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Metamath Proof Explorer

Theorem cdlemg16zz

Description: Eliminate P =/= Q from cdlemg16z . TODO: Use this only if needed. (Contributed by NM, 26-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 cdlemg16zz 
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) -> ( ( 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 id 
 |-  ( P = Q -> P = Q )
9 2fveq3 
 |-  ( P = Q -> ( F ` ( G ` P ) ) = ( F ` ( G ` Q ) ) )
10 8 9 oveq12d 
 |-  ( P = Q -> ( P .\/ ( F ` ( G ` P ) ) ) = ( Q .\/ ( F ` ( G ` Q ) ) ) )
11 10 oveq1d 
 |-  ( P = Q -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )
12 11 adantl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ P = Q ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )
13 simpl1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ P =/= Q ) -> ( K e. HL /\ W e. H ) )
14 simpl21 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ P =/= Q ) -> ( P e. A /\ -. P .<_ W ) )
15 simpl22 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ P =/= Q ) -> ( Q e. A /\ -. Q .<_ W ) )
16 simpl23 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ P =/= Q ) -> F e. T )
17 simpl31 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ P =/= Q ) -> G e. T )
18 simpr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ P =/= Q ) -> P =/= Q )
19 simpl32 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ P =/= Q ) -> -. ( R ` F ) .<_ ( P .\/ Q ) )
20 simpl33 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ P =/= Q ) -> -. ( R ` G ) .<_ ( P .\/ Q ) )
21 1 2 3 4 5 6 7 cdlemg16z 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )
22 13 14 15 16 17 18 19 20 21 syl332anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ P =/= Q ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )
23 12 22 pm2.61dane 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )