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Theorem osumclN 36143
Description: Closure of orthogonal sum. If  X and  Y are orthogonal closed projective subspaces, then their sum is closed. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
osumcl.p  |-  .+  =  ( +P `  K
)
osumcl.o  |-  ._|_  =  ( _|_P `  K
)
osumcl.c  |-  C  =  ( PSubCl `  K )
Assertion
Ref Expression
osumclN  |-  ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  -> 
( X  .+  Y
)  e.  C )

Proof of Theorem osumclN
StepHypRef Expression
1 simpl1 997 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  ->  K  e.  HL )
2 simpl2 998 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  ->  X  e.  C )
3 eqid 2396 . . . . 5  |-  ( Atoms `  K )  =  (
Atoms `  K )
4 osumcl.c . . . . 5  |-  C  =  ( PSubCl `  K )
53, 4psubclssatN 36117 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  C )  ->  X  C_  ( Atoms `  K ) )
61, 2, 5syl2anc 659 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  ->  X  C_  ( Atoms `  K
) )
7 simpl3 999 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  ->  Y  e.  C )
83, 4psubclssatN 36117 . . . 4  |-  ( ( K  e.  HL  /\  Y  e.  C )  ->  Y  C_  ( Atoms `  K ) )
91, 7, 8syl2anc 659 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  ->  Y  C_  ( Atoms `  K
) )
10 osumcl.p . . . 4  |-  .+  =  ( +P `  K
)
113, 10paddssat 35990 . . 3  |-  ( ( K  e.  HL  /\  X  C_  ( Atoms `  K
)  /\  Y  C_  ( Atoms `  K ) )  ->  ( X  .+  Y )  C_  ( Atoms `  K ) )
121, 6, 9, 11syl3anc 1226 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  -> 
( X  .+  Y
)  C_  ( Atoms `  K ) )
13 simpll1 1033 . . . 4  |-  ( ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  /\  X  =  (/) )  ->  K  e.  HL )
14 oveq1 6225 . . . . . 6  |-  ( X  =  (/)  ->  ( X 
.+  Y )  =  ( (/)  .+  Y ) )
153, 10padd02 35988 . . . . . . 7  |-  ( ( K  e.  HL  /\  Y  C_  ( Atoms `  K
) )  ->  ( (/)  .+  Y )  =  Y )
161, 9, 15syl2anc 659 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  -> 
( (/)  .+  Y )  =  Y )
1714, 16sylan9eqr 2459 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  /\  X  =  (/) )  ->  ( X 
.+  Y )  =  Y )
18 simpll3 1035 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  /\  X  =  (/) )  ->  Y  e.  C )
1917, 18eqeltrd 2484 . . . 4  |-  ( ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  /\  X  =  (/) )  ->  ( X 
.+  Y )  e.  C )
20 osumcl.o . . . . 5  |-  ._|_  =  ( _|_P `  K
)
2120, 4psubcli2N 36115 . . . 4  |-  ( ( K  e.  HL  /\  ( X  .+  Y )  e.  C )  -> 
(  ._|_  `  (  ._|_  `  ( X  .+  Y
) ) )  =  ( X  .+  Y
) )
2213, 19, 21syl2anc 659 . . 3  |-  ( ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  /\  X  =  (/) )  ->  (  ._|_  `  (  ._|_  `  ( X 
.+  Y ) ) )  =  ( X 
.+  Y ) )
2310, 20, 4osumcllem11N 36142 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  ( X  C_  (  ._|_  `  Y )  /\  X  =/=  (/) ) )  -> 
( X  .+  Y
)  =  (  ._|_  `  (  ._|_  `  ( X 
.+  Y ) ) ) )
2423anassrs 646 . . . 4  |-  ( ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  /\  X  =/=  (/) )  ->  ( X 
.+  Y )  =  (  ._|_  `  (  ._|_  `  ( X  .+  Y
) ) ) )
2524eqcomd 2404 . . 3  |-  ( ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  /\  X  =/=  (/) )  ->  (  ._|_  `  (  ._|_  `  ( X 
.+  Y ) ) )  =  ( X 
.+  Y ) )
2622, 25pm2.61dane 2714 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  -> 
(  ._|_  `  (  ._|_  `  ( X  .+  Y
) ) )  =  ( X  .+  Y
) )
273, 20, 4ispsubclN 36113 . . 3  |-  ( K  e.  HL  ->  (
( X  .+  Y
)  e.  C  <->  ( ( X  .+  Y )  C_  ( Atoms `  K )  /\  (  ._|_  `  (  ._|_  `  ( X  .+  Y ) ) )  =  ( X  .+  Y ) ) ) )
281, 27syl 16 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  -> 
( ( X  .+  Y )  e.  C  <->  ( ( X  .+  Y
)  C_  ( Atoms `  K )  /\  (  ._|_  `  (  ._|_  `  ( X  .+  Y ) ) )  =  ( X 
.+  Y ) ) ) )
2912, 26, 28mpbir2and 920 1  |-  ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  -> 
( X  .+  Y
)  e.  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1836    =/= wne 2591    C_ wss 3406   (/)c0 3728   ` cfv 5513  (class class class)co 6218   Atomscatm 35440   HLchlt 35527   +Pcpadd 35971   _|_PcpolN 36078   PSubClcpscN 36110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-rep 4495  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513  ax-riotaBAD 35136
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-fal 1405  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-nel 2594  df-ral 2751  df-rex 2752  df-reu 2753  df-rmo 2754  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-iun 4262  df-iin 4263  df-br 4385  df-opab 4443  df-mpt 4444  df-id 4726  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6180  df-ov 6221  df-oprab 6222  df-mpt2 6223  df-1st 6721  df-2nd 6722  df-undef 6942  df-preset 15697  df-poset 15715  df-plt 15728  df-lub 15744  df-glb 15745  df-join 15746  df-meet 15747  df-p0 15809  df-p1 15810  df-lat 15816  df-clat 15878  df-oposet 35353  df-ol 35355  df-oml 35356  df-covers 35443  df-ats 35444  df-atl 35475  df-cvlat 35499  df-hlat 35528  df-psubsp 35679  df-pmap 35680  df-padd 35972  df-polarityN 36079  df-psubclN 36111
This theorem is referenced by:  pmapojoinN  36144  pexmidN  36145
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