Metamath Proof Explorer

Theorem cdleme26fALTN

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 t <_ t \/ v, then f_t(s) <_ f(t) \/ v. TODO: FIX COMMENT. (Contributed by NM, 1-Feb-2013) (New usage is discouraged.)

Ref Expression
Hypotheses cdleme26.b 
|- B = ( Base ` K )
cdleme26.l 
|- .<_ = ( le ` K )
cdleme26.j 
|- .\/ = ( join ` K )
cdleme26.m 
|- ./\ = ( meet ` K )
cdleme26.a 
|- A = ( Atoms ` K )
cdleme26.h 
|- H = ( LHyp ` K )
cdleme26f.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdleme26f.f 
|- F = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
cdleme26f.n 
|- N = ( ( P .\/ Q ) ./\ ( F .\/ ( ( S .\/ t ) ./\ W ) ) )
cdleme26f.i 
|- I = ( iota_ u e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> u = N ) )
Assertion cdleme26fALTN 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P =/= Q /\ S .<_ ( P .\/ Q ) ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) /\ ( S =/= t /\ S .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> I .<_ ( F .\/ V ) )

Proof

Step Hyp Ref Expression
1 cdleme26.b 
 |-  B = ( Base ` K )
2 cdleme26.l 
 |-  .<_ = ( le ` K )
3 cdleme26.j 
 |-  .\/ = ( join ` K )
4 cdleme26.m 
 |-  ./\ = ( meet ` K )
5 cdleme26.a 
 |-  A = ( Atoms ` K )
6 cdleme26.h 
 |-  H = ( LHyp ` K )
7 cdleme26f.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
8 cdleme26f.f 
 |-  F = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
9 cdleme26f.n 
 |-  N = ( ( P .\/ Q ) ./\ ( F .\/ ( ( S .\/ t ) ./\ W ) ) )
10 cdleme26f.i 
 |-  I = ( iota_ u e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> u = N ) )
11 simp11 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P =/= Q /\ S .<_ ( P .\/ Q ) ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) /\ ( S =/= t /\ S .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( K e. HL /\ W e. H ) )
12 simp21 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P =/= Q /\ S .<_ ( P .\/ Q ) ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) /\ ( S =/= t /\ S .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( P e. A /\ -. P .<_ W ) )
13 simp22 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P =/= Q /\ S .<_ ( P .\/ Q ) ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) /\ ( S =/= t /\ S .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( Q e. A /\ -. Q .<_ W ) )
14 simp23l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P =/= Q /\ S .<_ ( P .\/ Q ) ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) /\ ( S =/= t /\ S .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> S e. A )
15 simp23r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P =/= Q /\ S .<_ ( P .\/ Q ) ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) /\ ( S =/= t /\ S .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> -. S .<_ W )
16 simp12l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P =/= Q /\ S .<_ ( P .\/ Q ) ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) /\ ( S =/= t /\ S .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> P =/= Q )
17 simp12r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P =/= Q /\ S .<_ ( P .\/ Q ) ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) /\ ( S =/= t /\ S .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> S .<_ ( P .\/ Q ) )
18 1 2 3 4 5 6 7 8 9 10 cdleme25cl 
 |-  ( ( ( ( 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 ) ) ) -> I e. B )
19 11 12 13 14 15 16 17 18 syl322anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P =/= Q /\ S .<_ ( P .\/ Q ) ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) /\ ( S =/= t /\ S .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> I e. B )
20 simp13l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P =/= Q /\ S .<_ ( P .\/ Q ) ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) /\ ( S =/= t /\ S .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> t e. A )
21 simp31 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P =/= Q /\ S .<_ ( P .\/ Q ) ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) /\ ( S =/= t /\ S .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) )
22 1 fvexi 
 |-  B e. _V
23 22 10 riotasv 
 |-  ( ( I e. B /\ t e. A /\ ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) ) -> I = N )
24 19 20 21 23 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P =/= Q /\ S .<_ ( P .\/ Q ) ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) /\ ( S =/= t /\ S .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> I = N )
25 simp23 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P =/= Q /\ S .<_ ( P .\/ Q ) ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) /\ ( S =/= t /\ S .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( S e. A /\ -. S .<_ W ) )
26 simp33 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P =/= Q /\ S .<_ ( P .\/ Q ) ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) /\ ( S =/= t /\ S .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( V e. A /\ V .<_ W ) )
27 simp32 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P =/= Q /\ S .<_ ( P .\/ Q ) ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) /\ ( S =/= t /\ S .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( S =/= t /\ S .<_ ( t .\/ V ) ) )
28 2 3 4 5 6 7 8 9 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 ) )
29 11 12 13 25 20 26 27 28 syl331anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P =/= Q /\ S .<_ ( P .\/ Q ) ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) /\ ( S =/= t /\ S .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> N .<_ ( F .\/ V ) )
30 24 29 eqbrtrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P =/= Q /\ S .<_ ( P .\/ Q ) ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) /\ ( S =/= t /\ S .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> I .<_ ( F .\/ V ) )