Metamath Proof Explorer

Theorem cdlemkoatnle

Description: Utility lemma. (Contributed by NM, 2-Jul-2013)

Ref Expression
Hypotheses cdlemk1.b 
|- B = ( Base ` K )
cdlemk1.l 
|- .<_ = ( le ` K )
cdlemk1.j 
|- .\/ = ( join ` K )
cdlemk1.m 
|- ./\ = ( meet ` K )
cdlemk1.a 
|- A = ( Atoms ` K )
cdlemk1.h 
|- H = ( LHyp ` K )
cdlemk1.t 
|- T = ( ( LTrn ` K ) ` W )
cdlemk1.r 
|- R = ( ( trL ` K ) ` W )
cdlemk1.s 
|- S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
cdlemk1.o 
|- O = ( S ` D )
Assertion cdlemkoatnle 
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> ( ( O ` P ) e. A /\ -. ( O ` P ) .<_ W ) )

Proof

Step Hyp Ref Expression
1 cdlemk1.b 
 |-  B = ( Base ` K )
2 cdlemk1.l 
 |-  .<_ = ( le ` K )
3 cdlemk1.j 
 |-  .\/ = ( join ` K )
4 cdlemk1.m 
 |-  ./\ = ( meet ` K )
5 cdlemk1.a 
 |-  A = ( Atoms ` K )
6 cdlemk1.h 
 |-  H = ( LHyp ` K )
7 cdlemk1.t 
 |-  T = ( ( LTrn ` K ) ` W )
8 cdlemk1.r 
 |-  R = ( ( trL ` K ) ` W )
9 cdlemk1.s 
 |-  S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
10 cdlemk1.o 
 |-  O = ( S ` D )
11 simp11 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> ( K e. HL /\ W e. H ) )
12 1 2 3 5 6 7 8 4 9 cdlemksel 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> ( S ` D ) e. T )
13 10 12 eqeltrid 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> O e. T )
14 simp22 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> ( P e. A /\ -. P .<_ W ) )
15 2 5 6 7 ltrnel 
 |-  ( ( ( K e. HL /\ W e. H ) /\ O e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( O ` P ) e. A /\ -. ( O ` P ) .<_ W ) )
16 11 13 14 15 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> ( ( O ` P ) e. A /\ -. ( O ` P ) .<_ W ) )