Metamath Proof Explorer

Theorem osumcllem10N

Description: Lemma for osumclN . Contradict osumcllem9N . (Contributed by NM, 25-Mar-2012) (New usage is discouraged.)

Ref Expression
Hypotheses 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 osumcllem10N 
|- ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. A /\ -. p e. ( X .+ Y ) ) -> M =/= X )

Proof

Step Hyp Ref Expression
1 osumcllem.l 
 |-  .<_ = ( le ` K )
2 osumcllem.j 
 |-  .\/ = ( join ` K )
3 osumcllem.a 
 |-  A = ( Atoms ` K )
4 osumcllem.p 
 |-  .+ = ( +P ` K )
5 osumcllem.o 
 |-  ._|_ = ( _|_P ` K )
6 osumcllem.c 
 |-  C = ( PSubCl ` K )
7 osumcllem.m 
 |-  M = ( X .+ { p } )
8 osumcllem.u 
 |-  U = ( ._|_ ` ( ._|_ ` ( X .+ Y ) ) )
9 simp11 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. A /\ -. p e. ( X .+ Y ) ) -> K e. HL )
10 simp2 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. A /\ -. p e. ( X .+ Y ) ) -> p e. A )
11 10 snssd 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. A /\ -. p e. ( X .+ Y ) ) -> { p } C_ A )
12 simp12 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. A /\ -. p e. ( X .+ Y ) ) -> X C_ A )
13 3 4 sspadd2 
 |-  ( ( K e. HL /\ { p } C_ A /\ X C_ A ) -> { p } C_ ( X .+ { p } ) )
14 9 11 12 13 syl3anc 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. A /\ -. p e. ( X .+ Y ) ) -> { p } C_ ( X .+ { p } ) )
15 vex 
 |-  p e. _V
16 15 snss 
 |-  ( p e. ( X .+ { p } ) <-> { p } C_ ( X .+ { p } ) )
17 14 16 sylibr 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. A /\ -. p e. ( X .+ Y ) ) -> p e. ( X .+ { p } ) )
18 17 7 eleqtrrdi 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. A /\ -. p e. ( X .+ Y ) ) -> p e. M )
19 3 4 sspadd1 
 |-  ( ( K e. HL /\ X C_ A /\ Y C_ A ) -> X C_ ( X .+ Y ) )
20 19 3ad2ant1 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. A /\ -. p e. ( X .+ Y ) ) -> X C_ ( X .+ Y ) )
21 simp3 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. A /\ -. p e. ( X .+ Y ) ) -> -. p e. ( X .+ Y ) )
22 20 21 ssneldd 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. A /\ -. p e. ( X .+ Y ) ) -> -. p e. X )
23 nelne1 
 |-  ( ( p e. M /\ -. p e. X ) -> M =/= X )
24 18 22 23 syl2anc 
 |-  ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. A /\ -. p e. ( X .+ Y ) ) -> M =/= X )