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Theorem blssp 26470
Description: A ball in the subspace metric. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 5-Jan-2014.)
Hypothesis
Ref Expression
blssp.2  |-  N  =  ( M  |`  ( S  X.  S ) )
Assertion
Ref Expression
blssp  |-  ( ( ( M  e.  ( Met `  X )  /\  S  C_  X
)  /\  ( Y  e.  S  /\  R  e.  RR+ ) )  ->  ( Y ( ball `  N
) R )  =  ( ( Y (
ball `  M ) R )  i^i  S
) )

Proof of Theorem blssp
StepHypRef Expression
1 metxmet 17899 . . 3  |-  ( M  e.  ( Met `  X
)  ->  M  e.  ( * Met `  X
) )
21ad2antrr 706 . 2  |-  ( ( ( M  e.  ( Met `  X )  /\  S  C_  X
)  /\  ( Y  e.  S  /\  R  e.  RR+ ) )  ->  M  e.  ( * Met `  X
) )
3 simprl 732 . . 3  |-  ( ( ( M  e.  ( Met `  X )  /\  S  C_  X
)  /\  ( Y  e.  S  /\  R  e.  RR+ ) )  ->  Y  e.  S )
4 simplr 731 . . . 4  |-  ( ( ( M  e.  ( Met `  X )  /\  S  C_  X
)  /\  ( Y  e.  S  /\  R  e.  RR+ ) )  ->  S  C_  X )
5 sseqin2 3388 . . . 4  |-  ( S 
C_  X  <->  ( X  i^i  S )  =  S )
64, 5sylib 188 . . 3  |-  ( ( ( M  e.  ( Met `  X )  /\  S  C_  X
)  /\  ( Y  e.  S  /\  R  e.  RR+ ) )  ->  ( X  i^i  S )  =  S )
73, 6eleqtrrd 2360 . 2  |-  ( ( ( M  e.  ( Met `  X )  /\  S  C_  X
)  /\  ( Y  e.  S  /\  R  e.  RR+ ) )  ->  Y  e.  ( X  i^i  S
) )
8 rpxr 10361 . . 3  |-  ( R  e.  RR+  ->  R  e. 
RR* )
98ad2antll 709 . 2  |-  ( ( ( M  e.  ( Met `  X )  /\  S  C_  X
)  /\  ( Y  e.  S  /\  R  e.  RR+ ) )  ->  R  e.  RR* )
10 blssp.2 . . 3  |-  N  =  ( M  |`  ( S  X.  S ) )
1110blres 17977 . 2  |-  ( ( M  e.  ( * Met `  X )  /\  Y  e.  ( X  i^i  S )  /\  R  e.  RR* )  ->  ( Y (
ball `  N ) R )  =  ( ( Y ( ball `  M ) R )  i^i  S ) )
122, 7, 9, 11syl3anc 1182 1  |-  ( ( ( M  e.  ( Met `  X )  /\  S  C_  X
)  /\  ( Y  e.  S  /\  R  e.  RR+ ) )  ->  ( Y ( ball `  N
) R )  =  ( ( Y (
ball `  M ) R )  i^i  S
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    i^i cin 3151    C_ wss 3152    X. cxp 4687    |` cres 4691   ` cfv 5255  (class class class)co 5858   RR*cxr 8866   RR+crp 10354   * Metcxmt 16369   Metcme 16370   ballcbl 16371
This theorem is referenced by:  bndss  26510
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-mulcl 8799  ax-i2m1 8805
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  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-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-rp 10355  df-xadd 10453  df-xmet 16373  df-met 16374  df-bl 16375
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