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Theorem nmcvcn 22191
Description: The norm of a normed complex vector space is a continuous function. (Contributed by NM, 16-May-2007.) (Proof shortened by Mario Carneiro, 10-Jan-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
nmcvcn.1  |-  N  =  ( normCV `  U )
nmcvcn.2  |-  C  =  ( IndMet `  U )
nmcvcn.j  |-  J  =  ( MetOpen `  C )
nmcvcn.k  |-  K  =  ( topGen `  ran  (,) )
Assertion
Ref Expression
nmcvcn  |-  ( U  e.  NrmCVec  ->  N  e.  ( J  Cn  K ) )

Proof of Theorem nmcvcn
Dummy variables  e 
d  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2436 . . 3  |-  ( BaseSet `  U )  =  (
BaseSet `  U )
2 nmcvcn.1 . . 3  |-  N  =  ( normCV `  U )
31, 2nvf 22147 . 2  |-  ( U  e.  NrmCVec  ->  N : (
BaseSet `  U ) --> RR )
4 simprr 734 . . . 4  |-  ( ( U  e.  NrmCVec  /\  (
x  e.  ( BaseSet `  U )  /\  e  e.  RR+ ) )  -> 
e  e.  RR+ )
51, 2nvcl 22148 . . . . . . . . . . . . . 14  |-  ( ( U  e.  NrmCVec  /\  x  e.  ( BaseSet `  U )
)  ->  ( N `  x )  e.  RR )
65ex 424 . . . . . . . . . . . . 13  |-  ( U  e.  NrmCVec  ->  ( x  e.  ( BaseSet `  U )  ->  ( N `  x
)  e.  RR ) )
71, 2nvcl 22148 . . . . . . . . . . . . . 14  |-  ( ( U  e.  NrmCVec  /\  y  e.  ( BaseSet `  U )
)  ->  ( N `  y )  e.  RR )
87ex 424 . . . . . . . . . . . . 13  |-  ( U  e.  NrmCVec  ->  ( y  e.  ( BaseSet `  U )  ->  ( N `  y
)  e.  RR ) )
96, 8anim12d 547 . . . . . . . . . . . 12  |-  ( U  e.  NrmCVec  ->  ( ( x  e.  ( BaseSet `  U
)  /\  y  e.  ( BaseSet `  U )
)  ->  ( ( N `  x )  e.  RR  /\  ( N `
 y )  e.  RR ) ) )
10 eqid 2436 . . . . . . . . . . . . . 14  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
1110remet 18821 . . . . . . . . . . . . 13  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  e.  ( Met `  RR )
12 metcl 18362 . . . . . . . . . . . . 13  |-  ( ( ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) )  e.  ( Met `  RR )  /\  ( N `  x )  e.  RR  /\  ( N `  y
)  e.  RR )  ->  ( ( N `
 x ) ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ( N `
 y ) )  e.  RR )
1311, 12mp3an1 1266 . . . . . . . . . . . 12  |-  ( ( ( N `  x
)  e.  RR  /\  ( N `  y )  e.  RR )  -> 
( ( N `  x ) ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ( N `
 y ) )  e.  RR )
149, 13syl6 31 . . . . . . . . . . 11  |-  ( U  e.  NrmCVec  ->  ( ( x  e.  ( BaseSet `  U
)  /\  y  e.  ( BaseSet `  U )
)  ->  ( ( N `  x )
( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ( N `  y
) )  e.  RR ) )
15143impib 1151 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  x  e.  ( BaseSet `  U )  /\  y  e.  ( BaseSet
`  U ) )  ->  ( ( N `
 x ) ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ( N `
 y ) )  e.  RR )
16 nmcvcn.2 . . . . . . . . . . . 12  |-  C  =  ( IndMet `  U )
171, 16imsmet 22183 . . . . . . . . . . 11  |-  ( U  e.  NrmCVec  ->  C  e.  ( Met `  ( BaseSet `  U ) ) )
18 metcl 18362 . . . . . . . . . . 11  |-  ( ( C  e.  ( Met `  ( BaseSet `  U )
)  /\  x  e.  ( BaseSet `  U )  /\  y  e.  ( BaseSet
`  U ) )  ->  ( x C y )  e.  RR )
