Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  osumcllem10N Structured version   Unicode version

Theorem osumcllem10N 30689
Description: Lemma for osumclN 30691. Contradict osumcllem9N 30688. (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 987 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  A  /\  -.  p  e.  ( X  .+  Y ) )  ->  K  e.  HL )
2 simp2 958 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  A  /\  -.  p  e.  ( X  .+  Y ) )  ->  p  e.  A
)
32snssd 3935 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  A  /\  -.  p  e.  ( X  .+  Y ) )  ->  { p }  C_  A )
4 simp12 988 . . . . 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 30540 . . . . 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  /\  Y  C_  A )  /\  p  e.  A  /\  -.  p  e.  ( X  .+  Y ) )  ->  { p }  C_  ( X  .+  {
p } ) )
9 vex 2951 . . . . 5  |-  p  e. 
_V
109snss 3918 . . . 4  |-  ( p  e.  ( X  .+  { p } )  <->  { p }  C_  ( X  .+  { p } ) )
118, 10sylibr 204 . . 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 2526 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  A  /\  -.  p  e.  ( X  .+  Y ) )  ->  p  e.  M
)
145, 6sspadd1 30539 . . . 4  |-  ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  ->  X  C_  ( X  .+  Y
) )
15143ad2ant1 978 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  A  /\  -.  p  e.  ( X  .+  Y ) )  ->  X  C_  ( X  .+  Y ) )
16 simp3 959 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  A  /\  -.  p  e.  ( X  .+  Y ) )  ->  -.  p  e.  ( X  .+  Y ) )
1715, 16ssneldd 3343 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  p  e.  A  /\  -.  p  e.  ( X  .+  Y ) )  ->  -.  p  e.  X )
18 nelne1 2687 . 2  |-  ( ( p  e.  M  /\  -.  p  e.  X
)  ->  M  =/=  X )
1913, 17, 18syl2anc 643 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 936    = wceq 1652    e. wcel 1725    =/= wne 2598    C_ wss 3312   {csn 3806   ` cfv 5446  (class class class)co 6073   lecple 13528   joincjn 14393   Atomscatm 29988   HLchlt 30075   + Pcpadd 30519   _|_ PcpolN 30626   PSubClcpscN 30658
This theorem is referenced by:  osumcllem11N  30690
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-padd 30520
  Copyright terms: Public domain W3C validator