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

Step	Hyp	Ref	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 )
5	3, 4	psubclssatN 33924	. . . 4 |- ( ( K e. HL /\ X e. C ) -> X C_ ( Atoms ` K ) )
6	1, 2, 5	syl2anc 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 )
8	3, 4	psubclssatN 33924	. . . 4 |- ( ( K e. HL /\ Y e. C ) -> Y C_ ( Atoms ` K ) )
9	1, 7, 8	syl2anc 661	. . 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 33797	. . 3 |- ( ( K e. HL /\ X C_ ( Atoms ` K ) /\ Y C_ ( Atoms ` K ) ) -> ( X .+ Y ) C_ ( Atoms ` K ) )
12	1, 6, 9, 11	syl3anc 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 ) )
15	3, 10	padd02 33795	. . . . . . 7 |- ( ( K e. HL /\ Y C_ ( Atoms ` K ) ) -> ( (/) .+ Y ) = Y )
16	1, 9, 15	syl2anc 661	. . . . . 6 |- ( ( ( K e. HL /\ X e. C /\ Y e. C ) /\ X C_ ( ._|_ ` Y ) ) -> ( (/) .+ Y ) = Y )
17	14, 16	sylan9eqr 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 )
19	17, 18	eqeltrd 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 )
21	20, 4	psubcli2N 33922	. . . 4 |- ( ( K e. HL /\ ( X .+ Y ) e. C ) -> ( ._|_ ` ( ._|_ ` ( X .+ Y ) ) ) = ( X .+ Y ) )
22	13, 19, 21	syl2anc 661	. . 3 |- ( ( ( ( K e. HL /\ X e. C /\ Y e. C ) /\ X C_ ( ._|_ ` Y ) ) /\ X = (/) ) -> ( ._|_ ` ( ._|_ ` ( X .+ Y ) ) ) = ( X .+ Y ) )
23	10, 20, 4	osumcllem11N 33949	. . . . 5 |- ( ( ( K e. HL /\ X e. C /\ Y e. C ) /\ ( X C_ ( ._|_ ` Y ) /\ X =/= (/) ) ) -> ( X .+ Y ) = ( ._|_ ` ( ._|_ ` ( X .+ Y ) ) ) )
24	23	anassrs 648	. . . 4 |- ( ( ( ( K e. HL /\ X e. C /\ Y e. C ) /\ X C_ ( ._|_ ` Y ) ) /\ X =/= (/) ) -> ( X .+ Y ) = ( ._|_ ` ( ._|_ ` ( X .+ Y ) ) ) )
25	24	eqcomd 2462	. . 3 |- ( ( ( ( K e. HL /\ X e. C /\ Y e. C ) /\ X C_ ( ._|_ ` Y ) ) /\ X =/= (/) ) -> ( ._|_ ` ( ._|_ ` ( X .+ Y ) ) ) = ( X .+ Y ) )
26	22, 25	pm2.61dane 2770	. 2 |- ( ( ( K e. HL /\ X e. C /\ Y e. C ) /\ X C_ ( ._|_ ` Y ) ) -> ( ._|_ ` ( ._|_ ` ( X .+ Y ) ) ) = ( X .+ Y ) )
27	3, 20, 4	ispsubclN 33920	. . 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 913	1 |- ( ( ( K e. HL /\ X e. C /\ Y e. C ) /\ X C_ ( ._|_ ` Y ) ) -> ( X .+ Y ) e. C )

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