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Theorem psubclinN 30759
Description: The intersection of two closed subspaces is closed. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
Hypothesis
Ref Expression
psubclin.c  |-  C  =  ( PSubCl `  K )
Assertion
Ref Expression
psubclinN  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  ( X  i^i  Y
)  e.  C )

Proof of Theorem psubclinN
StepHypRef Expression
1 simp1 955 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  K  e.  HL )
2 hlclat 30170 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  CLat )
323ad2ant1 976 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  K  e.  CLat )
4 eqid 2296 . . . . . . . 8  |-  ( Atoms `  K )  =  (
Atoms `  K )
5 psubclin.c . . . . . . . 8  |-  C  =  ( PSubCl `  K )
64, 5psubclssatN 30752 . . . . . . 7  |-  ( ( K  e.  HL  /\  X  e.  C )  ->  X  C_  ( Atoms `  K ) )
763adant3 975 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  X  C_  ( Atoms `  K ) )
8 eqid 2296 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
98, 4atssbase 30102 . . . . . 6  |-  ( Atoms `  K )  C_  ( Base `  K )
107, 9syl6ss 3204 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  X  C_  ( Base `  K ) )
11 eqid 2296 . . . . . 6  |-  ( lub `  K )  =  ( lub `  K )
128, 11clatlubcl 14233 . . . . 5  |-  ( ( K  e.  CLat  /\  X  C_  ( Base `  K
) )  ->  (
( lub `  K
) `  X )  e.  ( Base `  K
) )
133, 10, 12syl2anc 642 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  ( ( lub `  K
) `  X )  e.  ( Base `  K
) )
144, 5psubclssatN 30752 . . . . . . 7  |-  ( ( K  e.  HL  /\  Y  e.  C )  ->  Y  C_  ( Atoms `  K ) )
15143adant2 974 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  Y  C_  ( Atoms `  K ) )
1615, 9syl6ss 3204 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  Y  C_  ( Base `  K ) )
178, 11clatlubcl 14233 . . . . 5  |-  ( ( K  e.  CLat  /\  Y  C_  ( Base `  K
) )  ->  (
( lub `  K
) `  Y )  e.  ( Base `  K
) )
183, 16, 17syl2anc 642 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  ( ( lub `  K
) `  Y )  e.  ( Base `  K
) )
19 eqid 2296 . . . . 5  |-  ( meet `  K )  =  (
meet `  K )
20 eqid 2296 . . . . 5  |-  ( pmap `  K )  =  (
pmap `  K )
218, 19, 4, 20pmapmeet 30584 . . . 4  |-  ( ( K  e.  HL  /\  ( ( lub `  K
) `  X )  e.  ( Base `  K
)  /\  ( ( lub `  K ) `  Y )  e.  (
Base `  K )
)  ->  ( ( pmap `  K ) `  ( ( ( lub `  K ) `  X
) ( meet `  K
) ( ( lub `  K ) `  Y
) ) )  =  ( ( ( pmap `  K ) `  (
( lub `  K
) `  X )
)  i^i  ( ( pmap `  K ) `  ( ( lub `  K
) `  Y )
) ) )
221, 13, 18, 21syl3anc 1182 . . 3  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  ( ( pmap `  K
) `  ( (
( lub `  K
) `  X )
( meet `  K )
( ( lub `  K
) `  Y )
) )  =  ( ( ( pmap `  K
) `  ( ( lub `  K ) `  X ) )  i^i  ( ( pmap `  K
) `  ( ( lub `  K ) `  Y ) ) ) )
2311, 20, 5pmapidclN 30753 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  C )  ->  ( ( pmap `  K
) `  ( ( lub `  K ) `  X ) )  =  X )
24233adant3 975 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  ( ( pmap `  K
) `  ( ( lub `  K ) `  X ) )  =  X )
2511, 20, 5pmapidclN 30753 . . . . 5  |-  ( ( K  e.  HL  /\  Y  e.  C )  ->  ( ( pmap `  K
) `  ( ( lub `  K ) `  Y ) )  =  Y )
26253adant2 974 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  ( ( pmap `  K
) `  ( ( lub `  K ) `  Y ) )  =  Y )
2724, 26ineq12d 3384 . . 3  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  ( ( ( pmap `  K ) `  (
( lub `  K
) `  X )
)  i^i  ( ( pmap `  K ) `  ( ( lub `  K
) `  Y )
) )  =  ( X  i^i  Y ) )
2822, 27eqtrd 2328 . 2  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  ( ( pmap `  K
) `  ( (
( lub `  K
) `  X )
( meet `  K )
( ( lub `  K
) `  Y )
) )  =  ( X  i^i  Y ) )
29 hllat 30175 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
30293ad2ant1 976 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  K  e.  Lat )
318, 19latmcl 14173 . . . 4  |-  ( ( K  e.  Lat  /\  ( ( lub `  K
) `  X )  e.  ( Base `  K
)  /\  ( ( lub `  K ) `  Y )  e.  (
Base `  K )
)  ->  ( (
( lub `  K
) `  X )
( meet `  K )
( ( lub `  K
) `  Y )
)  e.  ( Base `  K ) )
3230, 13, 18, 31syl3anc 1182 . . 3  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  ( ( ( lub `  K ) `  X
) ( meet `  K
) ( ( lub `  K ) `  Y
) )  e.  (
Base `  K )
)
338, 20, 5pmapsubclN 30757 . . 3  |-  ( ( K  e.  HL  /\  ( ( ( lub `  K ) `  X
) ( meet `  K
) ( ( lub `  K ) `  Y
) )  e.  (
Base `  K )
)  ->  ( ( pmap `  K ) `  ( ( ( lub `  K ) `  X
) ( meet `  K
) ( ( lub `  K ) `  Y
) ) )  e.  C )
341, 32, 33syl2anc 642 . 2  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  ( ( pmap `  K
) `  ( (
( lub `  K
) `  X )
( meet `  K )
( ( lub `  K
) `  Y )
) )  e.  C
)
3528, 34eqeltrrd 2371 1  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  ( X  i^i  Y
)  e.  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1632    e. wcel 1696    i^i cin 3164    C_ wss 3165   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lubclub 14092   meetcmee 14095   Latclat 14167   CLatccla 14229   Atomscatm 30075   HLchlt 30162   pmapcpmap 30308   PSubClcpscN 30745
This theorem is referenced by:  osumcllem9N  30775
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-pmap 30315  df-polarityN 30714  df-psubclN 30746
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