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Theorem cssmre 16843
Description: The closed subspaces of a pre-Hilbert space are a Moore system. Unlike many of our other examples of closure systems, this one is not usually an algebraic closure system df-acs 13741: consider the Hilbert space of sequences  NN --> RR with convergent sum; the subspace of all sequences with finite support is the classic example of a non-closed subspace, but for every finite set of sequences of finite support, there is a finite-dimensional (and hence closed) subspace containing all of the sequences, so if closed subspaces were an algebraic closure system this would violate acsfiel 13806. (Contributed by Mario Carneiro, 13-Oct-2015.)
Hypotheses
Ref Expression
cssmre.v  |-  V  =  ( Base `  W
)
cssmre.c  |-  C  =  ( CSubSp `  W )
Assertion
Ref Expression
cssmre  |-  ( W  e.  PreHil  ->  C  e.  (Moore `  V ) )

Proof of Theorem cssmre
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cssmre.v . . . . . 6  |-  V  =  ( Base `  W
)
2 cssmre.c . . . . . 6  |-  C  =  ( CSubSp `  W )
31, 2cssss 16835 . . . . 5  |-  ( x  e.  C  ->  x  C_  V )
4 vex 2902 . . . . . 6  |-  x  e. 
_V
54elpw 3748 . . . . 5  |-  ( x  e.  ~P V  <->  x  C_  V
)
63, 5sylibr 204 . . . 4  |-  ( x  e.  C  ->  x  e.  ~P V )
76a1i 11 . . 3  |-  ( W  e.  PreHil  ->  ( x  e.  C  ->  x  e.  ~P V ) )
87ssrdv 3297 . 2  |-  ( W  e.  PreHil  ->  C  C_  ~P V )
91, 2css1 16840 . 2  |-  ( W  e.  PreHil  ->  V  e.  C
)
10 intss1 4007 . . . . . . . . . . . 12  |-  ( z  e.  x  ->  |^| x  C_  z )
11 eqid 2387 . . . . . . . . . . . . 13  |-  ( ocv `  W )  =  ( ocv `  W )
1211ocv2ss 16823 . . . . . . . . . . . 12  |-  ( |^| x  C_  z  ->  (
( ocv `  W
) `  z )  C_  ( ( ocv `  W
) `  |^| x ) )
1311ocv2ss 16823 . . . . . . . . . . . 12  |-  ( ( ( ocv `  W
) `  z )  C_  ( ( ocv `  W
) `  |^| x )  ->  ( ( ocv `  W ) `  (
( ocv `  W
) `  |^| x ) )  C_  ( ( ocv `  W ) `  ( ( ocv `  W
) `  z )
) )
1410, 12, 133syl 19 . . . . . . . . . . 11  |-  ( z  e.  x  ->  (
( ocv `  W
) `  ( ( ocv `  W ) `  |^| x ) )  C_  ( ( ocv `  W
) `  ( ( ocv `  W ) `  z ) ) )
1514ad2antll 710 . . . . . . . . . 10  |-  ( ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  /\  (
y  e.  ( ( ocv `  W ) `
 ( ( ocv `  W ) `  |^| x ) )  /\  z  e.  x )
)  ->  ( ( ocv `  W ) `  ( ( ocv `  W
) `  |^| x ) )  C_  ( ( ocv `  W ) `  ( ( ocv `  W
) `  z )
) )
16 simprl 733 . . . . . . . . . 10  |-  ( ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  /\  (
y  e.  ( ( ocv `  W ) `
 ( ( ocv `  W ) `  |^| x ) )  /\  z  e.  x )
)  ->  y  e.  ( ( ocv `  W
) `  ( ( ocv `  W ) `  |^| x ) ) )
1715, 16sseldd 3292 . . . . . . . . 9  |-  ( ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  /\  (
y  e.  ( ( ocv `  W ) `
 ( ( ocv `  W ) `  |^| x ) )  /\  z  e.  x )
)  ->  y  e.  ( ( ocv `  W
) `  ( ( ocv `  W ) `  z ) ) )
18 simpl2 961 . . . . . . . . . . 11  |-  ( ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  /\  (
y  e.  ( ( ocv `  W ) `
 ( ( ocv `  W ) `  |^| x ) )  /\  z  e.  x )
)  ->  x  C_  C
)
19 simprr 734 . . . . . . . . . . 11  |-  ( ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  /\  (
y  e.  ( ( ocv `  W ) `
 ( ( ocv `  W ) `  |^| x ) )  /\  z  e.  x )
)  ->  z  e.  x )
2018, 19sseldd 3292 . . . . . . . . . 10  |-  ( ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  /\  (
y  e.  ( ( ocv `  W ) `
 ( ( ocv `  W ) `  |^| x ) )  /\  z  e.  x )
)  ->  z  e.  C )
2111, 2cssi 16834 . . . . . . . . . 10  |-  ( z  e.  C  ->  z  =  ( ( ocv `  W ) `  (
( ocv `  W
) `  z )
) )
2220, 21syl 16 . . . . . . . . 9  |-  ( ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  /\  (
y  e.  ( ( ocv `  W ) `
 ( ( ocv `  W ) `  |^| x ) )  /\  z  e.  x )
)  ->  z  =  ( ( ocv `  W
) `  ( ( ocv `  W ) `  z ) ) )
2317, 22eleqtrrd 2464 . . . . . . . 8  |-  ( ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  /\  (
y  e.  ( ( ocv `  W ) `
 ( ( ocv `  W ) `  |^| x ) )  /\  z  e.  x )
)  ->  y  e.  z )
2423expr 599 . . . . . . 7  |-  ( ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  /\  y  e.  ( ( ocv `  W
) `  ( ( ocv `  W ) `  |^| x ) ) )  ->  ( z  e.  x  ->  y  e.  z ) )
2524alrimiv 1638 . . . . . 6  |-  ( ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  /\  y  e.  ( ( ocv `  W
) `  ( ( ocv `  W ) `  |^| x ) ) )  ->  A. z ( z  e.  x  ->  y  e.  z ) )
26 vex 2902 . . . . . . 7  |-  y  e. 
