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Theorem cssmre 18914
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 15095: 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 15160. (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 18906 . . . . 5  |-  ( x  e.  C  ->  x  C_  V )
4 selpw 3961 . . . . 5  |-  ( x  e.  ~P V  <->  x  C_  V
)
53, 4sylibr 212 . . . 4  |-  ( x  e.  C  ->  x  e.  ~P V )
65a1i 11 . . 3  |-  ( W  e.  PreHil  ->  ( x  e.  C  ->  x  e.  ~P V ) )
76ssrdv 3447 . 2  |-  ( W  e.  PreHil  ->  C  C_  ~P V )
81, 2css1 18911 . 2  |-  ( W  e.  PreHil  ->  V  e.  C
)
9 intss1 4241 . . . . . . . . . . . 12  |-  ( z  e.  x  ->  |^| x  C_  z )
10 eqid 2402 . . . . . . . . . . . . 13  |-  ( ocv `  W )  =  ( ocv `  W )
1110ocv2ss 18894 . . . . . . . . . . . 12  |-  ( |^| x  C_  z  ->  (
( ocv `  W
) `  z )  C_  ( ( ocv `  W
) `  |^| x ) )
1210ocv2ss 18894 . . . . . . . . . . . 12  |-  ( ( ( ocv `  W
) `  z )  C_  ( ( ocv `  W
) `  |^| x )  ->  ( ( ocv `  W ) `  (
( ocv `  W
) `  |^| x ) )  C_  ( ( ocv `  W ) `  ( ( ocv `  W
) `  z )
) )
139, 11, 123syl 20 . . . . . . . . . . 11  |-  ( z  e.  x  ->  (
( ocv `  W
) `  ( ( ocv `  W ) `  |^| x ) )  C_  ( ( ocv `  W
) `  ( ( ocv `  W ) `  z ) ) )
1413ad2antll 727 . . . . . . . . . 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 )
) )
15 simprl 756 . . . . . . . . . 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 ) ) )
1614, 15sseldd 3442 . . . . . . . . 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 ) ) )
17 simpl2 1001 . . . . . . . . . . 11  |-  ( ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  /\  (
y  e.  ( ( ocv `  W ) `
 ( ( ocv `  W ) `  |^| x ) )  /\  z  e.  x )
)  ->  x  C_  C
)
18 simprr 758 . . . . . . . . . . 11  |-  ( ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  /\  (
y  e.  ( ( ocv `  W ) `
 ( ( ocv `  W ) `  |^| x ) )  /\  z  e.  x )
)  ->  z  e.  x )
1917, 18sseldd 3442 . . . . . . . . . 10  |-  ( ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  /\  (
y  e.  ( ( ocv `  W ) `
 ( ( ocv `  W ) `  |^| x ) )  /\  z  e.  x )
)  ->  z  e.  C )
2010, 2cssi 18905 . . . . . . . . . 10  |-  ( z  e.  C  ->  z  =  ( ( ocv `  W ) `  (
( ocv `  W
) `  z )
) )
2119, 20syl 17 . . . . . . . . 9  |-  ( ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  /\  (
y  e.  ( ( ocv `  W ) `
 ( ( ocv `  W ) `  |^| x ) )  /\  z  e.  x )
)  ->  z  =  ( ( ocv `  W
) `  ( ( ocv `  W ) `  z ) ) )
2216, 21eleqtrrd 2493 . . . . . . . 8  |-  ( ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  /\  (
y  e.  ( ( ocv `  W ) `
 ( ( ocv `  W ) `  |^| x ) )  /\  z  e.  x )
)  ->  y  e.  z )
2322expr 613 . . . . . . 7  |-  ( ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  /\  y  e.  ( ( ocv `  W
) `  ( ( ocv `  W ) `  |^| x ) ) )  ->  ( z  e.  x  ->  y  e.  z ) )
2423alrimiv 1740 . . . . . 6  |-  ( ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  /\  y  e.  ( ( ocv `  W
) `  ( ( ocv `  W ) `  |^| x ) ) )  ->  A. z ( z  e.  x  ->  y  e.  z ) )
25 vex 3061 . . . . . . 7  |-  y  e. 
