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Theorem paddatclN 30844
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 30254 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  CLat )
213ad2ant1 979 . . . . 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 30836 . . . . . . 7  |-  ( ( K  e.  HL  /\  X  e.  C )  ->  X  C_  A )
6 eqid 2442 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
76, 3atssbase 30186 . . . . . . 7  |-  A  C_  ( Base `  K )
85, 7syl6ss 3346 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  C )  ->  X  C_  ( Base `  K ) )
983adant3 978 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  X  C_  ( Base `  K ) )
10 eqid 2442 . . . . . 6  |-  ( lub `  K )  =  ( lub `  K )
116, 10clatlubcl 14571 . . . . 5  |-  ( ( K  e.  CLat  /\  X  C_  ( Base `  K
) )  ->  (
( lub `  K
) `  X )  e.  ( Base `  K
) )
122, 9, 11syl2anc 644 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  ( ( lub `  K
) `  X )  e.  ( Base `  K
) )
13 eqid 2442 . . . . 5  |-  ( join `  K )  =  (
join `  K )
14 eqid 2442 . . . . 5  |-  ( pmap `  K )  =  (
pmap `  K )
15 paddatcl.p . . . . 5  |-  .+  =  ( + P `  K
)
166, 13, 3, 14, 15pmapjat1 30748 . . . 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 1232 . . 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 30837 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  C )  ->  ( ( pmap `  K
) `  ( ( lub `  K ) `  X ) )  =  X )
19183adant3 978 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  ( ( pmap `  K
) `  ( ( lub `  K ) `  X ) )  =  X )
203, 14pmapat 30658 . . . . 5  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( ( pmap `  K
) `  Q )  =  { Q } )
21203adant2 977 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  ( ( pmap `  K
) `  Q )  =  { Q } )
2219, 21oveq12d 6128 . . 3  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  ( ( ( pmap `  K ) `  (
( lub `  K
) `  X )
)  .+  ( ( pmap `  K ) `  Q ) )  =  ( X  .+  { Q } ) )
2317, 22eqtr2d 2475 . 2  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  ( X  .+  { Q } )  =  ( ( pmap `  K
) `  ( (
( lub `  K
) `  X )
( join `  K ) Q ) ) )
24 simp1 958 . . 3  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  K  e.  HL )
25 hllat 30259 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
26253ad2ant1 979 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  K  e.  Lat )
276, 3atbase 30185 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
28273ad2ant3 981 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  Q  e.  ( Base `  K ) )
296, 13latjcl 14510 . . . 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 1185 . . 3  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  ( ( ( lub `  K ) `  X
) ( join `  K
) Q )  e.  ( Base `  K
) )
316, 14, 4pmapsubclN 30841 . . 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 644 . 2  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  ( ( pmap `  K
) `  ( (
( lub `  K
) `  X )
( join `  K ) Q ) )  e.  C )
3323, 32eqeltrd 2516 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 360    /\ w3a 937    = wceq 1653    e. wcel 1727    C_ wss 3306   {csn 3838   ` cfv 5483  (class class class)co 6110   Basecbs 13500   lubclub 14430   joincjn 14432   Latclat 14505   CLatccla 14567   Atomscatm 30159   HLchlt 30246   pmapcpmap 30392   + Pcpadd 30690   PSubClcpscN 30829
This theorem is referenced by:  pclfinclN  30845  osumcllem9N  30859  pexmidlem6N  30870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-iin 4120  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-undef 6572  df-riota 6578  df-poset 14434  df-plt 14446  df-lub 14462  df-glb 14463  df-join 14464  df-meet 14465  df-p0 14499  df-p1 14500  df-lat 14506  df-clat 14568  df-oposet 30072  df-ol 30074  df-oml 30075  df-covers 30162  df-ats 30163  df-atl 30194  df-cvlat 30218  df-hlat 30247  df-pmap 30399  df-padd 30691  df-polarityN 30798  df-psubclN 30830
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