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Theorem pclssN 36034
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 3496 . . . . . 6  |-  ( X 
C_  Y  ->  ( Y  C_  y  ->  X  C_  y ) )
213ad2ant2 1016 . . . . 5  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  C_  A )  ->  ( Y  C_  y  ->  X  C_  y ) )
32adantr 463 . . . 4  |-  ( ( ( K  e.  V  /\  X  C_  Y  /\  Y  C_  A )  /\  y  e.  ( PSubSp `  K ) )  -> 
( Y  C_  y  ->  X  C_  y )
)
43ss2rabdv 3567 . . 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 4292 . . 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 994 . . 3  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  C_  A )  ->  K  e.  V )
8 sstr 3497 . . . 4  |-  ( ( X  C_  Y  /\  Y  C_  A )  ->  X  C_  A )
983adant1 1012 . . 3  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  C_  A )  ->  X  C_  A )
10 pclss.a . . . 4  |-  A  =  ( Atoms `  K )
11 eqid 2454 . . . 4  |-  ( PSubSp `  K )  =  (
PSubSp `  K )
12 pclss.c . . . 4  |-  U  =  ( PCl `  K
)
1310, 11, 12pclvalN 36030 . . 3  |-  ( ( K  e.  V  /\  X  C_  A )  -> 
( U `  X
)  =  |^| { y  e.  ( PSubSp `  K
)  |  X  C_  y } )
147, 9, 13syl2anc 659 . 2  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  C_  A )  ->  ( U `  X )  =  |^| { y  e.  ( PSubSp `  K )  |  X  C_  y } )
1510, 11, 12pclvalN 36030 . . 3  |-  ( ( K  e.  V  /\  Y  C_  A )  -> 
( U `  Y
)  =  |^| { y  e.  ( PSubSp `  K
)  |  Y  C_  y } )
16153adant2 1013 . 2  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  C_  A )  ->  ( U `  Y )  =  |^| { y  e.  ( PSubSp `  K )  |  Y  C_  y } )
176, 14, 163sstr4d 3532 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 971    = wceq 1398    e. wcel 1823   {crab 2808    C_ wss 3461   |^|cint 4271   ` cfv 5570   Atomscatm 35404   PSubSpcpsubsp 35636   PClcpclN 36027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-psubsp 35643  df-pclN 36028
This theorem is referenced by:  pclbtwnN  36037  pclunN  36038  pclfinN  36040  pclss2polN  36061  pclfinclN  36090
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