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Metamath Proof Explorer

Theorem cdlemkid2

Description: Lemma for cdlemkid . (Contributed by NM, 24-Jul-2013)

Ref Expression
Hypotheses cdlemk5.b 
|- B = ( Base ` K )
cdlemk5.l 
|- .<_ = ( le ` K )
cdlemk5.j 
|- .\/ = ( join ` K )
cdlemk5.m 
|- ./\ = ( meet ` K )
cdlemk5.a 
|- A = ( Atoms ` K )
cdlemk5.h 
|- H = ( LHyp ` K )
cdlemk5.t 
|- T = ( ( LTrn ` K ) ` W )
cdlemk5.r 
|- R = ( ( trL ` K ) ` W )
cdlemk5.z 
|- Z = ( ( P .\/ ( R ` b ) ) ./\ ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) )
cdlemk5.y 
|- Y = ( ( P .\/ ( R ` g ) ) ./\ ( Z .\/ ( R ` ( g o. `' b ) ) ) )
Assertion cdlemkid2 
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) /\ ( b e. T /\ b =/= ( _I |` B ) ) ) ) -> [_ G / g ]_ Y = P )

Proof

Step Hyp Ref Expression
1 cdlemk5.b 
 |-  B = ( Base ` K )
2 cdlemk5.l 
 |-  .<_ = ( le ` K )
3 cdlemk5.j 
 |-  .\/ = ( join ` K )
4 cdlemk5.m 
 |-  ./\ = ( meet ` K )
5 cdlemk5.a 
 |-  A = ( Atoms ` K )
6 cdlemk5.h 
 |-  H = ( LHyp ` K )
7 cdlemk5.t 
 |-  T = ( ( LTrn ` K ) ` W )
8 cdlemk5.r 
 |-  R = ( ( trL ` K ) ` W )
9 cdlemk5.z 
 |-  Z = ( ( P .\/ ( R ` b ) ) ./\ ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) )
10 cdlemk5.y 
 |-  Y = ( ( P .\/ ( R ` g ) ) ./\ ( Z .\/ ( R ` ( g o. `' b ) ) ) )
11 simp32 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) /\ ( b e. T /\ b =/= ( _I |` B ) ) ) ) -> G = ( _I |` B ) )
12 11 csbeq1d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) /\ ( b e. T /\ b =/= ( _I |` B ) ) ) ) -> [_ G / g ]_ Y = [_ ( _I |` B ) / g ]_ Y )
13 1 6 7 idltrn 
 |-  ( ( K e. HL /\ W e. H ) -> ( _I |` B ) e. T )
14 13 3ad2ant1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) /\ ( b e. T /\ b =/= ( _I |` B ) ) ) ) -> ( _I |` B ) e. T )
15 10 cdlemk41 
 |-  ( ( _I |` B ) e. T -> [_ ( _I |` B ) / g ]_ Y = ( ( P .\/ ( R ` ( _I |` B ) ) ) ./\ ( Z .\/ ( R ` ( ( _I |` B ) o. `' b ) ) ) ) )
16 14 15 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) /\ ( b e. T /\ b =/= ( _I |` B ) ) ) ) -> [_ ( _I |` B ) / g ]_ Y = ( ( P .\/ ( R ` ( _I |` B ) ) ) ./\ ( Z .\/ ( R ` ( ( _I |` B ) o. `' b ) ) ) ) )
17 eqid 
 |-  ( 0. ` K ) = ( 0. ` K )
18 1 17 6 8 trlid0 
 |-  ( ( K e. HL /\ W e. H ) -> ( R ` ( _I |` B ) ) = ( 0. ` K ) )
19 18 3ad2ant1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) /\ ( b e. T /\ b =/= ( _I |` B ) ) ) ) -> ( R ` ( _I |` B ) ) = ( 0. ` K ) )
20 19 oveq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) /\ ( b e. T /\ b =/= ( _I |` B ) ) ) ) -> ( P .\/ ( R ` ( _I |` B ) ) ) = ( P .\/ ( 0. ` K ) ) )
21 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) /\ ( b e. T /\ b =/= ( _I |` B ) ) ) ) -> K e. HL )
22 hlol 
 |-  ( K e. HL -> K e. OL )
23 21 22 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) /\ ( b e. T /\ b =/= ( _I |` B ) ) ) ) -> K e. OL )
24 simp31l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) /\ ( b e. T /\ b =/= ( _I |` B ) ) ) ) -> P e. A )
25 1 5 atbase 
 |-  ( P e. A -> P e. B )
26 24 25 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) /\ ( b e. T /\ b =/= ( _I |` B ) ) ) ) -> P e. B )
27 1 3 17 olj01 
 |-  ( ( K e. OL /\ P e. B ) -> ( P .\/ ( 0. ` K ) ) = P )
28 23 26 27 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) /\ ( b e. T /\ b =/= ( _I |` B ) ) ) ) -> ( P .\/ ( 0. ` K ) ) = P )
29 20 28 eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) /\ ( b e. T /\ b =/= ( _I |` B ) ) ) ) -> ( P .\/ ( R ` ( _I |` B ) ) ) = P )
30 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) /\ ( b e. T /\ b =/= ( _I |` B ) ) ) ) -> ( K e. HL /\ W e. H ) )
31 simp33l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) /\ ( b e. T /\ b =/= ( _I |` B ) ) ) ) -> b e. T )
