Metamath Proof Explorer

Theorem cdlemksv2

Description: Part of proof of Lemma K of Crawley p. 118. Value of the sigma(p) function S at the fixed P parameter. (Contributed by NM, 26-Jun-2013)

Ref Expression
Hypotheses cdlemk.b 
|- B = ( Base ` K )
cdlemk.l 
|- .<_ = ( le ` K )
cdlemk.j 
|- .\/ = ( join ` K )
cdlemk.a 
|- A = ( Atoms ` K )
cdlemk.h 
|- H = ( LHyp ` K )
cdlemk.t 
|- T = ( ( LTrn ` K ) ` W )
cdlemk.r 
|- R = ( ( trL ` K ) ` W )
cdlemk.m 
|- ./\ = ( meet ` K )
cdlemk.s 
|- S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
Assertion cdlemksv2 
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> ( ( S ` G ) ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) )

Proof

Step Hyp Ref Expression
1 cdlemk.b 
 |-  B = ( Base ` K )
2 cdlemk.l 
 |-  .<_ = ( le ` K )
3 cdlemk.j 
 |-  .\/ = ( join ` K )
4 cdlemk.a 
 |-  A = ( Atoms ` K )
5 cdlemk.h 
 |-  H = ( LHyp ` K )
6 cdlemk.t 
 |-  T = ( ( LTrn ` K ) ` W )
7 cdlemk.r 
 |-  R = ( ( trL ` K ) ` W )
8 cdlemk.m 
 |-  ./\ = ( meet ` K )
9 cdlemk.s 
 |-  S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
10 simp13 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> G e. T )
11 1 2 3 4 5 6 7 8 9 cdlemksv 
 |-  ( G e. T -> ( S ` G ) = ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) ) )
12 10 11 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> ( S ` G ) = ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) ) )
13 12 eqcomd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) ) = ( S ` G ) )
14 1 2 3 4 5 6 7 8 9 cdlemksel 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> ( S ` G ) e. T )
15 simp11 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> ( K e. HL /\ W e. H ) )
16 simp22 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> ( P e. A /\ -. P .<_ W ) )
17 simp1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) )
18 simp21 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> N e. T )
19 2 4 5 6 ltrnel 
 |-  ( ( ( K e. HL /\ W e. H ) /\ N e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( N ` P ) e. A /\ -. ( N ` P ) .<_ W ) )
20 15 18 16 19 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> ( ( N ` P ) e. A /\ -. ( N ` P ) .<_ W ) )
21 simp11l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> K e. HL )
22 simp22l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> P e. A )
23 20 simpld 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> ( N ` P ) e. A )
24 2 3 4 hlatlej2 
 |-  ( ( K e. HL /\ P e. A /\ ( N ` P ) e. A ) -> ( N ` P ) .<_ ( P .\/ ( N ` P ) ) )
25 21 22 23 24 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> ( N ` P ) .<_ ( P .\/ ( N ` P ) ) )
26 simp23 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> ( R ` F ) = ( R ` N ) )
27 26 oveq2d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> ( P .\/ ( R ` F ) ) = ( P .\/ ( R ` N ) ) )
28 2 3 4 5 6 7 trljat1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ N e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( P .\/ ( R ` N ) ) = ( P .\/ ( N ` P ) ) )
29 15 18 16 28 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> ( P .\/ ( R ` N ) ) = ( P .\/ ( N ` P ) ) )
30 27 29 eqtr2d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> ( P .\/ ( N ` P ) ) = ( P .\/ ( R ` F ) ) )
31 25 30 breqtrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> ( N ` P ) .<_ ( P .\/ ( R ` F ) ) )
32 simp31 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> F =/= ( _I |` B ) )
33 simp32 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> G =/= ( _I |` B ) )
34 simp33 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> ( R ` G ) =/= ( R ` F ) )
35 34 necomd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> ( R ` F ) =/= ( R ` G ) )
36 eqid 
 |-  ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) )
37 1 2 3 8 4 5 6 7 36 cdlemh 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( N ` P ) e. A /\ -. ( N ` P ) .<_ W ) /\ ( N ` P ) .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) e. A /\ -. ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) .<_ W ) )
38 17 16 20 31 32 33 35 37 syl133anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> ( ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) e. A /\ -. ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) .<_ W ) )
39 2 4 5 6 cdleme 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) e. A /\ -. ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) .<_ W ) ) -> E! i e. T ( i ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) )
40 15 16 38 39 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> E! i e. T ( i ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) )
41 nfcv 
 |-  F/_ i T
42 nfriota1 
 |-  F/_ i ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) )
43 41 42 nfmpt 
 |-  F/_ i ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
44 9 43 nfcxfr 
 |-  F/_ i S
45 nfcv 
 |-  F/_ i G
46 44 45 nffv 
 |-  F/_ i ( S ` G )
47 nfcv 
 |-  F/_ i P
48 46 47 nffv 
 |-  F/_ i ( ( S ` G ) ` P )
49 48 nfeq1 
 |-  F/ i ( ( S ` G ) ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) )
50 fveq1 
 |-  ( i = ( S ` G ) -> ( i ` P ) = ( ( S ` G ) ` P ) )
51 50 eqeq1d 
 |-  ( i = ( S ` G ) -> ( ( i ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) <-> ( ( S ` G ) ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) ) )
52 46 49 51 riota2f 
 |-  ( ( ( S ` G ) e. T /\ E! i e. T ( i ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) ) -> ( ( ( S ` G ) ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) <-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) ) = ( S ` G ) ) )
53 14 40 52 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> ( ( ( S ` G ) ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) <-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) ) = ( S ` G ) ) )
54 13 53 mpbird 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> ( ( S ` G ) ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) )