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Theorem axhcompl-zf 26636
Description: Derive axiom ax-hcompl 26840 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 458 . . . . . 6  |-  ( ( F  e.  ( Cau `  ( IndMet `  U )
)  /\  F  e.  ( ~H  ^m  NN ) )  ->  F  e.  ( Cau `  ( IndMet `  U ) ) )
3 eqid 2422 . . . . . . 7  |-  ( IndMet `  U )  =  (
IndMet `  U )
4 eqid 2422 . . . . . . 7  |-  ( MetOpen `  ( IndMet `  U )
)  =  ( MetOpen `  ( IndMet `  U )
)
53, 4hlcompl 26552 . . . . . 6  |-  ( ( U  e.  CHilOLD  /\  F  e.  ( Cau `  ( IndMet `  U )
) )  ->  F  e.  dom  ( ~~> t `  ( MetOpen `  ( IndMet `  U ) ) ) )
61, 2, 5sylancr 667 . . . . 5  |-  ( ( F  e.  ( Cau `  ( IndMet `  U )
)  /\  F  e.  ( ~H  ^m  NN ) )  ->  F  e.  dom  ( ~~> t `  ( MetOpen
`  ( IndMet `  U
) ) ) )
7 eldm2g 5046 . . . . . 6  |-  ( F  e.  ( Cau `  ( IndMet `
 U ) )  ->  ( F  e. 
dom  ( ~~> t `  ( MetOpen `  ( IndMet `  U ) ) )  <->  E. x <. F ,  x >.  e.  ( ~~> t `  ( MetOpen `  ( IndMet `  U ) ) ) ) )
87adantr 466 . . . . 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 213 . . . 4  |-  ( ( F  e.  ( Cau `  ( IndMet `  U )
)  /\  F  e.  ( ~H  ^m  NN ) )  ->  E. x <. F ,  x >.  e.  ( ~~> t `  ( MetOpen
`  ( IndMet `  U
) ) ) )
10 df-br 4421 . . . . . 6  |-  ( F ( ~~> t `  ( MetOpen
`  ( IndMet `  U
) ) ) x  <->  <. F ,  x >.  e.  ( ~~> t `  ( MetOpen
`  ( IndMet `  U
) ) ) )
111hlnvi 26529 . . . . . . . . . 10  |-  U  e.  NrmCVec
12 df-hba 26607 . . . . . . . . . . . 12  |-  ~H  =  ( BaseSet `  <. <.  +h  ,  .h  >. ,  normh >. )
13 axhil.1 . . . . . . . . . . . . 13  |-  U  = 
<. <.  +h  ,  .h  >. ,  normh >.
1413fveq2i 5880 . . . . . . . . . . . 12  |-  ( BaseSet `  U )  =  (
BaseSet `  <. <.  +h  ,  .h  >. ,  normh >. )
1512, 14eqtr4i 2454 . . . . . . . . . . 11  |-  ~H  =  ( BaseSet `  U )
1615, 3imsxmet 26309 . . . . . . . . . 10  |-  ( U  e.  NrmCVec  ->  ( IndMet `  U
)  e.  ( *Met `  ~H )
)
174mopntopon 21440 . . . . . . . . . 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 20299 . . . . . . . . 9  |-  ( ( ( MetOpen `  ( IndMet `  U ) )  e.  (TopOn `  ~H )  /\  F ( ~~> t `  ( MetOpen `  ( IndMet `  U ) ) ) x )  ->  x  e.  ~H )
2018, 19mpan 674 . . . . . . . 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 26618 . . . . . . . . . . . 12  |-  ~~>v  =  ( ( ~~> t `  ( MetOpen
`  ( IndMet `  U
) ) )  |`  ( ~H  ^m  NN ) )
2322breqi 4426 . . . . . . . . . . 11  |-  ( F 
~~>v  x  <->  F ( ( ~~> t `  ( MetOpen `  ( IndMet `  U ) ) )  |`  ( ~H  ^m  NN ) ) x )
24 vex 3084 . . . . . . . . . . . 12  |-  x  e. 
