Step |
Hyp |
Ref |
Expression |
1 |
|
cdleme40.b |
|- B = ( Base ` K ) |
2 |
|
cdleme40.l |
|- .<_ = ( le ` K ) |
3 |
|
cdleme40.j |
|- .\/ = ( join ` K ) |
4 |
|
cdleme40.m |
|- ./\ = ( meet ` K ) |
5 |
|
cdleme40.a |
|- A = ( Atoms ` K ) |
6 |
|
cdleme40.h |
|- H = ( LHyp ` K ) |
7 |
|
cdleme40.u |
|- U = ( ( P .\/ Q ) ./\ W ) |
8 |
|
cdleme40.e |
|- E = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) ) |
9 |
|
cdleme40.g |
|- G = ( ( P .\/ Q ) ./\ ( E .\/ ( ( s .\/ t ) ./\ W ) ) ) |
10 |
|
cdleme40.i |
|- I = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = G ) ) |
11 |
|
cdleme40.n |
|- N = if ( s .<_ ( P .\/ Q ) , I , D ) |
12 |
|
cdleme40a1.y |
|- Y = ( ( P .\/ Q ) ./\ ( E .\/ ( ( R .\/ t ) ./\ W ) ) ) |
13 |
|
cdleme40a1.c |
|- C = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = Y ) ) |
14 |
|
cdleme40.t |
|- T = ( ( v .\/ U ) ./\ ( Q .\/ ( ( P .\/ v ) ./\ W ) ) ) |
15 |
|
cdleme40.f |
|- F = ( ( P .\/ Q ) ./\ ( T .\/ ( ( S .\/ v ) ./\ W ) ) ) |
16 |
|
cdleme40a1.x |
|- X = ( ( P .\/ Q ) ./\ ( T .\/ ( ( u .\/ v ) ./\ W ) ) ) |
17 |
|
cdleme40.o |
|- O = ( iota_ z e. B A. v e. A ( ( -. v .<_ W /\ -. v .<_ ( P .\/ Q ) ) -> z = X ) ) |
18 |
|
cdleme40.v |
|- V = if ( u .<_ ( P .\/ Q ) , O , .< ) |
19 |
|
cdleme40a1.z |
|- Z = ( iota_ z e. B A. v e. A ( ( -. v .<_ W /\ -. v .<_ ( P .\/ Q ) ) -> z = F ) ) |
20 |
1
|
fvexi |
|- B e. _V |
21 |
|
nfv |
|- F/ v ( ( ( 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 ) ) |
22 |
|
nfcv |
|- F/_ v [_ R / s ]_ N |
23 |
|
nfra1 |
|- F/ v A. v e. A ( ( -. v .<_ W /\ -. v .<_ ( P .\/ Q ) ) -> z = F ) |
24 |
|
nfcv |
|- F/_ v B |
25 |
23 24
|
nfriota |
|- F/_ v ( iota_ z e. B A. v e. A ( ( -. v .<_ W /\ -. v .<_ ( P .\/ Q ) ) -> z = F ) ) |
26 |
19 25
|
nfcxfr |
|- F/_ v Z |
27 |
22 26
|
nfne |
|- F/ v [_ R / s ]_ N =/= Z |
28 |
27
|
a1i |
|- ( ( ( ( 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 ) ) -> F/ v [_ R / s ]_ N =/= Z ) |
29 |
19
|
a1i |
|- ( ( ( ( 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 ) ) -> Z = ( iota_ z e. B A. v e. A ( ( -. v .<_ W /\ -. v .<_ ( P .\/ Q ) ) -> z = F ) ) ) |
30 |
|
neeq2 |
|- ( F = Z -> ( [_ R / s ]_ N =/= F <-> [_ R / s ]_ N =/= Z ) ) |
31 |
30
|
adantl |
|- ( ( ( ( ( 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 ) ) /\ F = Z ) -> ( [_ R / s ]_ N =/= F <-> [_ R / s ]_ N =/= Z ) ) |
32 |
|
simpl11 |
|- ( ( ( ( ( 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 ) ) /\ ( v e. A /\ ( -. v .<_ W /\ -. v .<_ ( P .\/ Q ) ) ) ) -> ( K e. HL /\ W e. H ) ) |
33 |
|
simpl12 |
|- ( ( ( ( ( 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 ) ) /\ ( v e. A /\ ( -. v .<_ W /\ -. v .<_ ( P .\/ Q ) ) ) ) -> ( P e. A /\ -. P .<_ W ) ) |
34 |
|
simpl13 |
|- ( ( ( ( ( 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 ) ) /\ ( v e. A /\ ( -. v .<_ W /\ -. v .<_ ( P .\/ Q ) ) ) ) -> ( Q e. A /\ -. Q .<_ W ) ) |
35 |
|
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 .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ R =/= S ) ) /\ ( v e. A /\ ( -. v .<_ W /\ -. v .<_ ( P .\/ Q ) ) ) ) -> P =/= Q ) |
36 |
|
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 .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ R =/= S ) ) /\ ( v e. A /\ ( -. v .<_ W /\ -. v .<_ ( P .\/ Q ) ) ) ) -> ( R e. A /\ -. R .<_ W ) ) |
37 |
|
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 .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ R =/= S ) ) /\ ( v e. A /\ ( -. v .<_ W /\ -. v .<_ ( P .\/ Q ) ) ) ) -> ( S e. A /\ -. S .<_ W ) ) |