1917, 18syl3an1 1217 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  x  e.  ( BaseSet `  U )  /\  y  e.  ( BaseSet
`  U ) )  ->  ( x C y )  e.  RR )
20 eqid 2436 . . . . . . . . . . . 12  |-  ( +v
`  U )  =  ( +v `  U
)
21 eqid 2436 . . . . . . . . . . . 12  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
221, 20, 21, 2nvabs 22162 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  x  e.  ( BaseSet `  U )  /\  y  e.  ( BaseSet
`  U ) )  ->  ( abs `  (
( N `  x
)  -  ( N `
 y ) ) )  <_  ( N `  ( x ( +v
`  U ) (
-u 1 ( .s
OLD `  U )
y ) ) ) )
2393impib 1151 . . . . . . . . . . . 12  |-  ( ( U  e.  NrmCVec  /\  x  e.  ( BaseSet `  U )  /\  y  e.  ( BaseSet
`  U ) )  ->  ( ( N `
 x )  e.  RR  /\  ( N `
 y )  e.  RR ) )
2410remetdval 18820 . . . . . . . . . . . 12  |-  ( ( ( N `  x
)  e.  RR  /\  ( N `  y )  e.  RR )  -> 
( ( N `  x ) ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ( N `
 y ) )  =  ( abs `  (
( N `  x
)  -  ( N `
 y ) ) ) )
2523, 24syl 16 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  x  e.  ( BaseSet `  U )  /\  y  e.  ( BaseSet
`  U ) )  ->  ( ( N `
 x ) ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ( N `
 y ) )  =  ( abs `  (
( N `  x
)  -  ( N `
 y ) ) ) )
261, 20, 21, 2, 16imsdval2 22179 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  x  e.  ( BaseSet `  U )  /\  y  e.  ( BaseSet
`  U ) )  ->  ( x C y )  =  ( N `  ( x ( +v `  U
) ( -u 1
( .s OLD `  U
) y ) ) ) )
2722, 25, 263brtr4d 4242 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  x  e.  ( BaseSet `  U )  /\  y  e.  ( BaseSet
`  U ) )  ->  ( ( N `
 x ) ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ( N `
 y ) )  <_  ( x C y ) )
2815, 19, 27jca31 521 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  x  e.  ( BaseSet `  U )  /\  y  e.  ( BaseSet
`  U ) )  ->  ( ( ( ( N `  x
) ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) ( N `  y
) )  e.  RR  /\  ( x C y )  e.  RR )  /\  ( ( N `
 x ) ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ( N `
 y ) )  <_  ( x C y ) ) )
29283expa 1153 . . . . . . . 8  |-  ( ( ( U  e.  NrmCVec  /\  x  e.  ( BaseSet `  U ) )  /\  y  e.  ( BaseSet `  U ) )  -> 
( ( ( ( N `  x ) ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ( N `  y
) )  e.  RR  /\  ( x C y )  e.  RR )  /\  ( ( N `
 x ) ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ( N `
 y ) )  <_  ( x C y ) ) )
30 rpre 10618 . . . . . . . 8  |-  ( e  e.  RR+  ->  e  e.  RR )
31 lelttr 9165 . . . . . . . . . . 11  |-  ( ( ( ( N `  x ) ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ( N `
 y ) )  e.  RR  /\  (
x C y )  e.  RR  /\  e  e.  RR )  ->  (
( ( ( N `
 x ) ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ( N `
 y ) )  <_  ( x C y )  /\  (
x C y )  <  e )  -> 
( ( N `  x ) ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ( N `
 y ) )  <  e ) )
32313expa 1153 . . . . . . . . . 10  |-  ( ( ( ( ( N `
 x ) ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ( N `
 y ) )  e.  RR  /\  (
