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Metamath Proof Explorer

Theorem cdleme22f2

Description: Part of proof of Lemma E in Crawley p. 113. cdleme22f with s and t swapped (this case is not mentioned by them). If s <_ t \/ v, then f(s) <_ f_s(t) \/ v. (Contributed by NM, 7-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 )
cdleme22f2.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdleme22f2.f 
|- F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
cdleme22f2.n 
|- N = ( ( P .\/ Q ) ./\ ( F .\/ ( ( T .\/ S ) ./\ W ) ) )
Assertion cdleme22f2 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> F .<_ ( N .\/ 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 cdleme22f2.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
7 cdleme22f2.f 
 |-  F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
8 cdleme22f2.n 
 |-  N = ( ( P .\/ Q ) ./\ ( F .\/ ( ( T .\/ S ) ./\ W ) ) )
9 simp11 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( K e. HL /\ W e. H ) )
10 simp2l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( P e. A /\ -. P .<_ W ) )
11 simp2r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( Q e. A /\ -. Q .<_ W ) )
12 9 10 11 3jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) )
13 simp12 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( T e. A /\ -. T .<_ W ) )
14 simp31l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> S e. A )
15 simp33 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( V e. A /\ V .<_ W ) )
16 simp32l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> S =/= T )
17 16 necomd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> T =/= S )
18 simp32r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> S .<_ ( T .\/ V ) )
19 simp11l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> K e. HL )
20 hlcvl 
 |-  ( K e. HL -> K e. CvLat )
21 19 20 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> K e. CvLat )
22 simp12l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> T e. A )
23 simp33l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> V e. A )
24 simp33r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> V .<_ W )
25 simp31r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> -. S .<_ W )
26 nbrne2 
 |-  ( ( V .<_ W /\ -. S .<_ W ) -> V =/= S )
27 26 necomd 
 |-  ( ( V .<_ W /\ -. S .<_ W ) -> S =/= V )
28 24 25 27 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> S =/= V )
29 1 2 4 cvlatexch2 
 |-  ( ( K e. CvLat /\ ( S e. A /\ T e. A /\ V e. A ) /\ S =/= V ) -> ( S .<_ ( T .\/ V ) -> T .<_ ( S .\/ V ) ) )
30 21 14 22 23 28 29 syl131anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( S .<_ ( T .\/ V ) -> T .<_ ( S .\/ V ) ) )
31 18 30 mpd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> T .<_ ( S .\/ V ) )
32 1 2 3 4 5 6 7 8 cdleme22f 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( T e. A /\ -. T .<_ W ) /\ S e. A /\ ( V e. A /\ V .<_ W ) ) /\ ( T =/= S /\ T .<_ ( S .\/ V ) ) ) -> N .<_ ( F .\/ V ) )
33 12 13 14 15 17 31 32 syl132anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> N .<_ ( F .\/ V ) )
34 simp31 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( S e. A /\ -. S .<_ W ) )
35 simp133 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> P =/= Q )
36 simp132 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> T .<_ ( P .\/ Q ) )
37 simp131 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> -. S .<_ ( P .\/ Q ) )
38 1 2 3 4 5 6 7 8 cdleme7ga 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( T e. A /\ -. T .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ T .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> N e. A )
39 12 13 34 35 36 37 38 syl123anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> N e. A )
40 1 2 3 4 5 6 7 cdleme3fa 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> F e. A )
41 9 10 11 34 35 37 40 syl132anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> F e. A )
42 1 2 3 4 5 6 7 8 cdleme7 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( T e. A /\ -. T .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ T .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> -. N .<_ W )
43 12 13 34 35 36 37 42 syl123anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> -. N .<_ W )
44 nbrne2 
 |-  ( ( V .<_ W /\ -. N .<_ W ) -> V =/= N )
45 44 necomd 
 |-  ( ( V .<_ W /\ -. N .<_ W ) -> N =/= V )
46 24 43 45 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> N =/= V )
47 1 2 4 cvlatexch2 
 |-  ( ( K e. CvLat /\ ( N e. A /\ F e. A /\ V e. A ) /\ N =/= V ) -> ( N .<_ ( F .\/ V ) -> F .<_ ( N .\/ V ) ) )
48 21 39 41 23 46 47 syl131anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( N .<_ ( F .\/ V ) -> F .<_ ( N .\/ V ) ) )
49 33 48 mpd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. S .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> F .<_ ( N .\/ V ) )