Metamath Proof Explorer


Theorem cdlemg17ir

Description: TODO: fix comment. (Contributed by NM, 13-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 cdlemg17ir
|- ( ( ( ( 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 ` P ) =/= P /\ ( R ` G ) .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( F ` ( G ` P ) ) = ( F ` Q ) )

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 simp1
 |-  ( ( ( ( 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 ` P ) =/= P /\ ( R ` G ) .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) )
9 simp22
 |-  ( ( ( ( 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 ` P ) =/= P /\ ( R ` G ) .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> G e. T )
10 simp23
 |-  ( ( ( ( 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 ` P ) =/= P /\ ( R ` G ) .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> P =/= Q )
11 simp3
 |-  ( ( ( ( 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 ` P ) =/= P /\ ( R ` G ) .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( ( G ` P ) =/= P /\ ( R ` G ) .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) )
12 1 2 3 4 5 6 7 cdlemg17b
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( G e. T /\ P =/= Q ) /\ ( ( G ` P ) =/= P /\ ( R ` G ) .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( G ` P ) = Q )
13 8 9 10 11 12 syl121anc
 |-  ( ( ( ( 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 ` P ) =/= P /\ ( R ` G ) .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( G ` P ) = Q )
14 13 fveq2d
 |-  ( ( ( ( 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 ` P ) =/= P /\ ( R ` G ) .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( F ` ( G ` P ) ) = ( F ` Q ) )