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Theorem osumcllem10N 30776
Description: Lemma for osumclN 30778. Contradict osumcllem9N 30775. (Contributed by NM, 25-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
osumcllem10N  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  A  /\  -.  p  e.  ( X  .+  Y ) )  ->  M  =/=  X
)

Proof of Theorem osumcllem10N
StepHypRef Expression
1 simp11 985 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  A  /\  -.  p  e.  ( X  .+  Y ) )  ->  K  e.  HL )
2 simp2 956 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  A  /\  -.  p  e.  ( X  .+  Y ) )  ->  p  e.  A
)
32snssd 3776 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  A  /\  -.  p  e.  ( X  .+  Y ) )  ->  { p }  C_  A )
4 simp12 986 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  A  /\  -.  p  e.  ( X  .+  Y ) )  ->  X  C_  A
)
5 osumcllem.a . . . . . 6  |-  A  =  ( Atoms `  K )
6 osumcllem.p . . . . . 6  |-  .+  =  ( + P `  K
)
75, 6sspadd2 30627 . . . . 5  |-  ( ( K  e.  HL  /\  { p }  C_  A  /\  X  C_  A )  ->  { p }  C_  ( X  .+  {
p } ) )
81, 3, 4, 7syl3anc 1182 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  A  /\  -.  p  e.  ( X  .+  Y ) )  ->  { p }  C_  ( X  .+  {
p } ) )
9 vex 2804 . . . . 5  |-  p  e. 
_V
109snss 3761 . . . 4  |-  ( p  e.  ( X  .+  { p } )  <->  { p }  C_  ( X  .+  { p } ) )
118, 10sylibr 203 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  A  /\  -.  p  e.  ( X  .+  Y ) )  ->  p  e.  ( X  .+  { p } ) )
12 osumcllem.m . . 3  |-  M  =  ( X  .+  {
p } )
1311, 12syl6eleqr 2387 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  A  /\  -.  p  e.  ( X  .+  Y ) )  ->  p  e.  M
)
14 simp3 957 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  A  /\  -.  p  e.  ( X  .+  Y ) )  ->  -.  p  e.  ( X  .+  Y ) )
155, 6sspadd1 30626 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  ->  X  C_  ( X  .+  Y
) )
16153ad2ant1 976 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  A  /\  -.  p  e.  ( X  .+  Y ) )  ->  X  C_  ( X  .+  Y ) )
1716sseld 3192 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  A  /\  -.  p  e.  ( X  .+  Y ) )  ->  ( p  e.  X  ->  p  e.  ( X  .+  Y ) ) )
1814, 17mtod 168 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  A  /\  -.  p  e.  ( X  .+  Y ) )  ->  -.  p  e.  X )
19 nelne1 2548 . 2  |-  ( ( p  e.  M  /\  -.  p  e.  X
)  ->  M  =/=  X )
2013, 18, 19syl2anc 642 1  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  A  /\  -.  p  e.  ( X  .+  Y ) )  ->  M  =/=  X
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459    C_ wss 3165   {csn 3653   ` cfv 5271  (class class class)co 5874   lecple 13231   joincjn 14094   Atomscatm 30075   HLchlt 30162   + Pcpadd 30606   _|_ PcpolN 30713   PSubClcpscN 30745
This theorem is referenced by:  osumcllem11N  30777
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-padd 30607
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