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Theorem paddatclN 33956
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 33366 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  CLat )
213ad2ant1 1009 . . . . 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 33948 . . . . . . 7  |-  ( ( K  e.  HL  /\  X  e.  C )  ->  X  C_  A )
6 eqid 2454 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
76, 3atssbase 33298 . . . . . . 7  |-  A  C_  ( Base `  K )
85, 7syl6ss 3479 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  C )  ->  X  C_  ( Base `  K ) )
983adant3 1008 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  X  C_  ( Base `  K ) )
10 eqid 2454 . . . . . 6  |-  ( lub `  K )  =  ( lub `  K )
116, 10clatlubcl 15405 . . . . 5  |-  ( ( K  e.  CLat  /\  X  C_  ( Base `  K
) )  ->  (
( lub `  K
) `  X )  e.  ( Base `  K
) )
122, 9, 11syl2anc 661 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  ( ( lub `  K
) `  X )  e.  ( Base `  K
) )
13 eqid 2454 . . . . 5  |-  ( join `  K )  =  (
join `  K )
14 eqid 2454 . . . . 5  |-  ( pmap `  K )  =  (
pmap `  K )
15 paddatcl.p . . . . 5  |-  .+  =  ( +P `  K
)
166, 13, 3, 14, 15pmapjat1 33860 . . . 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 1266 . . 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 33949 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  C )  ->  ( ( pmap `  K
) `  ( ( lub `  K ) `  X ) )  =  X )
19183adant3 1008 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  ( ( pmap `  K
) `  ( ( lub `  K ) `  X ) )  =  X )
203, 14pmapat 33770 . . . . 5  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( ( pmap `  K
) `  Q )  =  { Q } )
21203adant2 1007 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  ( ( pmap `  K
) `  Q )  =  { Q } )
2219, 21oveq12d 6221 . . 3  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  ( ( ( pmap `  K ) `  (
( lub `  K
) `  X )
)  .+  ( ( pmap `  K ) `  Q ) )  =  ( X  .+  { Q } ) )
2317, 22eqtr2d 2496 . 2  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  ( X  .+  { Q } )  =  ( ( pmap `  K
) `  ( (
( lub `  K
) `  X )
( join `  K ) Q ) ) )
24 simp1 988 . . 3  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  K  e.  HL )
25 hllat 33371 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
26253ad2ant1 1009 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  K  e.  Lat )
276, 3atbase 33297 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
28273ad2ant3 1011 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  Q  e.  ( Base `  K ) )
296, 13latjcl 15344 . . . 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 1219 . . 3  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  ( ( ( lub `  K ) `  X
) ( join `  K
) Q )  e.  ( Base `  K
) )
316, 14, 4pmapsubclN 33953 . . 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 661 . 2  |-  ( ( K  e.  HL  /\  X  e.  C  /\  Q  e.  A )  ->  ( ( pmap `  K
) `  ( (
( lub `  K
) `  X )
( join `  K ) Q ) )  e.  C )
3323, 32eqeltrd 2542 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758    C_ wss 3439   {csn 3988   ` cfv 5529  (class class class)co 6203   Basecbs 14296   lubclub 15235   joincjn 15237   Latclat 15338   CLatccla 15400   Atomscatm 33271   HLchlt 33358   pmapcpmap 33504   +Pcpadd 33802   PSubClcpscN 33941
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 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-riotaBAD 32967
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 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-iin 4285  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-1st 6690  df-2nd 6691  df-undef 6905  df-poset 15239  df-plt 15251  df-lub 15267  df-glb 15268  df-join 15269  df-meet 15270  df-p0 15332  df-p1 15333  df-lat 15339  df-clat 15401  df-oposet 33184  df-ol 33186  df-oml 33187  df-covers 33274  df-ats 33275  df-atl 33306  df-cvlat 33330  df-hlat 33359  df-pmap 33511  df-padd 33803  df-polarityN 33910  df-psubclN 33942
This theorem is referenced by:  pclfinclN  33957  osumcllem9N  33971  pexmidlem6N  33982
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