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Theorem elpadd0 33762
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 2773 . . . 4  |-  ( ( X  =/=  (/)  /\  Y  =/=  (/) )  <->  -.  ( X  =  (/)  \/  Y  =  (/) ) )
21bicomi 202 . . 3  |-  ( -.  ( X  =  (/)  \/  Y  =  (/) )  <->  ( X  =/=  (/)  /\  Y  =/=  (/) ) )
32con1bii 331 . 2  |-  ( -.  ( X  =/=  (/)  /\  Y  =/=  (/) )  <->  ( X  =  (/)  \/  Y  =  (/) ) )
4 eqid 2451 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
5 eqid 2451 . . . 4  |-  ( join `  K )  =  (
join `  K )
6 padd0.a . . . 4  |-  A  =  ( Atoms `  K )
7 padd0.p . . . 4  |-  .+  =  ( +P `  K
)
84, 5, 6, 7elpadd 33752 . . 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 3752 . . . . . . . 8  |-  -.  E. q  e.  (/)  E. r  e.  Y  S ( le `  K ) ( q ( join `  K
) r )
10 rexeq 3017 . . . . . . . 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 3752 . . . . . . . . . 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 2917 . . . . . . . 8  |-  -.  E. q  e.  X  E. r  e.  (/)  S ( le `  K ) ( q ( join `  K ) r )
15 rexeq 3017 . . . . . . . . 9  |-  ( Y  =  (/)  ->  ( E. r  e.  Y  S
( le `  K
) ( q (
join `  K )
r )  <->  E. r  e.  (/)  S ( le
`  K ) ( q ( join `  K
) r ) ) )
1615rexbidv 2855 . . . . . . . 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 907 . . . . 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 965    = wceq 1370    e. wcel 1758    =/= wne 2644   E.wrex 2796    C_ wss 3429   (/)c0 3738   class class class wbr 4393   ` cfv 5519  (class class class)co 6193   lecple 14356   joincjn 15225   Atomscatm 33217   +Pcpadd 33748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-1st 6680  df-2nd 6681  df-padd 33749
This theorem is referenced by:  paddval0  33763
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