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Metamath Proof Explorer

Theorem cdlemg29

Description: Eliminate ( FP ) =/= P and ( GP ) =/= P from cdlemg28 . TODO: would it be better to do this later? (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 )
cdlemg31.n 
|- N = ( ( P .\/ v ) ./\ ( Q .\/ ( R ` F ) ) )
cdlemg33.o 
|- O = ( ( P .\/ v ) ./\ ( Q .\/ ( R ` G ) ) )
Assertion cdlemg29 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O ) /\ z .<_ ( P .\/ v ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ 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 cdlemg31.n 
 |-  N = ( ( P .\/ v ) ./\ ( Q .\/ ( R ` F ) ) )
9 cdlemg33.o 
 |-  O = ( ( P .\/ v ) ./\ ( Q .\/ ( R ` G ) ) )
10 simpl11 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O ) /\ z .<_ ( P .\/ v ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) /\ ( F ` P ) = P ) -> ( K e. HL /\ W e. H ) )
11 simpl12 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O ) /\ z .<_ ( P .\/ v ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) /\ ( F ` P ) = P ) -> ( P e. A /\ -. P .<_ W ) )
12 simpl13 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O ) /\ z .<_ ( P .\/ v ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) /\ ( F ` P ) = P ) -> ( Q e. A /\ -. Q .<_ W ) )
13 simp23l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O ) /\ z .<_ ( P .\/ v ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) -> F e. T )
14 13 adantr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O ) /\ z .<_ ( P .\/ v ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) /\ ( F ` P ) = P ) -> F e. T )
15 simp23r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O ) /\ z .<_ ( P .\/ v ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) -> G e. T )
16 15 adantr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O ) /\ z .<_ ( P .\/ v ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) /\ ( F ` P ) = P ) -> G e. T )
17 simpr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O ) /\ z .<_ ( P .\/ v ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) /\ ( F ` P ) = P ) -> ( F ` P ) = P )
18 1 2 3 4 5 6 7 cdlemg14f 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` P ) = P ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )
19 10 11 12 14 16 17 18 syl123anc 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O ) /\ z .<_ ( P .\/ v ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) /\ ( F ` P ) = P ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )
20 simpl11 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O ) /\ z .<_ ( P .\/ v ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) /\ ( G ` P ) = P ) -> ( K e. HL /\ W e. H ) )
21 simpl12 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O ) /\ z .<_ ( P .\/ v ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) /\ ( G ` P ) = P ) -> ( P e. A /\ -. P .<_ W ) )
22 simpl13 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O ) /\ z .<_ ( P .\/ v ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) /\ ( G ` P ) = P ) -> ( Q e. A /\ -. Q .<_ W ) )
23 13 adantr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O ) /\ z .<_ ( P .\/ v ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) /\ ( G ` P ) = P ) -> F e. T )
24 15 adantr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O ) /\ z .<_ ( P .\/ v ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) /\ ( G ` P ) = P ) -> G e. T )
25 simpr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O ) /\ z .<_ ( P .\/ v ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) /\ ( G ` P ) = P ) -> ( G ` P ) = P )
26 1 2 3 4 5 6 7 cdlemg14g 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( G ` P ) = P ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )
27 20 21 22 23 24 25 26 syl123anc 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O ) /\ z .<_ ( P .\/ v ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) /\ ( G ` P ) = P ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )
28 simpl1 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O ) /\ z .<_ ( P .\/ v ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) -> ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) )
29 simpl2 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O ) /\ z .<_ ( P .\/ v ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) -> ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) )
30 simp31l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O ) /\ z .<_ ( P .\/ v ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) -> z =/= N )
31 30 adantr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O ) /\ z .<_ ( P .\/ v ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) -> z =/= N )
32 simp31r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O ) /\ z .<_ ( P .\/ v ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) -> z =/= O )
33 32 adantr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O ) /\ z .<_ ( P .\/ v ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) -> z =/= O )
34 simpl32 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O ) /\ z .<_ ( P .\/ v ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) -> z .<_ ( P .\/ v ) )
35 31 33 34 3jca 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O ) /\ z .<_ ( P .\/ v ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) -> ( z =/= N /\ z =/= O /\ z .<_ ( P .\/ v ) ) )
36 simpl33 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O ) /\ z .<_ ( P .\/ v ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) -> ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) )
37 simpr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O ) /\ z .<_ ( P .\/ v ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) -> ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) )
38 1 2 3 4 5 6 7 8 9 cdlemg28 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O /\ z .<_ ( P .\/ v ) ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )
39 28 29 35 36 37 38 syl113anc 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O ) /\ z .<_ ( P .\/ v ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )
40 19 27 39 pm2.61da2ne 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( z e. A /\ -. z .<_ W ) /\ ( F e. T /\ G e. T ) ) /\ ( ( z =/= N /\ z =/= O ) /\ z .<_ ( P .\/ v ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )