Metamath Proof Explorer


Theorem cdlemg11a

Description: TODO: FIX COMMENT. (Contributed by NM, 4-May-2013)

Ref Expression
Hypotheses cdlemg8.l
|- .<_ = ( le ` K )
cdlemg8.j
|- .\/ = ( join ` K )
cdlemg8.m
|- ./\ = ( meet ` K )
cdlemg8.a
|- A = ( Atoms ` K )
cdlemg8.h
|- H = ( LHyp ` K )
cdlemg8.t
|- T = ( ( LTrn ` K ) ` W )
Assertion cdlemg11a
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> ( F ` ( G ` P ) ) =/= P )

Proof

Step Hyp Ref Expression
1 cdlemg8.l
 |-  .<_ = ( le ` K )
2 cdlemg8.j
 |-  .\/ = ( join ` K )
3 cdlemg8.m
 |-  ./\ = ( meet ` K )
4 cdlemg8.a
 |-  A = ( Atoms ` K )
5 cdlemg8.h
 |-  H = ( LHyp ` K )
6 cdlemg8.t
 |-  T = ( ( LTrn ` K ) ` W )
7 simp33
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) )
8 simpr
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) /\ ( F ` ( G ` P ) ) = P ) -> ( F ` ( G ` P ) ) = P )
9 simpl1
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) /\ ( F ` ( G ` P ) ) = P ) -> ( K e. HL /\ W e. H ) )
10 simpl2
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) /\ ( F ` ( G ` P ) ) = P ) -> ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) )
11 simpl31
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) /\ ( F ` ( G ` P ) ) = P ) -> F e. T )
12 simpl32
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) /\ ( F ` ( G ` P ) ) = P ) -> G e. T )
13 1 4 5 6 cdlemg6
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) -> ( F ` ( G ` Q ) ) = Q )
14 9 10 11 12 8 13 syl113anc
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) /\ ( F ` ( G ` P ) ) = P ) -> ( F ` ( G ` Q ) ) = Q )
15 8 14 oveq12d
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) /\ ( F ` ( G ` P ) ) = P ) -> ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) = ( P .\/ Q ) )
16 15 ex
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> ( ( F ` ( G ` P ) ) = P -> ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) = ( P .\/ Q ) ) )
17 16 necon3d
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> ( ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) -> ( F ` ( G ` P ) ) =/= P ) )
18 7 17 mpd
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> ( F ` ( G ` P ) ) =/= P )