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Theorem axhcompl-zf 21594
Description: Derive axiom ax-hcompl 21797 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 2296 . . . . . . 7  |-  ( IndMet `  U )  =  (
IndMet `  U )
4 eqid 2296 . . . . . . 7  |-  ( MetOpen `  ( IndMet `  U )
)  =  ( MetOpen `  ( IndMet `  U )
)
53, 4hlcompl 21510 . . . . . 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 4891 . . . . . 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 4040 . . . . . 6  |-  ( F ( ~~> t `  ( MetOpen
`  ( IndMet `  U
) ) ) x  <->  <. F ,  x >.  e.  ( ~~> t `  ( MetOpen
`  ( IndMet `  U
) ) ) )
111hlnvi 21487 . . . . . . . . . 10  |-  U  e.  NrmCVec
12 df-hba 21565 . . . . . . . . . . . 12  |-  ~H  =  ( BaseSet `  <. <.  +h  ,  .h  >. ,  normh >. )
13 axhil.1 . . . . . . . . . . . . 13  |-  U  = 
<. <.  +h  ,  .h  >. ,  normh >.
1413fveq2i 5544 . . . . . . . . . . . 12  |-  ( BaseSet `  U )  =  (
BaseSet `  <. <.  +h  ,  .h  >. ,  normh >. )
1512, 14eqtr4i 2319 . . . . . . . . . . 11  |-  ~H  =  ( BaseSet `  U )
1615, 3imsxmet 21277 . . . . . . . . . 10  |-  ( U  e.  NrmCVec  ->  ( IndMet `  U
)  e.  ( * Met `  ~H )
)
174mopntopon 18001 . . . . . . . . . 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 17041 . . . . . . . . 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 21576 . . . . . . . . . . . 12  |-  ~~>v  =  ( ( ~~> t `  ( MetOpen
`  ( IndMet `  U
) ) )  |`  ( ~H  ^m  NN ) )
2322breqi 4045 . . . . . . . . . . 11  |-  ( F 
~~>v  x  <->  F ( ( ~~> t `  ( MetOpen `  ( IndMet `  U ) ) )  |`  ( ~H  ^m  NN ) ) x )
24 vex 2804 . . . . . . . . . . . 12  |-  x  e. 
_V
2524brres 4977 . . . . . . . . . . 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 1612 . . . 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 3371 . . 3  |-  ( F  e.  ( ( Cau `  ( IndMet `  U )
)  i^i  ( ~H  ^m  NN ) )  <->  ( F  e.  ( Cau `  ( IndMet `
 U ) )  /\  F  e.  ( ~H  ^m  NN ) ) )
36 df-rex 2562 . . 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 21575 . 2  |-  Cauchy  =  ( ( Cau `  ( IndMet `
 U ) )  i^i  ( ~H  ^m  NN ) )
3937, 38eleq2s 2388 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 1531    = wceq 1632    e. wcel 1696   E.wrex 2557    i^i cin 3164   <.cop 3656   class class class wbr 4039   dom cdm 4705    |` cres 4707   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   NNcn 9762   * Metcxmt 16385   MetOpencmopn 16388  TopOnctopon 16648   ~~> tclm 16972   Caucca 18695   NrmCVeccnv 21156   BaseSetcba 21158   IndMetcims 21163   CHil OLDchlo 21480   ~Hchil 21515    +h cva 21516    .h csm 21517   normhcno 21519   Cauchyccau 21522    ~~>v chli 21523
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ico 10678  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-rest 13343  df-topgen 13360  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-top 16652  df-bases 16654  df-topon 16655  df-ntr 16773  df-nei 16851  df-lm 16975  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-cfil 18697  df-cau 18698  df-cmet 18699  df-grpo 20874  df-gid 20875  df-ginv 20876  df-gdiv 20877  df-ablo 20965  df-vc 21118  df-nv 21164  df-va 21167  df-ba 21168  df-sm 21169  df-0v 21170  df-vs 21171  df-nmcv 21172  df-ims 21173  df-cbn 21458  df-hlo 21481  df-hba 21565  df-hvsub 21567  df-hlim 21568  df-hcau 21569
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