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Theorem elpaddri 30600
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 986 . 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 984 . . 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 985 . . 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 987 . . 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 6089 . . . . 5  |-  ( q  =  Q  ->  (
q  .\/  r )  =  ( Q  .\/  r ) )
65breq2d 4225 . . . 4  |-  ( q  =  Q  ->  ( S  .<_  ( q  .\/  r )  <->  S  .<_  ( Q  .\/  r ) ) )
7 oveq2 6090 . . . . 5  |-  ( r  =  R  ->  ( Q  .\/  r )  =  ( Q  .\/  R
) )
87breq2d 4225 . . . 4  |-  ( r  =  R  ->  ( S  .<_  ( Q  .\/  r )  <->  S  .<_  ( Q  .\/  R ) ) )
96, 8rspc2ev 3061 . . 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 1185 . 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 3635 . . . . . 6  |-  ( Q  e.  X  ->  X  =/=  (/) )
12 ne0i 3635 . . . . . 6  |-  ( R  e.  Y  ->  Y  =/=  (/) )
1311, 12anim12i 551 . . . . 5  |-  ( ( Q  e.  X  /\  R  e.  Y )  ->  ( X  =/=  (/)  /\  Y  =/=  (/) ) )
1413anim2i 554 . . . 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 978 . . 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 30598 . . 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 16 . 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 890 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2600   E.wrex 2707    C_ wss 3321   (/)c0 3629   class class class wbr 4213   ` cfv 5455  (class class class)co 6082   lecple 13537   joincjn 14402   Latclat 14475   Atomscatm 30062   + Pcpadd 30593
This theorem is referenced by:  elpaddatriN  30601  paddasslem8  30625  paddasslem12  30629  paddasslem13  30630  pmodlem1  30644  osumcllem5N  30758  pexmidlem2N  30769
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-undef 6544  df-riota 6550  df-lub 14432  df-join 14434  df-lat 14476  df-ats 30066  df-padd 30594
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