Metamath Proof Explorer

Theorem cdlemg28a

Description: Part of proof of Lemma G of Crawley p. 116. First equality of the equation of line 14 on p. 117. (Contributed by NM, 29-May-2013)

Ref Expression
Hypotheses cdlemg12.l 
|- .<_ = ( le ` K )
cdlemg12.j 
|- .\/ = ( join ` K )
cdlemg12.m 
|- ./\ = ( meet ` K )
cdlemg12.a 
|- A = ( Atoms ` K )
cdlemg12.h 
|- H = ( LHyp ` K )
cdlemg12.t 
|- T = ( ( LTrn ` K ) ` W )
cdlemg12b.r 
|- R = ( ( trL ` K ) ` W )
Assertion cdlemg28a 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ z .<_ ( P .\/ v ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( z .\/ ( F ` ( G ` z ) ) ) ./\ W ) )

Proof

Step Hyp Ref Expression
1 cdlemg12.l 
 |-  .<_ = ( le ` K )
2 cdlemg12.j 
 |-  .\/ = ( join ` K )
3 cdlemg12.m 
 |-  ./\ = ( meet ` K )
4 cdlemg12.a 
 |-  A = ( Atoms ` K )
5 cdlemg12.h 
 |-  H = ( LHyp ` K )
6 cdlemg12.t 
 |-  T = ( ( LTrn ` K ) ` W )
7 cdlemg12b.r 
 |-  R = ( ( trL ` K ) ` W )
8 simp11 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ z .<_ ( P .\/ v ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> ( K e. HL /\ W e. H ) )
9 simp12 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ z .<_ ( P .\/ v ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> ( P e. A /\ -. P .<_ W ) )
10 simp21 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ z .<_ ( P .\/ v ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> ( z e. A /\ -. z .<_ W ) )
11 simp22 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ z .<_ ( P .\/ v ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> F e. T )
12 simp23 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ z .<_ ( P .\/ v ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> G e. T )
13 simp1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ z .<_ ( P .\/ v ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) )
14 simp21l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ z .<_ ( P .\/ v ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> z e. A )
15 simp31l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ z .<_ ( P .\/ v ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> v =/= ( R ` F ) )
16 simp32 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ z .<_ ( P .\/ v ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> z .<_ ( P .\/ v ) )
17 simp33l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ z .<_ ( P .\/ v ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> ( F ` P ) =/= P )
18 1 2 3 4 5 6 7 cdlemg27a 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( z e. A /\ F e. T ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> -. ( R ` F ) .<_ ( P .\/ z ) )
19 13 14 11 15 16 17 18 syl123anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ z .<_ ( P .\/ v ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> -. ( R ` F ) .<_ ( P .\/ z ) )
20 simp31r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ z .<_ ( P .\/ v ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> v =/= ( R ` G ) )
21 simp33r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ z .<_ ( P .\/ v ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> ( G ` P ) =/= P )
22 1 2 3 4 5 6 7 cdlemg27a 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( z e. A /\ G e. T ) /\ ( v =/= ( R ` G ) /\ z .<_ ( P .\/ v ) /\ ( G ` P ) =/= P ) ) -> -. ( R ` G ) .<_ ( P .\/ z ) )
23 13 14 12 20 16 21 22 syl123anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ z .<_ ( P .\/ v ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> -. ( R ` G ) .<_ ( P .\/ z ) )
24 1 2 3 4 5 6 7 cdlemg25zz 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ F e. T ) /\ ( G e. T /\ -. ( R ` F ) .<_ ( P .\/ z ) /\ -. ( R ` G ) .<_ ( P .\/ z ) ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( z .\/ ( F ` ( G ` z ) ) ) ./\ W ) )
25 8 9 10 11 12 19 23 24 syl133anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ z .<_ ( P .\/ v ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( z .\/ ( F ` ( G ` z ) ) ) ./\ W ) )