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Theorem osumclN 33950
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 991 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  ->  K  e.  HL )
2 simpl2 992 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  ->  X  e.  C )
3 eqid 2454 . . . . 5  |-  ( Atoms `  K )  =  (
Atoms `  K )
4 osumcl.c . . . . 5  |-  C  =  ( PSubCl `  K )
53, 4psubclssatN 33924 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  C )  ->  X  C_  ( Atoms `  K ) )
61, 2, 5syl2anc 661 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  ->  X  C_  ( Atoms `  K
) )
7 simpl3 993 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  ->  Y  e.  C )
83, 4psubclssatN 33924 . . . 4  |-  ( ( K  e.  HL  /\  Y  e.  C )  ->  Y  C_  ( Atoms `  K ) )
91, 7, 8syl2anc 661 . . 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 33797 . . 3  |-  ( ( K  e.  HL  /\  X  C_  ( Atoms `  K
)  /\  Y  C_  ( Atoms `  K ) )  ->  ( X  .+  Y )  C_  ( Atoms `  K ) )
121, 6, 9, 11syl3anc 1219 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  -> 
( X  .+  Y
)  C_  ( Atoms `  K ) )
13 simpll1 1027 . . . 4  |-  ( ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  /\  X  =  (/) )  ->  K  e.  HL )
14 oveq1 6208 . . . . . 6  |-  ( X  =  (/)  ->  ( X 
.+  Y )  =  ( (/)  .+  Y ) )
153, 10padd02 33795 . . . . . . 7  |-  ( ( K  e.  HL  /\  Y  C_  ( Atoms `  K
) )  ->  ( (/)  .+  Y )  =  Y )
161, 9, 15syl2anc 661 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  -> 
( (/)  .+  Y )  =  Y )
1714, 16sylan9eqr 2517 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  /\  X  =  (/) )  ->  ( X 
.+  Y )  =  Y )
18 simpll3 1029 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  /\  X  =  (/) )  ->  Y  e.  C )
1917, 18eqeltrd 2542 . . . 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 33922 . . . 4  |-  ( ( K  e.  HL  /\  ( X  .+  Y )  e.  C )  -> 
(  ._|_  `  (  ._|_  `  ( X  .+  Y
) ) )  =  ( X  .+  Y
) )
2213, 19, 21syl2anc 661 . . 3  |-  ( ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  /\  X  =  (/) )  ->  (  ._|_  `  (  ._|_  `  ( X 
.+  Y ) ) )  =  ( X 
.+  Y ) )
2310, 20, 4osumcllem11N 33949 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  ( X  C_  (  ._|_  `  Y )  /\  X  =/=  (/) ) )  -> 
( X  .+  Y
)  =  (  ._|_  `  (  ._|_  `  ( X 
.+  Y ) ) ) )
2423anassrs 648 . . . 4  |-  ( ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  /\  X  =/=  (/) )  ->  ( X 
.+  Y )  =  (  ._|_  `  (  ._|_  `  ( X  .+  Y
) ) ) )
2524eqcomd 2462 . . 3  |-  ( ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  /\  X  =/=  (/) )  ->  (  ._|_  `  (  ._|_  `  ( X 
.+  Y ) ) )  =  ( X 
.+  Y ) )
2622, 25pm2.61dane 2770 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  -> 
(  ._|_  `  (  ._|_  `  ( X  .+  Y
) ) )  =  ( X  .+  Y
) )
273, 20, 4ispsubclN 33920 . . 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 913 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648    C_ wss 3437   (/)c0 3746   ` cfv 5527  (class class class)co 6201   Atomscatm 33247   HLchlt 33334   +Pcpadd 33778   _|_PcpolN 33885   PSubClcpscN 33917
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 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-riotaBAD 32943
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 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-iin 4283  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-1st 6688  df-2nd 6689  df-undef 6903  df-poset 15236  df-plt 15248  df-lub 15264  df-glb 15265  df-join 15266  df-meet 15267  df-p0 15329  df-p1 15330  df-lat 15336  df-clat 15398  df-oposet 33160  df-ol 33162  df-oml 33163  df-covers 33250  df-ats 33251  df-atl 33282  df-cvlat 33306  df-hlat 33335  df-psubsp 33486  df-pmap 33487  df-padd 33779  df-polarityN 33886  df-psubclN 33918
This theorem is referenced by:  pmapojoinN  33951  pexmidN  33952
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