Metamath Proof Explorer

Theorem cdlemg40

Description: Eliminate P =/= Q conditions from cdlemg39 . TODO: Fix comment. (Contributed by NM, 31-May-2013)

Ref Expression
Hypotheses cdlemg35.l 
|- .<_ = ( le ` K )
cdlemg35.j 
|- .\/ = ( join ` K )
cdlemg35.m 
|- ./\ = ( meet ` K )
cdlemg35.a 
|- A = ( Atoms ` K )
cdlemg35.h 
|- H = ( LHyp ` K )
cdlemg35.t 
|- T = ( ( LTrn ` K ) ` W )
Assertion cdlemg40 
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )

Proof

Step Hyp Ref Expression
1 cdlemg35.l 
 |-  .<_ = ( le ` K )
2 cdlemg35.j 
 |-  .\/ = ( join ` K )
3 cdlemg35.m 
 |-  ./\ = ( meet ` K )
4 cdlemg35.a 
 |-  A = ( Atoms ` K )
5 cdlemg35.h 
 |-  H = ( LHyp ` K )
6 cdlemg35.t 
 |-  T = ( ( LTrn ` K ) ` W )
7 id 
 |-  ( P = Q -> P = Q )
8 2fveq3 
 |-  ( P = Q -> ( F ` ( G ` P ) ) = ( F ` ( G ` Q ) ) )
9 7 8 oveq12d 
 |-  ( P = Q -> ( P .\/ ( F ` ( G ` P ) ) ) = ( Q .\/ ( F ` ( G ` Q ) ) ) )
10 9 oveq1d 
 |-  ( P = Q -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )
11 10 adantl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T ) ) /\ P = Q ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )
12 simpl1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T ) ) /\ P =/= Q ) -> ( K e. HL /\ W e. H ) )
13 simpl2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T ) ) /\ P =/= Q ) -> ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) )
14 simpl3l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T ) ) /\ P =/= Q ) -> F e. T )
15 simpl3r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T ) ) /\ P =/= Q ) -> G e. T )
16 simpr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T ) ) /\ P =/= Q ) -> P =/= Q )
17 eqid 
 |-  ( ( trL ` K ) ` W ) = ( ( trL ` K ) ` W )
18 1 2 3 4 5 6 17 cdlemg39 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )
19 12 13 14 15 16 18 syl113anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T ) ) /\ P =/= Q ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )
20 11 19 pm2.61dane 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )