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Theorem cssmre 18484
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 14833: 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 14898. (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 18476 . . . . 5  |-  ( x  e.  C  ->  x  C_  V )
4 selpw 4010 . . . . 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 3503 . 2  |-  ( W  e.  PreHil  ->  C  C_  ~P V )
81, 2css1 18481 . 2  |-  ( W  e.  PreHil  ->  V  e.  C
)
9 intss1 4290 . . . . . . . . . . . 12  |-  ( z  e.  x  ->  |^| x  C_  z )
10 eqid 2460 . . . . . . . . . . . . 13  |-  ( ocv `  W )  =  ( ocv `  W )
1110ocv2ss 18464 . . . . . . . . . . . 12  |-  ( |^| x  C_  z  ->  (
( ocv `  W
) `  z )  C_  ( ( ocv `  W
) `  |^| x ) )
1210ocv2ss 18464 . . . . . . . . . . . 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 728 . . . . . . . . . 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 755 . . . . . . . . . 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 3498 . . . . . . . . 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 995 . . . . . . . . . . 11  |-  ( ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  /\  (
y  e.  ( ( ocv `  W ) `
 ( ( ocv `  W ) `  |^| x ) )  /\  z  e.  x )
)  ->  x  C_  C
)
18 simprr 756 . . . . . . . . . . 11  |-  ( ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  /\  (
y  e.  ( ( ocv `  W ) `
 ( ( ocv `  W ) `  |^| x ) )  /\  z  e.  x )
)  ->  z  e.  x )
1917, 18sseldd 3498 . . . . . . . . . 10  |-  ( ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  /\  (
y  e.  ( ( ocv `  W ) `
 ( ( ocv `  W ) `  |^| x ) )  /\  z  e.  x )
)  ->  z  e.  C )
2010, 2cssi 18475 . . . . . . . . . 10  |-  ( z  e.  C  ->  z  =  ( ( ocv `  W ) `  (
( ocv `  W
) `  z )
) )
2119, 20syl 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 ) ) )
2216, 21eleqtrrd 2551 . . . . . . . 8  |-  ( ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  /\  (
y  e.  ( ( ocv `  W ) `
 ( ( ocv `  W ) `  |^| x ) )  /\  z  e.  x )
)  ->  y  e.  z )
2322expr 615 . . . . . . 7  |-  ( ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  /\  y  e.  ( ( ocv `  W
) `  ( ( ocv `  W ) `  |^| x ) ) )  ->  ( z  e.  x  ->  y  e.  z ) )
2423alrimiv 1690 . . . . . 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 3109 . . . . . . 7  |-  y  e. 
_V
2625elint 4281 . . . . . 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 434 . . . 4  |-  ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  ->  ( y  e.  ( ( ocv `  W ) `  (
( ocv `  W
) `  |^| x ) )  ->  y  e.  |^| x ) )
2928ssrdv 3503 . . 3  |-  ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  ->  ( ( ocv `  W ) `
 ( ( ocv `  W ) `  |^| x ) )  C_  |^| x )
30 simp1 991 . . . 4  |-  ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  ->  W  e. 
PreHil )
31 intssuni 4297 . . . . . 6  |-  ( x  =/=  (/)  ->  |^| x  C_  U. x )
32313ad2ant3 1014 . . . . 5  |-  ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  ->  |^| x  C_ 
U. x )
33 simp2 992 . . . . . . 7  |-  ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  ->  x  C_  C )
3473ad2ant1 1012 . . . . . . 7  |-  ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  ->  C  C_  ~P V )
3533, 34sstrd 3507 . . . . . 6  |-  ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  ->  x  C_  ~P V )
36 sspwuni 4404 . . . . . 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 3507 . . . 4  |-  ( ( W  e.  PreHil  /\  x  C_  C  /\  x  =/=  (/) )  ->  |^| x  C_  V )
391, 2, 10iscss2 18477 . . . 4  |-  ( ( W  e.  PreHil  /\  |^| x  C_  V )  -> 
( |^| x  e.  C  <->  ( ( ocv `  W
) `  ( ( ocv `  W ) `  |^| x ) )  C_  |^| x ) )
4030, 38, 39syl2anc 661 . . 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 14846 1  |-  ( W  e.  PreHil  ->  C  e.  (Moore `  V ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968   A.wal 1372    = wceq 1374    e. wcel 1762    =/= wne 2655    C_ wss 3469   (/)c0 3778   ~Pcpw 4003   U.cuni 4238   |^|cint 4275   ` cfv 5579   Basecbs 14479  Moorecmre 14826   PreHilcphl 18419   ocvcocv 18451   CSubSpccss 18452
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-tpos 6945  df-recs 7032  df-rdg 7066  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-plusg 14557  df-mulr 14558  df-sca 14560  df-vsca 14561  df-ip 14562  df-0g 14686  df-mre 14830  df-mnd 15721  df-mhm 15770  df-grp 15851  df-ghm 16053  df-mgp 16925  df-ur 16937  df-rng 16981  df-oppr 17049  df-rnghom 17141  df-staf 17270  df-srng 17271  df-lmod 17290  df-lmhm 17444  df-lvec 17525  df-sra 17594  df-rgmod 17595  df-phl 18421  df-ocv 18454  df-css 18455
This theorem is referenced by:  mrccss  18485
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