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 ) )