Metamath Proof Explorer

Theorem cdlemk6u

Description: Part of proof of Lemma K of Crawley p. 118. Apply dalaw . (Contributed by NM, 4-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 cdlemk6u 
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( ( P .\/ ( G ` P ) ) ./\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) .<_ ( ( ( ( G ` P ) .\/ ( X ` P ) ) ./\ ( ( R ` ( G o. `' D ) ) .\/ ( R ` ( X o. `' D ) ) ) ) .\/ ( ( ( X ` P ) .\/ P ) ./\ ( ( R ` ( X o. `' D ) ) .\/ ( O ` P ) ) ) ) )

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 1 2 3 4 5 6 7 8 9 10 cdlemk5u 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( ( P .\/ ( O ` P ) ) ./\ ( ( G ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) .<_ ( ( X ` P ) .\/ ( R ` ( X o. `' D ) ) ) )
12 simp11l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> K e. HL )
13 simp22l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> P e. A )
14 simp11 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( K e. HL /\ W e. H ) )
15 simp212 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> G e. T )
16 2 5 6 7 ltrnat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ P e. A ) -> ( G ` P ) e. A )
17 14 15 13 16 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( G ` P ) e. A )
18 simp213 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> X e. T )
19 2 5 6 7 ltrnat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. T /\ P e. A ) -> ( X ` P ) e. A )
20 14 18 13 19 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( X ` P ) e. A )
21 simp1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) )
22 simp211 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> N e. T )
23 simp22 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( P e. A /\ -. P .<_ W ) )
24 simp23 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( R ` F ) = ( R ` N ) )
25 simp3l1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> F =/= ( _I |` B ) )
26 simp3l2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> D =/= ( _I |` B ) )
27 simp3r1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( R ` D ) =/= ( R ` F ) )
28 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 ) )
29 28 simpld 
 |-  ( ( ( ( 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 )
30 21 22 23 24 25 26 27 29 syl133anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( O ` P ) e. A )
31 simp13 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> D e. T )
32 simp3r2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( R ` G ) =/= ( R ` D ) )
33 5 6 7 8 trlcocnvat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( G e. T /\ D e. T ) /\ ( R ` G ) =/= ( R ` D ) ) -> ( R ` ( G o. `' D ) ) e. A )
34 14 15 31 32 33 syl121anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( R ` ( G o. `' D ) ) e. A )
35 simp3r3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( R ` X ) =/= ( R ` D ) )
36 5 6 7 8 trlcocnvat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. T /\ D e. T ) /\ ( R ` X ) =/= ( R ` D ) ) -> ( R ` ( X o. `' D ) ) e. A )
37 14 18 31 35 36 syl121anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( R ` ( X o. `' D ) ) e. A )
38 2 3 4 5 dalaw 
 |-  ( ( K e. HL /\ ( P e. A /\ ( G ` P ) e. A /\ ( X ` P ) e. A ) /\ ( ( O ` P ) e. A /\ ( R ` ( G o. `' D ) ) e. A /\ ( R ` ( X o. `' D ) ) e. A ) ) -> ( ( ( P .\/ ( O ` P ) ) ./\ ( ( G ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) .<_ ( ( X ` P ) .\/ ( R ` ( X o. `' D ) ) ) -> ( ( P .\/ ( G ` P ) ) ./\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) .<_ ( ( ( ( G ` P ) .\/ ( X ` P ) ) ./\ ( ( R ` ( G o. `' D ) ) .\/ ( R ` ( X o. `' D ) ) ) ) .\/ ( ( ( X ` P ) .\/ P ) ./\ ( ( R ` ( X o. `' D ) ) .\/ ( O ` P ) ) ) ) ) )
39 12 13 17 20 30 34 37 38 syl133anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( ( ( P .\/ ( O ` P ) ) ./\ ( ( G ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) .<_ ( ( X ` P ) .\/ ( R ` ( X o. `' D ) ) ) -> ( ( P .\/ ( G ` P ) ) ./\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) .<_ ( ( ( ( G ` P ) .\/ ( X ` P ) ) ./\ ( ( R ` ( G o. `' D ) ) .\/ ( R ` ( X o. `' D ) ) ) ) .\/ ( ( ( X ` P ) .\/ P ) ./\ ( ( R ` ( X o. `' D ) ) .\/ ( O ` P ) ) ) ) ) )
40 11 39 mpd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( ( P .\/ ( G ` P ) ) ./\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) .<_ ( ( ( ( G ` P ) .\/ ( X ` P ) ) ./\ ( ( R ` ( G o. `' D ) ) .\/ ( R ` ( X o. `' D ) ) ) ) .\/ ( ( ( X ` P ) .\/ P ) ./\ ( ( R ` ( X o. `' D ) ) .\/ ( O ` P ) ) ) ) )