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Theorem osumcllem8N 30211
Description: Lemma for osumclN 30215. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
osumcllem.l  |-  .<_  =  ( le `  K )
osumcllem.j  |-  .\/  =  ( join `  K )
osumcllem.a  |-  A  =  ( Atoms `  K )
osumcllem.p  |-  .+  =  ( + P `  K
)
osumcllem.o  |-  ._|_  =  ( _|_ P `  K
)
osumcllem.c  |-  C  =  ( PSubCl `  K )
osumcllem.m  |-  M  =  ( X  .+  {
p } )
osumcllem.u  |-  U  =  (  ._|_  `  (  ._|_  `  ( X  .+  Y
) ) )
Assertion
Ref Expression
osumcllem8N  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  ( X  C_  (  ._|_  `  Y )  /\  X  =/=  (/)  /\  p  e.  A )  /\  -.  p  e.  ( X  .+  Y ) )  -> 
( Y  i^i  M
)  =  (/) )

Proof of Theorem osumcllem8N
Dummy variable  q is distinct from all other variables.
StepHypRef Expression
1 n0 3552 . . . 4  |-  ( ( Y  i^i  M )  =/=  (/)  <->  E. q  q  e.  ( Y  i^i  M
) )
2 osumcllem.l . . . . . . 7  |-  .<_  =  ( le `  K )
3 osumcllem.j . . . . . . 7  |-  .\/  =  ( join `  K )
4 osumcllem.a . . . . . . 7  |-  A  =  ( Atoms `  K )
5 osumcllem.p . . . . . . 7  |-  .+  =  ( + P `  K
)
6 osumcllem.o . . . . . . 7  |-  ._|_  =  ( _|_ P `  K
)
7 osumcllem.c . . . . . . 7  |-  C  =  ( PSubCl `  K )
8 osumcllem.m . . . . . . 7  |-  M  =  ( X  .+  {
p } )
9 osumcllem.u . . . . . . 7  |-  U  =  (  ._|_  `  (  ._|_  `  ( X  .+  Y
) ) )
102, 3, 4, 5, 6, 7, 8, 9osumcllem7N 30210 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  ( X  C_  (  ._|_  `  Y )  /\  X  =/=  (/)  /\  p  e.  A )  /\  q  e.  ( Y  i^i  M
) )  ->  p  e.  ( X  .+  Y
) )
11103expia 1154 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  ( X  C_  (  ._|_  `  Y )  /\  X  =/=  (/)  /\  p  e.  A ) )  -> 
( q  e.  ( Y  i^i  M )  ->  p  e.  ( X  .+  Y ) ) )
1211exlimdv 1641 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  ( X  C_  (  ._|_  `  Y )  /\  X  =/=  (/)  /\  p  e.  A ) )  -> 
( E. q  q  e.  ( Y  i^i  M )  ->  p  e.  ( X  .+  Y ) ) )
131, 12syl5bi 208 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  ( X  C_  (  ._|_  `  Y )  /\  X  =/=  (/)  /\  p  e.  A ) )  -> 
( ( Y  i^i  M )  =/=  (/)  ->  p  e.  ( X  .+  Y
) ) )
1413necon1bd 2597 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  ( X  C_  (  ._|_  `  Y )  /\  X  =/=  (/)  /\  p  e.  A ) )  -> 
( -.  p  e.  ( X  .+  Y
)  ->  ( Y  i^i  M )  =  (/) ) )
15143impia 1149 1  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  ( X  C_  (  ._|_  `  Y )  /\  X  =/=  (/)  /\  p  e.  A )  /\  -.  p  e.  ( X  .+  Y ) )  -> 
( Y  i^i  M
)  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 935   E.wex 1546    = wceq 1647    e. wcel 1715    =/= wne 2529    i^i cin 3237    C_ wss 3238   (/)c0 3543   {csn 3729   ` cfv 5358  (class class class)co 5981   lecple 13423   joincjn 14288   Atomscatm 29512   HLchlt 29599   + Pcpadd 30043   _|_ PcpolN 30150   PSubClcpscN 30182
This theorem is referenced by:  osumcllem9N  30212
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-iin 4010  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-undef 6440  df-riota 6446  df-poset 14290  df-plt 14302  df-lub 14318  df-glb 14319  df-join 14320  df-meet 14321  df-p0 14355  df-p1 14356  df-lat 14362  df-clat 14424  df-oposet 29425  df-ol 29427  df-oml 29428  df-covers 29515  df-ats 29516  df-atl 29547  df-cvlat 29571  df-hlat 29600  df-pmap 29752  df-padd 30044  df-polarityN 30151
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