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Metamath Proof Explorer

Theorem pexmidlem5N

Description: Lemma for pexmidN . (Contributed by NM, 2-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 pexmidlem5N 
|- ( ( ( K e. HL /\ X C_ A /\ p e. A ) /\ ( X =/= (/) /\ -. p e. ( X .+ ( ._|_ ` X ) ) ) ) -> ( ( ._|_ ` X ) i^i M ) = (/) )

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 n0 
 |-  ( ( ( ._|_ ` X ) i^i M ) =/= (/) <-> E. q q e. ( ( ._|_ ` X ) i^i M ) )
8 1 2 3 4 5 6 pexmidlem4N 
 |-  ( ( ( K e. HL /\ X C_ A /\ p e. A ) /\ ( X =/= (/) /\ q e. ( ( ._|_ ` X ) i^i M ) ) ) -> p e. ( X .+ ( ._|_ ` X ) ) )
9 8 expr 
 |-  ( ( ( K e. HL /\ X C_ A /\ p e. A ) /\ X =/= (/) ) -> ( q e. ( ( ._|_ ` X ) i^i M ) -> p e. ( X .+ ( ._|_ ` X ) ) ) )
10 9 exlimdv 
 |-  ( ( ( K e. HL /\ X C_ A /\ p e. A ) /\ X =/= (/) ) -> ( E. q q e. ( ( ._|_ ` X ) i^i M ) -> p e. ( X .+ ( ._|_ ` X ) ) ) )
11 7 10 biimtrid 
 |-  ( ( ( K e. HL /\ X C_ A /\ p e. A ) /\ X =/= (/) ) -> ( ( ( ._|_ ` X ) i^i M ) =/= (/) -> p e. ( X .+ ( ._|_ ` X ) ) ) )
12 11 necon1bd 
 |-  ( ( ( K e. HL /\ X C_ A /\ p e. A ) /\ X =/= (/) ) -> ( -. p e. ( X .+ ( ._|_ ` X ) ) -> ( ( ._|_ ` X ) i^i M ) = (/) ) )
13 12 impr 
 |-  ( ( ( K e. HL /\ X C_ A /\ p e. A ) /\ ( X =/= (/) /\ -. p e. ( X .+ ( ._|_ ` X ) ) ) ) -> ( ( ._|_ ` X ) i^i M ) = (/) )