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Theorem axhcompl-zf 26042
Description: Derive axiom ax-hcompl 26246 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 2457 . . . . . . 7  |-  ( IndMet `  U )  =  (
IndMet `  U )
4 eqid 2457 . . . . . . 7  |-  ( MetOpen `  ( IndMet `  U )
)  =  ( MetOpen `  ( IndMet `  U )
)
53, 4hlcompl 25958 . . . . . 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 5209 . . . . . 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 4457 . . . . . 6  |-  ( F ( ~~> t `  ( MetOpen
`  ( IndMet `  U
) ) ) x  <->  <. F ,  x >.  e.  ( ~~> t `  ( MetOpen
`  ( IndMet `  U
) ) ) )
111hlnvi 25935 . . . . . . . . . 10  |-  U  e.  NrmCVec
12 df-hba 26013 . . . . . . . . . . . 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 2489 . . . . . . . . . . 11  |-  ~H  =  ( BaseSet `  U )
1615, 3imsxmet 25725 . . . . . . . . . 10  |-  ( U  e.  NrmCVec  ->  ( IndMet `  U
)  e.  ( *Met `  ~H )
)
174mopntopon 21068 . . . . . . . . . 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 19925 . . . . . . . . 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 26024 . . . . . . . . . . . 12  |-  ~~>v  =  ( ( ~~> t `  ( MetOpen
`  ( IndMet `  U
) ) )  |`  ( ~H  ^m  NN ) )
2322breqi 4462 . . . . . . . . . . 11  |-  ( F 
~~>v  x  <->  F ( ( ~~> t `  ( MetOpen `  ( IndMet `  U ) ) )  |`  ( ~H  ^m  NN ) ) x )
24 vex 3112 . . . . . . . . . . . 12  |-  x  e. 
_V
2524brres 5290 . . . . . . . . . . 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 903 . . . . . . . . 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 1711 . . . 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 3683 . . 3  |-  ( F  e.  ( ( Cau `  ( IndMet `  U )
)  i^i  ( ~H  ^m  NN ) )  <->  ( F  e.  ( Cau `  ( IndMet `
 U ) )  /\  F  e.  ( ~H  ^m  NN ) ) )
36 df-rex 2813 . . 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 26023 . 2  |-  Cauchy  =  ( ( Cau `  ( IndMet `
 U ) )  i^i  ( ~H  ^m  NN ) )
3937, 38eleq2s 2565 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 1395   E.wex 1613    e. wcel 1819   E.wrex 2808    i^i cin 3470   <.cop 4038   class class class wbr 4456   dom cdm 5008    |` cres 5010   ` cfv 5594  (class class class)co 6296    ^m cmap 7438   NNcn 10556   *Metcxmt 18530   MetOpencmopn 18535  TopOnctopon 19522   ~~> tclm 19854   Caucca 21818   NrmCVeccnv 25604   BaseSetcba 25606   IndMetcims 25611   CHilOLDchlo 25928   ~Hchil 25963    +h cva 25964    .h csm 25965   normhcno 25967   Cauchyccau 25970    ~~>v chli 25971
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588  ax-mulf 9589
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ico 11560  df-seq 12111  df-exp 12170  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-rest 14840  df-topgen 14861  df-psmet 18538  df-xmet 18539  df-met 18540  df-bl 18541  df-mopn 18542  df-fbas 18543  df-fg 18544  df-top 19526  df-bases 19528  df-topon 19529  df-ntr 19648  df-nei 19726  df-lm 19857  df-fil 20473  df-fm 20565  df-flim 20566  df-flf 20567  df-cfil 21820  df-cau 21821  df-cmet 21822  df-grpo 25320  df-gid 25321  df-ginv 25322  df-gdiv 25323  df-ablo 25411  df-vc 25566  df-nv 25612  df-va 25615  df-ba 25616  df-sm 25617  df-0v 25618  df-vs 25619  df-nmcv 25620  df-ims 25621  df-cbn 25906  df-hlo 25929  df-hba 26013  df-hvsub 26015  df-hlim 26016  df-hcau 26017
This theorem is referenced by: (None)
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