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Theorem psubclinN 30137
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 29548 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  CLat )
323ad2ant1 976 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  K  e.  CLat )
4 eqid 2283 . . . . . . . 8  |-  ( Atoms `  K )  =  (
Atoms `  K )
5 psubclin.c . . . . . . . 8  |-  C  =  ( PSubCl `  K )
64, 5psubclssatN 30130 . . . . . . 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 2283 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
98, 4atssbase 29480 . . . . . 6  |-  ( Atoms `  K )  C_  ( Base `  K )
107, 9syl6ss 3191 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  X  C_  ( Base `  K ) )
11 eqid 2283 . . . . . 6  |-  ( lub `  K )  =  ( lub `  K )
128, 11clatlubcl 14217 . . . . 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 30130 . . . . . . 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 3191 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  Y  C_  ( Base `  K ) )
178, 11clatlubcl 14217 . . . . 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 2283 . . . . 5  |-  ( meet `  K )  =  (
meet `  K )
20 eqid 2283 . . . . 5  |-  ( pmap `  K )  =  (
pmap `  K )
218, 19, 4, 20pmapmeet 29962 . . . 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 30131 . . . . 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 30131 . . . . 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 3371 . . 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 2315 . 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 29553 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
30293ad2ant1 976 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  K  e.  Lat )
318, 19latmcl 14157 . . . 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 30135 . . 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 2358 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 1623    e. wcel 1684    i^i cin 3151    C_ wss 3152   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lubclub 14076   meetcmee 14079   Latclat 14151   CLatccla 14213   Atomscatm 29453   HLchlt 29540   pmapcpmap 29686   PSubClcpscN 30123
This theorem is referenced by:  osumcllem9N  30153
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  ax-un 4512
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-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  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-iun 3907  df-iin 3908  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-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-pmap 29693  df-polarityN 30092  df-psubclN 30124
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