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Theorem elpadd0 30668
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.
StepHypRef Expression
1 neanior 2691 . . . 4  |-  ( ( X  =/=  (/)  /\  Y  =/=  (/) )  <->  -.  ( X  =  (/)  \/  Y  =  (/) ) )
21bicomi 195 . . 3  |-  ( -.  ( X  =  (/)  \/  Y  =  (/) )  <->  ( X  =/=  (/)  /\  Y  =/=  (/) ) )
32con1bii 323 . 2  |-  ( -.  ( X  =/=  (/)  /\  Y  =/=  (/) )  <->  ( X  =  (/)  \/  Y  =  (/) ) )
4 eqid 2438 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
5 eqid 2438 . . . 4  |-  ( join `  K )  =  (
join `  K )
6 padd0.a . . . 4  |-  A  =  ( Atoms `  K )
7 padd0.p . . . 4  |-  .+  =  ( + P `  K
)
84, 5, 6, 7elpadd 30658 . . 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 3643 . . . . . . . 8  |-  -.  E. q  e.  (/)  E. r  e.  Y  S ( le `  K ) ( q ( join `  K
) r )
10 rexeq 2907 . . . . . . . 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 ) ) )
119, 10mtbiri 296 . . . . . . 7  |-  ( X  =  (/)  ->  -.  E. q  e.  X  E. r  e.  Y  S
( le `  K
) ( q (
join `  K )
r ) )
12 rex0 3643 . . . . . . . . . 10  |-  -.  E. r  e.  (/)  S ( le `  K ) ( q ( join `  K ) r )
1312a1i 11 . . . . . . . . 9  |-  ( q  e.  X  ->  -.  E. r  e.  (/)  S ( le `  K ) ( q ( join `  K ) r ) )
1413nrex 2810 . . . . . . . 8  |-  -.  E. q  e.  X  E. r  e.  (/)  S ( le `  K ) ( q ( join `  K ) r )
15 rexeq 2907 . . . . . . . . 9  |-  ( Y  =  (/)  ->  ( E. r  e.  Y  S
( le `  K
) ( q (
join `  K )
r )  <->  E. r  e.  (/)  S ( le
`  K ) ( q ( join `  K
) r ) ) )
1615rexbidv 2728 . . . . . . . 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 ) ) )
1714, 16mtbiri 296 . . . . . . 7  |-  ( Y  =  (/)  ->  -.  E. q  e.  X  E. r  e.  Y  S
( le `  K
) ( q (
join `  K )
r ) )
1811, 17jaoi 370 . . . . . 6  |-  ( ( X  =  (/)  \/  Y  =  (/) )  ->  -.  E. q  e.  X  E. r  e.  Y  S
( le `  K
) ( q (
join `  K )
r ) )
1918intnand 884 . . . . 5  |-  ( ( X  =  (/)  \/  Y  =  (/) )  ->  -.  ( S  e.  A  /\  E. q  e.  X  E. r  e.  Y  S ( le `  K ) ( q ( join `  K
) r ) ) )
20 biorf 396 . . . . 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
) ) ) )
2119, 20syl 16 . . . 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 378 . . . 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 ) ) ) )
2321, 22syl6rbb 255 . . 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
) ) )
248, 23sylan9bb 682 . 2  |-  ( ( ( K  e.  B  /\  X  C_  A  /\  Y  C_  A )  /\  ( X  =  (/)  \/  Y  =  (/) ) )  -> 
( S  e.  ( X  .+  Y )  <-> 
( S  e.  X  \/  S  e.  Y
) ) )
253, 24sylan2b 463 1  |-  ( ( ( K  e.  B  /\  X  C_  A  /\  Y  C_  A )  /\  -.  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  -> 
( S  e.  ( X  .+  Y )  <-> 
( S  e.  X  \/  S  e.  Y
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708    C_ wss 3322   (/)c0 3630   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   lecple 13538   joincjn 14403   Atomscatm 30123   + Pcpadd 30654
This theorem is referenced by:  paddval0  30669
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-padd 30655
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