Metamath Proof Explorer

Theorem cdleme22f

Description: Part of proof of Lemma E in Crawley p. 113, 3rd paragraph, 6th and 7th lines on p. 115. F , N represent f(t), f_t(s) respectively. If s <_ t \/ v, then f_t(s) <_ f(t) \/ v. (Contributed by NM, 6-Dec-2012)

Ref Expression
Hypotheses cdleme22.l 
|- .<_ = ( le ` K )
cdleme22.j 
|- .\/ = ( join ` K )
cdleme22.m 
|- ./\ = ( meet ` K )
cdleme22.a 
|- A = ( Atoms ` K )
cdleme22.h 
|- H = ( LHyp ` K )
cdleme22f.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdleme22f.f 
|- F = ( ( T .\/ U ) ./\ ( Q .\/ ( ( P .\/ T ) ./\ W ) ) )
cdleme22f.n 
|- N = ( ( P .\/ Q ) ./\ ( F .\/ ( ( S .\/ T ) ./\ W ) ) )
Assertion cdleme22f 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ ( V e. A /\ V .<_ W ) ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) ) -> N .<_ ( F .\/ V ) )

Proof

Step Hyp Ref Expression
1 cdleme22.l 
 |-  .<_ = ( le ` K )
2 cdleme22.j 
 |-  .\/ = ( join ` K )
3 cdleme22.m 
 |-  ./\ = ( meet ` K )
4 cdleme22.a 
 |-  A = ( Atoms ` K )
5 cdleme22.h 
 |-  H = ( LHyp ` K )
6 cdleme22f.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
7 cdleme22f.f 
 |-  F = ( ( T .\/ U ) ./\ ( Q .\/ ( ( P .\/ T ) ./\ W ) ) )
8 cdleme22f.n 
 |-  N = ( ( P .\/ Q ) ./\ ( F .\/ ( ( S .\/ T ) ./\ W ) ) )
9 simp11l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ ( V e. A /\ V .<_ W ) ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) ) -> K e. HL )
10 9 hllatd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ ( V e. A /\ V .<_ W ) ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) ) -> K e. Lat )
11 simp12l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ ( V e. A /\ V .<_ W ) ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) ) -> P e. A )
12 simp13l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ ( V e. A /\ V .<_ W ) ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) ) -> Q e. A )
13 eqid 
 |-  ( Base ` K ) = ( Base ` K )
14 13 2 4 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) )
15 9 11 12 14 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ ( V e. A /\ V .<_ W ) ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) ) -> ( P .\/ Q ) e. ( Base ` K ) )
16 simp11r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ ( V e. A /\ V .<_ W ) ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) ) -> W e. H )
17 simp22 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ ( V e. A /\ V .<_ W ) ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) ) -> T e. A )
18 1 2 3 4 5 6 7 13 cdleme1b 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ T e. A ) ) -> F e. ( Base ` K ) )
19 9 16 11 12 17 18 syl23anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ ( V e. A /\ V .<_ W ) ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) ) -> F e. ( Base ` K ) )
20 simp21l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ ( V e. A /\ V .<_ W ) ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) ) -> S e. A )
21 13 2 4 hlatjcl 
 |-  ( ( K e. HL /\ S e. A /\ T e. A ) -> ( S .\/ T ) e. ( Base ` K ) )
22 9 20 17 21 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ ( V e. A /\ V .<_ W ) ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) ) -> ( S .\/ T ) e. ( Base ` K ) )
23 13 5 lhpbase 
 |-  ( W e. H -> W e. ( Base ` K ) )
24 16 23 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ ( V e. A /\ V .<_ W ) ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) ) -> W e. ( Base ` K ) )
25 13 3 latmcl 
 |-  ( ( K e. Lat /\ ( S .\/ T ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( S .\/ T ) ./\ W ) e. ( Base ` K ) )
26 10 22 24 25 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ ( V e. A /\ V .<_ W ) ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) ) -> ( ( S .\/ T ) ./\ W ) e. ( Base ` K ) )
27 13 2 latjcl 
 |-  ( ( K e. Lat /\ F e. ( Base ` K ) /\ ( ( S .\/ T ) ./\ W ) e. ( Base ` K ) ) -> ( F .\/ ( ( S .\/ T ) ./\ W ) ) e. ( Base ` K ) )
28 10 19 26 27 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ ( V e. A /\ V .<_ W ) ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) ) -> ( F .\/ ( ( S .\/ T ) ./\ W ) ) e. ( Base ` K ) )
29 13 1 3 latmle2 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ ( F .\/ ( ( S .\/ T ) ./\ W ) ) e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ ( F .\/ ( ( S .\/ T ) ./\ W ) ) ) .<_ ( F .\/ ( ( S .\/ T ) ./\ W ) ) )
30 10 15 28 29 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ ( V e. A /\ V .<_ W ) ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) ) -> ( ( P .\/ Q ) ./\ ( F .\/ ( ( S .\/ T ) ./\ W ) ) ) .<_ ( F .\/ ( ( S .\/ T ) ./\ W ) ) )
31 simp21 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ ( V e. A /\ V .<_ W ) ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) ) -> ( S e. A /\ -. S .<_ W ) )
32 simp3l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ ( V e. A /\ V .<_ W ) ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) ) -> S =/= T )
33 simp23l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ ( V e. A /\ V .<_ W ) ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) ) -> V e. A )
34 simp23r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ ( V e. A /\ V .<_ W ) ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) ) -> V .<_ W )
35 simp3r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ ( V e. A /\ V .<_ W ) ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) ) -> S .<_ ( T .\/ V ) )
36 2 4 hlatjcom 
 |-  ( ( K e. HL /\ T e. A /\ V e. A ) -> ( T .\/ V ) = ( V .\/ T ) )
37 9 17 33 36 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ ( V e. A /\ V .<_ W ) ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) ) -> ( T .\/ V ) = ( V .\/ T ) )
38 35 37 breqtrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ ( V e. A /\ V .<_ W ) ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) ) -> S .<_ ( V .\/ T ) )
39 hlcvl 
 |-  ( K e. HL -> K e. CvLat )
40 9 39 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ ( V e. A /\ V .<_ W ) ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) ) -> K e. CvLat )
41 1 2 4 cvlatexch2 
 |-  ( ( K e. CvLat /\ ( S e. A /\ V e. A /\ T e. A ) /\ S =/= T ) -> ( S .<_ ( V .\/ T ) -> V .<_ ( S .\/ T ) ) )
42 40 20 33 17 32 41 syl131anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ ( V e. A /\ V .<_ W ) ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) ) -> ( S .<_ ( V .\/ T ) -> V .<_ ( S .\/ T ) ) )
43 38 42 mpd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ ( V e. A /\ V .<_ W ) ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) ) -> V .<_ ( S .\/ T ) )
44 eqid 
 |-  ( ( S .\/ T ) ./\ W ) = ( ( S .\/ T ) ./\ W )
45 1 2 3 4 5 44 cdleme22aa 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ S =/= T ) /\ ( V e. A /\ V .<_ W /\ V .<_ ( S .\/ T ) ) ) -> V = ( ( S .\/ T ) ./\ W ) )
46 9 16 31 17 32 33 34 43 45 syl233anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ ( V e. A /\ V .<_ W ) ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) ) -> V = ( ( S .\/ T ) ./\ W ) )
47 46 oveq2d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ ( V e. A /\ V .<_ W ) ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) ) -> ( F .\/ V ) = ( F .\/ ( ( S .\/ T ) ./\ W ) ) )
48 30 47 breqtrrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ ( V e. A /\ V .<_ W ) ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) ) -> ( ( P .\/ Q ) ./\ ( F .\/ ( ( S .\/ T ) ./\ W ) ) ) .<_ ( F .\/ V ) )
49 8 48 eqbrtrid 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ ( V e. A /\ V .<_ W ) ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) ) -> N .<_ ( F .\/ V ) )