x C y )  e.  RR )  /\  e  e.  RR )  ->  ( ( ( ( N `  x ) ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ( N `  y
) )  <_  (
x C y )  /\  ( x C y )  <  e
)  ->  ( ( N `  x )
( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ( N `  y
) )  <  e
) )
3332expdimp 427 . . . . . . . . 9  |-  ( ( ( ( ( ( N `  x ) ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ( N `  y
) )  e.  RR  /\  ( x C y )  e.  RR )  /\  e  e.  RR )  /\  ( ( N `
 x ) ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ( N `
 y ) )  <_  ( x C y ) )  -> 
( ( x C y )  <  e  ->  ( ( N `  x ) ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ( N `
 y ) )  <  e ) )
3433an32s 780 . . . . . . . 8  |-  ( ( ( ( ( ( N `  x ) ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ( N `  y
) )  e.  RR  /\  ( x C y )  e.  RR )  /\  ( ( N `
 x ) ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ( N `
 y ) )  <_  ( x C y ) )  /\  e  e.  RR )  ->  ( ( x C y )  <  e  ->  ( ( N `  x ) ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ( N `
 y ) )  <  e ) )
3529, 30, 34syl2an 464 . . . . . . 7  |-  ( ( ( ( U  e.  NrmCVec 
/\  x  e.  (
BaseSet `  U ) )  /\  y  e.  (
BaseSet `  U ) )  /\  e  e.  RR+ )  ->  ( ( x C y )  < 
e  ->  ( ( N `  x )
( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ( N `  y
) )  <  e
) )
3635ex 424 . . . . . 6  |-  ( ( ( U  e.  NrmCVec  /\  x  e.  ( BaseSet `  U ) )  /\  y  e.  ( BaseSet `  U ) )  -> 
( e  e.  RR+  ->  ( ( x C y )  <  e  ->  ( ( N `  x ) ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ( N `
 y ) )  <  e ) ) )
3736ralrimdva 2796 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  x  e.  ( BaseSet `  U )
)  ->  ( e  e.  RR+  ->  A. y  e.  ( BaseSet `  U )
( ( x C y )  <  e  ->  ( ( N `  x ) ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ( N `
 y ) )  <  e ) ) )
3837impr 603 . . . 4  |-  ( ( U  e.  NrmCVec  /\  (
x  e.  ( BaseSet `  U )  /\  e  e.  RR+ ) )  ->  A. y  e.  ( BaseSet
`  U ) ( ( x C y )  <  e  -> 
( ( N `  x ) ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ( N `
 y ) )  <  e ) )
39 breq2 4216 . . . . . . 7  |-  ( d  =  e  ->  (
( x C y )  <  d  <->  ( x C y )  < 
e ) )
4039imbi1d 309 . . . . . 6  |-  ( d  =  e  ->  (
( ( x C y )  <  d  ->  ( ( N `  x ) ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ( N `
 y ) )  <  e )  <->  ( (
x C y )  <  e  ->  (
( N `  x
) ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) ( N `  y
) )  <  e
) ) )
4140ralbidv 2725 . . . . 5  |-  ( d  =  e  ->  ( A. y  e.  ( BaseSet
`  U ) ( ( x C y )  <  d  -> 
( ( N `  x ) ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ( N `
 y ) )  <  e )  <->  A. y  e.  ( BaseSet `  U )
( ( x C y )  <  e  ->  ( ( N `  x ) ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ( N `
 y ) )  <  e ) ) )
4241rspcev 3052 . . . 4  |-  ( ( e  e.  RR+  /\  A. y  e.  ( BaseSet `  U ) ( ( x C y )  <  e  ->  (
( N `  x
) ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) ( N `  y
) )  <  e
) )  ->  E. d  e.  RR+  A. y  e.  ( BaseSet `  U )
( ( x C y )  <  d  ->  ( ( N `  x ) ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ( N `
