HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  axhcompl-zf Unicode version

Theorem axhcompl-zf 21578
Description: Derive axiom ax-hcompl 21781 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. 
CHil OLD
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. 
CHil OLD
2 simpl 443 . . . . . 6  |-  ( ( F  e.  ( Cau `  ( IndMet `  U )
)  /\  F  e.  ( ~H  ^m  NN ) )  ->  F  e.  ( Cau `  ( IndMet `  U ) ) )
3 eqid 2283 . . . . . . 7  |-  ( IndMet `  U )  =  (
IndMet `  U )
4 eqid 2283 . . . . . . 7  |-  ( MetOpen `  ( IndMet `  U )
)  =  ( MetOpen `  ( IndMet `  U )
)
53, 4hlcompl 21494 . . . . . 6  |-  ( ( U  e.  CHil OLD  /\  F  e.  ( Cau `  ( IndMet `  U )
) )  ->  F  e.  dom  ( ~~> t `  ( MetOpen `  ( IndMet `  U ) ) ) )
61, 2, 5sylancr 644 . . . . 5  |-  ( ( F  e.  ( Cau `  ( IndMet `  U )
)  /\  F  e.  ( ~H  ^m  NN ) )  ->  F  e.  dom  ( ~~> t `  ( MetOpen
`  ( IndMet `  U
) ) ) )
7 eldm2g 4875 . . . . . 6  |-  ( F  e.  ( Cau `  ( IndMet `
 U ) )  ->  ( F  e. 
dom  ( ~~> t `  ( MetOpen `  ( IndMet `  U ) ) )  <->  E. x <. F ,  x >.  e.  ( ~~> t `  ( MetOpen `  ( IndMet `  U ) ) ) ) )
87adantr 451 . . . . 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 201 . . . 4  |-  ( ( F  e.  ( Cau `  ( IndMet `  U )
)  /\  F  e.  ( ~H  ^m  NN ) )  ->  E. x <. F ,  x >.  e.  ( ~~> t `  ( MetOpen
`  ( IndMet `  U
) ) ) )
10 df-br 4024 . . . . . 6  |-  ( F ( ~~> t `  ( MetOpen
`  ( IndMet `  U
) ) ) x  <->  <. F ,  x >.  e.  ( ~~> t `  ( MetOpen
`  ( IndMet `  U
) ) ) )
111hlnvi 21471 . . . . . . . . . 10  |-  U  e.  NrmCVec
12 df-hba 21549 . . . . . . . . . . . 12  |-  ~H  =  ( BaseSet `  <. <.  +h  ,  .h  >. ,  normh >. )
13 axhil.1 . . . . . . . . . . . . 13  |-  U  = 
<. <.  +h  ,  .h  >. ,  normh >.
1413fveq2i 5528 . . . . . . . . . . . 12  |-  ( BaseSet `  U )  =  (
BaseSet `  <. <.  +h  ,  .h  >. ,  normh >. )
1512, 14eqtr4i 2306 . . . . . . . . . . 11  |-  ~H  =  ( BaseSet `  U )
1615, 3imsxmet 21261 . . . . . . . . . 10  |-  ( U  e.  NrmCVec  ->  ( IndMet `  U
)  e.  ( * Met `  ~H )
)
174mopntopon 17985 . . . . . . . . . 10  |-  ( (
IndMet `  U )  e.  ( * Met `  ~H )  ->  ( MetOpen `  ( IndMet `
 U ) )  e.  (TopOn `  ~H ) )
1811, 16, 17mp2b 9 . . . . . . . . 9  |-  ( MetOpen `  ( IndMet `  U )
)  e.  (TopOn `  ~H )
19 lmcl 17025 . . . . . . . . 9  |-  ( ( ( MetOpen `  ( IndMet `  U ) )  e.  (TopOn `  ~H )  /\  F ( ~~> t `  ( MetOpen `  ( IndMet `  U ) ) ) x )  ->  x  e.  ~H )
2018, 19mpan 651 . . . . . . . 8  |-  ( F ( ~~> t `  ( MetOpen
`  ( IndMet `  U
) ) ) x  ->  x  e.  ~H )
2120a1i 10 . . . . . . 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 21560 . . . . . . . . . . . 12  |-  ~~>v  =  ( ( ~~> t `  ( MetOpen
`  ( IndMet `  U
) ) )  |`  ( ~H  ^m  NN ) )
2322breqi 4029 . . . . . . . . . . 11  |-  ( F 
~~>v  x  <->  F ( ( ~~> t `  ( MetOpen `  ( IndMet `  U ) ) )  |`  ( ~H  ^m  NN ) ) x )
24 vex 2791 . . . . . . . . . . . 12  |-  x  e. 
