Metamath Proof Explorer

Theorem cdlemg17dN

Description: TODO: fix comment. (Contributed by NM, 9-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 cdlemg17dN 
|- ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) /\ ( G ` P ) =/= P ) ) -> ( R ` G ) = ( ( P .\/ 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 simp1 
 |-  ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) /\ ( G ` P ) =/= P ) ) -> ( K e. HL /\ W e. H /\ G e. T ) )
9 simp21 
 |-  ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) /\ ( G ` P ) =/= P ) ) -> ( P e. A /\ -. P .<_ W ) )
10 simpl1 
 |-  ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> K e. HL )
11 simpl2 
 |-  ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> W e. H )
12 simpl3 
 |-  ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> G e. T )
13 simpr 
 |-  ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( P e. A /\ -. P .<_ W ) )
14 1 2 3 4 5 6 7 trlval2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( R ` G ) = ( ( P .\/ ( G ` P ) ) ./\ W ) )
15 10 11 12 13 14 syl211anc 
 |-  ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( R ` G ) = ( ( P .\/ ( G ` P ) ) ./\ W ) )
16 8 9 15 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) /\ ( G ` P ) =/= P ) ) -> ( R ` G ) = ( ( P .\/ ( G ` P ) ) ./\ W ) )
17 simp11 
 |-  ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) /\ ( G ` P ) =/= P ) ) -> K e. HL )
18 simp12 
 |-  ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) /\ ( G ` P ) =/= P ) ) -> W e. H )
19 17 18 jca 
 |-  ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) /\ ( G ` P ) =/= P ) ) -> ( K e. HL /\ W e. H ) )
20 simp22 
 |-  ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) /\ ( G ` P ) =/= P ) ) -> ( Q e. A /\ -. Q .<_ W ) )
21 simp13 
 |-  ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) /\ ( G ` P ) =/= P ) ) -> G e. T )
22 simp23 
 |-  ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) /\ ( G ` P ) =/= P ) ) -> P =/= Q )
23 simp33 
 |-  ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) /\ ( G ` P ) =/= P ) ) -> ( G ` P ) =/= P )
24 simp31 
 |-  ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) /\ ( G ` P ) =/= P ) ) -> ( R ` G ) .<_ ( P .\/ Q ) )
25 simp32 
 |-  ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) /\ ( G ` P ) =/= P ) ) -> -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) )
26 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 )
27 19 9 20 21 22 23 24 25 26 syl323anc 
 |-  ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) /\ ( G ` P ) =/= P ) ) -> ( G ` P ) = Q )
28 27 oveq2d 
 |-  ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) /\ ( G ` P ) =/= P ) ) -> ( P .\/ ( G ` P ) ) = ( P .\/ Q ) )
29 28 oveq1d 
 |-  ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) /\ ( G ` P ) =/= P ) ) -> ( ( P .\/ ( G ` P ) ) ./\ W ) = ( ( P .\/ Q ) ./\ W ) )
30 16 29 eqtrd 
 |-  ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) /\ ( G ` P ) =/= P ) ) -> ( R ` G ) = ( ( P .\/ Q ) ./\ W ) )