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Theorem osumclN 30082
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 960 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  ->  K  e.  HL )
2 simpl2 961 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  ->  X  e.  C )
3 eqid 2388 . . . . 5  |-  ( Atoms `  K )  =  (
Atoms `  K )
4 osumcl.c . . . . 5  |-  C  =  ( PSubCl `  K )
53, 4psubclssatN 30056 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  C )  ->  X  C_  ( Atoms `  K ) )
61, 2, 5syl2anc 643 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  ->  X  C_  ( Atoms `  K
) )
7 simpl3 962 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  ->  Y  e.  C )
83, 4psubclssatN 30056 . . . 4  |-  ( ( K  e.  HL  /\  Y  e.  C )  ->  Y  C_  ( Atoms `  K ) )
91, 7, 8syl2anc 643 . . 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 29929 . . 3  |-  ( ( K  e.  HL  /\  X  C_  ( Atoms `  K
)  /\  Y  C_  ( Atoms `  K ) )  ->  ( X  .+  Y )  C_  ( Atoms `  K ) )
121, 6, 9, 11syl3anc 1184 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  -> 
( X  .+  Y
)  C_  ( Atoms `  K ) )
13 simpll1 996 . . . 4  |-  ( ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  /\  X  =  (/) )  ->  K  e.  HL )
14 oveq1 6028 . . . . . 6  |-  ( X  =  (/)  ->  ( X 
.+  Y )  =  ( (/)  .+  Y ) )
153, 10padd02 29927 . . . . . . 7  |-  ( ( K  e.  HL  /\  Y  C_  ( Atoms `  K
) )  ->  ( (/)  .+  Y )  =  Y )
161, 9, 15syl2anc 643 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  -> 
( (/)  .+  Y )  =  Y )
1714, 16sylan9eqr 2442 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  /\  X  =  (/) )  ->  ( X 
.+  Y )  =  Y )
18 simpll3 998 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  /\  X  =  (/) )  ->  Y  e.  C )
1917, 18eqeltrd 2462 . . . 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 30054 . . . 4  |-  ( ( K  e.  HL  /\  ( X  .+  Y )  e.  C )  -> 
(  ._|_  `  (  ._|_  `  ( X  .+  Y
) ) )  =  ( X  .+  Y
) )
2213, 19, 21syl2anc 643 . . 3  |-  ( ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  /\  X  =  (/) )  ->  (  ._|_  `  (  ._|_  `  ( X 
.+  Y ) ) )  =  ( X 
.+  Y ) )
2310, 20, 4osumcllem11N 30081 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  ( X  C_  (  ._|_  `  Y )  /\  X  =/=  (/) ) )  -> 
( X  .+  Y
)  =  (  ._|_  `  (  ._|_  `  ( X 
.+  Y ) ) ) )
2423anassrs 630 . . . 4  |-  ( ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  /\  X  =/=  (/) )  ->  ( X 
.+  Y )  =  (  ._|_  `  (  ._|_  `  ( X  .+  Y
) ) ) )
2524eqcomd 2393 . . 3  |-  ( ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  /\  X  =/=  (/) )  ->  (  ._|_  `  (  ._|_  `  ( X 
.+  Y ) ) )  =  ( X 
.+  Y ) )
2622, 25pm2.61dane 2629 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  -> 
(  ._|_  `  (  ._|_  `  ( X  .+  Y
) ) )  =  ( X  .+  Y
) )
273, 20, 4ispsubclN 30052 . . 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 889 1  |-  ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  -> 
( X  .+  Y
)  e.  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2551    C_ wss 3264   (/)c0 3572   ` cfv 5395  (class class class)co 6021   Atomscatm 29379   HLchlt 29466   + Pcpadd 29910   _|_ PcpolN 30017   PSubClcpscN 30049
This theorem is referenced by:  pmapojoinN  30083  pexmidN  30084
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-iin 4039  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-undef 6480  df-riota 6486  df-poset 14331  df-plt 14343  df-lub 14359  df-glb 14360  df-join 14361  df-meet 14362  df-p0 14396  df-p1 14397  df-lat 14403  df-clat 14465  df-oposet 29292  df-ol 29294  df-oml 29295  df-covers 29382  df-ats 29383  df-atl 29414  df-cvlat 29438  df-hlat 29467  df-psubsp 29618  df-pmap 29619  df-padd 29911  df-polarityN 30018  df-psubclN 30050
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