Theorem psubspi2N 35034

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

Hypotheses

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

Assertion

Ref	Expression
psubspi2N	|- ( ( ( K e. D /\ X e. S /\ P e. A ) /\ ( Q e. X /\ R e. X /\ P .<_ ( Q .\/ R ) ) ) -> P e. X )

Proof of Theorem psubspi2N

Dummy variables r q are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 6657	. . . 4 |- ( q = Q -> ( q .\/ r ) = ( Q .\/ r ) )
2	1	breq2d 4665	. . 3 |- ( q = Q -> ( P .<_ ( q .\/ r ) <-> P .<_ ( Q .\/ r ) ) )
3		oveq2 6658	. . . 4 |- ( r = R -> ( Q .\/ r ) = ( Q .\/ R ) )
4	3	breq2d 4665	. . 3 |- ( r = R -> ( P .<_ ( Q .\/ r ) <-> P .<_ ( Q .\/ R ) ) )
5	2, 4	rspc2ev 3324	. 2 |- ( ( Q e. X /\ R e. X /\ P .<_ ( Q .\/ R ) ) -> E. q e. X E. r e. X P .<_ ( q .\/ r ) )
6		psubspset.l	. . 3 |- .<_ = ( le ` K )
7		psubspset.j	. . 3 |- .\/ = ( join ` K )
8		psubspset.a	. . 3 |- A = ( Atoms ` K )
9		psubspset.s	. . 3 |- S = ( PSubSp ` K )
10	6, 7, 8, 9	psubspi 35033	. 2 |- ( ( ( K e. D /\ X e. S /\ P e. A ) /\ E. q e. X E. r e. X P .<_ ( q .\/ r ) ) -> P e. X )
11	5, 10	sylan2 491	1 |- ( ( ( K e. D /\ X e. S /\ P e. A ) /\ ( Q e. X /\ R e. X /\ P .<_ ( Q .\/ R ) ) ) -> P e. X )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   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:  pclclN  35177  pclfinN  35186  pclfinclN  35236