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Theorem nvelbl2 22186
Description: Membership of an off-center vector in a ball. (Contributed by NM, 27-Dec-2007.) (Revised by Mario Carneiro, 10-Jan-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvelbl2.1  |-  X  =  ( BaseSet `  U )
nvelbl2.2  |-  G  =  ( +v `  U
)
nvelbl2.6  |-  N  =  ( normCV `  U )
nvelbl2.8  |-  D  =  ( IndMet `  U )
Assertion
Ref Expression
nvelbl2  |-  ( ( ( U  e.  NrmCVec  /\  R  e.  RR+ )  /\  ( P  e.  X  /\  A  e.  X
) )  ->  (
( P G A )  e.  ( P ( ball `  D
) R )  <->  ( N `  A )  <  R
) )

Proof of Theorem nvelbl2
StepHypRef Expression
1 simprl 733 . . . 4  |-  ( ( ( U  e.  NrmCVec  /\  R  e.  RR+ )  /\  ( P  e.  X  /\  A  e.  X
) )  ->  P  e.  X )
2 nvelbl2.1 . . . . . . 7  |-  X  =  ( BaseSet `  U )
3 nvelbl2.2 . . . . . . 7  |-  G  =  ( +v `  U
)
42, 3nvgcl 22099 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  P  e.  X  /\  A  e.  X )  ->  ( P G A )  e.  X )
543expb 1154 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( P  e.  X  /\  A  e.  X )
)  ->  ( P G A )  e.  X
)
65adantlr 696 . . . 4  |-  ( ( ( U  e.  NrmCVec  /\  R  e.  RR+ )  /\  ( P  e.  X  /\  A  e.  X
) )  ->  ( P G A )  e.  X )
71, 6jca 519 . . 3  |-  ( ( ( U  e.  NrmCVec  /\  R  e.  RR+ )  /\  ( P  e.  X  /\  A  e.  X
) )  ->  ( P  e.  X  /\  ( P G A )  e.  X ) )
8 eqid 2436 . . . 4  |-  ( -v
`  U )  =  ( -v `  U
)
9 nvelbl2.6 . . . 4  |-  N  =  ( normCV `  U )
10 nvelbl2.8 . . . 4  |-  D  =  ( IndMet `  U )
112, 8, 9, 10nvelbl 22185 . . 3  |-  ( ( ( U  e.  NrmCVec  /\  R  e.  RR+ )  /\  ( P  e.  X  /\  ( P G A )  e.  X ) )  ->  ( ( P G A )  e.  ( P ( ball `  D ) R )  <-> 
( N `  (
( P G A ) ( -v `  U ) P ) )  <  R ) )
127, 11syldan 457 . 2  |-  ( ( ( U  e.  NrmCVec  /\  R  e.  RR+ )  /\  ( P  e.  X  /\  A  e.  X
) )  ->  (
( P G A )  e.  ( P ( ball `  D
) R )  <->  ( N `  ( ( P G A ) ( -v
`  U ) P ) )  <  R
) )
132, 3, 8nvpncan2 22137 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  P  e.  X  /\  A  e.  X )  ->  (
( P G A ) ( -v `  U ) P )  =  A )
14133expb 1154 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( P  e.  X  /\  A  e.  X )
)  ->  ( ( P G A ) ( -v `  U ) P )  =  A )
1514adantlr 696 . . . 4  |-  ( ( ( U  e.  NrmCVec  /\  R  e.  RR+ )  /\  ( P  e.  X  /\  A  e.  X
) )  ->  (
( P G A ) ( -v `  U ) P )  =  A )
1615fveq2d 5732 . . 3  |-  ( ( ( U  e.  NrmCVec  /\  R  e.  RR+ )  /\  ( P  e.  X  /\  A  e.  X
) )  ->  ( N `  ( ( P G A ) ( -v `  U ) P ) )  =  ( N `  A
) )
1716breq1d 4222 . 2  |-  ( ( ( U  e.  NrmCVec  /\  R  e.  RR+ )  /\  ( P  e.  X  /\  A  e.  X
) )  ->  (
( N `  (
( P G A ) ( -v `  U ) P ) )  <  R  <->  ( N `  A )  <  R
) )
1812, 17bitrd 245 1  |-  ( ( ( U  e.  NrmCVec  /\  R  e.  RR+ )  /\  ( P  e.  X  /\  A  e.  X
) )  ->  (
( P G A )  e.  ( P ( ball `  D
) R )  <->  ( N `  A )  <  R
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   class class class wbr 4212   ` cfv 5454  (class class class)co 6081    < clt 9120   RR+crp 10612   ballcbl 16688   NrmCVeccnv 22063   +vcpv 22064   BaseSetcba 22065   -vcnsb 22068   normCVcnmcv 22069   IndMetcims 22070
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-addf 9069  ax-mulf 9070
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-xadd 10711  df-seq 11324  df-exp 11383  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-psmet 16694  df-xmet 16695  df-met 16696  df-bl 16697  df-grpo 21779  df-gid 21780  df-ginv 21781  df-gdiv 21782  df-ablo 21870  df-vc 22025  df-nv 22071  df-va 22074  df-ba 22075  df-sm 22076  df-0v 22077  df-vs 22078  df-nmcv 22079  df-ims 22080
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