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Theorem pexmidlem5N 30772
Description: Lemma for pexmidN 30767. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
pexmidlem.l  |-  .<_  =  ( le `  K )
pexmidlem.j  |-  .\/  =  ( join `  K )
pexmidlem.a  |-  A  =  ( Atoms `  K )
pexmidlem.p  |-  .+  =  ( + P `  K
)
pexmidlem.o  |-  ._|_  =  ( _|_ P `  K
)
pexmidlem.m  |-  M  =  ( X  .+  {
p } )
Assertion
Ref Expression
pexmidlem5N  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X
) ) ) )  ->  ( (  ._|_  `  X )  i^i  M
)  =  (/) )

Proof of Theorem pexmidlem5N
Dummy variable  q is distinct from all other variables.
StepHypRef Expression
1 n0 3638 . . . 4  |-  ( ( (  ._|_  `  X )  i^i  M )  =/=  (/) 
<->  E. q  q  e.  ( (  ._|_  `  X
)  i^i  M )
)
2 pexmidlem.l . . . . . . 7  |-  .<_  =  ( le `  K )
3 pexmidlem.j . . . . . . 7  |-  .\/  =  ( join `  K )
4 pexmidlem.a . . . . . . 7  |-  A  =  ( Atoms `  K )
5 pexmidlem.p . . . . . . 7  |-  .+  =  ( + P `  K
)
6 pexmidlem.o . . . . . . 7  |-  ._|_  =  ( _|_ P `  K
)
7 pexmidlem.m . . . . . . 7  |-  M  =  ( X  .+  {
p } )
82, 3, 4, 5, 6, 7pexmidlem4N 30771 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( X  =/=  (/)  /\  q  e.  ( (  ._|_  `  X
)  i^i  M )
) )  ->  p  e.  ( X  .+  (  ._|_  `  X ) ) )
98expr 600 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  X  =/=  (/) )  -> 
( q  e.  ( (  ._|_  `  X )  i^i  M )  ->  p  e.  ( X  .+  (  ._|_  `  X
) ) ) )
109exlimdv 1647 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  X  =/=  (/) )  -> 
( E. q  q  e.  ( (  ._|_  `  X )  i^i  M
)  ->  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )
111, 10syl5bi 210 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  X  =/=  (/) )  -> 
( ( (  ._|_  `  X )  i^i  M
)  =/=  (/)  ->  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )
1211necon1bd 2673 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  X  =/=  (/) )  -> 
( -.  p  e.  ( X  .+  (  ._|_  `  X ) )  ->  ( (  ._|_  `  X )  i^i  M
)  =  (/) ) )
1312impr 604 1  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X
) ) ) )  ->  ( (  ._|_  `  X )  i^i  M
)  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    /\ w3a 937   E.wex 1551    = wceq 1653    e. wcel 1726    =/= wne 2600    i^i cin 3320    C_ wss 3321   (/)c0 3629   {csn 3815   ` cfv 5455  (class class class)co 6082   lecple 13537   joincjn 14402   Atomscatm 30062   HLchlt 30149   + Pcpadd 30593   _|_ PcpolN 30700
This theorem is referenced by:  pexmidlem6N  30773
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-rmo 2714  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-iin 4097  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-poset 14404  df-plt 14416  df-lub 14432  df-glb 14433  df-join 14434  df-meet 14435  df-p0 14469  df-p1 14470  df-lat 14476  df-clat 14538  df-oposet 29975  df-ol 29977  df-oml 29978  df-covers 30065  df-ats 30066  df-atl 30097  df-cvlat 30121  df-hlat 30150  df-psubsp 30301  df-pmap 30302  df-padd 30594  df-polarityN 30701
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