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Theorem axhcompl-zf 25738
Description: Derive axiom ax-hcompl 25942 from Hilbert space under ZF set theory. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 13-May-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
axhil.1  |-  U  = 
<. <.  +h  ,  .h  >. ,  normh >.
axhil.2  |-  U  e. 
CHilOLD
Assertion
Ref Expression
axhcompl-zf  |-  ( F  e.  Cauchy  ->  E. x  e.  ~H  F  ~~>v  x )
Distinct variable groups:    x, F    x, U

Proof of Theorem axhcompl-zf
StepHypRef Expression
1 axhil.2 . . . . . 6  |-  U  e. 
CHilOLD
2 simpl 457 . . . . . 6  |-  ( ( F  e.  ( Cau `  ( IndMet `  U )
)  /\  F  e.  ( ~H  ^m  NN ) )  ->  F  e.  ( Cau `  ( IndMet `  U ) ) )
3 eqid 2467 . . . . . . 7  |-  ( IndMet `  U )  =  (
IndMet `  U )
4 eqid 2467 . . . . . . 7  |-  ( MetOpen `  ( IndMet `  U )
)  =  ( MetOpen `  ( IndMet `  U )
)
53, 4hlcompl 25654 . . . . . 6  |-  ( ( U  e.  CHilOLD  /\  F  e.  ( Cau `  ( IndMet `  U )
) )  ->  F  e.  dom  ( ~~> t `  ( MetOpen `  ( IndMet `  U ) ) ) )
61, 2, 5sylancr 663 . . . . 5  |-  ( ( F  e.  ( Cau `  ( IndMet `  U )
)  /\  F  e.  ( ~H  ^m  NN ) )  ->  F  e.  dom  ( ~~> t `  ( MetOpen
`  ( IndMet `  U
) ) ) )
7 eldm2g 5205 . . . . . 6  |-  ( F  e.  ( Cau `  ( IndMet `
 U ) )  ->  ( F  e. 
dom  ( ~~> t `  ( MetOpen `  ( IndMet `  U ) ) )  <->  E. x <. F ,  x >.  e.  ( ~~> t `  ( MetOpen `  ( IndMet `  U ) ) ) ) )
87adantr 465 . . . . 5  |-  ( ( F  e.  ( Cau `  ( IndMet `  U )
)  /\  F  e.  ( ~H  ^m  NN ) )  ->  ( F  e.  dom  ( ~~> t `  ( MetOpen `  ( IndMet `  U ) ) )  <->  E. x <. F ,  x >.  e.  ( ~~> t `  ( MetOpen `  ( IndMet `  U ) ) ) ) )
96, 8mpbid 210 . . . 4  |-  ( ( F  e.  ( Cau `  ( IndMet `  U )
)  /\  F  e.  ( ~H  ^m  NN ) )  ->  E. x <. F ,  x >.  e.  ( ~~> t `  ( MetOpen
`  ( IndMet `  U
) ) ) )
10 df-br 4454 . . . . . 6  |-  ( F ( ~~> t `  ( MetOpen
`  ( IndMet `  U
) ) ) x  <->  <. F ,  x >.  e.  ( ~~> t `  ( MetOpen
`  ( IndMet `  U
) ) ) )
111hlnvi 25631 . . . . . . . . . 10  |-  U  e.  NrmCVec
12 df-hba 25709 . . . . . . . . . . . 12  |-  ~H  =  ( BaseSet `  <. <.  +h  ,  .h  >. ,  normh >. )
13 axhil.1 . . . . . . . . . . . . 13  |-  U  = 
<. <.  +h  ,  .h  >. ,  normh >.
