Theorem elpadd2at 35927

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 994	. . 3 |- ( ( K e. Lat /\ Q e. A /\ R e. A ) -> K e. Lat )
2		simp2 995	. . . 4 |- ( ( K e. Lat /\ Q e. A /\ R e. A ) -> Q e. A )
3	2	snssd 4161	. . 3 |- ( ( K e. Lat /\ Q e. A /\ R e. A ) -> { Q } C_ A )
4		simp3 996	. . 3 |- ( ( K e. Lat /\ Q e. A /\ R e. A ) -> R e. A )
5		snnzg 4133	. . . 4 |- ( Q e. A -> { Q } =/= (/) )
6	5	3ad2ant2 1016	. . 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 35925	. . 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 1229	. 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 6277	. . . . . 6 |- ( r = Q -> ( r .\/ R ) = ( Q .\/ R ) )
14	13	breq2d 4451	. . . . 5 |- ( r = Q -> ( S .<_ ( r .\/ R ) <-> S .<_ ( Q .\/ R ) ) )
15	14	rexsng 4052	. . . 4 |- ( Q e. A -> ( E. r e. { Q } S .<_ ( r .\/ R ) <-> S .<_ ( Q .\/ R ) ) )
16	15	3ad2ant2 1016	. . 3 |- ( ( K e. Lat /\ Q e. A /\ R e. A ) -> ( E. r e. { Q } S .<_ ( r .\/ R ) <-> S .<_ ( Q .\/ R ) ) )
17	16	anbi2d 701	. 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 367   /\ w3a 971   = wceq 1398   e. wcel 1823   =/= wne 2649   E.wrex 2805   C_ wss 3461   (/)c0 3783   {csn 4016   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   lecple 14791   joincjn 15772   Latclat 15874   Atomscatm 35385   +Pcpadd 35916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-lub 15803  df-join 15805  df-lat 15875  df-ats 35389  df-padd 35917

This theorem is referenced by:  elpadd2at2  35928