Metamath Proof Explorer

Theorem cdlemg17bq

Description: cdlemg17b with P and Q swapped. Antecedent F e. ( TW ) is redundant for easier use. TODO: should we have redundant antecedent for cdlemg17b also? (Contributed by NM, 13-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 cdlemg17bq 
|- ( ( ( ( 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 ) ) ) ) -> ( G ` Q ) = 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 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 ) ) ) ) )
9 simp11 
 |-  ( ( ( ( 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 ) ) ) ) -> ( K e. HL /\ W e. H ) )
10 simp12 
 |-  ( ( ( ( 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 ) ) ) ) -> ( Q e. A /\ -. Q .<_ W ) )
11 simp13 
 |-  ( ( ( ( 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 ) ) ) ) -> ( P e. A /\ -. P .<_ W ) )
12 simp22 
 |-  ( ( ( ( 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 e. T )
13 simp23 
 |-  ( ( ( ( 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 ) ) ) ) -> Q =/= P )
14 simp3 
 |-  ( ( ( ( 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 ` Q ) =/= Q /\ ( R ` G ) .<_ ( Q .\/ P ) /\ -. E. r e. A ( -. r .<_ W /\ ( Q .\/ r ) = ( P .\/ r ) ) ) )
15 1 2 3 4 5 6 7 cdlemg17b 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( G e. T /\ Q =/= P ) /\ ( ( G ` Q ) =/= Q /\ ( R ` G ) .<_ ( Q .\/ P ) /\ -. E. r e. A ( -. r .<_ W /\ ( Q .\/ r ) = ( P .\/ r ) ) ) ) -> ( G ` Q ) = P )
16 9 10 11 12 13 14 15 syl321anc 
 |-  ( ( ( ( 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 ` Q ) = P )
17 8 16 syl 
 |-  ( ( ( ( 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 ) ) ) ) -> ( G ` Q ) = P )