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Theorem paddatclN 30197
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 29607 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  CLat )
213ad2ant1 977 . . . . 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 30189 . . . . . . 7  |-  ( ( K  e.  HL  /\  X  e.  C )  ->  X  C_  A )
6 eqid 2366 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
76, 3atssbase 29539 . . . . . . 7  |-  A  C_  ( Base `  K )
85, 7syl6ss 3277 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  C )  ->  X  C_  ( Base `  K ) )
983adant3 976 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  X  C_  ( Base `  K ) )
10 eqid 2366 . . . . . 6  |-  ( lub `  K )  =  ( lub `  K )
116, 10clatlubcl 14427 . . . . 5  |-  ( ( K  e.  CLat  /\  X  C_  ( Base `  K
) )  ->  (
( lub `  K
) `  X )  e.  ( Base `  K
) )
122, 9, 11syl2anc 642 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  ( ( lub `  K
) `  X )  e.  ( Base `  K
) )
13 eqid 2366 . . . . 5  |-  ( join `  K )  =  (
join `  K )
14 eqid 2366 . . . . 5  |-  ( pmap `  K )  =  (
pmap `  K )
15 paddatcl.p . . . . 5  |-  .+  =  ( + P `  K
)
166, 13, 3, 14, 15pmapjat1 30101 . . . 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 1230 . . 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 30190 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  C )  ->  ( ( pmap `  K
) `  ( ( lub `  K ) `  X ) )  =  X )
19183adant3 976 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  ( ( pmap `  K
) `  ( ( lub `  K ) `  X ) )  =  X )
203, 14pmapat 30011 . . . . 5  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( ( pmap `  K
) `  Q )  =  { Q } )
21203adant2 975 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  ( ( pmap `  K
) `  Q )  =  { Q } )
2219, 21oveq12d 5999 . . 3  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  ( ( ( pmap `  K ) `  (
( lub `  K
) `  X )
)  .+  ( ( pmap `  K ) `  Q ) )  =  ( X  .+  { Q } ) )
2317, 22eqtr2d 2399 . 2  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  ( X  .+  { Q } )  =  ( ( pmap `  K
) `  ( (
( lub `  K
) `  X )
( join `  K ) Q ) ) )
24 simp1 956 . . 3  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  K  e.  HL )
25 hllat 29612 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
26253ad2ant1 977 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  K  e.  Lat )
276, 3atbase 29538 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
28273ad2ant3 979 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  Q  e.  ( Base `  K ) )
296, 13latjcl 14366 . . . 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 1183 . . 3  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  ( ( ( lub `  K ) `  X
) ( join `  K
) Q )  e.  ( Base `  K
) )
316, 14, 4pmapsubclN 30194 . . 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 642 . 2  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  ( ( pmap `  K
) `  ( (
( lub `  K
) `  X )
( join `  K ) Q ) )  e.  C )
3323, 32eqeltrd 2440 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 358    /\ w3a 935    = wceq 1647    e. wcel 1715    C_ wss 3238   {csn 3729   ` cfv 5358  (class class class)co 5981   Basecbs 13356   lubclub 14286   joincjn 14288   Latclat 14361   CLatccla 14423   Atomscatm 29512   HLchlt 29599   pmapcpmap 29745   + Pcpadd 30043   PSubClcpscN 30182
This theorem is referenced by:  pclfinclN  30198  osumcllem9N  30212  pexmidlem6N  30223
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-iin 4010  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-undef 6440  df-riota 6446  df-poset 14290  df-plt 14302  df-lub 14318  df-glb 14319  df-join 14320  df-meet 14321  df-p0 14355  df-p1 14356  df-lat 14362  df-clat 14424  df-oposet 29425  df-ol 29427  df-oml 29428  df-covers 29515  df-ats 29516  df-atl 29547  df-cvlat 29571  df-hlat 29600  df-pmap 29752  df-padd 30044  df-polarityN 30151  df-psubclN 30183
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