Metamath Proof Explorer

Theorem cdlemk5

Description: Part of proof of Lemma K of Crawley p. 118. (Contributed by NM, 25-Jun-2013)

Ref Expression
Hypotheses cdlemk.b 
|- B = ( Base ` K )
cdlemk.l 
|- .<_ = ( le ` K )
cdlemk.j 
|- .\/ = ( join ` K )
cdlemk.a 
|- A = ( Atoms ` K )
cdlemk.h 
|- H = ( LHyp ` K )
cdlemk.t 
|- T = ( ( LTrn ` K ) ` W )
cdlemk.r 
|- R = ( ( trL ` K ) ` W )
cdlemk.m 
|- ./\ = ( meet ` K )
Assertion cdlemk5 
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> ( ( P .\/ ( N ` P ) ) ./\ ( ( G ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) .<_ ( ( X ` P ) .\/ ( R ` ( X o. `' F ) ) ) )

Proof

Step Hyp Ref Expression
1 cdlemk.b 
 |-  B = ( Base ` K )
2 cdlemk.l 
 |-  .<_ = ( le ` K )
3 cdlemk.j 
 |-  .\/ = ( join ` K )
4 cdlemk.a 
 |-  A = ( Atoms ` K )
5 cdlemk.h 
 |-  H = ( LHyp ` K )
6 cdlemk.t 
 |-  T = ( ( LTrn ` K ) ` W )
7 cdlemk.r 
 |-  R = ( ( trL ` K ) ` W )
8 cdlemk.m 
 |-  ./\ = ( meet ` K )
9 simp11l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> K e. HL )
10 simp11r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> W e. H )
11 simp12 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> F e. T )
12 simp21l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> N e. T )
13 simp23 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> ( R ` F ) = ( R ` N ) )
14 simp22 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> ( P e. A /\ -. P .<_ W ) )
15 1 2 3 4 5 6 7 cdlemk1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T ) /\ ( ( R ` F ) = ( R ` N ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( P .\/ ( N ` P ) ) = ( ( F ` P ) .\/ ( R ` F ) ) )
16 9 10 11 12 13 14 15 syl222anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> ( P .\/ ( N ` P ) ) = ( ( F ` P ) .\/ ( R ` F ) ) )
17 simp13 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> G e. T )
18 1 2 3 4 5 6 7 cdlemk2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( G ` P ) .\/ ( R ` ( G o. `' F ) ) ) = ( ( F ` P ) .\/ ( R ` ( G o. `' F ) ) ) )
19 9 10 11 17 14 18 syl221anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> ( ( G ` P ) .\/ ( R ` ( G o. `' F ) ) ) = ( ( F ` P ) .\/ ( R ` ( G o. `' F ) ) ) )
20 16 19 oveq12d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> ( ( P .\/ ( N ` P ) ) ./\ ( ( G ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) = ( ( ( F ` P ) .\/ ( R ` F ) ) ./\ ( ( F ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) )
21 simp21r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> X e. T )
22 simp33 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> ( R ` G ) =/= ( R ` F ) )
23 simp31 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> F =/= ( _I |` B ) )
24 simp32 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> G =/= ( _I |` B ) )
25 23 24 jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) )
26 1 2 3 4 5 6 7 8 cdlemk5a 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T /\ X e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( ( F ` P ) .\/ ( R ` F ) ) ./\ ( ( F ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) .<_ ( ( X ` P ) .\/ ( R ` ( X o. `' F ) ) ) )
27 9 10 11 17 21 22 25 14 26 syl233anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> ( ( ( F ` P ) .\/ ( R ` F ) ) ./\ ( ( F ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) .<_ ( ( X ` P ) .\/ ( R ` ( X o. `' F ) ) ) )
28 20 27 eqbrtrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> ( ( P .\/ ( N ` P ) ) ./\ ( ( G ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) .<_ ( ( X ` P ) .\/ ( R ` ( X o. `' F ) ) ) )