Metamath Proof Explorer


Theorem cdlemkoatnle-2N

Description: Utility lemma. (Contributed by NM, 2-Jul-2013) (New usage is discouraged.)

Ref Expression
Hypotheses cdlemk2.b
|- B = ( Base ` K )
cdlemk2.l
|- .<_ = ( le ` K )
cdlemk2.j
|- .\/ = ( join ` K )
cdlemk2.m
|- ./\ = ( meet ` K )
cdlemk2.a
|- A = ( Atoms ` K )
cdlemk2.h
|- H = ( LHyp ` K )
cdlemk2.t
|- T = ( ( LTrn ` K ) ` W )
cdlemk2.r
|- R = ( ( trL ` K ) ` W )
cdlemk2.s
|- S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
cdlemk2.q
|- Q = ( S ` C )
Assertion cdlemkoatnle-2N
|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ C e. T /\ N e. T ) /\ ( ( R ` C ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( Q ` P ) e. A /\ -. ( Q ` P ) .<_ W ) )

Proof

Step Hyp Ref Expression
1 cdlemk2.b
 |-  B = ( Base ` K )
2 cdlemk2.l
 |-  .<_ = ( le ` K )
3 cdlemk2.j
 |-  .\/ = ( join ` K )
4 cdlemk2.m
 |-  ./\ = ( meet ` K )
5 cdlemk2.a
 |-  A = ( Atoms ` K )
6 cdlemk2.h
 |-  H = ( LHyp ` K )
7 cdlemk2.t
 |-  T = ( ( LTrn ` K ) ` W )
8 cdlemk2.r
 |-  R = ( ( trL ` K ) ` W )
9 cdlemk2.s
 |-  S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
10 cdlemk2.q
 |-  Q = ( S ` C )
11 simp11
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ C e. T /\ N e. T ) /\ ( ( R ` C ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> K e. HL )
12 simp12
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ C e. T /\ N e. T ) /\ ( ( R ` C ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> W e. H )
13 11 12 jca
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ C e. T /\ N e. T ) /\ ( ( R ` C ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( K e. HL /\ W e. H ) )
14 simp21
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ C e. T /\ N e. T ) /\ ( ( R ` C ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> F e. T )
15 simp22
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ C e. T /\ N e. T ) /\ ( ( R ` C ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> C e. T )
16 simp23
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ C e. T /\ N e. T ) /\ ( ( R ` C ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> N e. T )
17 simp33
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ C e. T /\ N e. T ) /\ ( ( R ` C ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( P e. A /\ -. P .<_ W ) )
18 simp13
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ C e. T /\ N e. T ) /\ ( ( R ` C ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( R ` F ) = ( R ` N ) )
19 simp32l
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ C e. T /\ N e. T ) /\ ( ( R ` C ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> F =/= ( _I |` B ) )
20 simp32r
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ C e. T /\ N e. T ) /\ ( ( R ` C ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> C =/= ( _I |` B ) )
21 simp31
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ C e. T /\ N e. T ) /\ ( ( R ` C ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( R ` C ) =/= ( R ` F ) )
22 1 2 3 4 5 6 7 8 9 10 cdlemkoatnle
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ C e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ ( R ` C ) =/= ( R ` F ) ) ) -> ( ( Q ` P ) e. A /\ -. ( Q ` P ) .<_ W ) )
23 13 14 15 16 17 18 19 20 21 22 syl333anc
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ C e. T /\ N e. T ) /\ ( ( R ` C ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( Q ` P ) e. A /\ -. ( Q ` P ) .<_ W ) )