1413fveq2i 5875 . . . . . . . . . . . 12  |-  ( BaseSet `  U )  =  (
BaseSet `  <. <.  +h  ,  .h  >. ,  normh >. )
1512, 14eqtr4i 2499 . . . . . . . . . . 11  |-  ~H  =  ( BaseSet `  U )
1615, 3imsxmet 25421 . . . . . . . . . 10  |-  ( U  e.  NrmCVec  ->  ( IndMet `  U
)  e.  ( *Met `  ~H )
)
174mopntopon 20810 . . . . . . . . . 10  |-  ( (
IndMet `  U )  e.  ( *Met `  ~H )  ->  ( MetOpen `  ( IndMet `  U )
)  e.  (TopOn `  ~H ) )
1811, 16, 17mp2b 10 . . . . . . . . 9  |-  ( MetOpen `  ( IndMet `  U )
)  e.  (TopOn `  ~H )
19 lmcl 19666 . . . . . . . . 9  |-  ( ( ( MetOpen `  ( IndMet `  U ) )  e.  (TopOn `  ~H )  /\  F ( ~~> t `  ( MetOpen `  ( IndMet `  U ) ) ) x )  ->  x  e.  ~H )
2018, 19mpan 670 . . . . . . . 8  |-  ( F ( ~~> t `  ( MetOpen
`  ( IndMet `  U
) ) ) x  ->  x  e.  ~H )
2120a1i 11 . . . . . . 7  |-  ( ( F  e.  ( Cau `  ( IndMet `  U )
)  /\  F  e.  ( ~H  ^m  NN ) )  ->  ( F
( ~~> t `  ( MetOpen
`  ( IndMet `  U
) ) ) x  ->  x  e.  ~H ) )
2213, 11, 15, 3, 4h2hlm 25720 . . . . . . . . . . . 12  |-  ~~>v  =  ( ( ~~> t `  ( MetOpen
`  ( IndMet `  U
) ) )  |`  ( ~H  ^m  NN ) )
2322breqi 4459 . . . . . . . . . . 11  |-  ( F 
~~>v  x  <->  F ( ( ~~> t `  ( MetOpen `  ( IndMet `  U ) ) )  |`  ( ~H  ^m  NN ) ) x )
24 vex 3121 . . . . . . . . . . . 12  |-  x  e. 
_V
2524brres 5286 . . . . . . . . . . 11  |-  ( F ( ( ~~> t `  ( MetOpen `  ( IndMet `  U ) ) )  |`  ( ~H  ^m  NN ) ) x  <->  ( F
( ~~> t `  ( MetOpen
`  ( IndMet `  U
) ) ) x  /\  F  e.  ( ~H  ^m  NN ) ) )
26 ancom 450 . . . . . . . . . . 11  |-  ( ( F ( ~~> t `  ( MetOpen `  ( IndMet `  U ) ) ) x  /\  F  e.  ( ~H  ^m  NN ) )  <->  ( F  e.  ( ~H  ^m  NN )  /\  F ( ~~> t `  ( MetOpen `  ( IndMet `  U ) ) ) x ) )
2723, 25, 263bitri 271 . . . . . . . . . 10  |-  ( F 
~~>v  x  <->  ( F  e.  ( ~H  ^m  NN )  /\  F ( ~~> t `  ( MetOpen `  ( IndMet `  U ) ) ) x ) )
2827baib 901 . . . . . . . . 9  |-  ( F  e.  ( ~H  ^m  NN )  ->  ( F 
~~>v  x  <->  F ( ~~> t `  ( MetOpen `  ( IndMet `  U ) ) ) x ) )
2928adantl 466 . . . . . . . 8  |-  ( ( F  e.  ( Cau `  ( IndMet `  U )
)  /\  F  e.  ( ~H  ^m  NN ) )  ->  ( F  ~~>v  x 
<->  F ( ~~> t `  ( MetOpen `  ( IndMet `  U ) ) ) x ) )
3029biimprd 223 . . . . . . 7  |-  ( ( F  e.  ( Cau `  ( IndMet `  U )
)  /\  F  e.  ( ~H  ^m  NN ) )  ->  ( F
( ~~> t `  ( MetOpen
`  ( IndMet `  U
) ) ) x  ->  F  ~~>v  x ) )
3121, 30jcad 533 . . . . . 6  |-  ( ( F  e.  ( Cau `  ( IndMet `  U )
)  /\  F  e.  ( ~H  ^m  NN ) )  ->  ( F
( ~~> t `  ( MetOpen
`  ( IndMet `  U
) ) ) x  ->  ( x  e. 
