Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elpadd0 Structured version   Unicode version

Theorem elpadd0 35655
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 2782 . . . 4  |-  ( ( X  =/=  (/)  /\  Y  =/=  (/) )  <->  -.  ( X  =  (/)  \/  Y  =  (/) ) )
21bicomi 202 . . 3  |-  ( -.  ( X  =  (/)  \/  Y  =  (/) )  <->  ( X  =/=  (/)  /\  Y  =/=  (/) ) )
32con1bii 331 . 2  |-  ( -.  ( X  =/=  (/)  /\  Y  =/=  (/) )  <->  ( X  =  (/)  \/  Y  =  (/) ) )
4 eqid 2457 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
5 eqid 2457 . . . 4  |-  ( join `  K )  =  (
join `  K )
6 padd0.a . . . 4  |-  A  =  ( Atoms `  K )
7 padd0.p . . . 4  |-  .+  =  ( +P `  K
)
84, 5, 6, 7elpadd 35645 . . 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 3808 . . . . . . . 8  |-  -.  E. q  e.  (/)  E. r  e.  Y  S ( le `  K ) ( q ( join `  K
) r )
10 rexeq 3055 . . . . . . . 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 3808 . . . . . . . . . 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 3055 . . . . . . . . 9  |-  ( Y  =  (/)  ->  ( E. r  e.  Y  S
( le `  K
) ( q (
join `  K )
r )  <->  E. r  e.  (/)  S ( le
`  K ) ( q ( join `  K
) r ) ) )
1615rexbidv 2968 . . . . . . . 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 916 . . . . 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 973    = wceq 1395    e. wcel 1819    =/= wne 2652   E.wrex 2808    C_ wss 3471   (/)c0 3793   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   lecple 14719   joincjn 15700   Atomscatm 35110   +Pcpadd 35641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-padd 35642
This theorem is referenced by:  paddval0  35656
  Copyright terms: Public domain W3C validator