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Metamath Proof Explorer

Theorem pexmidlem2N

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 pexmidlem2N 
|- ( ( ( K e. HL /\ X C_ A /\ p e. A ) /\ ( r e. X /\ q e. ( ._|_ ` X ) /\ p .<_ ( r .\/ q ) ) ) -> p e. ( X .+ ( ._|_ ` 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 ) /\ ( r e. X /\ q e. ( ._|_ ` X ) /\ p .<_ ( r .\/ q ) ) ) -> K e. HL )
8 7 hllatd 
 |-  ( ( ( K e. HL /\ X C_ A /\ p e. A ) /\ ( r e. X /\ q e. ( ._|_ ` X ) /\ p .<_ ( r .\/ q ) ) ) -> K e. Lat )
9 simpl2 
 |-  ( ( ( K e. HL /\ X C_ A /\ p e. A ) /\ ( r e. X /\ q e. ( ._|_ ` X ) /\ p .<_ ( r .\/ q ) ) ) -> X C_ A )
10 3 5 polssatN 
 |-  ( ( K e. HL /\ X C_ A ) -> ( ._|_ ` X ) C_ A )
11 7 9 10 syl2anc 
 |-  ( ( ( K e. HL /\ X C_ A /\ p e. A ) /\ ( r e. X /\ q e. ( ._|_ ` X ) /\ p .<_ ( r .\/ q ) ) ) -> ( ._|_ ` X ) C_ A )
12 simpr1 
 |-  ( ( ( K e. HL /\ X C_ A /\ p e. A ) /\ ( r e. X /\ q e. ( ._|_ ` X ) /\ p .<_ ( r .\/ q ) ) ) -> r e. X )
13 simpr2 
 |-  ( ( ( K e. HL /\ X C_ A /\ p e. A ) /\ ( r e. X /\ q e. ( ._|_ ` X ) /\ p .<_ ( r .\/ q ) ) ) -> q e. ( ._|_ ` X ) )
14 simpl3 
 |-  ( ( ( K e. HL /\ X C_ A /\ p e. A ) /\ ( r e. X /\ q e. ( ._|_ ` X ) /\ p .<_ ( r .\/ q ) ) ) -> p e. A )
15 simpr3 
 |-  ( ( ( K e. HL /\ X C_ A /\ p e. A ) /\ ( r e. X /\ q e. ( ._|_ ` X ) /\ p .<_ ( r .\/ q ) ) ) -> p .<_ ( r .\/ q ) )
16 1 2 3 4 elpaddri 
 |-  ( ( ( K e. Lat /\ X C_ A /\ ( ._|_ ` X ) C_ A ) /\ ( r e. X /\ q e. ( ._|_ ` X ) ) /\ ( p e. A /\ p .<_ ( r .\/ q ) ) ) -> p e. ( X .+ ( ._|_ ` X ) ) )
17 8 9 11 12 13 14 15 16 syl322anc 
 |-  ( ( ( K e. HL /\ X C_ A /\ p e. A ) /\ ( r e. X /\ q e. ( ._|_ ` X ) /\ p .<_ ( r .\/ q ) ) ) -> p e. ( X .+ ( ._|_ ` X ) ) )