_V
2524brres 5126 . . . . . . . . . . 11  |-  ( F ( ( ~~> t `  ( MetOpen `  ( IndMet `  U ) ) )  |`  ( ~H  ^m  NN ) ) x  <->  ( F
( ~~> t `  ( MetOpen
`  ( IndMet `  U
) ) ) x  /\  F  e.  ( ~H  ^m  NN ) ) )
26 ancom 451 . . . . . . . . . . 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 274 . . . . . . . . . 10  |-  ( F 
~~>v  x  <->  ( F  e.  ( ~H  ^m  NN )  /\  F ( ~~> t `  ( MetOpen `  ( IndMet `  U ) ) ) x ) )
2827baib 911 . . . . . . . . 9  |-  ( F  e.  ( ~H  ^m  NN )  ->  ( F 
~~>v  x  <->  F ( ~~> t `  ( MetOpen `  ( IndMet `  U ) ) ) x ) )
2928adantl 467 . . . . . . . 8  |-  ( ( F  e.  ( Cau `  ( IndMet `  U )
)  /\  F  e.  ( ~H  ^m  NN ) )  ->  ( F  ~~>v  x 
<->  F ( ~~> t `  ( MetOpen `  ( IndMet `  U ) ) ) x ) )
3029biimprd 226 . . . . . . 7  |-  ( ( F  e.  ( Cau `  ( IndMet `  U )
)  /\  F  e.  ( ~H  ^m  NN ) )  ->  ( F
( ~~> t `  ( MetOpen
`  ( IndMet `  U
) ) ) x  ->  F  ~~>v  x ) )
3121, 30jcad 535 . . . . . 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 221 . . . . 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 1754 . . . 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 3649 . . 3  |-  ( F  e.  ( ( Cau `  ( IndMet `  U )
)  i^i  ( ~H  ^m  NN ) )  <->  ( F  e.  ( Cau `  ( IndMet `
 U ) )  /\  F  e.  ( ~H  ^m  NN ) ) )
36 df-rex 2781 . . 3  |-  ( E. x  e.  ~H  F  ~~>v  x 
<->  E. x ( x  e.  ~H  /\  F  ~~>v  x ) )
3734, 35, 363imtr4i 269 . 2  |-  ( F  e.  ( ( Cau `  ( IndMet `  U )
)  i^i  ( ~H  ^m  NN ) )  ->  E. x  e.  ~H  F  ~~>v  x )
3813, 11, 15, 3h2hcau 26617 . 2  |-  Cauchy  =  ( ( Cau `  ( IndMet `
 U ) )  i^i  ( ~H  ^m  NN ) )
3937, 38eleq2s 2530 1  |-  ( F  e.  Cauchy  ->  E. x  e.  ~H  F  ~~>v  x )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437   E.wex 1659    e. wcel 1868   E.wrex 2776    i^i cin 3435   <.cop 4002   class class class wbr 4420   dom cdm 4849    |` cres 4851   ` cfv 5597  (class class class)co 6301    ^m cmap 7476   NNcn 10609   *Metcxmt 18942   MetOpencmopn 18947  TopOnctopon 19904   ~~> tclm 20228   Caucca 22209   NrmCVeccnv 26188   BaseSetcba 26190   IndMetcims 26195   CHilOLDchlo 26522   ~Hchil 26557    +h cva 26558    .h csm 26559   normhcno 26561   Cauchyccau 26564    ~~>v chli 26565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-om 6703  df-1st 6803  df-2nd 6804  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-er 7367  df-map 7478  df-pm 7479  df-en 7574  df-dom 7575  df-sdom 7576  df-sup 7958  df-inf 7959  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ico 11641  df-seq 12213  df-exp 12272  df-cj 13150  df-re 13151  df-im 13152  df-sqrt 13286  df-abs 13287  df-rest 15308  df-topgen 15329  df-psmet 18949  df-xmet 18950  df-met 18951  df-bl 18952  df-mopn 18953  df-fbas 18954  df-fg 18955  df-top 19907  df-bases 19908  df-topon 19909  df-ntr 20021  df-nei 20100  df-lm 20231  df-fil 20847  df-fm 20939  df-flim 20940  df-flf 20941  df-cfil 22211  df-cau 22212  df-cmet 22213  df-grpo 25904  df-gid 25905  df-ginv 25906  df-gdiv 25907  df-ablo 25995  df-vc 26150  df-nv 26196  df-va 26199  df-ba 26200  df-sm 26201  df-0v 26202  df-vs 26203  df-nmcv 26204  df-ims 26205  df-cbn 26490  df-hlo 26523  df-hba 26607  df-hvsub 26609  df-hlim 26610  df-hcau 26611
This theorem is referenced by: (None)
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