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Theorem pclssN 30693
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 3357 . . . . . 6  |-  ( X 
C_  Y  ->  ( Y  C_  y  ->  X  C_  y ) )
213ad2ant2 980 . . . . 5  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  C_  A )  ->  ( Y  C_  y  ->  X  C_  y ) )
32adantr 453 . . . 4  |-  ( ( ( K  e.  V  /\  X  C_  Y  /\  Y  C_  A )  /\  y  e.  ( PSubSp `  K ) )  -> 
( Y  C_  y  ->  X  C_  y )
)
43ss2rabdv 3426 . . 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 4073 . . 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 958 . . 3  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  C_  A )  ->  K  e.  V )
8 sstr 3358 . . . 4  |-  ( ( X  C_  Y  /\  Y  C_  A )  ->  X  C_  A )
983adant1 976 . . 3  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  C_  A )  ->  X  C_  A )
10 pclss.a . . . 4  |-  A  =  ( Atoms `  K )
11 eqid 2438 . . . 4  |-  ( PSubSp `  K )  =  (
PSubSp `  K )
12 pclss.c . . . 4  |-  U  =  ( PCl `  K
)
1310, 11, 12pclvalN 30689 . . 3  |-  ( ( K  e.  V  /\  X  C_  A )  -> 
( U `  X
)  =  |^| { y  e.  ( PSubSp `  K
)  |  X  C_  y } )
147, 9, 13syl2anc 644 . 2  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  C_  A )  ->  ( U `  X )  =  |^| { y  e.  ( PSubSp `  K )  |  X  C_  y } )
1510, 11, 12pclvalN 30689 . . 3  |-  ( ( K  e.  V  /\  Y  C_  A )  -> 
( U `  Y
)  =  |^| { y  e.  ( PSubSp `  K
)  |  Y  C_  y } )
16153adant2 977 . 2  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  C_  A )  ->  ( U `  Y )  =  |^| { y  e.  ( PSubSp `  K )  |  Y  C_  y } )
176, 14, 163sstr4d 3393 1  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  C_  A )  ->  ( U `  X )  C_  ( U `  Y
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 937    = wceq 1653    e. wcel 1726   {crab 2711    C_ wss 3322   |^|cint 4052   ` cfv 5456   Atomscatm 30063   PSubSpcpsubsp 30295   PClcpclN 30686
This theorem is referenced by:  pclbtwnN  30696  pclunN  30697  pclfinN  30699  pclss2polN  30720  pclfinclN  30749
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-psubsp 30302  df-pclN 30687
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