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 .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ R =/= S ) ) /\ ( v e. A /\ ( -. v .<_ W /\ -. v .<_ ( P .\/ Q ) ) ) ) -> ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ R =/= S ) ) |
39 |
|
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 .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ R =/= S ) ) /\ ( v e. A /\ ( -. v .<_ W /\ -. v .<_ ( P .\/ Q ) ) ) ) -> v e. A ) |
40 |
|
simprrl |
|- ( ( ( ( ( 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 ) ) /\ ( v e. A /\ ( -. v .<_ W /\ -. v .<_ ( P .\/ Q ) ) ) ) -> -. v .<_ W ) |
41 |
|
simprrr |
|- ( ( ( ( ( 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 ) ) /\ ( v e. A /\ ( -. v .<_ W /\ -. v .<_ ( P .\/ Q ) ) ) ) -> -. v .<_ ( P .\/ Q ) ) |
42 |
39 40 41
|
3jca |
|- ( ( ( ( ( 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 ) ) /\ ( v e. A /\ ( -. v .<_ W /\ -. v .<_ ( P .\/ Q ) ) ) ) -> ( v e. A /\ -. v .<_ W /\ -. v .<_ ( P .\/ Q ) ) ) |
43 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
|
cdleme40m |
|- ( ( ( ( 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 ) /\ ( v e. A /\ -. v .<_ W /\ -. v .<_ ( P .\/ Q ) ) ) ) -> [_ R / s ]_ N =/= F ) |
44 |
32 33 34 35 36 37 38 42 43
|
syl332anc |
|- ( ( ( ( ( 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 ) ) /\ ( v e. A /\ ( -. v .<_ W /\ -. v .<_ ( P .\/ Q ) ) ) ) -> [_ R / s ]_ N =/= F ) |
45 |
44
|
ex |
|- ( ( ( ( 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 ) ) -> ( ( v e. A /\ ( -. v .<_ W /\ -. v .<_ ( P .\/ Q ) ) ) -> [_ R / s ]_ N =/= F ) ) |
46 |
|
simp1 |
|- ( ( ( ( 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 ) ) -> ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) ) |
47 |
|
simp23l |
|- ( ( ( ( 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 ) ) -> S e. A ) |
48 |
|
simp23r |
|- ( ( ( ( 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 ) ) -> -. S .<_ W ) |
49 |
|
simp21 |
|- ( ( ( ( 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 ) ) -> P =/= Q ) |
50 |
|
simp32 |
|- ( ( ( ( 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 ) ) -> S .<_ ( P .\/ Q ) ) |
51 |
1 2 3 4 5 6 7 14 15 19
|
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 ) ) ) -> Z e. B ) |
52 |
46 47 48 49 50 51
|
syl122anc |
|- ( ( ( ( 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 ) ) -> Z e. B ) |
53 |
|
simp11 |
|- ( ( ( ( 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 ) ) -> ( K e. HL /\ W e. H ) ) |
54 |
|
simp12 |
|- ( ( ( ( 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 ) ) -> ( P e. A /\ -. P .<_ W ) ) |
55 |
|
simp13 |
|- ( ( ( ( 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 ) ) -> ( Q e. A /\ -. Q .<_ W ) ) |
56 |
2 3 5 6
|
cdlemb2 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> E. v e. A ( -. v .<_ W /\ -. v .<_ ( P .\/ Q ) ) ) |
57 |
53 54 55 49 56
|
syl121anc |
|- ( ( ( ( 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 ) ) -> E. v e. A ( -. v .<_ W /\ -. v .<_ ( P .\/ Q ) ) ) |
58 |
21 28 29 31 45 52 57
|
riotasv3d |
|- ( ( ( ( ( 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 ) ) /\ B e. _V ) -> [_ R / s ]_ N =/= Z ) |
59 |
20 58
|
mpan2 |
|- ( ( ( ( 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 =/= Z ) |
60 |
16 17 18 15 19
|
cdleme31sn1c |
|- ( ( S e. A /\ S .<_ ( P .\/ Q ) ) -> [_ S / u ]_ V = Z ) |
61 |
47 50 60
|
syl2anc |
|- ( ( ( ( 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 ) ) -> [_ S / u ]_ V = Z ) |
62 |
59 61
|
neeqtrrd |
|- ( ( ( ( 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 / u ]_ V ) |