Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pclssN Unicode version

Theorem pclssN 30083
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 3186 . . . . . 6  |-  ( X 
C_  Y  ->  ( Y  C_  y  ->  X  C_  y ) )
213ad2ant2 977 . . . . 5  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  C_  A )  ->  ( Y  C_  y  ->  X  C_  y ) )
32adantr 451 . . . 4  |-  ( ( ( K  e.  V  /\  X  C_  Y  /\  Y  C_  A )  /\  y  e.  ( PSubSp `  K ) )  -> 
( Y  C_  y  ->  X  C_  y )
)
43ss2rabdv 3254 . . 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 3883 . . 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 15 . 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 955 . . 3  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  C_  A )  ->  K  e.  V )
8 sstr 3187 . . . 4  |-  ( ( X  C_  Y  /\  Y  C_  A )  ->  X  C_  A )
983adant1 973 . . 3  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  C_  A )  ->  X  C_  A )
10 pclss.a . . . 4  |-  A  =  ( Atoms `  K )
11 eqid 2283 . . . 4  |-  ( PSubSp `  K )  =  (
PSubSp `  K )
12 pclss.c . . . 4  |-  U  =  ( PCl `  K
)
1310, 11, 12pclvalN 30079 . . 3  |-  ( ( K  e.  V  /\  X  C_  A )  -> 
( U `  X
)  =  |^| { y  e.  ( PSubSp `  K
)  |  X  C_  y } )
147, 9, 13syl2anc 642 . 2  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  C_  A )  ->  ( U `  X )  =  |^| { y  e.  ( PSubSp `  K )  |  X  C_  y } )
1510, 11, 12pclvalN 30079 . . 3  |-  ( ( K  e.  V  /\  Y  C_  A )  -> 
( U `  Y
)  =  |^| { y  e.  ( PSubSp `  K
)  |  Y  C_  y } )
16153adant2 974 . 2  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  C_  A )  ->  ( U `  Y )  =  |^| { y  e.  ( PSubSp `  K )  |  Y  C_  y } )
176, 14, 163sstr4d 3221 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 934    = wceq 1623    e. wcel 1684   {crab 2547    C_ wss 3152   |^|cint 3862   ` cfv 5255   Atomscatm 29453   PSubSpcpsubsp 29685   PClcpclN 30076
This theorem is referenced by:  pclbtwnN  30086  pclunN  30087  pclfinN  30089  pclss2polN  30110  pclfinclN  30139
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-psubsp 29692  df-pclN 30077
  Copyright terms: Public domain W3C validator