_V
2625elint 4232 . . . . . 6  |-  ( y  e.  |^| x  <->  A. z
( z  e.  x  ->  y  e.  z ) )
2724, 26sylibr 212 . . . . 5  |-  ( ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  /\  y  e.  ( ( ocv `  W
) `  ( ( ocv `  W ) `  |^| x ) ) )  ->  y  e.  |^| x )
2827ex 432 . . . 4  |-  ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  ->  ( y  e.  ( ( ocv `  W ) `  (
( ocv `  W
) `  |^| x ) )  ->  y  e.  |^| x ) )
2928ssrdv 3447 . . 3  |-  ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  ->  ( ( ocv `  W ) `
 ( ( ocv `  W ) `  |^| x ) )  C_  |^| x )
30 simp1 997 . . . 4  |-  ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  ->  W  e. 
PreHil )
31 intssuni 4249 . . . . . 6  |-  ( x  =/=  (/)  ->  |^| x  C_  U. x )
32313ad2ant3 1020 . . . . 5  |-  ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  ->  |^| x  C_ 
U. x )
33 simp2 998 . . . . . . 7  |-  ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  ->  x  C_  C )
3473ad2ant1 1018 . . . . . . 7  |-  ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  ->  C  C_  ~P V )
3533, 34sstrd 3451 . . . . . 6  |-  ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  ->  x  C_  ~P V )
36 sspwuni 4359 . . . . . 6  |-  ( x 
C_  ~P V  <->  U. x  C_  V )
3735, 36sylib 196 . . . . 5  |-  ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  ->  U. x  C_  V )
3832, 37sstrd 3451 . . . 4  |-  ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  ->  |^| x  C_  V )
391, 2, 10iscss2 18907 . . . 4  |-  ( ( W  e.  PreHil  /\  |^| x  C_  V )  -> 
( |^| x  e.  C  <->  ( ( ocv `  W
) `  ( ( ocv `  W ) `  |^| x ) )  C_  |^| x ) )
4030, 38, 39syl2anc 659 . . 3  |-  ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  ->  ( |^| x  e.  C  <->  ( ( ocv `  W ) `  ( ( ocv `  W
) `  |^| x ) )  C_  |^| x ) )
4129, 40mpbird 232 . 2  |-  ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  ->  |^| x  e.  C )
427, 8, 41ismred 15108 1  |-  ( W  e.  PreHil  ->  C  e.  (Moore `  V ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974   A.wal 1403    = wceq 1405    e. wcel 1842    =/= wne 2598    C_ wss 3413   (/)c0 3737   ~Pcpw 3954   U.cuni 4190   |^|cint 4226   ` cfv 5525   Basecbs 14733  Moorecmre 15088   PreHilcphl 18849   ocvcocv 18881   CSubSpccss 18882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-cnex 9498  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-mulcom 9506  ax-addass 9507  ax-mulass 9508  ax-distr 9509  ax-i2m1 9510  ax-1ne0 9511  ax-1rid 9512  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515  ax-pre-lttri 9516  ax-pre-lttrn 9517  ax-pre-ltadd 9518  ax-pre-mulgt0 9519
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-ord 4824  df-on 4825  df-lim 4826  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-om 6639  df-tpos 6912  df-recs 6999  df-rdg 7033  df-er 7268  df-map 7379  df-en 7475  df-dom 7476  df-sdom 7477  df-pnf 9580  df-mnf 9581  df-xr 9582  df-ltxr 9583  df-le 9584  df-sub 9763  df-neg 9764  df-nn 10497  df-2 10555  df-3 10556  df-4 10557  df-5 10558  df-6 10559  df-7 10560  df-8 10561  df-ndx 14736  df-slot 14737  df-base 14738  df-sets 14739  df-plusg 14814  df-mulr 14815  df-sca 14817  df-vsca 14818  df-ip 14819  df-0g 14948  df-mre 15092  df-mgm 16088  df-sgrp 16127  df-mnd 16137  df-mhm 16182  df-grp 16273  df-ghm 16481  df-mgp 17354  df-ur 17366  df-ring 17412  df-oppr 17484  df-rnghom 17576  df-staf 17706  df-srng 17707  df-lmod 17726  df-lmhm 17880  df-lvec 17961  df-sra 18030  df-rgmod 18031  df-phl 18851  df-ocv 18884  df-css 18885
This theorem is referenced by:  mrccss  18915
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