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

Step	Hyp	Ref	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 )
5	3, 4	psubclssatN 36117	. . . 4 |- ( ( K e. HL /\ X e. C ) -> X C_ ( Atoms ` K ) )
6	1, 2, 5	syl2anc 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 )
8	3, 4	psubclssatN 36117	. . . 4 |- ( ( K e. HL /\ Y e. C ) -> Y C_ ( Atoms ` K ) )
9	1, 7, 8	syl2anc 659	. . 3 |- ( ( ( K e. HL /\ X e. C /\ Y e. C ) /\ X C_ ( ._|_ ` Y ) ) -> Y C_ ( Atoms ` K ) )
10		osumcl.p	. . . 4 |- .+ = ( +P ` K )
11	3, 10	paddssat 35990	. . 3 |- ( ( K e. HL /\ X C_ ( Atoms ` K ) /\ Y C_ ( Atoms ` K ) ) -> ( X .+ Y ) C_ ( Atoms ` K ) )
12	1, 6, 9, 11	syl3anc 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 ) )
15	3, 10	padd02 35988	. . . . . . 7 |- ( ( K e. HL /\ Y C_ ( Atoms ` K ) ) -> ( (/) .+ Y ) = Y )
16	1, 9, 15	syl2anc 659	. . . . . 6 |- ( ( ( K e. HL /\ X e. C /\ Y e. C ) /\ X C_ ( ._|_ ` Y ) ) -> ( (/) .+ Y ) = Y )
17	14, 16	sylan9eqr 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 )
19	17, 18	eqeltrd 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 )
21	20, 4	psubcli2N 36115	. . . 4 |- ( ( K e. HL /\ ( X .+ Y ) e. C ) -> ( ._|_ ` ( ._|_ ` ( X .+ Y ) ) ) = ( X .+ Y ) )
22	13, 19, 21	syl2anc 659	. . 3 |- ( ( ( ( K e. HL /\ X e. C /\ Y e. C ) /\ X C_ ( ._|_ ` Y ) ) /\ X = (/) ) -> ( ._|_ ` ( ._|_ ` ( X .+ Y ) ) ) = ( X .+ Y ) )
23	10, 20, 4	osumcllem11N 36142	. . . . 5 |- ( ( ( K e. HL /\ X e. C /\ Y e. C ) /\ ( X C_ ( ._|_ ` Y ) /\ X =/= (/) ) ) -> ( X .+ Y ) = ( ._|_ ` ( ._|_ ` ( X .+ Y ) ) ) )
24	23	anassrs 646	. . . 4 |- ( ( ( ( K e. HL /\ X e. C /\ Y e. C ) /\ X C_ ( ._|_ ` Y ) ) /\ X =/= (/) ) -> ( X .+ Y ) = ( ._|_ ` ( ._|_ ` ( X .+ Y ) ) ) )
25	24	eqcomd 2404	. . 3 |- ( ( ( ( K e. HL /\ X e. C /\ Y e. C ) /\ X C_ ( ._|_ ` Y ) ) /\ X =/= (/) ) -> ( ._|_ ` ( ._|_ ` ( X .+ Y ) ) ) = ( X .+ Y ) )
26	22, 25	pm2.61dane 2714	. 2 |- ( ( ( K e. HL /\ X e. C /\ Y e. C ) /\ X C_ ( ._|_ ` Y ) ) -> ( ._|_ ` ( ._|_ ` ( X .+ Y ) ) ) = ( X .+ Y ) )
27	3, 20, 4	ispsubclN 36113	. . 3 |- ( K e. HL -> ( ( X .+ Y ) e. C <-> ( ( X .+ Y ) C_ ( Atoms ` K ) /\ ( ._|_ ` ( ._|_ ` ( X .+ Y ) ) ) = ( X .+ Y ) ) ) )
28	1, 27	syl 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 ) ) ) )
29	12, 26, 28	mpbir2and 920	1 |- ( ( ( K e. HL /\ X e. C /\ Y e. C ) /\ X C_ ( ._|_ ` Y ) ) -> ( X .+ Y ) e. C )

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