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Theorem psubclinN 30646
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 957 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  K  e.  HL )
2 hlclat 30057 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  CLat )
323ad2ant1 978 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  K  e.  CLat )
4 eqid 2435 . . . . . . . 8  |-  ( Atoms `  K )  =  (
Atoms `  K )
5 psubclin.c . . . . . . . 8  |-  C  =  ( PSubCl `  K )
64, 5psubclssatN 30639 . . . . . . 7  |-  ( ( K  e.  HL  /\  X  e.  C )  ->  X  C_  ( Atoms `  K ) )
763adant3 977 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  X  C_  ( Atoms `  K ) )
8 eqid 2435 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
98, 4atssbase 29989 . . . . . 6  |-  ( Atoms `  K )  C_  ( Base `  K )
107, 9syl6ss 3352 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  X  C_  ( Base `  K ) )
11 eqid 2435 . . . . . 6  |-  ( lub `  K )  =  ( lub `  K )
128, 11clatlubcl 14530 . . . . 5  |-  ( ( K  e.  CLat  /\  X  C_  ( Base `  K
) )  ->  (
( lub `  K
) `  X )  e.  ( Base `  K
) )
133, 10, 12syl2anc 643 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  ( ( lub `  K
) `  X )  e.  ( Base `  K
) )
144, 5psubclssatN 30639 . . . . . . 7  |-  ( ( K  e.  HL  /\  Y  e.  C )  ->  Y  C_  ( Atoms `  K ) )
15143adant2 976 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  Y  C_  ( Atoms `  K ) )
1615, 9syl6ss 3352 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  Y  C_  ( Base `  K ) )
178, 11clatlubcl 14530 . . . . 5  |-  ( ( K  e.  CLat  /\  Y  C_  ( Base `  K
) )  ->  (
( lub `  K
) `  Y )  e.  ( Base `  K
) )
183, 16, 17syl2anc 643 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  ( ( lub `  K
) `  Y )  e.  ( Base `  K
) )
19 eqid 2435 . . . . 5  |-  ( meet `  K )  =  (
meet `  K )
20 eqid 2435 . . . . 5  |-  ( pmap `  K )  =  (
pmap `  K )
218, 19, 4, 20pmapmeet 30471 . . . 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 1184 . . 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 30640 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  C )  ->  ( ( pmap `  K
) `  ( ( lub `  K ) `  X ) )  =  X )
24233adant3 977 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  ( ( pmap `  K
) `  ( ( lub `  K ) `  X ) )  =  X )
2511, 20, 5pmapidclN 30640 . . . . 5  |-  ( ( K  e.  HL  /\  Y  e.  C )  ->  ( ( pmap `  K
) `  ( ( lub `  K ) `  Y ) )  =  Y )
26253adant2 976 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  ( ( pmap `  K
) `  ( ( lub `  K ) `  Y ) )  =  Y )
2724, 26ineq12d 3535 . . 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 2467 . 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 30062 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
30293ad2ant1 978 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  K  e.  Lat )
318, 19latmcl 14470 . . . 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 1184 . . 3  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  ( ( ( lub `  K ) `  X
) ( meet `  K
) ( ( lub `  K ) `  Y
) )  e.  (
Base `  K )
)
338, 20, 5pmapsubclN 30644 . . 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 643 . 2  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  ( ( pmap `  K
) `  ( (
( lub `  K
) `  X )
( meet `  K )
( ( lub `  K
) `  Y )
) )  e.  C
)
3528, 34eqeltrrd 2510 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 936    = wceq 1652    e. wcel 1725    i^i cin 3311    C_ wss 3312   ` cfv 5446  (class class class)co 6073   Basecbs 13459   lubclub 14389   meetcmee 14392   Latclat 14464   CLatccla 14526   Atomscatm 29962   HLchlt 30049   pmapcpmap 30195   PSubClcpscN 30632
This theorem is referenced by:  osumcllem9N  30662
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-poset 14393  df-plt 14405  df-lub 14421  df-glb 14422  df-join 14423  df-meet 14424  df-p0 14458  df-p1 14459  df-lat 14465  df-clat 14527  df-oposet 29875  df-ol 29877  df-oml 29878  df-covers 29965  df-ats 29966  df-atl 29997  df-cvlat 30021  df-hlat 30050  df-pmap 30202  df-polarityN 30601  df-psubclN 30633
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