 y ) )  <  e ) )
434, 38, 42syl2anc 643 . . 3  |-  ( ( U  e.  NrmCVec  /\  (
x  e.  ( BaseSet `  U )  /\  e  e.  RR+ ) )  ->  E. d  e.  RR+  A. y  e.  ( BaseSet `  U )
( ( x C y )  <  d  ->  ( ( N `  x ) ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ( N `
 y ) )  <  e ) )
4443ralrimivva 2798 . 2  |-  ( U  e.  NrmCVec  ->  A. x  e.  (
BaseSet `  U ) A. e  e.  RR+  E. d  e.  RR+  A. y  e.  ( BaseSet `  U )
( ( x C y )  <  d  ->  ( ( N `  x ) ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ( N `
 y ) )  <  e ) )
451, 16imsxmet 22184 . . 3  |-  ( U  e.  NrmCVec  ->  C  e.  ( * Met `  ( BaseSet
`  U ) ) )
4610rexmet 18822 . . 3  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  e.  ( * Met `  RR )
47 nmcvcn.j . . . 4  |-  J  =  ( MetOpen `  C )
48 nmcvcn.k . . . . 5  |-  K  =  ( topGen `  ran  (,) )
49 eqid 2436 . . . . . 6  |-  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )
5010, 49tgioo 18827 . . . . 5  |-  ( topGen ` 
ran  (,) )  =  (
MetOpen `  ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) )
5148, 50eqtri 2456 . . . 4  |-  K  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) )
5247, 51metcn 18573 . . 3  |-  ( ( C  e.  ( * Met `  ( BaseSet `  U ) )  /\  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) )  e.  ( * Met `  RR ) )  -> 
( N  e.  ( J  Cn  K )  <-> 
( N : (
BaseSet `  U ) --> RR 
/\  A. x  e.  (
BaseSet `  U ) A. e  e.  RR+  E. d  e.  RR+  A. y  e.  ( BaseSet `  U )
( ( x C y )  <  d  ->  ( ( N `  x ) ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ( N `
 y ) )  <  e ) ) ) )
5345, 46, 52sylancl 644 . 2  |-  ( U  e.  NrmCVec  ->  ( N  e.  ( J  Cn  K
)  <->  ( N :
( BaseSet `  U ) --> RR  /\  A. x  e.  ( BaseSet `  U ) A. e  e.  RR+  E. d  e.  RR+  A. y  e.  ( BaseSet `  U )
( ( x C y )  <  d  ->  ( ( N `  x ) ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ( N `
 y ) )  <  e ) ) ) )
543, 44, 53mpbir2and 889 1  |-  ( U  e.  NrmCVec  ->  N  e.  ( J  Cn  K ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705   E.wrex 2706   class class class wbr 4212    X. cxp 4876   ran crn 4879    |` cres 4880    o. ccom 4882   -->wf 5450   ` cfv 5454  (class class class)co 6081   RRcr 8989   1c1 8991    < clt 9120    <_ cle 9121    - cmin 9291   -ucneg 9292   RR+crp 10612   (,)cioo 10916   abscabs 12039   topGenctg 13665   * Metcxmt 16686   Metcme 16687   MetOpencmopn 16691    Cn ccn 17288   NrmCVeccnv 22063   +vcpv 22064   BaseSetcba 22065   .s
OLDcns 22066   normCVcnmcv 22069   IndMetcims 22070
This theorem is referenced by:  nmcnc  22192
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-q 10575  df-rp 10613  df-xneg 10710  df-xadd 10711  df-xmul 10712  df-ioo 10920  df-seq 11324  df-exp 11383  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-topgen 13667  df-psmet 16694  df-xmet 16695  df-met 16696  df-bl 16697  df-mopn 16698  df-top 16963  df-bases 16965  df-topon 16966  df-cn 17291  df-cnp 17292  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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