Theorem psubclsubN 35226

Description: A closed projective subspace is a projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
psubclsub.s	|- S = ( PSubSp ` K )
psubclsub.c	|- C = ( PSubCl ` K )

Assertion

Ref	Expression
psubclsubN	|- ( ( K e. HL /\ X e. C ) -> X e. S )

Proof of Theorem psubclsubN

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- ( _|_P ` K ) = ( _|_P ` K )
2		psubclsub.c	. . 3 |- C = ( PSubCl ` K )
3	1, 2	psubcli2N 35225	. 2 |- ( ( K e. HL /\ X e. C ) -> ( ( _|_P ` K ) ` ( ( _|_P ` K ) ` X ) ) = X )
4		eqid 2622	. . . . . . 7 |- ( Atoms ` K ) = ( Atoms ` K )
5	4, 1, 2	psubcliN 35224	. . . . . 6 |- ( ( K e. HL /\ X e. C ) -> ( X C_ ( Atoms ` K ) /\ ( ( _|_P ` K ) ` ( ( _|_P ` K ) ` X ) ) = X ) )
6	5	simpld 475	. . . . 5 |- ( ( K e. HL /\ X e. C ) -> X C_ ( Atoms ` K ) )
7		psubclsub.s	. . . . . 6 |- S = ( PSubSp ` K )
8	4, 7, 1	polsubN 35193	. . . . 5 |- ( ( K e. HL /\ X C_ ( Atoms ` K ) ) -> ( ( _|_P ` K ) ` X ) e. S )
9	6, 8	syldan 487	. . . 4 |- ( ( K e. HL /\ X e. C ) -> ( ( _|_P ` K ) ` X ) e. S )
10	4, 7	psubssat 35040	. . . 4 |- ( ( K e. HL /\ ( ( _|_P ` K ) ` X ) e. S ) -> ( ( _|_P ` K ) ` X ) C_ ( Atoms ` K ) )
11	9, 10	syldan 487	. . 3 |- ( ( K e. HL /\ X e. C ) -> ( ( _|_P ` K ) ` X ) C_ ( Atoms ` K ) )
12	4, 7, 1	polsubN 35193	. . 3 |- ( ( K e. HL /\ ( ( _|_P ` K ) ` X ) C_ ( Atoms ` K ) ) -> ( ( _|_P ` K ) ` ( ( _|_P ` K ) ` X ) ) e. S )
13	11, 12	syldan 487	. 2 |- ( ( K e. HL /\ X e. C ) -> ( ( _|_P ` K ) ` ( ( _|_P ` K ) ` X ) ) e. S )
14	3, 13	eqeltrrd 2702	1 |- ( ( K e. HL /\ X e. C ) -> X e. S )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   ` cfv 5888   Atomscatm 34550   HLchlt 34637   PSubSpcpsubsp 34782   _|_PcpolN 35188   PSubClcpscN 35220

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-riotaBAD 34239

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-undef 7399  df-preset 16928  df-poset 16946  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p1 17040  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-psubsp 34789  df-pmap 34790  df-polarityN 35189  df-psubclN 35221

This theorem is referenced by:  pclfinclN  35236