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Metamath Proof Explorer

Theorem pexmidlem7N

Description: Lemma for pexmidN . Contradict pexmidlem6N . (Contributed by NM, 3-Feb-2012) (New usage is discouraged.)

Ref Expression
Hypotheses 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 pexmidlem7N 
|- ( ( ( K e. HL /\ X C_ A /\ p e. A ) /\ ( ( ._|_ ` ( ._|_ ` X ) ) = X /\ X =/= (/) /\ -. p e. ( X .+ ( ._|_ ` X ) ) ) ) -> M =/= X )

Proof

Step Hyp Ref Expression
1 pexmidlem.l 
 |-  .<_ = ( le ` K )
2 pexmidlem.j 
 |-  .\/ = ( join ` K )
3 pexmidlem.a 
 |-  A = ( Atoms ` K )
4 pexmidlem.p 
 |-  .+ = ( +P ` K )
5 pexmidlem.o 
 |-  ._|_ = ( _|_P ` K )
6 pexmidlem.m 
 |-  M = ( X .+ { p } )
7 simpl1 
 |-  ( ( ( K e. HL /\ X C_ A /\ p e. A ) /\ ( ( ._|_ ` ( ._|_ ` X ) ) = X /\ X =/= (/) /\ -. p e. ( X .+ ( ._|_ ` X ) ) ) ) -> K e. HL )
8 simpl3 
 |-  ( ( ( K e. HL /\ X C_ A /\ p e. A ) /\ ( ( ._|_ ` ( ._|_ ` X ) ) = X /\ X =/= (/) /\ -. p e. ( X .+ ( ._|_ ` X ) ) ) ) -> p e. A )
9 8 snssd 
 |-  ( ( ( K e. HL /\ X C_ A /\ p e. A ) /\ ( ( ._|_ ` ( ._|_ ` X ) ) = X /\ X =/= (/) /\ -. p e. ( X .+ ( ._|_ ` X ) ) ) ) -> { p } C_ A )
10 simpl2 
 |-  ( ( ( K e. HL /\ X C_ A /\ p e. A ) /\ ( ( ._|_ ` ( ._|_ ` X ) ) = X /\ X =/= (/) /\ -. p e. ( X .+ ( ._|_ ` X ) ) ) ) -> X C_ A )
11 3 4 sspadd2 
 |-  ( ( K e. HL /\ { p } C_ A /\ X C_ A ) -> { p } C_ ( X .+ { p } ) )
12 7 9 10 11 syl3anc 
 |-  ( ( ( K e. HL /\ X C_ A /\ p e. A ) /\ ( ( ._|_ ` ( ._|_ ` X ) ) = X /\ X =/= (/) /\ -. p e. ( X .+ ( ._|_ ` X ) ) ) ) -> { p } C_ ( X .+ { p } ) )
13 vex 
 |-  p e. _V
14 13 snss 
 |-  ( p e. ( X .+ { p } ) <-> { p } C_ ( X .+ { p } ) )
15 12 14 sylibr 
 |-  ( ( ( K e. HL /\ X C_ A /\ p e. A ) /\ ( ( ._|_ ` ( ._|_ ` X ) ) = X /\ X =/= (/) /\ -. p e. ( X .+ ( ._|_ ` X ) ) ) ) -> p e. ( X .+ { p } ) )
16 15 6 eleqtrrdi 
 |-  ( ( ( K e. HL /\ X C_ A /\ p e. A ) /\ ( ( ._|_ ` ( ._|_ ` X ) ) = X /\ X =/= (/) /\ -. p e. ( X .+ ( ._|_ ` X ) ) ) ) -> p e. M )
17 3 5 polssatN 
 |-  ( ( K e. HL /\ X C_ A ) -> ( ._|_ ` X ) C_ A )
18 7 10 17 syl2anc 
 |-  ( ( ( K e. HL /\ X C_ A /\ p e. A ) /\ ( ( ._|_ ` ( ._|_ ` X ) ) = X /\ X =/= (/) /\ -. p e. ( X .+ ( ._|_ ` X ) ) ) ) -> ( ._|_ ` X ) C_ A )
19 3 4 sspadd1 
 |-  ( ( K e. HL /\ X C_ A /\ ( ._|_ ` X ) C_ A ) -> X C_ ( X .+ ( ._|_ ` X ) ) )
20 7 10 18 19 syl3anc 
 |-  ( ( ( K e. HL /\ X C_ A /\ p e. A ) /\ ( ( ._|_ ` ( ._|_ ` X ) ) = X /\ X =/= (/) /\ -. p e. ( X .+ ( ._|_ ` X ) ) ) ) -> X C_ ( X .+ ( ._|_ ` X ) ) )
21 simpr3 
 |-  ( ( ( K e. HL /\ X C_ A /\ p e. A ) /\ ( ( ._|_ ` ( ._|_ ` X ) ) = X /\ X =/= (/) /\ -. p e. ( X .+ ( ._|_ ` X ) ) ) ) -> -. p e. ( X .+ ( ._|_ ` X ) ) )
22 20 21 ssneldd 
 |-  ( ( ( K e. HL /\ X C_ A /\ p e. A ) /\ ( ( ._|_ ` ( ._|_ ` X ) ) = X /\ X =/= (/) /\ -. p e. ( X .+ ( ._|_ ` X ) ) ) ) -> -. p e. X )
23 nelne1 
 |-  ( ( p e. M /\ -. p e. X ) -> M =/= X )
24 16 22 23 syl2anc 
 |-  ( ( ( K e. HL /\ X C_ A /\ p e. A ) /\ ( ( ._|_ ` ( ._|_ ` X ) ) = X /\ X =/= (/) /\ -. p e. ( X .+ ( ._|_ ` X ) ) ) ) -> M =/= X )