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Theorem pclssN 33841
Description: Ordering is preserved by subspace closure. (Contributed by NM, 8-Sep-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
pclss.a  |-  A  =  ( Atoms `  K )
pclss.c  |-  U  =  ( PCl `  K
)
Assertion
Ref Expression
pclssN  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  C_  A )  ->  ( U `  X )  C_  ( U `  Y
) )

Proof of Theorem pclssN
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 sstr2 3458 . . . . . 6  |-  ( X 
C_  Y  ->  ( Y  C_  y  ->  X  C_  y ) )
213ad2ant2 1010 . . . . 5  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  C_  A )  ->  ( Y  C_  y  ->  X  C_  y ) )
32adantr 465 . . . 4  |-  ( ( ( K  e.  V  /\  X  C_  Y  /\  Y  C_  A )  /\  y  e.  ( PSubSp `  K ) )  -> 
( Y  C_  y  ->  X  C_  y )
)
43ss2rabdv 3528 . . 3  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  C_  A )  ->  { y  e.  ( PSubSp `  K
)  |  Y  C_  y }  C_  { y  e.  ( PSubSp `  K
)  |  X  C_  y } )
5 intss 4244 . . 3  |-  ( { y  e.  ( PSubSp `  K )  |  Y  C_  y }  C_  { y  e.  ( PSubSp `  K
)  |  X  C_  y }  ->  |^| { y  e.  ( PSubSp `  K
)  |  X  C_  y }  C_  |^| { y  e.  ( PSubSp `  K
)  |  Y  C_  y } )
64, 5syl 16 . 2  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  C_  A )  ->  |^| { y  e.  ( PSubSp `  K
)  |  X  C_  y }  C_  |^| { y  e.  ( PSubSp `  K
)  |  Y  C_  y } )
7 simp1 988 . . 3  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  C_  A )  ->  K  e.  V )
8 sstr 3459 . . . 4  |-  ( ( X  C_  Y  /\  Y  C_  A )  ->  X  C_  A )
983adant1 1006 . . 3  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  C_  A )  ->  X  C_  A )
10 pclss.a . . . 4  |-  A  =  ( Atoms `  K )
11 eqid 2451 . . . 4  |-  ( PSubSp `  K )  =  (
PSubSp `  K )
12 pclss.c . . . 4  |-  U  =  ( PCl `  K
)
1310, 11, 12pclvalN 33837 . . 3  |-  ( ( K  e.  V  /\  X  C_  A )  -> 
( U `  X
)  =  |^| { y  e.  ( PSubSp `  K
)  |  X  C_  y } )
147, 9, 13syl2anc 661 . 2  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  C_  A )  ->  ( U `  X )  =  |^| { y  e.  ( PSubSp `  K )  |  X  C_  y } )
1510, 11, 12pclvalN 33837 . . 3  |-  ( ( K  e.  V  /\  Y  C_  A )  -> 
( U `  Y
)  =  |^| { y  e.  ( PSubSp `  K
)  |  Y  C_  y } )
16153adant2 1007 . 2  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  C_  A )  ->  ( U `  Y )  =  |^| { y  e.  ( PSubSp `  K )  |  Y  C_  y } )
176, 14, 163sstr4d 3494 1  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  C_  A )  ->  ( U `  X )  C_  ( U `  Y
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1370    e. wcel 1758   {crab 2797    C_ wss 3423   |^|cint 4223   ` cfv 5513   Atomscatm 33211   PSubSpcpsubsp 33443   PClcpclN 33834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-int 4224  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-ov 6190  df-psubsp 33450  df-pclN 33835
This theorem is referenced by:  pclbtwnN  33844  pclunN  33845  pclfinN  33847  pclss2polN  33868  pclfinclN  33897
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