Theorem elpadd0 35095

Description: Member of projective subspace sum with at least one empty set. (Contributed by NM, 29-Dec-2011.)

Hypotheses

Ref	Expression
padd0.a	|- A = ( Atoms ` K )
padd0.p	|- .+ = ( +P ` K )

Assertion

Ref	Expression
elpadd0	|- ( ( ( K e. B /\ X C_ A /\ Y C_ A ) /\ -. ( X =/= (/) /\ Y =/= (/) ) ) -> ( S e. ( X .+ Y ) <-> ( S e. X \/ S e. Y ) ) )

Proof of Theorem elpadd0

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

Step	Hyp	Ref	Expression
1		neanior 2886	. . . 4 |- ( ( X =/= (/) /\ Y =/= (/) ) <-> -. ( X = (/) \/ Y = (/) ) )
2	1	bicomi 214	. . 3 |- ( -. ( X = (/) \/ Y = (/) ) <-> ( X =/= (/) /\ Y =/= (/) ) )
3	2	con1bii 346	. 2 |- ( -. ( X =/= (/) /\ Y =/= (/) ) <-> ( X = (/) \/ Y = (/) ) )
4		eqid 2622	. . . 4 |- ( le ` K ) = ( le ` K )
5		eqid 2622	. . . 4 |- ( join ` K ) = ( join ` K )
6		padd0.a	. . . 4 |- A = ( Atoms ` K )
7		padd0.p	. . . 4 |- .+ = ( +P ` K )
8	4, 5, 6, 7	elpadd 35085	. . 3 |- ( ( K e. B /\ X C_ A /\ Y C_ A ) -> ( S e. ( X .+ Y ) <-> ( ( S e. X \/ S e. Y ) \/ ( S e. A /\ E. q e. X E. r e. Y S ( le ` K ) ( q ( join ` K ) r ) ) ) ) )
9		rex0 3938	. . . . . . . 8 |- -. E. q e. (/) E. r e. Y S ( le ` K ) ( q ( join ` K ) r )
10		rexeq 3139	. . . . . . . 8 |- ( X = (/) -> ( E. q e. X E. r e. Y S ( le ` K ) ( q ( join ` K ) r ) <-> E. q e. (/) E. r e. Y S ( le ` K ) ( q ( join ` K ) r ) ) )
11	9, 10	mtbiri 317	. . . . . . 7 |- ( X = (/) -> -. E. q e. X E. r e. Y S ( le ` K ) ( q ( join ` K ) r ) )
12		rex0 3938	. . . . . . . . . 10 |- -. E. r e. (/) S ( le ` K ) ( q ( join ` K ) r )
13	12	a1i 11	. . . . . . . . 9 |- ( q e. X -> -. E. r e. (/) S ( le ` K ) ( q ( join ` K ) r ) )
14	13	nrex 3000	. . . . . . . 8 |- -. E. q e. X E. r e. (/) S ( le ` K ) ( q ( join ` K ) r )
15		rexeq 3139	. . . . . . . . 9 |- ( Y = (/) -> ( E. r e. Y S ( le ` K ) ( q ( join ` K ) r ) <-> E. r e. (/) S ( le ` K ) ( q ( join ` K ) r ) ) )
16	15	rexbidv 3052	. . . . . . . 8 |- ( Y = (/) -> ( E. q e. X E. r e. Y S ( le ` K ) ( q ( join ` K ) r ) <-> E. q e. X E. r e. (/) S ( le ` K ) ( q ( join ` K ) r ) ) )
17	14, 16	mtbiri 317	. . . . . . 7 |- ( Y = (/) -> -. E. q e. X E. r e. Y S ( le ` K ) ( q ( join ` K ) r ) )
18	11, 17	jaoi 394	. . . . . 6 |- ( ( X = (/) \/ Y = (/) ) -> -. E. q e. X E. r e. Y S ( le ` K ) ( q ( join ` K ) r ) )
19	18	intnand 962	. . . . 5 |- ( ( X = (/) \/ Y = (/) ) -> -. ( S e. A /\ E. q e. X E. r e. Y S ( le ` K ) ( q ( join ` K ) r ) ) )
20		biorf 420	. . . . 5 |- ( -. ( S e. A /\ E. q e. X E. r e. Y S ( le ` K ) ( q ( join ` K ) r ) ) -> ( ( S e. X \/ S e. Y ) <-> ( ( S e. A /\ E. q e. X E. r e. Y S ( le ` K ) ( q ( join ` K ) r ) ) \/ ( S e. X \/ S e. Y ) ) ) )
21	19, 20	syl 17	. . . 4 |- ( ( X = (/) \/ Y = (/) ) -> ( ( S e. X \/ S e. Y ) <-> ( ( S e. A /\ E. q e. X E. r e. Y S ( le ` K ) ( q ( join ` K ) r ) ) \/ ( S e. X \/ S e. Y ) ) ) )
22		orcom 402	. . . 4 |- ( ( ( S e. A /\ E. q e. X E. r e. Y S ( le ` K ) ( q ( join ` K ) r ) ) \/ ( S e. X \/ S e. Y ) ) <-> ( ( S e. X \/ S e. Y ) \/ ( S e. A /\ E. q e. X E. r e. Y S ( le ` K ) ( q ( join ` K ) r ) ) ) )
23	21, 22	syl6rbb 277	. . 3 |- ( ( X = (/) \/ Y = (/) ) -> ( ( ( S e. X \/ S e. Y ) \/ ( S e. A /\ E. q e. X E. r e. Y S ( le ` K ) ( q ( join ` K ) r ) ) ) <-> ( S e. X \/ S e. Y ) ) )
24	8, 23	sylan9bb 736	. 2 |- ( ( ( K e. B /\ X C_ A /\ Y C_ A ) /\ ( X = (/) \/ Y = (/) ) ) -> ( S e. ( X .+ Y ) <-> ( S e. X \/ S e. Y ) ) )
25	3, 24	sylan2b 492	1 |- ( ( ( K e. B /\ X C_ A /\ Y C_ A ) /\ -. ( X =/= (/) /\ Y =/= (/) ) ) -> ( S e. ( X .+ Y ) <-> ( S e. X \/ S e. Y ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ 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   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-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-padd 35082

This theorem is referenced by:  paddval0  35096