Theorem elpadd2at 33372

Description: Membership in a projective subspace sum of two points. (Contributed by NM, 29-Jan-2012.)

Hypotheses

Ref	Expression
paddfval.l	|- .<_ = ( le ` K )
paddfval.j	|- .\/ = ( join ` K )
paddfval.a	|- A = ( Atoms ` K )
paddfval.p	|- .+ = ( +P ` K )

Assertion

Ref	Expression
elpadd2at	|- ( ( K e. Lat /\ Q e. A /\ R e. A ) -> ( S e. ( { Q } .+ { R } ) <-> ( S e. A /\ S .<_ ( Q .\/ R ) ) ) )

Proof of Theorem elpadd2at

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1009	. . 3 |- ( ( K e. Lat /\ Q e. A /\ R e. A ) -> K e. Lat )
2		simp2 1010	. . . 4 |- ( ( K e. Lat /\ Q e. A /\ R e. A ) -> Q e. A )
3	2	snssd 4085	. . 3 |- ( ( K e. Lat /\ Q e. A /\ R e. A ) -> { Q } C_ A )
4		simp3 1011	. . 3 |- ( ( K e. Lat /\ Q e. A /\ R e. A ) -> R e. A )
5		snnzg 4057	. . . 4 |- ( Q e. A -> { Q } =/= (/) )
6	5	3ad2ant2 1031	. . 3 |- ( ( K e. Lat /\ Q e. A /\ R e. A ) -> { Q } =/= (/) )
7		paddfval.l	. . . 4 |- .<_ = ( le ` K )
8		paddfval.j	. . . 4 |- .\/ = ( join ` K )
9		paddfval.a	. . . 4 |- A = ( Atoms ` K )
10		paddfval.p	. . . 4 |- .+ = ( +P ` K )
11	7, 8, 9, 10	elpaddat 33370	. . 3 |- ( ( ( K e. Lat /\ { Q } C_ A /\ R e. A ) /\ { Q } =/= (/) ) -> ( S e. ( { Q } .+ { R } ) <-> ( S e. A /\ E. r e. { Q } S .<_ ( r .\/ R ) ) ) )
12	1, 3, 4, 6, 11	syl31anc 1274	. 2 |- ( ( K e. Lat /\ Q e. A /\ R e. A ) -> ( S e. ( { Q } .+ { R } ) <-> ( S e. A /\ E. r e. { Q } S .<_ ( r .\/ R ) ) ) )
13		oveq1 6282	. . . . . 6 |- ( r = Q -> ( r .\/ R ) = ( Q .\/ R ) )
14	13	breq2d 4385	. . . . 5 |- ( r = Q -> ( S .<_ ( r .\/ R ) <-> S .<_ ( Q .\/ R ) ) )
15	14	rexsng 3974	. . . 4 |- ( Q e. A -> ( E. r e. { Q } S .<_ ( r .\/ R ) <-> S .<_ ( Q .\/ R ) ) )
16	15	3ad2ant2 1031	. . 3 |- ( ( K e. Lat /\ Q e. A /\ R e. A ) -> ( E. r e. { Q } S .<_ ( r .\/ R ) <-> S .<_ ( Q .\/ R ) ) )
17	16	anbi2d 715	. 2 |- ( ( K e. Lat /\ Q e. A /\ R e. A ) -> ( ( S e. A /\ E. r e. { Q } S .<_ ( r .\/ R ) ) <-> ( S e. A /\ S .<_ ( Q .\/ R ) ) ) )
18	12, 17	bitrd 261	1 |- ( ( K e. Lat /\ Q e. A /\ R e. A ) -> ( S e. ( { Q } .+ { R } ) <-> ( S e. A /\ S .<_ ( Q .\/ R ) ) ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   /\ w3a 986   = wceq 1447   e. wcel 1890   =/= wne 2621   E.wrex 2737   C_ wss 3371   (/)c0 3698   {csn 3935   class class class wbr 4373   ` cfv 5560  (class class class)co 6275   lecple 15207   joincjn 16199   Latclat 16301   Atomscatm 32830   +Pcpadd 33361

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1672  ax-4 1685  ax-5 1761  ax-6 1808  ax-7 1854  ax-8 1892  ax-9 1899  ax-10 1918  ax-11 1923  ax-12 1936  ax-13 2091  ax-ext 2431  ax-rep 4486  ax-sep 4496  ax-nul 4505  ax-pow 4553  ax-pr 4611  ax-un 6570

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 988  df-tru 1450  df-ex 1667  df-nf 1671  df-sb 1801  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3014  df-sbc 3235  df-csb 3331  df-dif 3374  df-un 3376  df-in 3378  df-ss 3385  df-nul 3699  df-if 3849  df-pw 3920  df-sn 3936  df-pr 3938  df-op 3942  df-uni 4168  df-iun 4249  df-br 4374  df-opab 4433  df-mpt 4434  df-id 4726  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5524  df-fun 5562  df-fn 5563  df-f 5564  df-f1 5565  df-fo 5566  df-f1o 5567  df-fv 5568  df-riota 6237  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6780  df-2nd 6781  df-lub 16230  df-join 16232  df-lat 16302  df-ats 32834  df-padd 33362

This theorem is referenced by:  elpadd2at2  33373