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Metamath Proof Explorer

Theorem cdleme27N

Description: Part of proof of Lemma E in Crawley p. 113. Eliminate the s =/= t antecedent in cdleme27a . (Contributed by NM, 3-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 )
cdleme27.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdleme27.f 
|- F = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
cdleme27.z 
|- Z = ( ( z .\/ U ) ./\ ( Q .\/ ( ( P .\/ z ) ./\ W ) ) )
cdleme27.n 
|- N = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( s .\/ z ) ./\ W ) ) )
cdleme27.d 
|- D = ( iota_ u e. B A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = N ) )
cdleme27.c 
|- C = if ( s .<_ ( P .\/ Q ) , D , F )
cdleme27.g 
|- G = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
cdleme27.o 
|- O = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( t .\/ z ) ./\ W ) ) )
cdleme27.e 
|- E = ( iota_ u e. B A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = O ) )
cdleme27.y 
|- Y = if ( t .<_ ( P .\/ Q ) , E , G )
Assertion cdleme27N 
|- ( ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s .<_ ( t .\/ V ) /\ ( V e. A /\ V .<_ W ) ) ) -> C .<_ ( Y .\/ 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 cdleme27.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
8 cdleme27.f 
 |-  F = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
9 cdleme27.z 
 |-  Z = ( ( z .\/ U ) ./\ ( Q .\/ ( ( P .\/ z ) ./\ W ) ) )
10 cdleme27.n 
 |-  N = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( s .\/ z ) ./\ W ) ) )
11 cdleme27.d 
 |-  D = ( iota_ u e. B A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = N ) )
12 cdleme27.c 
 |-  C = if ( s .<_ ( P .\/ Q ) , D , F )
13 cdleme27.g 
 |-  G = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
14 cdleme27.o 
 |-  O = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( t .\/ z ) ./\ W ) ) )
15 cdleme27.e 
 |-  E = ( iota_ u e. B A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = O ) )
16 cdleme27.y 
 |-  Y = if ( t .<_ ( P .\/ Q ) , E , G )
17 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 cdleme27b 
 |-  ( s = t -> C = Y )
18 17 adantl 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s .<_ ( t .\/ V ) /\ ( V e. A /\ V .<_ W ) ) ) /\ s = t ) -> C = Y )
19 simp11l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s .<_ ( t .\/ V ) /\ ( V e. A /\ V .<_ W ) ) ) -> K e. HL )
20 19 hllatd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s .<_ ( t .\/ V ) /\ ( V e. A /\ V .<_ W ) ) ) -> K e. Lat )
21 simp11r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s .<_ ( t .\/ V ) /\ ( V e. A /\ V .<_ W ) ) ) -> W e. H )
22 simp21 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s .<_ ( t .\/ V ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( P e. A /\ -. P .<_ W ) )
23 simp22 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s .<_ ( t .\/ V ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( Q e. A /\ -. Q .<_ W ) )
24 simp23 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s .<_ ( t .\/ V ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( t e. A /\ -. t .<_ W ) )
25 simp12 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s .<_ ( t .\/ V ) /\ ( V e. A /\ V .<_ W ) ) ) -> P =/= Q )
26 1 2 3 4 5 6 7 13 9 14 15 16 cdleme27cl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( t e. A /\ -. t .<_ W ) /\ P =/= Q ) ) -> Y e. B )
27 19 21 22 23 24 25 26 syl222anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s .<_ ( t .\/ V ) /\ ( V e. A /\ V .<_ W ) ) ) -> Y e. B )
28 simp3rl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s .<_ ( t .\/ V ) /\ ( V e. A /\ V .<_ W ) ) ) -> V e. A )
29 1 5 atbase 
 |-  ( V e. A -> V e. B )
30 28 29 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s .<_ ( t .\/ V ) /\ ( V e. A /\ V .<_ W ) ) ) -> V e. B )
31 1 2 3 latlej1 
 |-  ( ( K e. Lat /\ Y e. B /\ V e. B ) -> Y .<_ ( Y .\/ V ) )
32 20 27 30 31 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s .<_ ( t .\/ V ) /\ ( V e. A /\ V .<_ W ) ) ) -> Y .<_ ( Y .\/ V ) )
33 32 adantr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s .<_ ( t .\/ V ) /\ ( V e. A /\ V .<_ W ) ) ) /\ s = t ) -> Y .<_ ( Y .\/ V ) )
34 18 33 eqbrtrd 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s .<_ ( t .\/ V ) /\ ( V e. A /\ V .<_ W ) ) ) /\ s = t ) -> C .<_ ( Y .\/ V ) )
35 simpl11 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s .<_ ( t .\/ V ) /\ ( V e. A /\ V .<_ W ) ) ) /\ s =/= t ) -> ( K e. HL /\ W e. H ) )
36 simpl12 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s .<_ ( t .\/ V ) /\ ( V e. A /\ V .<_ W ) ) ) /\ s =/= t ) -> P =/= Q )
37 simpl13 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s .<_ ( t .\/ V ) /\ ( V e. A /\ V .<_ W ) ) ) /\ s =/= t ) -> ( s e. A /\ -. s .<_ W ) )
38 simpl21 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s .<_ ( t .\/ V ) /\ ( V e. A /\ V .<_ W ) ) ) /\ s =/= t ) -> ( P e. A /\ -. P .<_ W ) )
39 simpl22 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s .<_ ( t .\/ V ) /\ ( V e. A /\ V .<_ W ) ) ) /\ s =/= t ) -> ( Q e. A /\ -. Q .<_ W ) )
40 simpl23 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s .<_ ( t .\/ V ) /\ ( V e. A /\ V .<_ W ) ) ) /\ s =/= t ) -> ( t e. A /\ -. t .<_ W ) )
41 simpr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s .<_ ( t .\/ V ) /\ ( V e. A /\ V .<_ W ) ) ) /\ s =/= t ) -> s =/= t )
42 simpl3l 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s .<_ ( t .\/ V ) /\ ( V e. A /\ V .<_ W ) ) ) /\ s =/= t ) -> s .<_ ( t .\/ V ) )
43 41 42 jca 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s .<_ ( t .\/ V ) /\ ( V e. A /\ V .<_ W ) ) ) /\ s =/= t ) -> ( s =/= t /\ s .<_ ( t .\/ V ) ) )
44 simpl3r 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s .<_ ( t .\/ V ) /\ ( V e. A /\ V .<_ W ) ) ) /\ s =/= t ) -> ( V e. A /\ V .<_ W ) )
45 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 cdleme27a 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> C .<_ ( Y .\/ V ) )
46 35 36 37 38 39 40 43 44 45 syl332anc 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s .<_ ( t .\/ V ) /\ ( V e. A /\ V .<_ W ) ) ) /\ s =/= t ) -> C .<_ ( Y .\/ V ) )
47 34 46 pm2.61dane 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s .<_ ( t .\/ V ) /\ ( V e. A /\ V .<_ W ) ) ) -> C .<_ ( Y .\/ V ) )