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Theorem elpadd0 34480
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 2785 . . . 4  |-  ( ( X  =/=  (/)  /\  Y  =/=  (/) )  <->  -.  ( X  =  (/)  \/  Y  =  (/) ) )
21bicomi 202 . . 3  |-  ( -.  ( X  =  (/)  \/  Y  =  (/) )  <->  ( X  =/=  (/)  /\  Y  =/=  (/) ) )
32con1bii 331 . 2  |-  ( -.  ( X  =/=  (/)  /\  Y  =/=  (/) )  <->  ( X  =  (/)  \/  Y  =  (/) ) )
4 eqid 2460 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
5 eqid 2460 . . . 4  |-  ( join `  K )  =  (
join `  K )
6 padd0.a . . . 4  |-  A  =  ( Atoms `  K )
7 padd0.p . . . 4  |-  .+  =  ( +P `  K
)
84, 5, 6, 7elpadd 34470 . . 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 3792 . . . . . . . 8  |-  -.  E. q  e.  (/)  E. r  e.  Y  S ( le `  K ) ( q ( join `  K
) r )
10 rexeq 3052 . . . . . . . 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 303 . . . . . . 7  |-  ( X  =  (/)  ->  -.  E. q  e.  X  E. r  e.  Y  S
( le `  K
) ( q (
join `  K )
r ) )
12 rex0 3792 . . . . . . . . . 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 2912 . . . . . . . 8  |-  -.  E. q  e.  X  E. r  e.  (/)  S ( le `  K ) ( q ( join `  K ) r )
15 rexeq 3052 . . . . . . . . 9  |-  ( Y  =  (/)  ->  ( E. r  e.  Y  S
( le `  K
) ( q (
join `  K )
r )  <->  E. r  e.  (/)  S ( le
`  K ) ( q ( join `  K
) r ) ) )
1615rexbidv 2966 . . . . . . . 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 303 . . . . . . 7  |-  ( Y  =  (/)  ->  -.  E. q  e.  X  E. r  e.  Y  S
( le `  K
) ( q (
join `  K )
r ) )
1811, 17jaoi 379 . . . . . 6  |-  ( ( X  =  (/)  \/  Y  =  (/) )  ->  -.  E. q  e.  X  E. r  e.  Y  S
( le `  K
) ( q (
join `  K )
r ) )
1918intnand 909 . . . . 5  |-  ( ( X  =  (/)  \/  Y  =  (/) )  ->  -.  ( S  e.  A  /\  E. q  e.  X  E. r  e.  Y  S ( le `  K ) ( q ( join `  K
) r ) ) )
20 biorf 405 . . . . 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 387 . . . 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 262 . . 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 699 . 2  |-  ( ( ( K  e.  B  /\  X  C_  A  /\  Y  C_  A )  /\  ( X  =  (/)  \/  Y  =  (/) ) )  -> 
( S  e.  ( X  .+  Y )  <-> 
( S  e.  X  \/  S  e.  Y
) ) )
253, 24sylan2b 475 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2655   E.wrex 2808    C_ wss 3469   (/)c0 3778   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   lecple 14551   joincjn 15420   Atomscatm 33935   +Pcpadd 34466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-padd 34467
This theorem is referenced by:  paddval0  34481
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