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Metamath Proof Explorer

Theorem cdlemg21

Description: Version of cdlemg19 with ( RF ) .<_ ( P .\/ Q ) instead of ( RG ) .<_ ( P .\/ Q ) as a condition. (Contributed by NM, 23-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 cdlemg21 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( F e. T /\ G e. T ) /\ P =/= Q /\ ( F ` P ) =/= P ) /\ ( ( R ` F ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( ( 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 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 /\ ( F ` P ) =/= P ) /\ ( ( R ` F ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( 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 simp21r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( F e. T /\ G e. T ) /\ P =/= Q /\ ( F ` P ) =/= P ) /\ ( ( R ` F ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> G e. T )
10 simp21l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( F e. T /\ G e. T ) /\ P =/= Q /\ ( F ` P ) =/= P ) /\ ( ( R ` F ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> F e. T )
11 9 10 jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( F e. T /\ G e. T ) /\ P =/= Q /\ ( F ` P ) =/= P ) /\ ( ( R ` F ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( G e. T /\ F e. T ) )
12 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 /\ ( F ` P ) =/= P ) /\ ( ( R ` F ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> P =/= Q )
13 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 /\ ( F ` P ) =/= P ) /\ ( ( R ` F ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( F ` P ) =/= P )
14 simp31 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( F e. T /\ G e. T ) /\ P =/= Q /\ ( F ` P ) =/= P ) /\ ( ( R ` F ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( R ` F ) .<_ ( P .\/ Q ) )
15 simp33 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( F e. T /\ G e. T ) /\ P =/= Q /\ ( F ` P ) =/= P ) /\ ( ( R ` F ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) )
16 1 2 3 4 5 6 7 cdlemg17j 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( G e. T /\ F e. T /\ P =/= Q ) /\ ( ( F ` P ) =/= P /\ ( R ` F ) .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( F ` ( G ` P ) ) = ( G ` ( F ` P ) ) )
17 8 9 10 12 13 14 15 16 syl133anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( F e. T /\ G e. T ) /\ P =/= Q /\ ( F ` P ) =/= P ) /\ ( ( R ` F ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( F ` ( G ` P ) ) = ( G ` ( F ` P ) ) )
18 simp11 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( F e. T /\ G e. T ) /\ P =/= Q /\ ( F ` P ) =/= P ) /\ ( ( R ` F ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( K e. HL /\ W e. H ) )
19 simp13 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( F e. T /\ G e. T ) /\ P =/= Q /\ ( F ` P ) =/= P ) /\ ( ( R ` F ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( Q e. A /\ -. Q .<_ W ) )
20 simp12 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( F e. T /\ G e. T ) /\ P =/= Q /\ ( F ` P ) =/= P ) /\ ( ( R ` F ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( P e. A /\ -. P .<_ W ) )
21 12 necomd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( F e. T /\ G e. T ) /\ P =/= Q /\ ( F ` P ) =/= P ) /\ ( ( R ` F ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> Q =/= P )
22 1 4 5 6 ltrnatneq 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) =/= P ) -> ( F ` Q ) =/= Q )
23 18 10 20 19 13 22 syl131anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( F e. T /\ G e. T ) /\ P =/= Q /\ ( F ` P ) =/= P ) /\ ( ( R ` F ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( F ` Q ) =/= Q )
24 simp11l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( F e. T /\ G e. T ) /\ P =/= Q /\ ( F ` P ) =/= P ) /\ ( ( R ` F ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> K e. HL )
25 simp12l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( F e. T /\ G e. T ) /\ P =/= Q /\ ( F ` P ) =/= P ) /\ ( ( R ` F ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> P e. A )
26 simp13l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( F e. T /\ G e. T ) /\ P =/= Q /\ ( F ` P ) =/= P ) /\ ( ( R ` F ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> Q e. A )
