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Theorem pexmidlem7N 30090
Description: Lemma for pexmidN 30083. Contradict pexmidlem6N 30089. (Contributed by NM, 3-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
pexmidlem7N  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  M  =/=  X )

Proof of Theorem pexmidlem7N
StepHypRef Expression
1 simpl1 960 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  K  e.  HL )
2 simpl3 962 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  p  e.  A )
32snssd 3886 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  { p }  C_  A )
4 simpl2 961 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  X  C_  A )
5 pexmidlem.a . . . . . 6  |-  A  =  ( Atoms `  K )
6 pexmidlem.p . . . . . 6  |-  .+  =  ( + P `  K
)
75, 6sspadd2 29930 . . . . 5  |-  ( ( K  e.  HL  /\  { p }  C_  A  /\  X  C_  A )  ->  { p }  C_  ( X  .+  {
p } ) )
81, 3, 4, 7syl3anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  { p }  C_  ( X  .+  { p } ) )
9 vex 2902 . . . . 5  |-  p  e. 
_V
109snss 3869 . . . 4  |-  ( p  e.  ( X  .+  { p } )  <->  { p }  C_  ( X  .+  { p } ) )
118, 10sylibr 204 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  p  e.  ( X  .+  {
p } ) )
12 pexmidlem.m . . 3  |-  M  =  ( X  .+  {
p } )
1311, 12syl6eleqr 2478 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  p  e.  M )
14 pexmidlem.o . . . . . 6  |-  ._|_  =  ( _|_ P `  K
)
155, 14polssatN 30022 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  X )  C_  A )
161, 4, 15syl2anc 643 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  (  ._|_  `  X )  C_  A )
175, 6sspadd1 29929 . . . 4  |-  ( ( K  e.  HL  /\  X  C_  A  /\  (  ._|_  `  X )  C_  A )  ->  X  C_  ( X  .+  (  ._|_  `  X ) ) )
181, 4, 16, 17syl3anc 1184 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  X  C_  ( X  .+  (  ._|_  `  X ) ) )
19 simpr3 965 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  -.  p  e.  ( X  .+  (  ._|_  `  X
) ) )
2018, 19ssneldd 3294 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  -.  p  e.  X )
21 nelne1 2639 . 2  |-  ( ( p  e.  M  /\  -.  p  e.  X
)  ->  M  =/=  X )
2213, 20, 21syl2anc 643 1  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  p  e.  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/)  /\  -.  p  e.  ( X  .+  (  ._|_  `  X ) ) ) )  ->  M  =/=  X )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2550    C_ wss 3263   (/)c0 3571   {csn 3757   ` cfv 5394  (class class class)co 6020   lecple 13463   joincjn 14328   Atomscatm 29378   HLchlt 29465   + Pcpadd 29909   _|_ PcpolN 30016
This theorem is referenced by:  pexmidlem8N  30091
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-iin 4038  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-undef 6479  df-riota 6485  df-poset 14330  df-lub 14358  df-glb 14359  df-join 14360  df-meet 14361  df-p1 14396  df-lat 14402  df-clat 14464  df-oposet 29291  df-ol 29293  df-oml 29294  df-ats 29382  df-atl 29413  df-cvlat 29437  df-hlat 29466  df-psubsp 29617  df-pmap 29618  df-padd 29910  df-polarityN 30017
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