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Theorem cssmre 16912
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 13806: 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 13871. (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 16904 . . . . 5  |-  ( x  e.  C  ->  x  C_  V )
4 vex 2951 . . . . . 6  |-  x  e. 
_V
54elpw 3797 . . . . 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 3346 . 2  |-  ( W  e.  PreHil  ->  C  C_  ~P V )
91, 2css1 16909 . 2  |-  ( W  e.  PreHil  ->  V  e.  C
)
10 intss1 4057 . . . . . . . . . . . 12  |-  ( z  e.  x  ->  |^| x  C_  z )
11 eqid 2435 . . . . . . . . . . . . 13  |-  ( ocv `  W )  =  ( ocv `  W )
1211ocv2ss 16892 . . . . . . . . . . . 12  |-  ( |^| x  C_  z  ->  (
( ocv `  W
) `  z )  C_  ( ( ocv `  W
) `  |^| x ) )
1311ocv2ss 16892 . . . . . . . . . . . 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 3341 . . . . . . . . 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 3341 . . . . . . . . . 10  |-  ( ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  /\  (
y  e.  ( ( ocv `  W ) `
 ( ( ocv `  W ) `  |^| x ) )  /\  z  e.  x )
)  ->  z  e.  C )
2111, 2cssi 16903 . . . . . . . . . 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 2512 . . . . . . . 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 1641 . . . . . 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 2951 . . . . . . 7  |-  y  e. 
_V
2726elint 4048 . . . . . 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 3346 . . 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 4064 . . . . . 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 3350 . . . . . 6  |-  ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  ->  x  C_  ~P V )
37 sspwuni 4168 . . . . . 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 3350 . . . 4  |-  ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  ->  |^| x  C_  V )
401, 2, 11iscss2 16905 . . . 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 13819 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 1549    = wceq 1652    e. wcel 1725    =/= wne 2598    C_ wss 3312   (/)c0 3620   ~Pcpw 3791   U.cuni 4007   |^|cint 4042   ` cfv 5446   Basecbs 13461  Moorecmre 13799   PreHilcphl 16847   ocvcocv 16879   CSubSpccss 16880
This theorem is referenced by:  mrccss  16913
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-tpos 6471  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-plusg 13534  df-mulr 13535  df-sca 13537  df-vsca 13538  df-0g 13719  df-mre 13803  df-mnd 14682  df-mhm 14730  df-grp 14804  df-ghm 14996  df-mgp 15641  df-rng 15655  df-ur 15657  df-oppr 15720  df-rnghom 15811  df-staf 15925  df-srng 15926  df-lmod 15944  df-lmhm 16090  df-lvec 16167  df-sra 16236  df-rgmod 16237  df-phl 16849  df-ocv 16882  df-css 16883
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