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Theorem paddatclN 30443
Description: The projective sum of a closed subspace and an atom is a closed projective subspace. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
paddatcl.a  |-  A  =  ( Atoms `  K )
paddatcl.p  |-  .+  =  ( + P `  K
)
paddatcl.c  |-  C  =  ( PSubCl `  K )
Assertion
Ref Expression
paddatclN  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  ( X  .+  { Q } )  e.  C
)

Proof of Theorem paddatclN
StepHypRef Expression
1 hlclat 29853 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  CLat )
213ad2ant1 978 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  K  e.  CLat )
3 paddatcl.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
4 paddatcl.c . . . . . . . 8  |-  C  =  ( PSubCl `  K )
53, 4psubclssatN 30435 . . . . . . 7  |-  ( ( K  e.  HL  /\  X  e.  C )  ->  X  C_  A )
6 eqid 2412 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
76, 3atssbase 29785 . . . . . . 7  |-  A  C_  ( Base `  K )
85, 7syl6ss 3328 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  C )  ->  X  C_  ( Base `  K ) )
983adant3 977 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  X  C_  ( Base `  K ) )
10 eqid 2412 . . . . . 6  |-  ( lub `  K )  =  ( lub `  K )
116, 10clatlubcl 14503 . . . . 5  |-  ( ( K  e.  CLat  /\  X  C_  ( Base `  K
) )  ->  (
( lub `  K
) `  X )  e.  ( Base `  K
) )
122, 9, 11syl2anc 643 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  ( ( lub `  K
) `  X )  e.  ( Base `  K
) )
13 eqid 2412 . . . . 5  |-  ( join `  K )  =  (
join `  K )
14 eqid 2412 . . . . 5  |-  ( pmap `  K )  =  (
pmap `  K )
15 paddatcl.p . . . . 5  |-  .+  =  ( + P `  K
)
166, 13, 3, 14, 15pmapjat1 30347 . . . 4  |-  ( ( K  e.  HL  /\  ( ( lub `  K
) `  X )  e.  ( Base `  K
)  /\  Q  e.  A )  ->  (
( pmap `  K ) `  ( ( ( lub `  K ) `  X
) ( join `  K
) Q ) )  =  ( ( (
pmap `  K ) `  ( ( lub `  K
) `  X )
)  .+  ( ( pmap `  K ) `  Q ) ) )
1712, 16syld3an2 1231 . . 3  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  ( ( pmap `  K
) `  ( (
( lub `  K
) `  X )
( join `  K ) Q ) )  =  ( ( ( pmap `  K ) `  (
( lub `  K
) `  X )
)  .+  ( ( pmap `  K ) `  Q ) ) )
1810, 14, 4pmapidclN 30436 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  C )  ->  ( ( pmap `  K
) `  ( ( lub `  K ) `  X ) )  =  X )
19183adant3 977 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  ( ( pmap `  K
) `  ( ( lub `  K ) `  X ) )  =  X )
203, 14pmapat 30257 . . . . 5  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( ( pmap `  K
) `  Q )  =  { Q } )
21203adant2 976 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  ( ( pmap `  K
) `  Q )  =  { Q } )
2219, 21oveq12d 6066 . . 3  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  ( ( ( pmap `  K ) `  (
( lub `  K
) `  X )
)  .+  ( ( pmap `  K ) `  Q ) )  =  ( X  .+  { Q } ) )
2317, 22eqtr2d 2445 . 2  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  ( X  .+  { Q } )  =  ( ( pmap `  K
) `  ( (
( lub `  K
) `  X )
( join `  K ) Q ) ) )
24 simp1 957 . . 3  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  K  e.  HL )
25 hllat 29858 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
26253ad2ant1 978 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  K  e.  Lat )
276, 3atbase 29784 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
28273ad2ant3 980 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  Q  e.  ( Base `  K ) )
296, 13latjcl 14442 . . . 4  |-  ( ( K  e.  Lat  /\  ( ( lub `  K
) `  X )  e.  ( Base `  K
)  /\  Q  e.  ( Base `  K )
)  ->  ( (
( lub `  K
) `  X )
( join `  K ) Q )  e.  (
Base `  K )
)
3026, 12, 28, 29syl3anc 1184 . . 3  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  ( ( ( lub `  K ) `  X
) ( join `  K
) Q )  e.  ( Base `  K
) )
316, 14, 4pmapsubclN 30440 . . 3  |-  ( ( K  e.  HL  /\  ( ( ( lub `  K ) `  X
) ( join `  K
) Q )  e.  ( Base `  K
) )  ->  (
( pmap `  K ) `  ( ( ( lub `  K ) `  X
) ( join `  K
) Q ) )  e.  C )
3224, 30, 31syl2anc 643 . 2  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  ( ( pmap `  K
) `  ( (
( lub `  K
) `  X )
( join `  K ) Q ) )  e.  C )
3323, 32eqeltrd 2486 1  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  ( X  .+  { Q } )  e.  C
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    C_ wss 3288   {csn 3782   ` cfv 5421  (class class class)co 6048   Basecbs 13432   lubclub 14362   joincjn 14364   Latclat 14437   CLatccla 14499   Atomscatm 29758   HLchlt 29845   pmapcpmap 29991   + Pcpadd 30289   PSubClcpscN 30428
This theorem is referenced by:  pclfinclN  30444  osumcllem9N  30458  pexmidlem6N  30469
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-iin 4064  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-undef 6510  df-riota 6516  df-poset 14366  df-plt 14378  df-lub 14394  df-glb 14395  df-join 14396  df-meet 14397  df-p0 14431  df-p1 14432  df-lat 14438  df-clat 14500  df-oposet 29671  df-ol 29673  df-oml 29674  df-covers 29761  df-ats 29762  df-atl 29793  df-cvlat 29817  df-hlat 29846  df-pmap 29998  df-padd 30290  df-polarityN 30397  df-psubclN 30429
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