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Theorem cssmre 16609
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 13507: 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 13572. (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 16601 . . . . 5  |-  ( x  e.  C  ->  x  C_  V )
4 vex 2804 . . . . . 6  |-  x  e. 
_V
54elpw 3644 . . . . 5  |-  ( x  e.  ~P V  <->  x  C_  V
)
63, 5sylibr 203 . . . 4  |-  ( x  e.  C  ->  x  e.  ~P V )
76a1i 10 . . 3  |-  ( W  e.  PreHil  ->  ( x  e.  C  ->  x  e.  ~P V ) )
87ssrdv 3198 . 2  |-  ( W  e.  PreHil  ->  C  C_  ~P V )
91, 2css1 16606 . 2  |-  ( W  e.  PreHil  ->  V  e.  C
)
10 intss1 3893 . . . . . . . . . . . 12  |-  ( z  e.  x  ->  |^| x  C_  z )
11 eqid 2296 . . . . . . . . . . . . 13  |-  ( ocv `  W )  =  ( ocv `  W )
1211ocv2ss 16589 . . . . . . . . . . . 12  |-  ( |^| x  C_  z  ->  (
( ocv `  W
) `  z )  C_  ( ( ocv `  W
) `  |^| x ) )
1311ocv2ss 16589 . . . . . . . . . . . 12  |-  ( ( ( ocv `  W
) `  z )  C_  ( ( ocv `  W
) `  |^| x )  ->  ( ( ocv `  W ) `  (
( ocv `  W
) `  |^| x ) )  C_  ( ( ocv `  W ) `  ( ( ocv `  W
) `  z )
) )
1410, 12, 133syl 18 . . . . . . . . . . 11  |-  ( z  e.  x  ->  (
( ocv `  W
) `  ( ( ocv `  W ) `  |^| x ) )  C_  ( ( ocv `  W
) `  ( ( ocv `  W ) `  z ) ) )
1514ad2antll 709 . . . . . . . . . 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 732 . . . . . . . . . 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 3194 . . . . . . . . 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 959 . . . . . . . . . . 11  |-  ( ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  /\  (
y  e.  ( ( ocv `  W ) `
 ( ( ocv `  W ) `  |^| x ) )  /\  z  e.  x )
)  ->  x  C_  C
)
19 simprr 733 . . . . . . . . . . 11  |-  ( ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  /\  (
y  e.  ( ( ocv `  W ) `
 ( ( ocv `  W ) `  |^| x ) )  /\  z  e.  x )
)  ->  z  e.  x )
2018, 19sseldd 3194 . . . . . . . . . 10  |-  ( ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  /\  (
y  e.  ( ( ocv `  W ) `
 ( ( ocv `  W ) `  |^| x ) )  /\  z  e.  x )
)  ->  z  e.  C )
2111, 2cssi 16600 . . . . . . . . . 10  |-  ( z  e.  C  ->  z  =  ( ( ocv `  W ) `  (
( ocv `  W
) `  z )
) )
2220, 21syl 15 . . . . . . . . 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 2373 . . . . . . . 8  |-  ( ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  /\  (
y  e.  ( ( ocv `  W ) `
 ( ( ocv `  W ) `  |^| x ) )  /\  z  e.  x )
)  ->  y  e.  z )
2423expr 598 . . . . . . 7  |-  ( ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  /\  y  e.  ( ( ocv `  W
) `  ( ( ocv `  W ) `  |^| x ) ) )  ->  ( z  e.  x  ->  y  e.  z ) )
2524alrimiv 1621 . . . . . 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 2804 . . . . . . 7  |-  y  e. 
_V
2726elint 3884 . . . . . 6  |-  ( y  e.  |^| x  <->  A. z
( z  e.  x  ->  y  e.  z ) )
2825, 27sylibr 203 . . . . 5  |-  ( ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  /\  y  e.  ( ( ocv `  W
) `  ( ( ocv `  W ) `  |^| x ) ) )  ->  y  e.  |^| x )
2928ex 423 . . . 4  |-  ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  ->  ( y  e.  ( ( ocv `  W ) `  (
( ocv `  W
) `  |^| x ) )  ->  y  e.  |^| x ) )
3029ssrdv 3198 . . 3  |-  ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  ->  ( ( ocv `  W ) `
 ( ( ocv `  W ) `  |^| x ) )  C_  |^| x )
31 simp1 955 . . . 4  |-  ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  ->  W  e. 
PreHil )
32 intssuni 3900 . . . . . 6  |-  ( x  =/=  (/)  ->  |^| x  C_  U. x )
33323ad2ant3 978 . . . . 5  |-  ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  ->  |^| x  C_ 
U. x )
34 simp2 956 . . . . . . 7  |-  ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  ->  x  C_  C )
3583ad2ant1 976 . . . . . . 7  |-  ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  ->  C  C_  ~P V )
3634, 35sstrd 3202 . . . . . 6  |-  ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  ->  x  C_  ~P V )
37 sspwuni 4003 . . . . . 6  |-  ( x 
C_  ~P V  <->  U. x  C_  V )
3836, 37sylib 188 . . . . 5  |-  ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  ->  U. x  C_  V )
3933, 38sstrd 3202 . . . 4  |-  ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  ->  |^| x  C_  V )
401, 2, 11iscss2 16602 . . . 4  |-  ( ( W  e.  PreHil  /\  |^| x  C_  V )  -> 
( |^| x  e.  C  <->  ( ( ocv `  W
) `  ( ( ocv `  W ) `  |^| x ) )  C_  |^| x ) )
4131, 39, 40syl2anc 642 . . 3  |-  ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  ->  ( |^| x  e.  C  <->  ( ( ocv `  W ) `  ( ( ocv `  W
) `  |^| x ) )  C_  |^| x ) )
4230, 41mpbird 223 . 2  |-  ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  ->  |^| x  e.  C )
438, 9, 42ismred 13520 1  |-  ( W  e.  PreHil  ->  C  e.  (Moore `  V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   A.wal 1530    = wceq 1632    e. wcel 1696    =/= wne 2459    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   U.cuni 3843   |^|cint 3878   ` cfv 5271   Basecbs 13164  Moorecmre 13500   PreHilcphl 16544   ocvcocv 16576   CSubSpccss 16577
This theorem is referenced by:  mrccss  16610
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
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-int 3879  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-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-0g 13420  df-mre 13504  df-mnd 14383  df-mhm 14431  df-grp 14505  df-ghm 14697  df-mgp 15342  df-rng 15356  df-ur 15358  df-oppr 15421  df-rnghom 15512  df-staf 15626  df-srng 15627  df-lmod 15645  df-lmhm 15795  df-lvec 15872  df-sra 15941  df-rgmod 15942  df-phl 16546  df-ocv 16579  df-css 16580
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