Theorem psubspi 35033

Description: Property of a projective subspace. (Contributed by NM, 13-Jan-2012.)

Hypotheses

Ref	Expression
psubspset.l	|- .<_ = ( le ` K )
psubspset.j	|- .\/ = ( join ` K )
psubspset.a	|- A = ( Atoms ` K )
psubspset.s	|- S = ( PSubSp ` K )

Assertion

Ref	Expression
psubspi	|- ( ( ( K e. D /\ X e. S /\ P e. A ) /\ E. q e. X E. r e. X P .<_ ( q .\/ r ) ) -> P e. X )

Distinct variable groups:   A, r, q   K, q, r   X, q, r   A, q   P, q, r

Allowed substitution hints:   D( r, q)   S( r, q)   .\/ ( r, q)   .<_ ( r, q)

Proof of Theorem psubspi

Dummy variable p is distinct from all other variables.

Step	Hyp	Ref	Expression
1		psubspset.l	. . . . . 6 |- .<_ = ( le ` K )
2		psubspset.j	. . . . . 6 |- .\/ = ( join ` K )
3		psubspset.a	. . . . . 6 |- A = ( Atoms ` K )
4		psubspset.s	. . . . . 6 |- S = ( PSubSp ` K )
5	1, 2, 3, 4	ispsubsp2 35032	. . . . 5 |- ( K e. D -> ( X e. S <-> ( X C_ A /\ A. p e. A ( E. q e. X E. r e. X p .<_ ( q .\/ r ) -> p e. X ) ) ) )
6	5	simplbda 654	. . . 4 |- ( ( K e. D /\ X e. S ) -> A. p e. A ( E. q e. X E. r e. X p .<_ ( q .\/ r ) -> p e. X ) )
7	6	ex 450	. . 3 |- ( K e. D -> ( X e. S -> A. p e. A ( E. q e. X E. r e. X p .<_ ( q .\/ r ) -> p e. X ) ) )
8		breq1 4656	. . . . . 6 |- ( p = P -> ( p .<_ ( q .\/ r ) <-> P .<_ ( q .\/ r ) ) )
9	8	2rexbidv 3057	. . . . 5 |- ( p = P -> ( E. q e. X E. r e. X p .<_ ( q .\/ r ) <-> E. q e. X E. r e. X P .<_ ( q .\/ r ) ) )
10		eleq1 2689	. . . . 5 |- ( p = P -> ( p e. X <-> P e. X ) )
11	9, 10	imbi12d 334	. . . 4 |- ( p = P -> ( ( E. q e. X E. r e. X p .<_ ( q .\/ r ) -> p e. X ) <-> ( E. q e. X E. r e. X P .<_ ( q .\/ r ) -> P e. X ) ) )
12	11	rspccv 3306	. . 3 |- ( A. p e. A ( E. q e. X E. r e. X p .<_ ( q .\/ r ) -> p e. X ) -> ( P e. A -> ( E. q e. X E. r e. X P .<_ ( q .\/ r ) -> P e. X ) ) )
13	7, 12	syl6 35	. 2 |- ( K e. D -> ( X e. S -> ( P e. A -> ( E. q e. X E. r e. X P .<_ ( q .\/ r ) -> P e. X ) ) ) )
14	13	3imp1 1280	1 |- ( ( ( K e. D /\ X e. S /\ P e. A ) /\ E. q e. X E. r e. X P .<_ ( q .\/ r ) ) -> P e. X )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   lecple 15948   joincjn 16944   Atomscatm 34550   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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-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-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-psubsp 34789

This theorem is referenced by:  psubspi2N  35034  paddidm  35127