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Theorem osumcllem8N 30449
Description: Lemma for osumclN 30453. (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 3601 . . . 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 30448 . . . . . 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 1155 . . . . 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 1643 . . . 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 209 . . 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 2639 . 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 1150 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 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1721    =/= wne 2571    i^i cin 3283    C_ wss 3284   (/)c0 3592   {csn 3778   ` cfv 5417  (class class class)co 6044   lecple 13495   joincjn 14360   Atomscatm 29750   HLchlt 29837   + Pcpadd 30281   _|_ PcpolN 30388   PSubClcpscN 30420
This theorem is referenced by:  osumcllem9N  30450
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-iin 4060  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-undef 6506  df-riota 6512  df-poset 14362  df-plt 14374  df-lub 14390  df-glb 14391  df-join 14392  df-meet 14393  df-p0 14427  df-p1 14428  df-lat 14434  df-clat 14496  df-oposet 29663  df-ol 29665  df-oml 29666  df-covers 29753  df-ats 29754  df-atl 29785  df-cvlat 29809  df-hlat 29838  df-pmap 29990  df-padd 30282  df-polarityN 30389
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