Theorem elpaddri 35088

Description: Condition implying membership in a projective subspace sum. (Contributed by NM, 8-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
elpaddri	|- ( ( ( K e. Lat /\ X C_ A /\ Y C_ A ) /\ ( Q e. X /\ R e. Y ) /\ ( S e. A /\ S .<_ ( Q .\/ R ) ) ) -> S e. ( X .+ Y ) )

Proof of Theorem elpaddri

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

Step	Hyp	Ref	Expression
1		simp3l 1089	. 2 |- ( ( ( K e. Lat /\ X C_ A /\ Y C_ A ) /\ ( Q e. X /\ R e. Y ) /\ ( S e. A /\ S .<_ ( Q .\/ R ) ) ) -> S e. A )
2		simp2l 1087	. . 3 |- ( ( ( K e. Lat /\ X C_ A /\ Y C_ A ) /\ ( Q e. X /\ R e. Y ) /\ ( S e. A /\ S .<_ ( Q .\/ R ) ) ) -> Q e. X )
3		simp2r 1088	. . 3 |- ( ( ( K e. Lat /\ X C_ A /\ Y C_ A ) /\ ( Q e. X /\ R e. Y ) /\ ( S e. A /\ S .<_ ( Q .\/ R ) ) ) -> R e. Y )
4		simp3r 1090	. . 3 |- ( ( ( K e. Lat /\ X C_ A /\ Y C_ A ) /\ ( Q e. X /\ R e. Y ) /\ ( S e. A /\ S .<_ ( Q .\/ R ) ) ) -> S .<_ ( Q .\/ R ) )
5		oveq1 6657	. . . . 5 |- ( q = Q -> ( q .\/ r ) = ( Q .\/ r ) )
6	5	breq2d 4665	. . . 4 |- ( q = Q -> ( S .<_ ( q .\/ r ) <-> S .<_ ( Q .\/ r ) ) )
7		oveq2 6658	. . . . 5 |- ( r = R -> ( Q .\/ r ) = ( Q .\/ R ) )
8	7	breq2d 4665	. . . 4 |- ( r = R -> ( S .<_ ( Q .\/ r ) <-> S .<_ ( Q .\/ R ) ) )
9	6, 8	rspc2ev 3324	. . 3 |- ( ( Q e. X /\ R e. Y /\ S .<_ ( Q .\/ R ) ) -> E. q e. X E. r e. Y S .<_ ( q .\/ r ) )
10	2, 3, 4, 9	syl3anc 1326	. 2 |- ( ( ( K e. Lat /\ X C_ A /\ Y C_ A ) /\ ( Q e. X /\ R e. Y ) /\ ( S e. A /\ S .<_ ( Q .\/ R ) ) ) -> E. q e. X E. r e. Y S .<_ ( q .\/ r ) )
11		ne0i 3921	. . . . . 6 |- ( Q e. X -> X =/= (/) )
12		ne0i 3921	. . . . . 6 |- ( R e. Y -> Y =/= (/) )
13	11, 12	anim12i 590	. . . . 5 |- ( ( Q e. X /\ R e. Y ) -> ( X =/= (/) /\ Y =/= (/) ) )
14	13	anim2i 593	. . . 4 |- ( ( ( K e. Lat /\ X C_ A /\ Y C_ A ) /\ ( Q e. X /\ R e. Y ) ) -> ( ( K e. Lat /\ X C_ A /\ Y C_ A ) /\ ( X =/= (/) /\ Y =/= (/) ) ) )
15	14	3adant3 1081	. . 3 |- ( ( ( K e. Lat /\ X C_ A /\ Y C_ A ) /\ ( Q e. X /\ R e. Y ) /\ ( S e. A /\ S .<_ ( Q .\/ R ) ) ) -> ( ( K e. Lat /\ X C_ A /\ Y C_ A ) /\ ( X =/= (/) /\ Y =/= (/) ) ) )
16		paddfval.l	. . . 4 |- .<_ = ( le ` K )
17		paddfval.j	. . . 4 |- .\/ = ( join ` K )
18		paddfval.a	. . . 4 |- A = ( Atoms ` K )
19		paddfval.p	. . . 4 |- .+ = ( +P ` K )
20	16, 17, 18, 19	elpaddn0 35086	. . 3 |- ( ( ( K e. Lat /\ X C_ A /\ Y C_ A ) /\ ( X =/= (/) /\ Y =/= (/) ) ) -> ( S e. ( X .+ Y ) <-> ( S e. A /\ E. q e. X E. r e. Y S .<_ ( q .\/ r ) ) ) )
21	15, 20	syl 17	. 2 |- ( ( ( K e. Lat /\ X C_ A /\ Y C_ A ) /\ ( Q e. X /\ R e. Y ) /\ ( S e. A /\ S .<_ ( Q .\/ R ) ) ) -> ( S e. ( X .+ Y ) <-> ( S e. A /\ E. q e. X E. r e. Y S .<_ ( q .\/ r ) ) ) )
22	1, 10, 21	mpbir2and 957	1 |- ( ( ( K e. Lat /\ X C_ A /\ Y C_ A ) /\ ( Q e. X /\ R e. Y ) /\ ( S e. A /\ S .<_ ( Q .\/ R ) ) ) -> S e. ( X .+ Y ) )

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   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:  elpaddatriN  35089  paddasslem8  35113  paddasslem12  35117  paddasslem13  35118  pmodlem1  35132  osumcllem5N  35246  pexmidlem2N  35257