Metamath Proof Explorer

Theorem cdlemg17iqN

Description: cdlemg17i with P and Q swapped. (Contributed by NM, 13-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 cdlemg17iqN 
|- ( ( ( K e. HL /\ W e. H /\ ( F e. T /\ 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 ` ( F ` Q ) ) = ( F ` P ) )

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 simp11 
 |-  ( ( ( K e. HL /\ W e. H /\ ( F e. T /\ 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 )
9 simp12 
 |-  ( ( ( K e. HL /\ W e. H /\ ( F e. T /\ 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 )
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 ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) /\ ( G ` P ) =/= P ) ) -> ( K e. HL /\ W e. H ) )
11 simp21 
 |-  ( ( ( K e. HL /\ W e. H /\ ( F e. T /\ 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 ) )
12 simp22 
 |-  ( ( ( K e. HL /\ W e. H /\ ( F e. T /\ 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 ) )
13 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 ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) /\ ( G ` P ) =/= P ) ) -> F e. T )
14 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 ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) /\ ( G ` P ) =/= P ) ) -> G e. T )
15 simp23 
 |-  ( ( ( K e. HL /\ W e. H /\ ( F e. T /\ 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 )
16 simp33 
 |-  ( ( ( K e. HL /\ W e. H /\ ( F e. T /\ 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 )
17 simp31 
 |-  ( ( ( K e. HL /\ W e. H /\ ( F e. T /\ 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 ) )
18 simp32 
 |-  ( ( ( K e. HL /\ W e. H /\ ( F e. T /\ 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 ) ) )
19 1 2 3 4 5 6 7 cdlemg17pq 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( ( G ` P ) =/= P /\ ( R ` G ) .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( F e. T /\ G e. T /\ Q =/= P ) /\ ( ( G ` Q ) =/= Q /\ ( R ` G ) .<_ ( Q .\/ P ) /\ -. E. r e. A ( -. r .<_ W /\ ( Q .\/ r ) = ( P .\/ r ) ) ) ) )
20 10 11 12 13 14 15 16 17 18 19 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 ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) /\ ( G ` P ) =/= P ) ) -> ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( F e. T /\ G e. T /\ Q =/= P ) /\ ( ( G ` Q ) =/= Q /\ ( R ` G ) .<_ ( Q .\/ P ) /\ -. E. r e. A ( -. r .<_ W /\ ( Q .\/ r ) = ( P .\/ r ) ) ) ) )
21 1 2 3 4 5 6 7 cdlemg17i 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( F e. T /\ G e. T /\ Q =/= P ) /\ ( ( G ` Q ) =/= Q /\ ( R ` G ) .<_ ( Q .\/ P ) /\ -. E. r e. A ( -. r .<_ W /\ ( Q .\/ r ) = ( P .\/ r ) ) ) ) -> ( G ` ( F ` Q ) ) = ( F ` P ) )
22 20 21 syl 
 |-  ( ( ( K e. HL /\ W e. H /\ ( F e. T /\ 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 ` ( F ` Q ) ) = ( F ` P ) )