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Metamath Proof Explorer

Theorem cdlemg33d

Description: TODO: Fix comment. (Contributed by NM, 30-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 )
cdlemg31.n 
|- N = ( ( P .\/ v ) ./\ ( Q .\/ ( R ` F ) ) )
cdlemg33.o 
|- O = ( ( P .\/ v ) ./\ ( Q .\/ ( R ` G ) ) )
Assertion cdlemg33d 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( N = ( 0. ` K ) /\ O e. A ) /\ ( F e. T /\ G e. T ) ) /\ ( P =/= Q /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> E. z e. A ( -. z .<_ W /\ ( z =/= N /\ z =/= O /\ z .<_ ( P .\/ v ) ) ) )

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 cdlemg31.n 
 |-  N = ( ( P .\/ v ) ./\ ( Q .\/ ( R ` F ) ) )
9 cdlemg33.o 
 |-  O = ( ( P .\/ v ) ./\ ( Q .\/ ( R ` G ) ) )
10 simp1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( N = ( 0. ` K ) /\ O e. A ) /\ ( F e. T /\ G e. T ) ) /\ ( P =/= Q /\ v =/= ( R ` G ) /\ 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 ) ) )
11 simp21 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( N = ( 0. ` K ) /\ O e. A ) /\ ( F e. T /\ G e. T ) ) /\ ( P =/= Q /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( v e. A /\ v .<_ W ) )
12 simp22r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( N = ( 0. ` K ) /\ O e. A ) /\ ( F e. T /\ G e. T ) ) /\ ( P =/= Q /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> O e. A )
13 simp22l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( N = ( 0. ` K ) /\ O e. A ) /\ ( F e. T /\ G e. T ) ) /\ ( P =/= Q /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> N = ( 0. ` K ) )
14 12 13 jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( N = ( 0. ` K ) /\ O e. A ) /\ ( F e. T /\ G e. T ) ) /\ ( P =/= Q /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( O e. A /\ N = ( 0. ` K ) ) )
15 simp23r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( N = ( 0. ` K ) /\ O e. A ) /\ ( F e. T /\ G e. T ) ) /\ ( P =/= Q /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> G e. T )
16 simp23l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( N = ( 0. ` K ) /\ O e. A ) /\ ( F e. T /\ G e. T ) ) /\ ( P =/= Q /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> F e. T )
17 15 16 jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( N = ( 0. ` K ) /\ O e. A ) /\ ( F e. T /\ G e. T ) ) /\ ( P =/= Q /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( G e. T /\ F e. T ) )
18 simp3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( N = ( 0. ` K ) /\ O e. A ) /\ ( F e. T /\ G e. T ) ) /\ ( P =/= Q /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( P =/= Q /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) )
19 1 2 3 4 5 6 7 9 8 cdlemg33c 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( O e. A /\ N = ( 0. ` K ) ) /\ ( G e. T /\ F e. T ) ) /\ ( P =/= Q /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> E. z e. A ( -. z .<_ W /\ ( z =/= O /\ z =/= N /\ z .<_ ( P .\/ v ) ) ) )
20 10 11 14 17 18 19 syl131anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( N = ( 0. ` K ) /\ O e. A ) /\ ( F e. T /\ G e. T ) ) /\ ( P =/= Q /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> E. z e. A ( -. z .<_ W /\ ( z =/= O /\ z =/= N /\ z .<_ ( P .\/ v ) ) ) )
21 3ancoma 
 |-  ( ( z =/= O /\ z =/= N /\ z .<_ ( P .\/ v ) ) <-> ( z =/= N /\ z =/= O /\ z .<_ ( P .\/ v ) ) )
22 21 anbi2i 
 |-  ( ( -. z .<_ W /\ ( z =/= O /\ z =/= N /\ z .<_ ( P .\/ v ) ) ) <-> ( -. z .<_ W /\ ( z =/= N /\ z =/= O /\ z .<_ ( P .\/ v ) ) ) )
23 22 rexbii 
 |-  ( E. z e. A ( -. z .<_ W /\ ( z =/= O /\ z =/= N /\ z .<_ ( P .\/ v ) ) ) <-> E. z e. A ( -. z .<_ W /\ ( z =/= N /\ z =/= O /\ z .<_ ( P .\/ v ) ) ) )
24 20 23 sylib 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( N = ( 0. ` K ) /\ O e. A ) /\ ( F e. T /\ G e. T ) ) /\ ( P =/= Q /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> E. z e. A ( -. z .<_ W /\ ( z =/= N /\ z =/= O /\ z .<_ ( P .\/ v ) ) ) )