27 2 4 hlatjcom 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) = ( Q .\/ P ) )
28 24 25 26 27 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( F e. T /\ G e. T ) /\ P =/= Q /\ ( F ` P ) =/= P ) /\ ( ( R ` F ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( P .\/ Q ) = ( Q .\/ P ) )
29 14 28 breqtrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( F e. T /\ G e. T ) /\ P =/= Q /\ ( F ` P ) =/= P ) /\ ( ( R ` F ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( R ` F ) .<_ ( Q .\/ P ) )
30 eqcom 
 |-  ( ( P .\/ r ) = ( Q .\/ r ) <-> ( Q .\/ r ) = ( P .\/ r ) )
31 30 anbi2i 
 |-  ( ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) <-> ( -. r .<_ W /\ ( Q .\/ r ) = ( P .\/ r ) ) )
32 31 rexbii 
 |-  ( E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) <-> E. r e. A ( -. r .<_ W /\ ( Q .\/ r ) = ( P .\/ r ) ) )
33 15 32 sylnib 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( F e. T /\ G e. T ) /\ P =/= Q /\ ( F ` P ) =/= P ) /\ ( ( R ` F ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> -. E. r e. A ( -. r .<_ W /\ ( Q .\/ r ) = ( P .\/ r ) ) )
34 1 2 3 4 5 6 7 cdlemg17j 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( G e. T /\ F e. T /\ Q =/= P ) /\ ( ( F ` Q ) =/= Q /\ ( R ` F ) .<_ ( Q .\/ P ) /\ -. E. r e. A ( -. r .<_ W /\ ( Q .\/ r ) = ( P .\/ r ) ) ) ) -> ( F ` ( G ` Q ) ) = ( G ` ( F ` Q ) ) )
35 18 19 20 9 10 21 23 29 33 34 syl333anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( F e. T /\ G e. T ) /\ P =/= Q /\ ( F ` P ) =/= P ) /\ ( ( R ` F ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( F ` ( G ` Q ) ) = ( G ` ( F ` Q ) ) )
36 17 35 oveq12d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( F e. T /\ G e. T ) /\ P =/= Q /\ ( F ` P ) =/= P ) /\ ( ( R ` F ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) = ( ( G ` ( F ` P ) ) .\/ ( G ` ( F ` Q ) ) ) )
37 simp32 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( F e. T /\ G e. T ) /\ P =/= Q /\ ( F ` P ) =/= P ) /\ ( ( R ` F ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) )
38 36 37 eqnetrrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( F e. T /\ G e. T ) /\ P =/= Q /\ ( F ` P ) =/= P ) /\ ( ( R ` F ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( ( G ` ( F ` P ) ) .\/ ( G ` ( F ` Q ) ) ) =/= ( P .\/ Q ) )
39 1 2 3 4 5 6 7 cdlemg19 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( G e. T /\ F e. T ) /\ P =/= Q /\ ( F ` P ) =/= P ) /\ ( ( R ` F ) .<_ ( P .\/ Q ) /\ ( ( G ` ( F ` P ) ) .\/ ( G ` ( F ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( ( P .\/ ( G ` ( F ` P ) ) ) ./\ W ) = ( ( Q .\/ ( G ` ( F ` Q ) ) ) ./\ W ) )
40 8 11 12 13 14 38 15 39 syl133anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( F e. T /\ G e. T ) /\ P =/= Q /\ ( F ` P ) =/= P ) /\ ( ( R ` F ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( ( P .\/ ( G ` ( F ` P ) ) ) ./\ W ) = ( ( Q .\/ ( G ` ( F ` Q ) ) ) ./\ W ) )
41 17 oveq2d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( F e. T /\ G e. T ) /\ P =/= Q /\ ( F ` P ) =/= P ) /\ ( ( R ` F ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( P .\/ ( F ` ( G ` P ) ) ) = ( P .\/ ( G ` ( F ` P ) ) ) )
42 41 oveq1d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( F e. T /\ G e. T ) /\ P =/= Q /\ ( F ` P ) =/= P ) /\ ( ( R ` F ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( P .\/ ( G ` ( F ` P ) ) ) ./\ W ) )
43 35 oveq2d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( F e. T /\ G e. T ) /\ P =/= Q /\ ( F ` P ) =/= P ) /\ ( ( R ` F ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( Q .\/ ( F ` ( G ` Q ) ) ) = ( Q .\/ ( G ` ( F ` Q ) ) ) )
44 43 oveq1d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( F e. T /\ G e. T ) /\ P =/= Q /\ ( F ` P ) =/= P ) /\ ( ( R ` F ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) = ( ( Q .\/ ( G ` ( F ` Q ) ) ) ./\ W ) )
45 40 42 44 3eqtr4d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( F e. T /\ G e. T ) /\ P =/= Q /\ ( F ` P ) =/= P ) /\ ( ( R ` F ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )