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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemg33 Structured version   Unicode version
Theorem cdlemg33 33743
Description: Combine cdlemg33b 33739, cdlemg33c 33740, cdlemg33d 33741, cdlemg33e 33742. TODO: Fix comment. (Contributed by NM, 30-May-2013.)
Hypotheses
Ref Expression
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
Ref Expression
cdlemg33 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ 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 ) ) ) )
Distinct variable groups:   A, r   G, r   .\/ , r   .<_ , r   P, r   Q, r   W, r   F, r   z, A   z, F, r   H, r, z   z, .\/   K, r, z   z, .<_   N, r, z   z, P   z, Q   z, R   z, T   z, W   z, v, r   z, G   z, O, r
Allowed substitution hints:   A( v)   P( v)   Q( v)   R( v, r)   T( v, r)   F( v)   G( v)   H( v)   .\/ ( v)   K( v)   .<_ ( v)   ./\ ( z, v, r)   N( v)   O( v)   W( v)
Proof of Theorem cdlemg33
StepHypRef Expression
1 simp11 1029 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( K e. HL /\ W e. H ) )
2 simp12 1030 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( P e. A /\ -. P .<_ W ) )
3 simp13 1031 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( Q e. A /\ -. Q .<_ W ) )
4 simp21 1032 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( v e. A /\ v .<_ W ) )
5 simp22l 1118 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> F e. T )
6 simp31 1035 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> v =/= ( R ` F ) )
7 cdlemg12.l . . . 4 |- .<_ = ( le ` K )
8 cdlemg12.j . . . 4 |- .\/ = ( join ` K )
9 cdlemg12.m . . . 4 |- ./\ = ( meet ` K )
10 cdlemg12.a . . . 4 |- A = ( Atoms ` K )
11 cdlemg12.h . . . 4 |- H = ( LHyp ` K )
12 cdlemg12.t . . . 4 |- T = ( ( LTrn ` K ) ` W )
13 cdlemg12b.r . . . 4 |- R = ( ( trL ` K ) ` W )
14 cdlemg31.n . . . 4 |- N = ( ( P .\/ v ) ./\ ( Q .\/ ( R ` F ) ) )
157, 8, 9, 10, 11, 12, 13, 14cdlemg31b0a 33727 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) ) ) -> ( N e. A \/ N = ( 0. ` K ) ) )
161, 2, 3, 4, 5, 6, 15syl132anc 1250 . 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( N e. A \/ N = ( 0. ` K ) ) )
17 simp22r 1119 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> G e. T )
18 simp32 1036 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> v =/= ( R ` G ) )
19 cdlemg33.o . . . 4 |- O = ( ( P .\/ v ) ./\ ( Q .\/ ( R ` G ) ) )
207, 8, 9, 10, 11, 12, 13, 19cdlemg31b0a 33727 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( G e. T /\ v =/= ( R ` G ) ) ) -> ( O e. A \/ O = ( 0. ` K ) ) )
211, 2, 3, 4, 17, 18, 20syl132anc 1250 . 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( O e. A \/ O = ( 0. ` K ) ) )
22 simpl1 1002 . . . . 5 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( N e. A /\ O e. A ) ) -> ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) )
23 simpl21 1077 . . . . 5 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( N e. A /\ O e. A ) ) -> ( v e. A /\ v .<_ W ) )
24 simpr 461 . . . . 5 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( N e. A /\ O e. A ) ) -> ( N e. A /\ O e. A ) )
25 simpl22 1078 . . . . 5 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( N e. A /\ O e. A ) ) -> ( F e. T /\ G e. T ) )
26 simpl23 1079 . . . . 5 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( N e. A /\ O e. A ) ) -> P =/= Q )
27 simpl31 1080 . . . . 5 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( N e. A /\ O e. A ) ) -> v =/= ( R ` F ) )
28 simpl33 1082 . . . . 5 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( N e. A /\ O e. A ) ) -> E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) )
