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Metamath Proof Explorer

Theorem cdlemg8

Description: TODO: FIX COMMENT. (Contributed by NM, 29-Apr-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 cdlemg8 
|- ( ( ( 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 ) ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )

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 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 ) )
8 simpl21 
 |-  ( ( ( ( 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 ) )
9 simpl22 
 |-  ( ( ( ( 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 ) -> ( Q e. A /\ -. Q .<_ W ) )
10 simpl23 
 |-  ( ( ( ( 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 )
11 simpl3l 
 |-  ( ( ( ( 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 )
12 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 )
13 1 2 3 4 5 6 cdlemg8a 
 |-  ( ( ( 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 ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )
14 7 8 9 10 11 12 13 syl123anc 
 |-  ( ( ( ( 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 .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )
15 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 ) )
16 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 ) /\ F e. T ) )
17 simpl3l 
 |-  ( ( ( ( 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 )
18 simpl3r 
 |-  ( ( ( ( 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 ) )
19 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 )
20 1 2 3 4 5 6 cdlemg8d 
 |-  ( ( ( 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 .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )
21 15 16 17 18 19 20 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 ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )
22 14 21 pm2.61dane 
 |-  ( ( ( 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 ) ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )