Metamath Proof Explorer


Theorem cdlemk16a

Description: Part of proof of Lemma K of Crawley p. 118. (Contributed by NM, 3-Jul-2013)

Ref Expression
Hypotheses cdlemk1.b
|- B = ( Base ` K )
cdlemk1.l
|- .<_ = ( le ` K )
cdlemk1.j
|- .\/ = ( join ` K )
cdlemk1.m
|- ./\ = ( meet ` K )
cdlemk1.a
|- A = ( Atoms ` K )
cdlemk1.h
|- H = ( LHyp ` K )
cdlemk1.t
|- T = ( ( LTrn ` K ) ` W )
cdlemk1.r
|- R = ( ( trL ` K ) ` W )
cdlemk1.s
|- S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
cdlemk1.o
|- O = ( S ` D )
Assertion cdlemk16a
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) /\ G e. T ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( ( ( R ` D ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` G ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( ( P .\/ ( R ` G ) ) ./\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) e. A /\ -. ( ( P .\/ ( R ` G ) ) ./\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) .<_ W ) )

Proof

Step Hyp Ref Expression
1 cdlemk1.b
 |-  B = ( Base ` K )
2 cdlemk1.l
 |-  .<_ = ( le ` K )
3 cdlemk1.j
 |-  .\/ = ( join ` K )
4 cdlemk1.m
 |-  ./\ = ( meet ` K )
5 cdlemk1.a
 |-  A = ( Atoms ` K )
6 cdlemk1.h
 |-  H = ( LHyp ` K )
7 cdlemk1.t
 |-  T = ( ( LTrn ` K ) ` W )
8 cdlemk1.r
 |-  R = ( ( trL ` K ) ` W )
9 cdlemk1.s
 |-  S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
10 cdlemk1.o
 |-  O = ( S ` D )
11 simp11
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) /\ G e. T ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( ( ( R ` D ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` G ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( K e. HL /\ W e. H ) )
12 simp22
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) /\ G e. T ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( ( ( R ` D ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` G ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> D e. T )
13 simp13
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) /\ G e. T ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( ( ( R ` D ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` G ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> G e. T )
14 simp33
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) /\ G e. T ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( ( ( R ` D ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` G ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( P e. A /\ -. P .<_ W ) )
15 simp21
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) /\ G e. T ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( ( ( R ` D ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` G ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> F e. T )
16 simp23
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) /\ G e. T ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( ( ( R ` D ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` G ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> N e. T )
17 simp12
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) /\ G e. T ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( ( ( R ` D ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` G ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( R ` F ) = ( R ` N ) )
18 simp321
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) /\ G e. T ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( ( ( R ` D ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` G ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> F =/= ( _I |` B ) )
19 simp323
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) /\ G e. T ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( ( ( R ` D ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` G ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> D =/= ( _I |` B ) )
20 simp31l
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) /\ G e. T ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( ( ( R ` D ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` G ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( R ` D ) =/= ( R ` F ) )
21 1 2 3 4 5 6 7 8 9 10 cdlemkoatnle
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> ( ( O ` P ) e. A /\ -. ( O ` P ) .<_ W ) )
22 11 15 12 16 14 17 18 19 20 21 syl333anc
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) /\ G e. T ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( ( ( R ` D ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` G ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( O ` P ) e. A /\ -. ( O ` P ) .<_ W ) )
23 1 2 3 4 5 6 7 8 9 10 cdlemkole
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> ( O ` P ) .<_ ( P .\/ ( R ` D ) ) )
24 11 15 12 16 14 17 18 19 20 23 syl333anc
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) /\ G e. T ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( ( ( R ` D ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` G ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( O ` P ) .<_ ( P .\/ ( R ` D ) ) )
25 simp322
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) /\ G e. T ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( ( ( R ` D ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` G ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> G =/= ( _I |` B ) )
26 simp31r
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) /\ G e. T ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( ( ( R ` D ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` G ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( R ` D ) =/= ( R ` G ) )
27 eqid
 |-  ( ( P .\/ ( R ` G ) ) ./\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) )
28 1 2 3 4 5 6 7 8 27 cdlemh
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ D e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( O ` P ) e. A /\ -. ( O ` P ) .<_ W ) /\ ( O ` P ) .<_ ( P .\/ ( R ` D ) ) ) /\ ( D =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` D ) =/= ( R ` G ) ) ) -> ( ( ( P .\/ ( R ` G ) ) ./\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) e. A /\ -. ( ( P .\/ ( R ` G ) ) ./\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) .<_ W ) )
29 11 12 13 14 22 24 19 25 26 28 syl333anc
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) /\ G e. T ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( ( ( R ` D ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` G ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( ( P .\/ ( R ` G ) ) ./\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) e. A /\ -. ( ( P .\/ ( R ` G ) ) ./\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) .<_ W ) )