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Theorem psubclinN 33874
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 988 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  K  e.  HL )
2 hlclat 33285 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  CLat )
323ad2ant1 1009 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  K  e.  CLat )
4 eqid 2450 . . . . . . . 8  |-  ( Atoms `  K )  =  (
Atoms `  K )
5 psubclin.c . . . . . . . 8  |-  C  =  ( PSubCl `  K )
64, 5psubclssatN 33867 . . . . . . 7  |-  ( ( K  e.  HL  /\  X  e.  C )  ->  X  C_  ( Atoms `  K ) )
763adant3 1008 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  X  C_  ( Atoms `  K ) )
8 eqid 2450 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
98, 4atssbase 33217 . . . . . 6  |-  ( Atoms `  K )  C_  ( Base `  K )
107, 9syl6ss 3452 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  X  C_  ( Base `  K ) )
11 eqid 2450 . . . . . 6  |-  ( lub `  K )  =  ( lub `  K )
128, 11clatlubcl 15370 . . . . 5  |-  ( ( K  e.  CLat  /\  X  C_  ( Base `  K
) )  ->  (
( lub `  K
) `  X )  e.  ( Base `  K
) )
133, 10, 12syl2anc 661 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  ( ( lub `  K
) `  X )  e.  ( Base `  K
) )
144, 5psubclssatN 33867 . . . . . . 7  |-  ( ( K  e.  HL  /\  Y  e.  C )  ->  Y  C_  ( Atoms `  K ) )
15143adant2 1007 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  Y  C_  ( Atoms `  K ) )
1615, 9syl6ss 3452 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  Y  C_  ( Base `  K ) )
178, 11clatlubcl 15370 . . . . 5  |-  ( ( K  e.  CLat  /\  Y  C_  ( Base `  K
) )  ->  (
( lub `  K
) `  Y )  e.  ( Base `  K
) )
183, 16, 17syl2anc 661 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  ( ( lub `  K
) `  Y )  e.  ( Base `  K
) )
19 eqid 2450 . . . . 5  |-  ( meet `  K )  =  (
meet `  K )
20 eqid 2450 . . . . 5  |-  ( pmap `  K )  =  (
pmap `  K )
218, 19, 4, 20pmapmeet 33699 . . . 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 1219 . . 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 33868 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  C )  ->  ( ( pmap `  K
) `  ( ( lub `  K ) `  X ) )  =  X )
24233adant3 1008 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  ( ( pmap `  K
) `  ( ( lub `  K ) `  X ) )  =  X )
2511, 20, 5pmapidclN 33868 . . . . 5  |-  ( ( K  e.  HL  /\  Y  e.  C )  ->  ( ( pmap `  K
) `  ( ( lub `  K ) `  Y ) )  =  Y )
26253adant2 1007 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  ( ( pmap `  K
) `  ( ( lub `  K ) `  Y ) )  =  Y )
2724, 26ineq12d 3637 . . 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 2490 . 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 33290 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
30293ad2ant1 1009 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  K  e.  Lat )
318, 19latmcl 15310 . . . 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 1219 . . 3  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  ( ( ( lub `  K ) `  X
) ( meet `  K
) ( ( lub `  K ) `  Y
) )  e.  (
Base `  K )
)
338, 20, 5pmapsubclN 33872 . . 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 661 . 2  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  ( ( pmap `  K
) `  ( (
( lub `  K
) `  X )
( meet `  K )
( ( lub `  K
) `  Y )
) )  e.  C
)
3528, 34eqeltrrd 2537 1  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  ->  ( X  i^i  Y
)  e.  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1370    e. wcel 1757    i^i cin 3411    C_ wss 3412   ` cfv 5502  (class class class)co 6176   Basecbs 14262   lubclub 15200   meetcmee 15203   Latclat 15303   CLatccla 15365   Atomscatm 33190   HLchlt 33277   pmapcpmap 33423   PSubClcpscN 33860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4487  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458  ax-riotaBAD 32886
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-nel 2644  df-ral 2797  df-rex 2798  df-reu 2799  df-rmo 2800  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4176  df-iun 4257  df-iin 4258  df-br 4377  df-opab 4435  df-mpt 4436  df-id 4720  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-riota 6137  df-ov 6179  df-oprab 6180  df-undef 6878  df-poset 15204  df-plt 15216  df-lub 15232  df-glb 15233  df-join 15234  df-meet 15235  df-p0 15297  df-p1 15298  df-lat 15304  df-clat 15366  df-oposet 33103  df-ol 33105  df-oml 33106  df-covers 33193  df-ats 33194  df-atl 33225  df-cvlat 33249  df-hlat 33278  df-pmap 33430  df-polarityN 33829  df-psubclN 33861
This theorem is referenced by:  osumcllem9N  33890
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