Theorem elpadd2at 35092

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 1061	. . 3 |- ( ( K e. Lat /\ Q e. A /\ R e. A ) -> K e. Lat )
2		simp2 1062	. . . 4 |- ( ( K e. Lat /\ Q e. A /\ R e. A ) -> Q e. A )
3	2	snssd 4340	. . 3 |- ( ( K e. Lat /\ Q e. A /\ R e. A ) -> { Q } C_ A )
4		simp3 1063	. . 3 |- ( ( K e. Lat /\ Q e. A /\ R e. A ) -> R e. A )
5		snnzg 4308	. . . 4 |- ( Q e. A -> { Q } =/= (/) )
6	5	3ad2ant2 1083	. . 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 35090	. . 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 1329	. 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 6657	. . . . . 6 |- ( r = Q -> ( r .\/ R ) = ( Q .\/ R ) )
14	13	breq2d 4665	. . . . 5 |- ( r = Q -> ( S .<_ ( r .\/ R ) <-> S .<_ ( Q .\/ R ) ) )
15	14	rexsng 4219	. . . 4 |- ( Q e. A -> ( E. r e. { Q } S .<_ ( r .\/ R ) <-> S .<_ ( Q .\/ R ) ) )
16	15	3ad2ant2 1083	. . 3 |- ( ( K e. Lat /\ Q e. A /\ R e. A ) -> ( E. r e. { Q } S .<_ ( r .\/ R ) <-> S .<_ ( Q .\/ R ) ) )
17	16	anbi2d 740	. 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 268	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 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   C_ wss 3574   (/)c0 3915   {csn 4177   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   lecple 15948   joincjn 16944   Latclat 17045   Atomscatm 34550   +Pcpadd 35081

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  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-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-lub 16974  df-join 16976  df-lat 17046  df-ats 34554  df-padd 35082

This theorem is referenced by:  elpadd2at2  35093