Metamath Proof Explorer

Theorem dihjustlem

Description: Part of proof after Lemma N of Crawley p. 122 line 4, "the definition of phi(x) is independent of the atom q." (Contributed by NM, 2-Mar-2014)

Ref Expression
Hypotheses dihjust.b 
|- B = ( Base ` K )
dihjust.l 
|- .<_ = ( le ` K )
dihjust.j 
|- .\/ = ( join ` K )
dihjust.m 
|- ./\ = ( meet ` K )
dihjust.a 
|- A = ( Atoms ` K )
dihjust.h 
|- H = ( LHyp ` K )
dihjust.i 
|- I = ( ( DIsoB ` K ) ` W )
dihjust.J 
|- J = ( ( DIsoC ` K ) ` W )
dihjust.u 
|- U = ( ( DVecH ` K ) ` W )
dihjust.s 
|- .(+) = ( LSSum ` U )
Assertion dihjustlem 
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> ( ( J ` Q ) .(+) ( I ` ( X ./\ W ) ) ) C_ ( ( J ` R ) .(+) ( I ` ( X ./\ W ) ) ) )

Proof

Step Hyp Ref Expression
1 dihjust.b 
 |-  B = ( Base ` K )
2 dihjust.l 
 |-  .<_ = ( le ` K )
3 dihjust.j 
 |-  .\/ = ( join ` K )
4 dihjust.m 
 |-  ./\ = ( meet ` K )
5 dihjust.a 
 |-  A = ( Atoms ` K )
6 dihjust.h 
 |-  H = ( LHyp ` K )
7 dihjust.i 
 |-  I = ( ( DIsoB ` K ) ` W )
8 dihjust.J 
 |-  J = ( ( DIsoC ` K ) ` W )
9 dihjust.u 
 |-  U = ( ( DVecH ` K ) ` W )
10 dihjust.s 
 |-  .(+) = ( LSSum ` U )
11 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> K e. HL )
12 11 hllatd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> K e. Lat )
13 simp21l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> Q e. A )
14 1 5 atbase 
 |-  ( Q e. A -> Q e. B )
15 13 14 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> Q e. B )
16 simp23 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> X e. B )
17 simp1r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> W e. H )
18 1 6 lhpbase 
 |-  ( W e. H -> W e. B )
19 17 18 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> W e. B )
20 1 4 latmcl 
 |-  ( ( K e. Lat /\ X e. B /\ W e. B ) -> ( X ./\ W ) e. B )
21 12 16 19 20 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> ( X ./\ W ) e. B )
22 1 2 3 latlej1 
 |-  ( ( K e. Lat /\ Q e. B /\ ( X ./\ W ) e. B ) -> Q .<_ ( Q .\/ ( X ./\ W ) ) )
23 12 15 21 22 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> Q .<_ ( Q .\/ ( X ./\ W ) ) )
24 simp3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) )
25 23 24 breqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> Q .<_ ( R .\/ ( X ./\ W ) ) )
26 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> ( K e. HL /\ W e. H ) )
27 simp22 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> ( R e. A /\ -. R .<_ W ) )
28 simp21 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> ( Q e. A /\ -. Q .<_ W ) )
29 1 2 4 latmle2 
 |-  ( ( K e. Lat /\ X e. B /\ W e. B ) -> ( X ./\ W ) .<_ W )
30 12 16 19 29 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> ( X ./\ W ) .<_ W )
31 21 30 jca 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> ( ( X ./\ W ) e. B /\ ( X ./\ W ) .<_ W ) )
32 1 2 3 5 6 7 8 9 10 cdlemn 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( ( X ./\ W ) e. B /\ ( X ./\ W ) .<_ W ) ) ) -> ( Q .<_ ( R .\/ ( X ./\ W ) ) <-> ( J ` Q ) C_ ( ( J ` R ) .(+) ( I ` ( X ./\ W ) ) ) ) )
