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Theorem elpaddri 29991
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 5865 . . . . 5  |-  ( q  =  Q  ->  (
q  .\/  r )  =  ( Q  .\/  r ) )
65breq2d 4035 . . . 4  |-  ( q  =  Q  ->  ( S  .<_  ( q  .\/  r )  <->  S  .<_  ( Q  .\/  r ) ) )
7 oveq2 5866 . . . . 5  |-  ( r  =  R  ->  ( Q  .\/  r )  =  ( Q  .\/  R
) )
87breq2d 4035 . . . 4  |-  ( r  =  R  ->  ( S  .<_  ( Q  .\/  r )  <->  S  .<_  ( Q  .\/  R ) ) )
96, 8rspc2ev 2892 . . 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 3461 . . . . . 6  |-  ( Q  e.  X  ->  X  =/=  (/) )
12 ne0i 3461 . . . . . 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 29989 . . 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 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544    C_ wss 3152   (/)c0 3455   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   lecple 13215   joincjn 14078   Latclat 14151   Atomscatm 29453   + Pcpadd 29984
This theorem is referenced by:  elpaddatriN  29992  paddasslem8  30016  paddasslem12  30020  paddasslem13  30021  pmodlem1  30035  osumcllem5N  30149  pexmidlem2N  30160
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-lub 14108  df-join 14110  df-lat 14152  df-ats 29457  df-padd 29985
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