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Theorem pclssN 33371
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 3414 . . . . . 6  |-  ( X 
C_  Y  ->  ( Y  C_  y  ->  X  C_  y ) )
213ad2ant2 1027 . . . . 5  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  C_  A )  ->  ( Y  C_  y  ->  X  C_  y ) )
32adantr 466 . . . 4  |-  ( ( ( K  e.  V  /\  X  C_  Y  /\  Y  C_  A )  /\  y  e.  ( PSubSp `  K ) )  -> 
( Y  C_  y  ->  X  C_  y )
)
43ss2rabdv 3485 . . 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 4219 . . 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 17 . 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 1005 . . 3  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  C_  A )  ->  K  e.  V )
8 sstr 3415 . . . 4  |-  ( ( X  C_  Y  /\  Y  C_  A )  ->  X  C_  A )
983adant1 1023 . . 3  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  C_  A )  ->  X  C_  A )
10 pclss.a . . . 4  |-  A  =  ( Atoms `  K )
11 eqid 2428 . . . 4  |-  ( PSubSp `  K )  =  (
PSubSp `  K )
12 pclss.c . . . 4  |-  U  =  ( PCl `  K
)
1310, 11, 12pclvalN 33367 . . 3  |-  ( ( K  e.  V  /\  X  C_  A )  -> 
( U `  X
)  =  |^| { y  e.  ( PSubSp `  K
)  |  X  C_  y } )
147, 9, 13syl2anc 665 . 2  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  C_  A )  ->  ( U `  X )  =  |^| { y  e.  ( PSubSp `  K )  |  X  C_  y } )
1510, 11, 12pclvalN 33367 . . 3  |-  ( ( K  e.  V  /\  Y  C_  A )  -> 
( U `  Y
)  =  |^| { y  e.  ( PSubSp `  K
)  |  Y  C_  y } )
16153adant2 1024 . 2  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  C_  A )  ->  ( U `  Y )  =  |^| { y  e.  ( PSubSp `  K )  |  Y  C_  y } )
176, 14, 163sstr4d 3450 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 982    = wceq 1437    e. wcel 1872   {crab 2718    C_ wss 3379   |^|cint 4198   ` cfv 5544   Atomscatm 32741   PSubSpcpsubsp 32973   PClcpclN 33364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-int 4199  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-id 4711  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-ov 6252  df-psubsp 32980  df-pclN 33365
This theorem is referenced by:  pclbtwnN  33374  pclunN  33375  pclfinN  33377  pclss2polN  33398  pclfinclN  33427
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