297, 8, 9, 10, 11, 12, 13, 14, 19cdlemg33b 33739 . . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( N e. A /\ O e. A ) /\ ( F e. T /\ G e. T ) ) /\ ( P =/= Q /\ v =/= ( R ` F ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> E. z e. A ( -. z .<_ W /\ ( z =/= N /\ z =/= O /\ z .<_ ( P .\/ v ) ) ) )
3022, 23, 24, 25, 26, 27, 28, 29syl133anc 1255 . . . 4 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( N e. A /\ O e. A ) ) -> E. z e. A ( -. z .<_ W /\ ( z =/= N /\ z =/= O /\ z .<_ ( P .\/ v ) ) ) )
3130ex 434 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( ( N e. A /\ O e. A ) -> E. z e. A ( -. z .<_ W /\ ( z =/= N /\ z =/= O /\ z .<_ ( P .\/ v ) ) ) ) )
32 simpl1 1002 . . . . 5 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( N = ( 0. ` K ) /\ O e. A ) ) -> ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) )
33 simpl21 1077 . . . . 5 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( N = ( 0. ` K ) /\ O e. A ) ) -> ( v e. A /\ v .<_ W ) )
34 simpr 461 . . . . 5 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( N = ( 0. ` K ) /\ O e. A ) ) -> ( N = ( 0. ` K ) /\ O e. A ) )
35 simpl22 1078 . . . . 5 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( N = ( 0. ` K ) /\ O e. A ) ) -> ( F e. T /\ G e. T ) )
36 simpl23 1079 . . . . 5 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( N = ( 0. ` K ) /\ O e. A ) ) -> P =/= Q )
37 simpl32 1081 . . . . 5 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( N = ( 0. ` K ) /\ O e. A ) ) -> v =/= ( R ` G ) )
38 simpl33 1082 . . . . 5 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( N = ( 0. ` K ) /\ O e. A ) ) -> E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) )
397, 8, 9, 10, 11, 12, 13, 14, 19cdlemg33d 33741 . . . . 5 |- ( ( ( ( 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 ) ) ) )
4032, 33, 34, 35, 36, 37, 38, 39syl133anc 1255 . . . 4 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( N = ( 0. ` K ) /\ O e. A ) ) -> E. z e. A ( -. z .<_ W /\ ( z =/= N /\ z =/= O /\ z .<_ ( P .\/ v ) ) ) )
4140ex 434 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( ( N = ( 0. ` K ) /\ O e. A ) -> E. z e. A ( -. z .<_ W /\ ( z =/= N /\ z =/= O /\ z .<_ ( P .\/ v ) ) ) ) )
42 simpl1 1002 . . . . 5 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( N e. A /\ O = ( 0. ` K ) ) ) -> ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) )
43 simpl21 1077 . . . . 5 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( N e. A /\ O = ( 0. ` K ) ) ) -> ( v e. A /\ v .<_ W ) )
44 simpr 461 . . . . 5 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( N e. A /\ O = ( 0. ` K ) ) ) -> ( N e. A /\ O = ( 0. ` K ) ) )
45 simpl22 1078 . . . . 5 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( N e. A /\ O = ( 0. ` K ) ) ) -> ( F e. T /\ G e. T ) )
46 simpl23 1079 . . . . 5 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( N e. A /\ O = ( 0. ` K ) ) ) -> P =/= Q )
47 simpl31 1080 . . . . 5 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( N e. A /\ O = ( 0. ` K ) ) ) -> v =/= ( R ` F ) )
48 simpl33 1082 . . . . 5 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( N e. A /\ O = ( 0. ` K ) ) ) -> E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) )
497, 8, 9, 10, 11, 12, 13, 14, 19cdlemg33c 33740 . . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( N e. A /\ O = ( 0. ` K ) ) /\ ( F e. T /\ G e. T ) ) /\ ( P =/= Q /\ v =/= ( R ` F ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> E. z e. A ( -. z .<_ W /\ ( z =/= N /\ z =/= O /\ z .<_ ( P .\/ v ) ) ) )
