Theorem elpadd2at 34620

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 996	. . 3 |- ( ( K e. Lat /\ Q e. A /\ R e. A ) -> K e. Lat )
2		simp2 997	. . . 4 |- ( ( K e. Lat /\ Q e. A /\ R e. A ) -> Q e. A )
3	2	snssd 4172	. . 3 |- ( ( K e. Lat /\ Q e. A /\ R e. A ) -> { Q } C_ A )
4		simp3 998	. . 3 |- ( ( K e. Lat /\ Q e. A /\ R e. A ) -> R e. A )
5		snnzg 4144	. . . 4 |- ( Q e. A -> { Q } =/= (/) )
6	5	3ad2ant2 1018	. . 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 34618	. . 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 1231	. 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 6291	. . . . . 6 |- ( r = Q -> ( r .\/ R ) = ( Q .\/ R ) )
14	13	breq2d 4459	. . . . 5 |- ( r = Q -> ( S .<_ ( r .\/ R ) <-> S .<_ ( Q .\/ R ) ) )
15	14	rexsng 4063	. . . 4 |- ( Q e. A -> ( E. r e. { Q } S .<_ ( r .\/ R ) <-> S .<_ ( Q .\/ R ) ) )
16	15	3ad2ant2 1018	. . 3 |- ( ( K e. Lat /\ Q e. A /\ R e. A ) -> ( E. r e. { Q } S .<_ ( r .\/ R ) <-> S .<_ ( Q .\/ R ) ) )
17	16	anbi2d 703	. 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 253	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 184   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   =/= wne 2662   E.wrex 2815   C_ wss 3476   (/)c0 3785   {csn 4027   class class class wbr 4447   ` cfv 5588  (class class class)co 6284   lecple 14562   joincjn 15431   Latclat 15532   Atomscatm 34078   +Pcpadd 34609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785  df-lub 15461  df-join 15463  df-lat 15533  df-ats 34082  df-padd 34610

This theorem is referenced by:  elpadd2at2  34621