Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  osumclN Structured version   Visualization version   Unicode version

Theorem osumclN 33578
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 1017 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  ->  K  e.  HL )
2 simpl2 1018 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  ->  X  e.  C )
3 eqid 2462 . . . . 5  |-  ( Atoms `  K )  =  (
Atoms `  K )
4 osumcl.c . . . . 5  |-  C  =  ( PSubCl `  K )
53, 4psubclssatN 33552 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  C )  ->  X  C_  ( Atoms `  K ) )
61, 2, 5syl2anc 671 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  ->  X  C_  ( Atoms `  K
) )
7 simpl3 1019 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  ->  Y  e.  C )
83, 4psubclssatN 33552 . . . 4  |-  ( ( K  e.  HL  /\  Y  e.  C )  ->  Y  C_  ( Atoms `  K ) )
91, 7, 8syl2anc 671 . . 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 33425 . . 3  |-  ( ( K  e.  HL  /\  X  C_  ( Atoms `  K
)  /\  Y  C_  ( Atoms `  K ) )  ->  ( X  .+  Y )  C_  ( Atoms `  K ) )
121, 6, 9, 11syl3anc 1276 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  -> 
( X  .+  Y
)  C_  ( Atoms `  K ) )
13 simpll1 1053 . . . 4  |-  ( ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  /\  X  =  (/) )  ->  K  e.  HL )
14 oveq1 6327 . . . . . 6  |-  ( X  =  (/)  ->  ( X 
.+  Y )  =  ( (/)  .+  Y ) )
153, 10padd02 33423 . . . . . . 7  |-  ( ( K  e.  HL  /\  Y  C_  ( Atoms `  K
) )  ->  ( (/)  .+  Y )  =  Y )
161, 9, 15syl2anc 671 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  -> 
( (/)  .+  Y )  =  Y )
1714, 16sylan9eqr 2518 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  /\  X  =  (/) )  ->  ( X 
.+  Y )  =  Y )
18 simpll3 1055 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  /\  X  =  (/) )  ->  Y  e.  C )
1917, 18eqeltrd 2540 . . . 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 33550 . . . 4  |-  ( ( K  e.  HL  /\  ( X  .+  Y )  e.  C )  -> 
(  ._|_  `  (  ._|_  `  ( X  .+  Y
) ) )  =  ( X  .+  Y
) )
2213, 19, 21syl2anc 671 . . 3  |-  ( ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  /\  X  =  (/) )  ->  (  ._|_  `  (  ._|_  `  ( X 
.+  Y ) ) )  =  ( X 
.+  Y ) )
2310, 20, 4osumcllem11N 33577 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  ( X  C_  (  ._|_  `  Y )  /\  X  =/=  (/) ) )  -> 
( X  .+  Y
)  =  (  ._|_  `  (  ._|_  `  ( X 
.+  Y ) ) ) )
2423anassrs 658 . . . 4  |-  ( ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  /\  X  =/=  (/) )  ->  ( X 
.+  Y )  =  (  ._|_  `  (  ._|_  `  ( X  .+  Y
) ) ) )
2524eqcomd 2468 . . 3  |-  ( ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  /\  X  =/=  (/) )  ->  (  ._|_  `  (  ._|_  `  ( X 
.+  Y ) ) )  =  ( X 
.+  Y ) )
2622, 25pm2.61dane 2723 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  C  /\  Y  e.  C )  /\  X  C_  (  ._|_  `  Y ) )  -> 
(  ._|_  `  (  ._|_  `  ( X  .+  Y
) ) )  =  ( X  .+  Y
) )
273, 20, 4ispsubclN 33548 . . 3  |-  ( K  e.  HL  ->  (
( X  .+  Y
)  e.  C  <->  ( ( X  .+  Y )  C_  ( Atoms `  K )  /\  (  ._|_  `  (  ._|_  `  ( X  .+  Y ) ) )  =  ( X  .+  Y ) ) ) )
281, 27syl 17 . 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 938 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 189    /\ wa 375    /\ w3a 991    = wceq 1455    e. wcel 1898    =/= wne 2633    C_ wss 3416   (/)c0 3743   ` cfv 5605  (class class class)co 6320   Atomscatm 32875   HLchlt 32962   +Pcpadd 33406   _|_PcpolN 33513   PSubClcpscN 33545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4531  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615  ax-riotaBAD 32571
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-fal 1461  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-iun 4294  df-iin 4295  df-br 4419  df-opab 4478  df-mpt 4479  df-id 4771  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-riota 6282  df-ov 6323  df-oprab 6324  df-mpt2 6325  df-1st 6825  df-2nd 6826  df-undef 7051  df-preset 16228  df-poset 16246  df-plt 16259  df-lub 16275  df-glb 16276  df-join 16277  df-meet 16278  df-p0 16340  df-p1 16341  df-lat 16347  df-clat 16409  df-oposet 32788  df-ol 32790  df-oml 32791  df-covers 32878  df-ats 32879  df-atl 32910  df-cvlat 32934  df-hlat 32963  df-psubsp 33114  df-pmap 33115  df-padd 33407  df-polarityN 33514  df-psubclN 33546
This theorem is referenced by:  pmapojoinN  33579  pexmidN  33580
  Copyright terms: Public domain W3C validator