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Theorem osumclN 30701
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 2435 . . . . 5  |-  ( Atoms `  K )  =  (
Atoms `  K )
4 osumcl.c . . . . 5  |-  C  =  ( PSubCl `  K )
53, 4psubclssatN 30675 . . . 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 30675 . . . 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 30548 . . 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 6080 . . . . . 6  |-  ( X  =  (/)  ->  ( X 
.+  Y )  =  ( (/)  .+  Y ) )
153, 10padd02 30546 . . . . . . 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 2489 . . . . 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 2509 . . . 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 30673 . . . 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 30700 . . . . 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 2440 . . 3  |-  ( ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  /\  X  =/=  (/) )  ->  (  ._|_  `  (  ._|_  `  ( X 
.+  Y ) ) )  =  ( X 
.+  Y ) )
2622, 25pm2.61dane 2676 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  -> 
(  ._|_  `  (  ._|_  `  ( X  .+  Y
) ) )  =  ( X  .+  Y
) )
273, 20, 4ispsubclN 30671 . . 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 1652    e. wcel 1725    =/= wne 2598    C_ wss 3312   (/)c0 3620   ` cfv 5446  (class class class)co 6073   Atomscatm 29998   HLchlt 30085   + Pcpadd 30529   _|_ PcpolN 30636   PSubClcpscN 30668
This theorem is referenced by:  pmapojoinN  30702  pexmidN  30703
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-p1 14461  df-lat 14467  df-clat 14529  df-oposet 29911  df-ol 29913  df-oml 29914  df-covers 30001  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086  df-psubsp 30237  df-pmap 30238  df-padd 30530  df-polarityN 30637  df-psubclN 30669
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