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

Theorem elpaddri 30613
Description: Condition implying membership in a projective subspace sum. (Contributed by NM, 8-Jan-2012.)
Hypotheses
Ref Expression
paddfval.l  |-  .<_  =  ( le `  K )
paddfval.j  |-  .\/  =  ( join `  K )
paddfval.a  |-  A  =  ( Atoms `  K )
paddfval.p  |-  .+  =  ( + P `  K
)
Assertion
Ref Expression
elpaddri  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( Q  e.  X  /\  R  e.  Y
)  /\  ( S  e.  A  /\  S  .<_  ( Q  .\/  R ) ) )  ->  S  e.  ( X  .+  Y
) )

Proof of Theorem elpaddri
Dummy variables  q 
r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp3l 983 . 2  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( Q  e.  X  /\  R  e.  Y
)  /\  ( S  e.  A  /\  S  .<_  ( Q  .\/  R ) ) )  ->  S  e.  A )
2 simp2l 981 . . 3  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( Q  e.  X  /\  R  e.  Y
)  /\  ( S  e.  A  /\  S  .<_  ( Q  .\/  R ) ) )  ->  Q  e.  X )
3 simp2r 982 . . 3  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( Q  e.  X  /\  R  e.  Y
)  /\  ( S  e.  A  /\  S  .<_  ( Q  .\/  R ) ) )  ->  R  e.  Y )
4 simp3r 984 . . 3  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( Q  e.  X  /\  R  e.  Y
)  /\  ( S  e.  A  /\  S  .<_  ( Q  .\/  R ) ) )  ->  S  .<_  ( Q  .\/  R
) )
5 oveq1 5881 . . . . 5  |-  ( q  =  Q  ->  (
q  .\/  r )  =  ( Q  .\/  r ) )
65breq2d 4051 . . . 4  |-  ( q  =  Q  ->  ( S  .<_  ( q  .\/  r )  <->  S  .<_  ( Q  .\/  r ) ) )
7 oveq2 5882 . . . . 5  |-  ( r  =  R  ->  ( Q  .\/  r )  =  ( Q  .\/  R
) )
87breq2d 4051 . . . 4  |-  ( r  =  R  ->  ( S  .<_  ( Q  .\/  r )  <->  S  .<_  ( Q  .\/  R ) ) )
96, 8rspc2ev 2905 . . 3  |-  ( ( Q  e.  X  /\  R  e.  Y  /\  S  .<_  ( Q  .\/  R ) )  ->  E. q  e.  X  E. r  e.  Y  S  .<_  ( q  .\/  r ) )
102, 3, 4, 9syl3anc 1182 . 2  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( Q  e.  X  /\  R  e.  Y
)  /\  ( S  e.  A  /\  S  .<_  ( Q  .\/  R ) ) )  ->  E. q  e.  X  E. r  e.  Y  S  .<_  ( q  .\/  r ) )
11 ne0i 3474 . . . . . 6  |-  ( Q  e.  X  ->  X  =/=  (/) )
12 ne0i 3474 . . . . . 6  |-  ( R  e.  Y  ->  Y  =/=  (/) )
1311, 12anim12i 549 . . . . 5  |-  ( ( Q  e.  X  /\  R  e.  Y )  ->  ( X  =/=  (/)  /\  Y  =/=  (/) ) )
1413anim2i 552 . . . 4  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( Q  e.  X  /\  R  e.  Y
) )  ->  (
( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) ) )
15143adant3 975 . . 3  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( Q  e.  X  /\  R  e.  Y
)  /\  ( S  e.  A  /\  S  .<_  ( Q  .\/  R ) ) )  ->  (
( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) ) )
16 paddfval.l . . . 4  |-  .<_  =  ( le `  K )
17 paddfval.j . . . 4  |-  .\/  =  ( join `  K )
18 paddfval.a . . . 4  |-  A  =  ( Atoms `  K )
19 paddfval.p . . . 4  |-  .+  =  ( + P `  K
)
2016, 17, 18, 19elpaddn0 30611 . . 3  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  -> 
( S  e.  ( X  .+  Y )  <-> 
( S  e.  A  /\  E. q  e.  X  E. r  e.  Y  S  .<_  ( q  .\/  r ) ) ) )
2115, 20syl 15 . 2  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( Q  e.  X  /\  R  e.  Y
)  /\  ( S  e.  A  /\  S  .<_  ( Q  .\/  R ) ) )  ->  ( S  e.  ( X  .+  Y )  <->  ( S  e.  A  /\  E. q  e.  X  E. r  e.  Y  S  .<_  ( q  .\/  r ) ) ) )
221, 10, 21mpbir2and 888 1  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( Q  e.  X  /\  R  e.  Y
)  /\  ( S  e.  A  /\  S  .<_  ( Q  .\/  R ) ) )  ->  S  e.  ( X  .+  Y
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557    C_ wss 3165   (/)c0 3468   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   lecple 13231   joincjn 14094   Latclat 14167   Atomscatm 30075   + Pcpadd 30606
This theorem is referenced by:  elpaddatriN  30614  paddasslem8  30638  paddasslem12  30642  paddasslem13  30643  pmodlem1  30657  osumcllem5N  30771  pexmidlem2N  30782
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-lub 14124  df-join 14126  df-lat 14168  df-ats 30079  df-padd 30607
  Copyright terms: Public domain W3C validator