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Theorem paddatclN 32947
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 32357 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  CLat )
213ad2ant1 1018 . . . . 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 32939 . . . . . . 7  |-  ( ( K  e.  HL  /\  X  e.  C )  ->  X  C_  A )
6 eqid 2402 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
76, 3atssbase 32289 . . . . . . 7  |-  A  C_  ( Base `  K )
85, 7syl6ss 3453 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  C )  ->  X  C_  ( Base `  K ) )
983adant3 1017 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  X  C_  ( Base `  K ) )
10 eqid 2402 . . . . . 6  |-  ( lub `  K )  =  ( lub `  K )
116, 10clatlubcl 15958 . . . . 5  |-  ( ( K  e.  CLat  /\  X  C_  ( Base `  K
) )  ->  (
( lub `  K
) `  X )  e.  ( Base `  K
) )
122, 9, 11syl2anc 659 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  ( ( lub `  K
) `  X )  e.  ( Base `  K
) )
13 eqid 2402 . . . . 5  |-  ( join `  K )  =  (
join `  K )
14 eqid 2402 . . . . 5  |-  ( pmap `  K )  =  (
pmap `  K )
15 paddatcl.p . . . . 5  |-  .+  =  ( +P `  K
)
166, 13, 3, 14, 15pmapjat1 32851 . . . 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 1277 . . 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 32940 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  C )  ->  ( ( pmap `  K
) `  ( ( lub `  K ) `  X ) )  =  X )
19183adant3 1017 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  ( ( pmap `  K
) `  ( ( lub `  K ) `  X ) )  =  X )
203, 14pmapat 32761 . . . . 5  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( ( pmap `  K
) `  Q )  =  { Q } )
21203adant2 1016 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  ( ( pmap `  K
) `  Q )  =  { Q } )
2219, 21oveq12d 6252 . . 3  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  ( ( ( pmap `  K ) `  (
( lub `  K
) `  X )
)  .+  ( ( pmap `  K ) `  Q ) )  =  ( X  .+  { Q } ) )
2317, 22eqtr2d 2444 . 2  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  ( X  .+  { Q } )  =  ( ( pmap `  K
) `  ( (
( lub `  K
) `  X )
( join `  K ) Q ) ) )
24 simp1 997 . . 3  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  K  e.  HL )
25 hllat 32362 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
26253ad2ant1 1018 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  K  e.  Lat )
276, 3atbase 32288 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
28273ad2ant3 1020 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  Q  e.  ( Base `  K ) )
296, 13latjcl 15897 . . . 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 1230 . . 3  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  ( ( ( lub `  K ) `  X
) ( join `  K
) Q )  e.  ( Base `  K
) )
316, 14, 4pmapsubclN 32944 . . 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 659 . 2  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  ( ( pmap `  K
) `  ( (
( lub `  K
) `  X )
( join `  K ) Q ) )  e.  C )
3323, 32eqeltrd 2490 1  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  ( X  .+  { Q } )  e.  C
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    C_ wss 3413   {csn 3971   ` cfv 5525  (class class class)co 6234   Basecbs 14733   lubclub 15787   joincjn 15789   Latclat 15891   CLatccla 15953   Atomscatm 32262   HLchlt 32349   pmapcpmap 32495   +Pcpadd 32793   PSubClcpscN 32932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-riotaBAD 31958
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-iin 4273  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-1st 6738  df-2nd 6739  df-undef 6959  df-preset 15773  df-poset 15791  df-plt 15804  df-lub 15820  df-glb 15821  df-join 15822  df-meet 15823  df-p0 15885  df-p1 15886  df-lat 15892  df-clat 15954  df-oposet 32175  df-ol 32177  df-oml 32178  df-covers 32265  df-ats 32266  df-atl 32297  df-cvlat 32321  df-hlat 32350  df-pmap 32502  df-padd 32794  df-polarityN 32901  df-psubclN 32933
This theorem is referenced by:  pclfinclN  32948  osumcllem9N  32962  pexmidlem6N  32973
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