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Theorem elpadd0 33343
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 2745 . . . 4  |-  ( ( X  =/=  (/)  /\  Y  =/=  (/) )  <->  -.  ( X  =  (/)  \/  Y  =  (/) ) )
21bicomi 205 . . 3  |-  ( -.  ( X  =  (/)  \/  Y  =  (/) )  <->  ( X  =/=  (/)  /\  Y  =/=  (/) ) )
32con1bii 332 . 2  |-  ( -.  ( X  =/=  (/)  /\  Y  =/=  (/) )  <->  ( X  =  (/)  \/  Y  =  (/) ) )
4 eqid 2422 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
5 eqid 2422 . . . 4  |-  ( join `  K )  =  (
join `  K )
6 padd0.a . . . 4  |-  A  =  ( Atoms `  K )
7 padd0.p . . . 4  |-  .+  =  ( +P `  K
)
84, 5, 6, 7elpadd 33333 . . 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 3776 . . . . . . . 8  |-  -.  E. q  e.  (/)  E. r  e.  Y  S ( le `  K ) ( q ( join `  K
) r )
10 rexeq 3023 . . . . . . . 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 304 . . . . . . 7  |-  ( X  =  (/)  ->  -.  E. q  e.  X  E. r  e.  Y  S
( le `  K
) ( q (
join `  K )
r ) )
12 rex0 3776 . . . . . . . . . 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 2877 . . . . . . . 8  |-  -.  E. q  e.  X  E. r  e.  (/)  S ( le `  K ) ( q ( join `  K ) r )
15 rexeq 3023 . . . . . . . . 9  |-  ( Y  =  (/)  ->  ( E. r  e.  Y  S
( le `  K
) ( q (
join `  K )
r )  <->  E. r  e.  (/)  S ( le
`  K ) ( q ( join `  K
) r ) ) )
1615rexbidv 2936 . . . . . . . 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 304 . . . . . . 7  |-  ( Y  =  (/)  ->  -.  E. q  e.  X  E. r  e.  Y  S
( le `  K
) ( q (
join `  K )
r ) )
1811, 17jaoi 380 . . . . . 6  |-  ( ( X  =  (/)  \/  Y  =  (/) )  ->  -.  E. q  e.  X  E. r  e.  Y  S
( le `  K
) ( q (
join `  K )
r ) )
1918intnand 924 . . . . 5  |-  ( ( X  =  (/)  \/  Y  =  (/) )  ->  -.  ( S  e.  A  /\  E. q  e.  X  E. r  e.  Y  S ( le `  K ) ( q ( join `  K
) r ) ) )
20 biorf 406 . . . . 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 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 388 . . . 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 265 . . 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 704 . 2  |-  ( ( ( K  e.  B  /\  X  C_  A  /\  Y  C_  A )  /\  ( X  =  (/)  \/  Y  =  (/) ) )  -> 
( S  e.  ( X  .+  Y )  <-> 
( S  e.  X  \/  S  e.  Y
) ) )
253, 24sylan2b 477 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 187    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2614   E.wrex 2772    C_ wss 3436   (/)c0 3761   class class class wbr 4423   ` cfv 5601  (class class class)co 6305   lecple 15196   joincjn 16188   Atomscatm 32798   +Pcpadd 33329
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-padd 33330
This theorem is referenced by:  paddval0  33344
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