~H  /\  F  ~~>v  x ) ) )
3210, 31syl5bir 218 . . . . 5  |-  ( ( F  e.  ( Cau `  ( IndMet `  U )
)  /\  F  e.  ( ~H  ^m  NN ) )  ->  ( <. F ,  x >.  e.  ( ~~> t `  ( MetOpen `  ( IndMet `  U )
) )  ->  (
x  e.  ~H  /\  F  ~~>v  x ) ) )
3332eximdv 1686 . . . 4  |-  ( ( F  e.  ( Cau `  ( IndMet `  U )
)  /\  F  e.  ( ~H  ^m  NN ) )  ->  ( E. x <. F ,  x >.  e.  ( ~~> t `  ( MetOpen `  ( IndMet `  U ) ) )  ->  E. x ( x  e.  ~H  /\  F  ~~>v  x ) ) )
349, 33mpd 15 . . 3  |-  ( ( F  e.  ( Cau `  ( IndMet `  U )
)  /\  F  e.  ( ~H  ^m  NN ) )  ->  E. x
( x  e.  ~H  /\  F  ~~>v  x ) )
35 elin 3692 . . 3  |-  ( F  e.  ( ( Cau `  ( IndMet `  U )
)  i^i  ( ~H  ^m  NN ) )  <->  ( F  e.  ( Cau `  ( IndMet `
 U ) )  /\  F  e.  ( ~H  ^m  NN ) ) )
36 df-rex 2823 . . 3  |-  ( E. x  e.  ~H  F  ~~>v  x 
<->  E. x ( x  e.  ~H  /\  F  ~~>v  x ) )
3734, 35, 363imtr4i 266 . 2  |-  ( F  e.  ( ( Cau `  ( IndMet `  U )
)  i^i  ( ~H  ^m  NN ) )  ->  E. x  e.  ~H  F  ~~>v  x )
3813, 11, 15, 3h2hcau 25719 . 2  |-  Cauchy  =  ( ( Cau `  ( IndMet `
 U ) )  i^i  ( ~H  ^m  NN ) )
3937, 38eleq2s 2575 1  |-  ( F  e.  Cauchy  ->  E. x  e.  ~H  F  ~~>v  x )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   E.wrex 2818    i^i cin 3480   <.cop 4039   class class class wbr 4453   dom cdm 5005    |` cres 5007   ` cfv 5594  (class class class)co 6295    ^m cmap 7432   NNcn 10548   *Metcxmt 18273   MetOpencmopn 18278  TopOnctopon 19264   ~~> tclm 19595   Caucca 21560   NrmCVeccnv 25300   BaseSetcba 25302   IndMetcims 25307   CHilOLDchlo 25624   ~Hchil 25659    +h cva 25660    .h csm 25661   normhcno 25663   Cauchyccau 25666    ~~>v chli 25667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582  ax-addf 9583  ax-mulf 9584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-map 7434  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-q 11195  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-ico 11547  df-seq 12088  df-exp 12147  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-rest 14695  df-topgen 14716  df-psmet 18281  df-xmet 18282  df-met 18283  df-bl 18284  df-mopn 18285  df-fbas 18286  df-fg 18287  df-top 19268  df-bases 19270  df-topon 19271  df-ntr 19389  df-nei 19467  df-lm 19598  df-fil 20215  df-fm 20307  df-flim 20308  df-flf 20309  df-cfil 21562  df-cau 21563  df-cmet 21564  df-grpo 25016  df-gid 25017  df-ginv 25018  df-gdiv 25019  df-ablo 25107  df-vc 25262  df-nv 25308  df-va 25311  df-ba 25312  df-sm 25313  df-0v 25314  df-vs 25315  df-nmcv 25316  df-ims 25317  df-cbn 25602  df-hlo 25625  df-hba 25709  df-hvsub 25711  df-hlim 25712  df-hcau 25713
This theorem is referenced by: (None)
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