Metamath Proof Explorer

Theorem cdleme41snaw

Description: Part of proof of Lemma E in Crawley p. 113. Show that f(r) is for combined cases; compare cdleme32snaw . TODO: FIX COMMENT. (Contributed by NM, 18-Mar-2013)

Ref Expression
Hypotheses cdleme41.b 
|- B = ( Base ` K )
cdleme41.l 
|- .<_ = ( le ` K )
cdleme41.j 
|- .\/ = ( join ` K )
cdleme41.m 
|- ./\ = ( meet ` K )
cdleme41.a 
|- A = ( Atoms ` K )
cdleme41.h 
|- H = ( LHyp ` K )
cdleme41.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdleme41.d 
|- D = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
cdleme41.e 
|- E = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
cdleme41.g 
|- G = ( ( P .\/ Q ) ./\ ( E .\/ ( ( s .\/ t ) ./\ W ) ) )
cdleme41.i 
|- I = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = G ) )
cdleme41.n 
|- N = if ( s .<_ ( P .\/ Q ) , I , D )
Assertion cdleme41snaw 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ R =/= S ) -> [_ R / s ]_ N =/= [_ S / s ]_ N )

Proof

Step Hyp Ref Expression
1 cdleme41.b 
 |-  B = ( Base ` K )
2 cdleme41.l 
 |-  .<_ = ( le ` K )
3 cdleme41.j 
 |-  .\/ = ( join ` K )
4 cdleme41.m 
 |-  ./\ = ( meet ` K )
5 cdleme41.a 
 |-  A = ( Atoms ` K )
6 cdleme41.h 
 |-  H = ( LHyp ` K )
7 cdleme41.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
8 cdleme41.d 
 |-  D = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
9 cdleme41.e 
 |-  E = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
10 cdleme41.g 
 |-  G = ( ( P .\/ Q ) ./\ ( E .\/ ( ( s .\/ t ) ./\ W ) ) )
11 cdleme41.i 
 |-  I = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = G ) )
12 cdleme41.n 
 |-  N = if ( s .<_ ( P .\/ Q ) , I , D )
13 simpl1 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ R =/= S ) /\ ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) ) ) -> ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) )
14 simpl21 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ R =/= S ) /\ ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) ) ) -> P =/= Q )
15 simpl22 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ R =/= S ) /\ ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) ) ) -> ( R e. A /\ -. R .<_ W ) )
16 simpl23 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ R =/= S ) /\ ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) ) ) -> ( S e. A /\ -. S .<_ W ) )
17 simprl 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ R =/= S ) /\ ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) ) ) -> R .<_ ( P .\/ Q ) )
18 simprr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ R =/= S ) /\ ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) ) ) -> S .<_ ( P .\/ Q ) )
19 simpl3 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ R =/= S ) /\ ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) ) ) -> R =/= S )
20 eqid 
 |-  ( ( u .\/ U ) ./\ ( Q .\/ ( ( P .\/ u ) ./\ W ) ) ) = ( ( u .\/ U ) ./\ ( Q .\/ ( ( P .\/ u ) ./\ W ) ) )
21 1 2 3 4 5 6 7 9 10 11 12 8 20 cdleme40w 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ R =/= S ) ) -> [_ R / s ]_ N =/= [_ S / s ]_ N )
22 13 14 15 16 17 18 19 21 syl133anc 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ R =/= S ) /\ ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) ) ) -> [_ R / s ]_ N =/= [_ S / s ]_ N )
23 simpl1 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ R =/= S ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) )
24 simpl21 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ R =/= S ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> P =/= Q )
25 simpl22 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ R =/= S ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( R e. A /\ -. R .<_ W ) )
26 simpl23 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ R =/= S ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( S e. A /\ -. S .<_ W ) )
27 simprl 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ R =/= S ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> R .<_ ( P .\/ Q ) )
28 simprr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ R =/= S ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> -. S .<_ ( P .\/ Q ) )
29 simpl3 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ R =/= S ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> R =/= S )
30 1 2 3 4 5 6 7 8 9 10 11 12 cdleme41sn3aw 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ R =/= S ) ) -> [_ R / s ]_ N =/= [_ S / s ]_ N )
31 23 24 25 26 27 28 29 30 syl133anc 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ R =/= S ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> [_ R / s ]_ N =/= [_ S / s ]_ N )
32 simpl1 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ R =/= S ) /\ ( -. R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) ) ) -> ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) )
33 simpl21 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ R =/= S ) /\ ( -. R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) ) ) -> P =/= Q )
34 simpl22 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ R =/= S ) /\ ( -. R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) ) ) -> ( R e. A /\ -. R .<_ W ) )
35 simpl23 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ R =/= S ) /\ ( -. R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) ) ) -> ( S e. A /\ -. S .<_ W ) )
36 simprl 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ R =/= S ) /\ ( -. R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) ) ) -> -. R .<_ ( P .\/ Q ) )
37 simprr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ R =/= S ) /\ ( -. R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) ) ) -> S .<_ ( P .\/ Q ) )
38 simpl3 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ R =/= S ) /\ ( -. R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) ) ) -> R =/= S )
39 1 2 3 4 5 6 7 8 9 10 11 12 cdleme41sn4aw 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ R =/= S ) ) -> [_ R / s ]_ N =/= [_ S / s ]_ N )
40 32 33 34 35 36 37 38 39 syl133anc 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ R =/= S ) /\ ( -. R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) ) ) -> [_ R / s ]_ N =/= [_ S / s ]_ N )
41 simpl1 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ R =/= S ) /\ ( -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) )
42 simpl21 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ R =/= S ) /\ ( -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> P =/= Q )
43 simpl22 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ R =/= S ) /\ ( -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( R e. A /\ -. R .<_ W ) )
44 simpl23 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ R =/= S ) /\ ( -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( S e. A /\ -. S .<_ W ) )
45 simprl 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ R =/= S ) /\ ( -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> -. R .<_ ( P .\/ Q ) )
46 simprr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ R =/= S ) /\ ( -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> -. S .<_ ( P .\/ Q ) )
47 simpl3 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ R =/= S ) /\ ( -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> R =/= S )
48 1 2 3 4 5 6 7 8 12 cdleme35sn2aw 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ R =/= S ) ) -> [_ R / s ]_ N =/= [_ S / s ]_ N )
49 41 42 43 44 45 46 47 48 syl133anc 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ R =/= S ) /\ ( -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> [_ R / s ]_ N =/= [_ S / s ]_ N )
50 22 31 40 49 4casesdan 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ R =/= S ) -> [_ R / s ]_ N =/= [_ S / s ]_ N )