_V
2524brres 4961 . . . . . . . . . . 11  |-  ( F ( ( ~~> t `  ( MetOpen `  ( IndMet `  U ) ) )  |`  ( ~H  ^m  NN ) ) x  <->  ( F
( ~~> t `  ( MetOpen
`  ( IndMet `  U
) ) ) x  /\  F  e.  ( ~H  ^m  NN ) ) )
26 ancom 437 . . . . . . . . . . 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 262 . . . . . . . . . 10  |-  ( F 
~~>v  x  <->  ( F  e.  ( ~H  ^m  NN )  /\  F ( ~~> t `  ( MetOpen `  ( IndMet `  U ) ) ) x ) )
2827baib 871 . . . . . . . . 9  |-  ( F  e.  ( ~H  ^m  NN )  ->  ( F 
~~>v  x  <->  F ( ~~> t `  ( MetOpen `  ( IndMet `  U ) ) ) x ) )
2928adantl 452 . . . . . . . 8  |-  ( ( F  e.  ( Cau `  ( IndMet `  U )
)  /\  F  e.  ( ~H  ^m  NN ) )  ->  ( F  ~~>v  x 
<->  F ( ~~> t `  ( MetOpen `  ( IndMet `  U ) ) ) x ) )
3029biimprd 214 . . . . . . 7  |-  ( ( F  e.  ( Cau `  ( IndMet `  U )
)  /\  F  e.  ( ~H  ^m  NN ) )  ->  ( F
( ~~> t `  ( MetOpen
`  ( IndMet `  U
) ) ) x  ->  F  ~~>v  x ) )
3121, 30jcad 519 . . . . . 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 209 . . . . 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 1608 . . . 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 14 . . 3  |-  ( ( F  e.  ( Cau `  ( IndMet `  U )
)  /\  F  e.  ( ~H  ^m  NN ) )  ->  E. x
( x  e.  ~H  /\  F  ~~>v  x ) )
35 elin 3358 . . 3  |-  ( F  e.  ( ( Cau `  ( IndMet `  U )
)  i^i  ( ~H  ^m  NN ) )  <->  ( F  e.  ( Cau `  ( IndMet `
 U ) )  /\  F  e.  ( ~H  ^m  NN ) ) )
36 df-rex 2549 . . 3  |-  ( E. x  e.  ~H  F  ~~>v  x 
<->  E. x ( x  e.  ~H  /\  F  ~~>v  x ) )
3734, 35, 363imtr4i 257 . 2  |-  ( F  e.  ( ( Cau `  ( IndMet `  U )
)  i^i  ( ~H  ^m  NN ) )  ->  E. x  e.  ~H  F  ~~>v  x )
3813, 11, 15, 3h2hcau 21559 . 2  |-  Cauchy  =  ( ( Cau `  ( IndMet `
 U ) )  i^i  ( ~H  ^m  NN ) )
3937, 38eleq2s 2375 1  |-  ( F  e.  Cauchy  ->  E. x  e.  ~H  F  ~~>v  x )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   E.wrex 2544    i^i cin 3151   <.cop 3643   class class class wbr 4023   dom cdm 4689    |` cres 4691   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   NNcn 9746   * Metcxmt 16369   MetOpencmopn 16372  TopOnctopon 16632   ~~> tclm 16956   Caucca 18679   NrmCVeccnv 21140   BaseSetcba 21142   IndMetcims 21147   CHil OLDchlo 21464   ~Hchil 21499    +h cva 21500    .h csm 21501   normhcno 21503   Cauchyccau 21506    ~~>v chli 21507
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-rep 4131  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-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-reu 2550  df-rmo 2551  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-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  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-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ico 10662  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-rest 13327  df-topgen 13344  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-top 16636  df-bases 16638  df-topon 16639  df-ntr 16757  df-nei 16835  df-lm 16959  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-cfil 18681  df-cau 18682  df-cmet 18683  df-grpo 20858  df-gid 20859  df-ginv 20860  df-gdiv 20861  df-ablo 20949  df-vc 21102  df-nv 21148  df-va 21151  df-ba 21152  df-sm 21153  df-0v 21154  df-vs 21155  df-nmcv 21156  df-ims 21157  df-cbn 21442  df-hlo 21465  df-hba 21549  df-hvsub 21551  df-hlim 21552  df-hcau 21553
  Copyright terms: Public domain W3C validator