Theorem pointpsubN 35037

Description: A point (singleton of an atom) is a projective subspace. Remark below Definition 15.1 of [MaedaMaeda] p. 61. (Contributed by NM, 13-Oct-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
pointpsub.p	|- P = ( Points ` K )
pointpsub.s	|- S = ( PSubSp ` K )

Assertion

Ref	Expression
pointpsubN	|- ( ( K e. AtLat /\ X e. P ) -> X e. S )

Proof of Theorem pointpsubN

Dummy variable q is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . 4 |- ( Atoms ` K ) = ( Atoms ` K )
2		pointpsub.p	. . . 4 |- P = ( Points ` K )
3	1, 2	ispointN 35028	. . 3 |- ( K e. AtLat -> ( X e. P <-> E. q e. ( Atoms ` K ) X = { q } ) )
4		pointpsub.s	. . . . . . 7 |- S = ( PSubSp ` K )
5	1, 4	snatpsubN 35036	. . . . . 6 |- ( ( K e. AtLat /\ q e. ( Atoms ` K ) ) -> { q } e. S )
6	5	ex 450	. . . . 5 |- ( K e. AtLat -> ( q e. ( Atoms ` K ) -> { q } e. S ) )
7		eleq1a 2696	. . . . 5 |- ( { q } e. S -> ( X = { q } -> X e. S ) )
8	6, 7	syl6 35	. . . 4 |- ( K e. AtLat -> ( q e. ( Atoms ` K ) -> ( X = { q } -> X e. S ) ) )
9	8	rexlimdv 3030	. . 3 |- ( K e. AtLat -> ( E. q e. ( Atoms ` K ) X = { q } -> X e. S ) )
10	3, 9	sylbid 230	. 2 |- ( K e. AtLat -> ( X e. P -> X e. S ) )
11	10	imp 445	1 |- ( ( K e. AtLat /\ X e. P ) -> X e. S )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   {csn 4177   ` cfv 5888   Atomscatm 34550   AtLatcal 34551   PointscpointsN 34781   PSubSpcpsubsp 34782

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

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-ral 2917  df-rex 2918  df-reu 2919  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-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-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-lat 17046  df-covers 34553  df-ats 34554  df-atl 34585  df-pointsN 34788  df-psubsp 34789

This theorem is referenced by: (None)