Metamath Proof Explorer

Theorem cdleme26f2

Description: Part of proof of Lemma E in Crawley p. 113. cdleme26fALTN with s and t swapped (this case is not mentioned by them). If s <_ t \/ v, then f(s) <_ f_s(t) \/ v. TODO: FIX COMMENT. (Contributed by NM, 1-Feb-2013)

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 )
cdleme26f2.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdleme26f2.f 
|- G = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
cdleme26f2.n 
|- O = ( ( P .\/ Q ) ./\ ( G .\/ ( ( T .\/ s ) ./\ W ) ) )
cdleme26f2.e 
|- E = ( iota_ u e. B A. s e. A ( ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) -> u = O ) )
Assertion cdleme26f2 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P =/= Q /\ T .<_ ( P .\/ Q ) ) /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( -. s .<_ ( P .\/ Q ) /\ ( s =/= T /\ s .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> G .<_ ( E .\/ 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 cdleme26f2.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
8 cdleme26f2.f 
 |-  G = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
9 cdleme26f2.n 
 |-  O = ( ( P .\/ Q ) ./\ ( G .\/ ( ( T .\/ s ) ./\ W ) ) )
10 cdleme26f2.e 
 |-  E = ( iota_ u e. B A. s e. A ( ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) -> u = O ) )
11 simp11 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P =/= Q /\ T .<_ ( P .\/ Q ) ) /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( -. s .<_ ( P .\/ Q ) /\ ( s =/= T /\ s .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( K e. HL /\ W e. H ) )
12 simp23 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P =/= Q /\ T .<_ ( P .\/ Q ) ) /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( -. s .<_ ( P .\/ Q ) /\ ( s =/= T /\ s .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( T e. A /\ -. T .<_ W ) )
13 simp31 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P =/= Q /\ T .<_ ( P .\/ Q ) ) /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( -. s .<_ ( P .\/ Q ) /\ ( s =/= T /\ s .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> -. s .<_ ( P .\/ Q ) )
14 simp12r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P =/= Q /\ T .<_ ( P .\/ Q ) ) /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( -. s .<_ ( P .\/ Q ) /\ ( s =/= T /\ s .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> T .<_ ( P .\/ Q ) )
15 simp12l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P =/= Q /\ T .<_ ( P .\/ Q ) ) /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( -. s .<_ ( P .\/ Q ) /\ ( s =/= T /\ s .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> P =/= Q )
16 13 14 15 3jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P =/= Q /\ T .<_ ( P .\/ Q ) ) /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( -. s .<_ ( P .\/ Q ) /\ ( s =/= T /\ s .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( -. s .<_ ( P .\/ Q ) /\ T .<_ ( P .\/ Q ) /\ P =/= Q ) )
17 simp21 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P =/= Q /\ T .<_ ( P .\/ Q ) ) /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( -. s .<_ ( P .\/ Q ) /\ ( s =/= T /\ s .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( P e. A /\ -. P .<_ W ) )
18 simp22 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P =/= Q /\ T .<_ ( P .\/ Q ) ) /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( -. s .<_ ( P .\/ Q ) /\ ( s =/= T /\ s .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( Q e. A /\ -. Q .<_ W ) )
19 simp13 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P =/= Q /\ T .<_ ( P .\/ Q ) ) /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( -. s .<_ ( P .\/ Q ) /\ ( s =/= T /\ s .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( s e. A /\ -. s .<_ W ) )
20 simp32 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P =/= Q /\ T .<_ ( P .\/ Q ) ) /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( -. s .<_ ( P .\/ Q ) /\ ( s =/= T /\ s .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( s =/= T /\ s .<_ ( T .\/ V ) ) )
21 simp33 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P =/= Q /\ T .<_ ( P .\/ Q ) ) /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( -. s .<_ ( P .\/ Q ) /\ ( s =/= T /\ s .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( V e. A /\ V .<_ W ) )
22 2 3 4 5 6 7 8 9 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 ) ) ) -> G .<_ ( O .\/ V ) )
23 11 12 16 17 18 19 20 21 22 syl323anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P =/= Q /\ T .<_ ( P .\/ Q ) ) /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( -. s .<_ ( P .\/ Q ) /\ ( s =/= T /\ s .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> G .<_ ( O .\/ V ) )
24 simp23l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P =/= Q /\ T .<_ ( P .\/ Q ) ) /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( -. s .<_ ( P .\/ Q ) /\ ( s =/= T /\ s .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> T e. A )
25 simp23r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P =/= Q /\ T .<_ ( P .\/ Q ) ) /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( -. s .<_ ( P .\/ Q ) /\ ( s =/= T /\ s .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> -. T .<_ W )
26 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 ) ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ T .<_ ( P .\/ Q ) ) ) -> E e. B )
27 11 17 18 24 25 15 14 26 syl322anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P =/= Q /\ T .<_ ( P .\/ Q ) ) /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( -. s .<_ ( P .\/ Q ) /\ ( s =/= T /\ s .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> E e. B )
28 simp13l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P =/= Q /\ T .<_ ( P .\/ Q ) ) /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( -. s .<_ ( P .\/ Q ) /\ ( s =/= T /\ s .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> s e. A )
29 simp13r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P =/= Q /\ T .<_ ( P .\/ Q ) ) /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( -. s .<_ ( P .\/ Q ) /\ ( s =/= T /\ s .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> -. s .<_ W )
30 1 fvexi 
 |-  B e. _V
31 30 10 riotasv 
 |-  ( ( E e. B /\ s e. A /\ ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) ) -> E = O )
32 27 28 29 13 31 syl112anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P =/= Q /\ T .<_ ( P .\/ Q ) ) /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( -. s .<_ ( P .\/ Q ) /\ ( s =/= T /\ s .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> E = O )
33 32 oveq1d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P =/= Q /\ T .<_ ( P .\/ Q ) ) /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( -. s .<_ ( P .\/ Q ) /\ ( s =/= T /\ s .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( E .\/ V ) = ( O .\/ V ) )
34 23 33 breqtrrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P =/= Q /\ T .<_ ( P .\/ Q ) ) /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( -. s .<_ ( P .\/ Q ) /\ ( s =/= T /\ s .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> G .<_ ( E .\/ V ) )