This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem cdlemc1

Description: Part of proof of Lemma C in Crawley p. 112. TODO: shorten with atmod3i1 ? (Contributed by NM, 29-May-2012)

Ref Expression
Hypotheses cdlemc1.b 
|- B = ( Base ` K )
cdlemc1.l 
|- .<_ = ( le ` K )
cdlemc1.j 
|- .\/ = ( join ` K )
cdlemc1.m 
|- ./\ = ( meet ` K )
cdlemc1.a 
|- A = ( Atoms ` K )
cdlemc1.h 
|- H = ( LHyp ` K )
Assertion cdlemc1 
|- ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ ( P e. A /\ -. P .<_ W ) ) -> ( P .\/ ( ( P .\/ X ) ./\ W ) ) = ( P .\/ X ) )

Proof

Step Hyp Ref Expression
1 cdlemc1.b 
 |-  B = ( Base ` K )
2 cdlemc1.l 
 |-  .<_ = ( le ` K )
3 cdlemc1.j 
 |-  .\/ = ( join ` K )
4 cdlemc1.m 
 |-  ./\ = ( meet ` K )
5 cdlemc1.a 
 |-  A = ( Atoms ` K )
6 cdlemc1.h 
 |-  H = ( LHyp ` K )
7 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ ( P e. A /\ -. P .<_ W ) ) -> K e. HL )
8 7 hllatd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ ( P e. A /\ -. P .<_ W ) ) -> K e. Lat )
9 simp3l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ ( P e. A /\ -. P .<_ W ) ) -> P e. A )
10 1 5 atbase 
 |-  ( P e. A -> P e. B )
11 9 10 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ ( P e. A /\ -. P .<_ W ) ) -> P e. B )
12 simp2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ ( P e. A /\ -. P .<_ W ) ) -> X e. B )
13 1 3 latjcl 
 |-  ( ( K e. Lat /\ P e. B /\ X e. B ) -> ( P .\/ X ) e. B )
14 8 11 12 13 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ ( P e. A /\ -. P .<_ W ) ) -> ( P .\/ X ) e. B )
15 simp1r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ ( P e. A /\ -. P .<_ W ) ) -> W e. H )
16 1 6 lhpbase 
 |-  ( W e. H -> W e. B )
17 15 16 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ ( P e. A /\ -. P .<_ W ) ) -> W e. B )
18 1 4 latmcl 
 |-  ( ( K e. Lat /\ ( P .\/ X ) e. B /\ W e. B ) -> ( ( P .\/ X ) ./\ W ) e. B )
19 8 14 17 18 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( P .\/ X ) ./\ W ) e. B )
20 1 3 latjcom 
 |-  ( ( K e. Lat /\ P e. B /\ ( ( P .\/ X ) ./\ W ) e. B ) -> ( P .\/ ( ( P .\/ X ) ./\ W ) ) = ( ( ( P .\/ X ) ./\ W ) .\/ P ) )
21 8 11 19 20 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ ( P e. A /\ -. P .<_ W ) ) -> ( P .\/ ( ( P .\/ X ) ./\ W ) ) = ( ( ( P .\/ X ) ./\ W ) .\/ P ) )
22 1 2 3 latlej1 
 |-  ( ( K e. Lat /\ P e. B /\ X e. B ) -> P .<_ ( P .\/ X ) )
23 8 11 12 22 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ ( P e. A /\ -. P .<_ W ) ) -> P .<_ ( P .\/ X ) )
24 1 2 3 4 5 atmod2i1 
 |-  ( ( K e. HL /\ ( P e. A /\ ( P .\/ X ) e. B /\ W e. B ) /\ P .<_ ( P .\/ X ) ) -> ( ( ( P .\/ X ) ./\ W ) .\/ P ) = ( ( P .\/ X ) ./\ ( W .\/ P ) ) )
25 7 9 14 17 23 24 syl131anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( ( P .\/ X ) ./\ W ) .\/ P ) = ( ( P .\/ X ) ./\ ( W .\/ P ) ) )
26 eqid 
 |-  ( 1. ` K ) = ( 1. ` K )
27 2 3 26 5 6 lhpjat1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( W .\/ P ) = ( 1. ` K ) )
28 27 3adant2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ ( P e. A /\ -. P .<_ W ) ) -> ( W .\/ P ) = ( 1. ` K ) )
29 28 oveq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( P .\/ X ) ./\ ( W .\/ P ) ) = ( ( P .\/ X ) ./\ ( 1. ` K ) ) )
30 hlol 
 |-  ( K e. HL -> K e. OL )
31 7 30 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ ( P e. A /\ -. P .<_ W ) ) -> K e. OL )
32 1 4 26 olm11 
 |-  ( ( K e. OL /\ ( P .\/ X ) e. B ) -> ( ( P .\/ X ) ./\ ( 1. ` K ) ) = ( P .\/ X ) )
33 31 14 32 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( P .\/ X ) ./\ ( 1. ` K ) ) = ( P .\/ X ) )
34 29 33 eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( P .\/ X ) ./\ ( W .\/ P ) ) = ( P .\/ X ) )
35 21 25 34 3eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ ( P e. A /\ -. P .<_ W ) ) -> ( P .\/ ( ( P .\/ X ) ./\ W ) ) = ( P .\/ X ) )