnf.d . . . . 5  |-  D  =  ( ( abs  o.  -  )  |`  ( S  X.  S ) )
4 eqid 2400 . . . . . . 7  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
54remet 21477 . . . . . 6  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  e.  ( Met `  RR )
6 xpeq12 4959 . . . . . . . . 9  |-  ( ( S  =  RR  /\  S  =  RR )  ->  ( S  X.  S
)  =  ( RR 
X.  RR ) )
76anidms 643 . . . . . . . 8  |-  ( S  =  RR  ->  ( S  X.  S )  =  ( RR  X.  RR ) )
87reseq2d 5213 . . . . . . 7  |-  ( S  =  RR  ->  (
( abs  o.  -  )  |`  ( S  X.  S
) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) )
9 fveq2 5803 . . . . . . 7  |-  ( S  =  RR  ->  ( Met `  S )  =  ( Met `  RR ) )
108, 9eleq12d 2482 . . . . . 6  |-  ( S  =  RR  ->  (
( ( abs  o.  -  )  |`  ( S  X.  S ) )  e.  ( Met `  S
)  <->  ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) )  e.  ( Met `  RR ) ) )
115, 10mpbiri 233 . . . . 5  |-  ( S  =  RR  ->  (
( abs  o.  -  )  |`  ( S  X.  S
) )  e.  ( Met `  S ) )
123, 11syl5eqel 2492 . . . 4  |-  ( S  =  RR  ->  D  e.  ( Met `  S
) )
13 relco 5440 . . . . . . . . 9  |-  Rel  ( abs  o.  -  )
14 resdm 5254 . . . . . . . . 9  |-  ( Rel  ( abs  o.  -  )  ->  ( ( abs 
o.  -  )  |`  dom  ( abs  o.  -  ) )  =  ( abs  o.  -  ) )
1513, 14ax-mp 5 . . . . . . . 8  |-  ( ( abs  o.  -  )  |` 
dom  ( abs  o.  -  ) )  =  ( abs  o.  -  )
16 absf 13224 . . . . . . . . . . . 12  |-  abs : CC
--> RR
17 ax-resscn 9497 . . . . . . . . . . . 12  |-  RR  C_  CC
18 fss 5676 . . . . . . . . . . . 12  |-  ( ( abs : CC --> RR  /\  RR  C_  CC )  ->  abs : CC --> CC )
1916, 17, 18mp2an 670 . . . . . . . . . . 11  |-  abs : CC
--> CC
20 subf 9776 . . . . . . . . . . 11  |-  -  :
( CC  X.  CC )
--> CC
21 fco 5678 . . . . . . . . . . 11  |-  ( ( abs : CC --> CC  /\  -  : ( CC  X.  CC ) --> CC )  -> 
( abs  o.  -  ) : ( CC  X.  CC ) --> CC )
2219, 20, 21mp2an 670 . . . . . . . . . 10  |-  ( abs 
o.  -  ) :
( CC  X.  CC )
--> CC
2322fdmi 5673 . . . . . . . . 9  |-  dom  ( abs  o.  -  )  =  ( CC  X.  CC )
2423reseq2i 5210 . . . . . . . 8  |-  ( ( abs  o.  -  )  |` 
dom  ( abs  o.  -  ) )  =  ( ( abs  o.  -  )  |`  ( CC 
X.  CC ) )
2515, 24eqtr3i 2431 . . . . . . 7  |-  ( abs 
o.  -  )  =  ( ( abs  o.  -  )  |`  ( CC 
X.  CC ) )
26 cnmet 21461 . . . . . . 7  |-  ( abs 
o.  -  )  e.  ( Met `  CC )
2725, 26eqeltrri 2485 . . . . . 6  |-  ( ( abs  o.  -  )  |`  ( CC  X.  CC ) )  e.  ( Met `  CC )
28 xpeq12 4959 . . . . . . . . 9  |-  ( ( S  =  CC  /\  S  =  CC )  ->  ( S  X.  S
)  =  ( CC 
X.  CC ) )
2928anidms 643 . . . . . . . 8  |-  ( S  =  CC  ->  ( S  X.  S )  =  ( CC  X.  CC ) )
3029reseq2d 5213 . . . . . . 7  |-  ( S  =  CC  ->  (
( abs  o.  -  )  |`  ( S  X.  S
) )  =  ( ( abs  o.  -  )  |`  ( CC  X.  CC ) ) )
31 fveq2 5803 . . . . . . 7  |-  ( S  =  CC  ->  ( Met `  S )  =  ( Met `  CC ) )
3230, 31eleq12d 2482 . . . . . 6  |-  ( S  =  CC  ->  (
( ( abs  o.  -  )  |`  ( S  X.  S ) )  e.  ( Met `  S
)  <->  ( ( abs 
o.  -  )  |`  ( CC  X.  CC ) )  e.  ( Met `  CC ) ) )
3327, 32mpbiri 233 . . . . 5  |-  ( S  =  CC  ->  (
( abs  o.  -  )  |`  ( S  X.  S
) )  e.  ( Met `  S ) )
343, 33syl5eqel 2492 . . . 4  |-  ( S  =  CC  ->  D  e.  ( Met `  S
) )
3512, 34jaoi 377 . . 3  |-  ( ( S  =  RR  \/  S  =  CC )  ->  D  e.  ( Met `  S ) )
361, 2, 353syl 20 . 2  |-  ( ph  ->  D  e.  ( Met `  S ) )
37 blpnf 21082 . 2  |-  ( ( D  e.  ( Met `  S )  /\  P  e.  S )  ->  ( P ( ball `  D
) +oo )  =  S )
3836, 37sylan 469 1  |-  ( (
ph  /\  P  e.  S )  ->  ( P ( ball `  D
) +oo )  =  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 366    /\ wa 367    = wceq 1403    e. wcel 1840    C_ wss 3411   {cpr 3971    X. cxp 4938   dom cdm 4940    |` cres 4942    o. ccom 4944   Rel wrel 4945   -->wf 5519   ` cfv 5523  (class class class)co 6232   CCcc 9438   RRcr 9439   +oocpnf 9573    - cmin 9759   abscabs 13121   Metcme 18614   ballcbl 18615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517  ax-pre-sup 9518
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-om 6637  df-1st 6736  df-2nd 6737  df-recs 6997  df-rdg 7031  df-er 7266  df-map 7377  df-en 7473  df-dom 7474  df-sdom 7475  df-sup 7853  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-div 10166  df-nn 10495  df-2 10553  df-3 10554  df-n0 10755  df-z 10824  df-uz 11044  df-rp 11182  df-xneg 11287  df-xadd 11288  df-xmul 11289  df-seq 12060  df-exp 12119  df-cj 12986  df-re 12987  df-im 12988  df-sqrt 13122  df-abs 13123  df-psmet 18621  df-xmet 18622  df-met 18623  df-bl 18624
This theorem is referenced by:  dvconstbi  36051
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