33 26 27 28 31 32 syl13anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> ( Q .<_ ( R .\/ ( X ./\ W ) ) <-> ( J ` Q ) C_ ( ( J ` R ) .(+) ( I ` ( X ./\ W ) ) ) ) )
34 25 33 mpbid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> ( J ` Q ) C_ ( ( J ` R ) .(+) ( I ` ( X ./\ W ) ) ) )
35 6 9 26 dvhlmod 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> U e. LMod )
36 eqid 
 |-  ( LSubSp ` U ) = ( LSubSp ` U )
37 36 lsssssubg 
 |-  ( U e. LMod -> ( LSubSp ` U ) C_ ( SubGrp ` U ) )
38 35 37 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> ( LSubSp ` U ) C_ ( SubGrp ` U ) )
39 2 5 6 9 8 36 diclss 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) ) -> ( J ` R ) e. ( LSubSp ` U ) )
40 26 27 39 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> ( J ` R ) e. ( LSubSp ` U ) )
41 38 40 sseldd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> ( J ` R ) e. ( SubGrp ` U ) )
42 1 2 6 9 7 36 diblss 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( X ./\ W ) e. B /\ ( X ./\ W ) .<_ W ) ) -> ( I ` ( X ./\ W ) ) e. ( LSubSp ` U ) )
43 26 21 30 42 syl12anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> ( I ` ( X ./\ W ) ) e. ( LSubSp ` U ) )
44 38 43 sseldd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> ( I ` ( X ./\ W ) ) e. ( SubGrp ` U ) )
45 10 lsmub2 
 |-  ( ( ( J ` R ) e. ( SubGrp ` U ) /\ ( I ` ( X ./\ W ) ) e. ( SubGrp ` U ) ) -> ( I ` ( X ./\ W ) ) C_ ( ( J ` R ) .(+) ( I ` ( X ./\ W ) ) ) )
46 41 44 45 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> ( I ` ( X ./\ W ) ) C_ ( ( J ` R ) .(+) ( I ` ( X ./\ W ) ) ) )
47 2 5 6 9 8 36 diclss 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( J ` Q ) e. ( LSubSp ` U ) )
48 26 28 47 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> ( J ` Q ) e. ( LSubSp ` U ) )
49 38 48 sseldd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> ( J ` Q ) e. ( SubGrp ` U ) )
50 36 10 lsmcl 
 |-  ( ( U e. LMod /\ ( J ` R ) e. ( LSubSp ` U ) /\ ( I ` ( X ./\ W ) ) e. ( LSubSp ` U ) ) -> ( ( J ` R ) .(+) ( I ` ( X ./\ W ) ) ) e. ( LSubSp ` U ) )
51 35 40 43 50 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> ( ( J ` R ) .(+) ( I ` ( X ./\ W ) ) ) e. ( LSubSp ` U ) )
52 38 51 sseldd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> ( ( J ` R ) .(+) ( I ` ( X ./\ W ) ) ) e. ( SubGrp ` U ) )
53 10 lsmlub 
 |-  ( ( ( J ` Q ) e. ( SubGrp ` U ) /\ ( I ` ( X ./\ W ) ) e. ( SubGrp ` U ) /\ ( ( J ` R ) .(+) ( I ` ( X ./\ W ) ) ) e. ( SubGrp ` U ) ) -> ( ( ( J ` Q ) C_ ( ( J ` R ) .(+) ( I ` ( X ./\ W ) ) ) /\ ( I ` ( X ./\ W ) ) C_ ( ( J ` R ) .(+) ( I ` ( X ./\ W ) ) ) ) <-> ( ( J ` Q ) .(+) ( I ` ( X ./\ W ) ) ) C_ ( ( J ` R ) .(+) ( I ` ( X ./\ W ) ) ) ) )
54 49 44 52 53 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> ( ( ( J ` Q ) C_ ( ( J ` R ) .(+) ( I ` ( X ./\ W ) ) ) /\ ( I ` ( X ./\ W ) ) C_ ( ( J ` R ) .(+) ( I ` ( X ./\ W ) ) ) ) <-> ( ( J ` Q ) .(+) ( I ` ( X ./\ W ) ) ) C_ ( ( J ` R ) .(+) ( I ` ( X ./\ W ) ) ) ) )
55 34 46 54 mpbi2and 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> ( ( J ` Q ) .(+) ( I ` ( X ./\ W ) ) ) C_ ( ( J ` R ) .(+) ( I ` ( X ./\ W ) ) ) )