_V
2726elint 3998 . . . . . 6  |-  ( y  e.  |^| x  <->  A. z
( z  e.  x  ->  y  e.  z ) )
2825, 27sylibr 204 . . . . 5  |-  ( ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  /\  y  e.  ( ( ocv `  W
) `  ( ( ocv `  W ) `  |^| x ) ) )  ->  y  e.  |^| x )
2928ex 424 . . . 4  |-  ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  ->  ( y  e.  ( ( ocv `  W ) `  (
( ocv `  W
) `  |^| x ) )  ->  y  e.  |^| x ) )
3029ssrdv 3297 . . 3  |-  ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  ->  ( ( ocv `  W ) `
 ( ( ocv `  W ) `  |^| x ) )  C_  |^| x )
31 simp1 957 . . . 4  |-  ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  ->  W  e. 
PreHil )
32 intssuni 4014 . . . . . 6  |-  ( x  =/=  (/)  ->  |^| x  C_  U. x )
33323ad2ant3 980 . . . . 5  |-  ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  ->  |^| x  C_ 
U. x )
34 simp2 958 . . . . . . 7  |-  ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  ->  x  C_  C )
3583ad2ant1 978 . . . . . . 7  |-  ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  ->  C  C_  ~P V )
3634, 35sstrd 3301 . . . . . 6  |-  ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  ->  x  C_  ~P V )
37 sspwuni 4117 . . . . . 6  |-  ( x 
C_  ~P V  <->  U. x  C_  V )
3836, 37sylib 189 . . . . 5  |-  ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  ->  U. x  C_  V )
3933, 38sstrd 3301 . . . 4  |-  ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  ->  |^| x  C_  V )
401, 2, 11iscss2 16836 . . . 4  |-  ( ( W  e.  PreHil  /\  |^| x  C_  V )  -> 
( |^| x  e.  C  <->  ( ( ocv `  W
) `  ( ( ocv `  W ) `  |^| x ) )  C_  |^| x ) )
4131, 39, 40syl2anc 643 . . 3  |-  ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  ->  ( |^| x  e.  C  <->  ( ( ocv `  W ) `  ( ( ocv `  W
) `  |^| x ) )  C_  |^| x ) )
4230, 41mpbird 224 . 2  |-  ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  ->  |^| x  e.  C )
438, 9, 42ismred 13754 1  |-  ( W  e.  PreHil  ->  C  e.  (Moore `  V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   A.wal 1546    = wceq 1649    e. wcel 1717    =/= wne 2550    C_ wss 3263   (/)c0 3571   ~Pcpw 3742   U.cuni 3957   |^|cint 3992   ` cfv 5394   Basecbs 13396  Moorecmre 13734   PreHilcphl 16778   ocvcocv 16810   CSubSpccss 16811
This theorem is referenced by:  mrccss  16844
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-tpos 6415  df-riota 6485  df-recs 6569  df-rdg 6604  df-er 6841  df-map 6956  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-plusg 13469  df-mulr 13470  df-sca 13472  df-vsca 13473  df-0g 13654  df-mre 13738  df-mnd 14617  df-mhm 14665  df-grp 14739  df-ghm 14931  df-mgp 15576  df-rng 15590  df-ur 15592  df-oppr 15655  df-rnghom 15746  df-staf 15860  df-srng 15861  df-lmod 15879  df-lmhm 16025  df-lvec 16102  df-sra 16171  df-rgmod 16172  df-phl 16780  df-ocv 16813  df-css 16814
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