Metamath Proof Explorer

Theorem cdlemk17

Description: Part of proof of Lemma K of Crawley p. 118. Line 21 on p. 119. O , D are k_1, f_1. (Contributed by NM, 1-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 cdlemk17 
|- ( ( ( ( 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 ) ) ) -> ( N ` P ) = ( ( P .\/ ( R ` F ) ) ./\ ( ( O ` P ) .\/ ( R ` ( F o. `' D ) ) ) ) )

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 cdlemk15 
 |-  ( ( ( ( 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 ) ) ) -> ( N ` P ) .<_ ( ( P .\/ ( R ` F ) ) ./\ ( ( O ` P ) .\/ ( R ` ( F o. `' D ) ) ) ) )
12 simp11l 
 |-  ( ( ( ( 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 ) ) ) -> K e. HL )
13 hlatl 
 |-  ( K e. HL -> K e. AtLat )
14 12 13 syl 
 |-  ( ( ( ( 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 ) ) ) -> K e. AtLat )
15 simp11 
 |-  ( ( ( ( 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 ) ) ) -> ( K e. HL /\ W e. H ) )
16 simp21 
 |-  ( ( ( ( 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 ) ) ) -> N e. T )
17 simp22l 
 |-  ( ( ( ( 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 ) ) ) -> P e. A )
18 2 5 6 7 ltrnat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ N e. T /\ P e. A ) -> ( N ` P ) e. A )
19 15 16 17 18 syl3anc 
 |-  ( ( ( ( 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 ) ) ) -> ( N ` P ) e. A )
20 1 2 3 4 5 6 7 8 9 10 cdlemk16 
 |-  ( ( ( ( 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 ) ) ) -> ( ( ( P .\/ ( R ` F ) ) ./\ ( ( O ` P ) .\/ ( R ` ( F o. `' D ) ) ) ) e. A /\ -. ( ( P .\/ ( R ` F ) ) ./\ ( ( O ` P ) .\/ ( R ` ( F o. `' D ) ) ) ) .<_ W ) )
21 20 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 ) ) ) -> ( ( P .\/ ( R ` F ) ) ./\ ( ( O ` P ) .\/ ( R ` ( F o. `' D ) ) ) ) e. A )
22 2 5 atcmp 
 |-  ( ( K e. AtLat /\ ( N ` P ) e. A /\ ( ( P .\/ ( R ` F ) ) ./\ ( ( O ` P ) .\/ ( R ` ( F o. `' D ) ) ) ) e. A ) -> ( ( N ` P ) .<_ ( ( P .\/ ( R ` F ) ) ./\ ( ( O ` P ) .\/ ( R ` ( F o. `' D ) ) ) ) <-> ( N ` P ) = ( ( P .\/ ( R ` F ) ) ./\ ( ( O ` P ) .\/ ( R ` ( F o. `' D ) ) ) ) ) )
23 14 19 21 22 syl3anc 
 |-  ( ( ( ( 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 ) ) ) -> ( ( N ` P ) .<_ ( ( P .\/ ( R ` F ) ) ./\ ( ( O ` P ) .\/ ( R ` ( F o. `' D ) ) ) ) <-> ( N ` P ) = ( ( P .\/ ( R ` F ) ) ./\ ( ( O ` P ) .\/ ( R ` ( F o. `' D ) ) ) ) ) )
24 11 23 mpbid 
 |-  ( ( ( ( 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 ) ) ) -> ( N ` P ) = ( ( P .\/ ( R ` F ) ) ./\ ( ( O ` P ) .\/ ( R ` ( F o. `' D ) ) ) ) )