5042, 43, 44, 45, 46, 47, 48, 49syl133anc 1255 . . . 4 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( N e. A /\ O = ( 0. ` K ) ) ) -> E. z e. A ( -. z .<_ W /\ ( z =/= N /\ z =/= O /\ z .<_ ( P .\/ v ) ) ) )
5150ex 434 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( ( N e. A /\ O = ( 0. ` K ) ) -> E. z e. A ( -. z .<_ W /\ ( z =/= N /\ z =/= O /\ z .<_ ( P .\/ v ) ) ) ) )
52 simpl1 1002 . . . . 5 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( N = ( 0. ` K ) /\ O = ( 0. ` K ) ) ) -> ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) )
53 simpl21 1077 . . . . 5 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( N = ( 0. ` K ) /\ O = ( 0. ` K ) ) ) -> ( v e. A /\ v .<_ W ) )
54 simpr 461 . . . . 5 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( N = ( 0. ` K ) /\ O = ( 0. ` K ) ) ) -> ( N = ( 0. ` K ) /\ O = ( 0. ` K ) ) )
55 simpl22 1078 . . . . 5 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( N = ( 0. ` K ) /\ O = ( 0. ` K ) ) ) -> ( F e. T /\ G e. T ) )
56 simpl23 1079 . . . . 5 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( N = ( 0. ` K ) /\ O = ( 0. ` K ) ) ) -> P =/= Q )
57 simpl31 1080 . . . . 5 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( N = ( 0. ` K ) /\ O = ( 0. ` K ) ) ) -> v =/= ( R ` F ) )
58 simpl33 1082 . . . . 5 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( N = ( 0. ` K ) /\ O = ( 0. ` K ) ) ) -> E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) )
597, 8, 9, 10, 11, 12, 13, 14, 19cdlemg33e 33742 . . . . 5 |- ( ( ( ( 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 = ( 0. ` K ) ) /\ ( F e. T /\ G e. T ) ) /\ ( P =/= Q /\ v =/= ( R ` F ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> E. z e. A ( -. z .<_ W /\ ( z =/= N /\ z =/= O /\ z .<_ ( P .\/ v ) ) ) )
6052, 53, 54, 55, 56, 57, 58, 59syl133anc 1255 . . . 4 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( N = ( 0. ` K ) /\ O = ( 0. ` K ) ) ) -> E. z e. A ( -. z .<_ W /\ ( z =/= N /\ z =/= O /\ z .<_ ( P .\/ v ) ) ) )
6160ex 434 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( ( N = ( 0. ` K ) /\ O = ( 0. ` K ) ) -> E. z e. A ( -. z .<_ W /\ ( z =/= N /\ z =/= O /\ z .<_ ( P .\/ v ) ) ) ) )
6231, 41, 51, 61ccased 950 . 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( ( ( N e. A \/ N = ( 0. ` K ) ) /\ ( O e. A \/ O = ( 0. ` K ) ) ) -> E. z e. A ( -. z .<_ W /\ ( z =/= N /\ z =/= O /\ z .<_ ( P .\/ v ) ) ) ) )
6316, 21, 62mp2and 679 1 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ 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 ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   \/ wo 368   /\ wa 369   /\ w3a 976   = wceq 1407   e. wcel 1844   =/= wne 2600   E.wrex 2757   class class class wbr 4397   ` cfv 5571  (class class class)co 6280   lecple 14918   joincjn 15899   meetcmee 15900   0.cp0 15993   Atomscatm 32294   HLchlt 32381   LHypclh 33014   LTrncltrn 33131   trLctrl 33189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-iun 4275  df-iin 4276  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-1st 6786  df-2nd 6787  df-map 7461  df-preset 15883  df-poset 15901  df-plt 15914  df-lub 15930  df-glb 15931  df-join 15932  df-meet 15933  df-p0 15995  df-p1 15996  df-lat 16002  df-clat 16064  df-oposet 32207  df-ol 32209  df-oml 32210  df-covers 32297  df-ats 32298  df-atl 32329  df-cvlat 32353  df-hlat 32382  df-llines 32528  df-lplanes 32529  df-psubsp 32533  df-pmap 32534  df-padd 32826  df-lhyp 33018  df-laut 33019  df-ldil 33134  df-ltrn 33135  df-trl 33190
This theorem is referenced by:  cdlemg34  33744
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