Metamath Proof Explorer

Theorem cdlemg16ALTN

Description: This version of cdlemg16 uses cdlemg15a instead of cdlemg15 , in case cdlemg15 ends up not being needed. TODO: FIX COMMENT. (Contributed by NM, 6-May-2013) (New usage is discouraged.)

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 cdlemg16ALTN 
|- ( ( ( K e. HL /\ W e. H /\ ( F e. T /\ G e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )

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 simpl11 
 |-  ( ( ( ( K e. HL /\ W e. H /\ ( F e. T /\ G e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( R ` F ) = ( R ` G ) ) -> K e. HL )
9 simpl12 
 |-  ( ( ( ( K e. HL /\ W e. H /\ ( F e. T /\ G e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( R ` F ) = ( R ` G ) ) -> W e. H )
10 8 9 jca 
 |-  ( ( ( ( K e. HL /\ W e. H /\ ( F e. T /\ G e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( R ` F ) = ( R ` G ) ) -> ( K e. HL /\ W e. H ) )
11 simpl21 
 |-  ( ( ( ( K e. HL /\ W e. H /\ ( F e. T /\ G e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( R ` F ) = ( R ` G ) ) -> ( P e. A /\ -. P .<_ W ) )
12 simpl22 
 |-  ( ( ( ( K e. HL /\ W e. H /\ ( F e. T /\ G e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( R ` F ) = ( R ` G ) ) -> ( Q e. A /\ -. Q .<_ W ) )
13 simpl13 
 |-  ( ( ( ( K e. HL /\ W e. H /\ ( F e. T /\ G e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( R ` F ) = ( R ` G ) ) -> ( F e. T /\ G e. T ) )
14 simpr 
 |-  ( ( ( ( K e. HL /\ W e. H /\ ( F e. T /\ G e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( R ` F ) = ( R ` G ) ) -> ( R ` F ) = ( R ` G ) )
15 simpl31 
 |-  ( ( ( ( K e. HL /\ W e. H /\ ( F e. T /\ G e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( R ` F ) = ( R ` G ) ) -> ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) )
16 1 2 3 4 5 6 7 cdlemg15a 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T ) /\ ( ( R ` F ) = ( R ` G ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )
17 10 11 12 13 14 15 16 syl312anc 
 |-  ( ( ( ( K e. HL /\ W e. H /\ ( F e. T /\ G e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( R ` F ) = ( R ` G ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )
18 simpl11 
 |-  ( ( ( ( K e. HL /\ W e. H /\ ( F e. T /\ G e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( R ` F ) =/= ( R ` G ) ) -> K e. HL )
19 simpl12 
 |-  ( ( ( ( K e. HL /\ W e. H /\ ( F e. T /\ G e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( R ` F ) =/= ( R ` G ) ) -> W e. H )
20 18 19 jca 
 |-  ( ( ( ( K e. HL /\ W e. H /\ ( F e. T /\ G e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( R ` F ) =/= ( R ` G ) ) -> ( K e. HL /\ W e. H ) )
21 simpl21 
 |-  ( ( ( ( K e. HL /\ W e. H /\ ( F e. T /\ G e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( R ` F ) =/= ( R ` G ) ) -> ( P e. A /\ -. P .<_ W ) )
22 simpl22 
 |-  ( ( ( ( K e. HL /\ W e. H /\ ( F e. T /\ G e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( R ` F ) =/= ( R ` G ) ) -> ( Q e. A /\ -. Q .<_ W ) )
23 simp13l 
 |-  ( ( ( K e. HL /\ W e. H /\ ( F e. T /\ G e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) -> F e. T )
24 23 adantr 
 |-  ( ( ( ( K e. HL /\ W e. H /\ ( F e. T /\ G e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( R ` F ) =/= ( R ` G ) ) -> F e. T )
25 simp13r 
 |-  ( ( ( K e. HL /\ W e. H /\ ( F e. T /\ G e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) -> G e. T )
26 25 adantr 
 |-  ( ( ( ( K e. HL /\ W e. H /\ ( F e. T /\ G e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( R ` F ) =/= ( R ` G ) ) -> G e. T )
27 simpl23 
 |-  ( ( ( ( K e. HL /\ W e. H /\ ( F e. T /\ G e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( R ` F ) =/= ( R ` G ) ) -> P =/= Q )
28 simpl32 
 |-  ( ( ( ( K e. HL /\ W e. H /\ ( F e. T /\ G e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( R ` F ) =/= ( R ` G ) ) -> -. ( R ` F ) .<_ ( P .\/ Q ) )
29 simpl33 
 |-  ( ( ( ( K e. HL /\ W e. H /\ ( F e. T /\ G e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( R ` F ) =/= ( R ` G ) ) -> -. ( R ` G ) .<_ ( P .\/ Q ) )
30 28 29 jca 
 |-  ( ( ( ( K e. HL /\ W e. H /\ ( F e. T /\ G e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( R ` F ) =/= ( R ` G ) ) -> ( -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) )
31 simpr 
 |-  ( ( ( ( K e. HL /\ W e. H /\ ( F e. T /\ G e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( R ` F ) =/= ( R ` G ) ) -> ( R ` F ) =/= ( R ` G ) )
32 simpl31 
 |-  ( ( ( ( K e. HL /\ W e. H /\ ( F e. T /\ G e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( R ` F ) =/= ( R ` G ) ) -> ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) )
33 1 2 3 4 5 6 7 cdlemg12 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( ( -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) /\ ( R ` F ) =/= ( R ` G ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )
34 20 21 22 24 26 27 30 31 32 33 syl333anc 
 |-  ( ( ( ( K e. HL /\ W e. H /\ ( F e. T /\ G e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( R ` F ) =/= ( R ` G ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )
35 17 34 pm2.61dane 
 |-  ( ( ( K e. HL /\ W e. H /\ ( F e. T /\ G e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )