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Metamath Proof Explorer

Theorem cdlemk5a

Description: Part of proof of Lemma K of Crawley p. 118. (Contributed by NM, 3-Jul-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 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 ) ) ) )

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 simp1l 
 |-  ( ( ( 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 ) ) ) -> K e. HL )
10 simp1r 
 |-  ( ( ( 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 ) ) ) -> W e. H )
11 simp21 
 |-  ( ( ( 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 e. T )
12 simp22 
 |-  ( ( ( 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 ) ) ) -> G e. T )
13 simp3 
 |-  ( ( ( 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 ) ) ) -> ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) )
14 1 2 3 4 5 6 7 8 cdlemk3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G 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 ) ) ) ) = ( F ` P ) )
15 9 10 11 12 13 14 syl221anc 
 |-  ( ( ( 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 ) ) ) ) = ( F ` P ) )
16 simp23 
 |-  ( ( ( 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 ) ) ) -> X e. T )
17 simp33l 
 |-  ( ( ( 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 ) ) ) -> P e. A )
18 simp33r 
 |-  ( ( ( 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 ) ) ) -> -. P .<_ W )
19 1 2 3 4 5 6 7 8 cdlemk4 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( F ` P ) .<_ ( ( X ` P ) .\/ ( R ` ( X o. `' F ) ) ) )
20 9 10 11 16 17 18 19 syl222anc 
 |-  ( ( ( 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 ) .<_ ( ( X ` P ) .\/ ( R ` ( X o. `' F ) ) ) )
21 15 20 eqbrtrd 
 |-  ( ( ( 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 ) ) ) )