32 6 7 ltrncnv 
 |-  ( ( ( K e. HL /\ W e. H ) /\ b e. T ) -> `' b e. T )
33 30 31 32 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) /\ ( b e. T /\ b =/= ( _I |` B ) ) ) ) -> `' b e. T )
34 1 6 7 ltrn1o 
 |-  ( ( ( K e. HL /\ W e. H ) /\ `' b e. T ) -> `' b : B -1-1-onto-> B )
35 30 33 34 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) /\ ( b e. T /\ b =/= ( _I |` B ) ) ) ) -> `' b : B -1-1-onto-> B )
36 f1of 
 |-  ( `' b : B -1-1-onto-> B -> `' b : B --> B )
37 fcoi2 
 |-  ( `' b : B --> B -> ( ( _I |` B ) o. `' b ) = `' b )
38 35 36 37 3syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) /\ ( b e. T /\ b =/= ( _I |` B ) ) ) ) -> ( ( _I |` B ) o. `' b ) = `' b )
39 38 fveq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) /\ ( b e. T /\ b =/= ( _I |` B ) ) ) ) -> ( R ` ( ( _I |` B ) o. `' b ) ) = ( R ` `' b ) )
40 6 7 8 trlcnv 
 |-  ( ( ( K e. HL /\ W e. H ) /\ b e. T ) -> ( R ` `' b ) = ( R ` b ) )
41 30 31 40 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) /\ ( b e. T /\ b =/= ( _I |` B ) ) ) ) -> ( R ` `' b ) = ( R ` b ) )
42 39 41 eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) /\ ( b e. T /\ b =/= ( _I |` B ) ) ) ) -> ( R ` ( ( _I |` B ) o. `' b ) ) = ( R ` b ) )
43 42 oveq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) /\ ( b e. T /\ b =/= ( _I |` B ) ) ) ) -> ( Z .\/ ( R ` ( ( _I |` B ) o. `' b ) ) ) = ( Z .\/ ( R ` b ) ) )
44 simp31 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) /\ ( b e. T /\ b =/= ( _I |` B ) ) ) ) -> ( P e. A /\ -. P .<_ W ) )
45 simp33 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) /\ ( b e. T /\ b =/= ( _I |` B ) ) ) ) -> ( b e. T /\ b =/= ( _I |` B ) ) )
46 44 45 jca 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) /\ ( b e. T /\ b =/= ( _I |` B ) ) ) ) -> ( ( P e. A /\ -. P .<_ W ) /\ ( b e. T /\ b =/= ( _I |` B ) ) ) )
47 1 2 3 4 5 6 7 8 9 cdlemkid1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( b e. T /\ b =/= ( _I |` B ) ) ) ) -> ( Z .\/ ( R ` b ) ) = ( P .\/ ( R ` b ) ) )
48 46 47 syld3an3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) /\ ( b e. T /\ b =/= ( _I |` B ) ) ) ) -> ( Z .\/ ( R ` b ) ) = ( P .\/ ( R ` b ) ) )
49 43 48 eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) /\ ( b e. T /\ b =/= ( _I |` B ) ) ) ) -> ( Z .\/ ( R ` ( ( _I |` B ) o. `' b ) ) ) = ( P .\/ ( R ` b ) ) )
50 29 49 oveq12d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) /\ ( b e. T /\ b =/= ( _I |` B ) ) ) ) -> ( ( P .\/ ( R ` ( _I |` B ) ) ) ./\ ( Z .\/ ( R ` ( ( _I |` B ) o. `' b ) ) ) ) = ( P ./\ ( P .\/ ( R ` b ) ) ) )
51 21 hllatd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) /\ ( b e. T /\ b =/= ( _I |` B ) ) ) ) -> K e. Lat )
52 1 6 7 8 trlcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ b e. T ) -> ( R ` b ) e. B )
53 30 31 52 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) /\ ( b e. T /\ b =/= ( _I |` B ) ) ) ) -> ( R ` b ) e. B )
54 1 3 4 latabs2 
 |-  ( ( K e. Lat /\ P e. B /\ ( R ` b ) e. B ) -> ( P ./\ ( P .\/ ( R ` b ) ) ) = P )
55 51 26 53 54 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) /\ ( b e. T /\ b =/= ( _I |` B ) ) ) ) -> ( P ./\ ( P .\/ ( R ` b ) ) ) = P )
56 50 55 eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) /\ ( b e. T /\ b =/= ( _I |` B ) ) ) ) -> ( ( P .\/ ( R ` ( _I |` B ) ) ) ./\ ( Z .\/ ( R ` ( ( _I |` B ) o. `' b ) ) ) ) = P )
57 16 56 eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) /\ ( b e. T /\ b =/= ( _I |` B ) ) ) ) -> [_ ( _I |` B ) / g ]_ Y = P )
58 12 57 eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) /\ ( b e. T /\ b =/= ( _I |` B ) ) ) ) -> [_ G / g ]_ Y = P )