Step |
Hyp |
Ref |
Expression |
1 |
|
lhpex1.l |
|- .<_ = ( le ` K ) |
2 |
|
lhpex1.a |
|- A = ( Atoms ` K ) |
3 |
|
lhpex1.h |
|- H = ( LHyp ` K ) |
4 |
|
simpl1 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X = Y ) -> ( K e. HL /\ W e. H ) ) |
5 |
1 2 3
|
lhpexle2 |
|- ( ( K e. HL /\ W e. H ) -> E. p e. A ( p .<_ W /\ p =/= X /\ p =/= Z ) ) |
6 |
4 5
|
syl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X = Y ) -> E. p e. A ( p .<_ W /\ p =/= X /\ p =/= Z ) ) |
7 |
|
simp31 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X = Y ) /\ p e. A /\ ( p .<_ W /\ p =/= X /\ p =/= Z ) ) -> p .<_ W ) |
8 |
|
simp32 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X = Y ) /\ p e. A /\ ( p .<_ W /\ p =/= X /\ p =/= Z ) ) -> p =/= X ) |
9 |
|
simp1r |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X = Y ) /\ p e. A /\ ( p .<_ W /\ p =/= X /\ p =/= Z ) ) -> X = Y ) |
10 |
8 9
|
neeqtrd |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X = Y ) /\ p e. A /\ ( p .<_ W /\ p =/= X /\ p =/= Z ) ) -> p =/= Y ) |
11 |
|
simp33 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X = Y ) /\ p e. A /\ ( p .<_ W /\ p =/= X /\ p =/= Z ) ) -> p =/= Z ) |
12 |
8 10 11
|
3jca |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X = Y ) /\ p e. A /\ ( p .<_ W /\ p =/= X /\ p =/= Z ) ) -> ( p =/= X /\ p =/= Y /\ p =/= Z ) ) |
13 |
7 12
|
jca |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X = Y ) /\ p e. A /\ ( p .<_ W /\ p =/= X /\ p =/= Z ) ) -> ( p .<_ W /\ ( p =/= X /\ p =/= Y /\ p =/= Z ) ) ) |
14 |
13
|
3exp |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X = Y ) -> ( p e. A -> ( ( p .<_ W /\ p =/= X /\ p =/= Z ) -> ( p .<_ W /\ ( p =/= X /\ p =/= Y /\ p =/= Z ) ) ) ) ) |
15 |
14
|
reximdvai |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X = Y ) -> ( E. p e. A ( p .<_ W /\ p =/= X /\ p =/= Z ) -> E. p e. A ( p .<_ W /\ ( p =/= X /\ p =/= Y /\ p =/= Z ) ) ) ) |
16 |
6 15
|
mpd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X = Y ) -> E. p e. A ( p .<_ W /\ ( p =/= X /\ p =/= Y /\ p =/= Z ) ) ) |
17 |
|
simprrr |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( -. p .<_ ( X ( join ` K ) Y ) /\ p .<_ W ) ) ) -> p .<_ W ) |
18 |
|
simp11l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) -> K e. HL ) |
19 |
18
|
adantr |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( -. p .<_ ( X ( join ` K ) Y ) /\ p .<_ W ) ) ) -> K e. HL ) |
20 |
19
|
hllatd |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( -. p .<_ ( X ( join ` K ) Y ) /\ p .<_ W ) ) ) -> K e. Lat ) |
21 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
22 |
21 2
|
atbase |
|- ( p e. A -> p e. ( Base ` K ) ) |
23 |
22
|
ad2antrl |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( -. p .<_ ( X ( join ` K ) Y ) /\ p .<_ W ) ) ) -> p e. ( Base ` K ) ) |
24 |
|
simp121 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) -> X e. A ) |
25 |
24
|
adantr |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( -. p .<_ ( X ( join ` K ) Y ) /\ p .<_ W ) ) ) -> X e. A ) |
26 |
21 2
|
atbase |
|- ( X e. A -> X e. ( Base ` K ) ) |
27 |
25 26
|
syl |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( -. p .<_ ( X ( join ` K ) Y ) /\ p .<_ W ) ) ) -> X e. ( Base ` K ) ) |
28 |
|
simp122 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) -> Y e. A ) |
29 |
28
|
adantr |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( -. p .<_ ( X ( join ` K ) Y ) /\ p .<_ W ) ) ) -> Y e. A ) |
30 |
21 2
|
atbase |
|- ( Y e. A -> Y e. ( Base ` K ) ) |
31 |
29 30
|
syl |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( -. p .<_ ( X ( join ` K ) Y ) /\ p .<_ W ) ) ) -> Y e. ( Base ` K ) ) |
32 |
|
simprrl |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( -. p .<_ ( X ( join ` K ) Y ) /\ p .<_ W ) ) ) -> -. p .<_ ( X ( join ` K ) Y ) ) |
33 |
|
eqid |
|- ( join ` K ) = ( join ` K ) |
34 |
21 1 33
|
latnlej1l |
|- ( ( K e. Lat /\ ( p e. ( Base ` K ) /\ X e. ( Base ` K ) /\ Y e. ( Base ` K ) ) /\ -. p .<_ ( X ( join ` K ) Y ) ) -> p =/= X ) |
35 |
20 23 27 31 32 34
|
syl131anc |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( -. p .<_ ( X ( join ` K ) Y ) /\ p .<_ W ) ) ) -> p =/= X ) |
36 |
21 1 33
|
latnlej1r |
|- ( ( K e. Lat /\ ( p e. ( Base ` K ) /\ X e. ( Base ` K ) /\ Y e. ( Base ` K ) ) /\ -. p .<_ ( X ( join ` K ) Y ) ) -> p =/= Y ) |
37 |
20 23 27 31 32 36
|
syl131anc |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( -. p .<_ ( X ( join ` K ) Y ) /\ p .<_ W ) ) ) -> p =/= Y ) |
38 |
|
simpl3 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( -. p .<_ ( X ( join ` K ) Y ) /\ p .<_ W ) ) ) -> Z .<_ ( X ( join ` K ) Y ) ) |
39 |
|
nbrne2 |
|- ( ( Z .<_ ( X ( join ` K ) Y ) /\ -. p .<_ ( X ( join ` K ) Y ) ) -> Z =/= p ) |
40 |
39
|
necomd |
|- ( ( Z .<_ ( X ( join ` K ) Y ) /\ -. p .<_ ( X ( join ` K ) Y ) ) -> p =/= Z ) |
41 |
38 32 40
|
syl2anc |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( -. p .<_ ( X ( join ` K ) Y ) /\ p .<_ W ) ) ) -> p =/= Z ) |
42 |
35 37 41
|
3jca |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( -. p .<_ ( X ( join ` K ) Y ) /\ p .<_ W ) ) ) -> ( p =/= X /\ p =/= Y /\ p =/= Z ) ) |
43 |
17 42
|
jca |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( -. p .<_ ( X ( join ` K ) Y ) /\ p .<_ W ) ) ) -> ( p .<_ W /\ ( p =/= X /\ p =/= Y /\ p =/= Z ) ) ) |
44 |
|
simp11 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) -> ( K e. HL /\ W e. H ) ) |
45 |
|
simp131 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) -> X .<_ W ) |
46 |
|
simp132 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) -> Y .<_ W ) |
47 |
|
eqid |
|- ( lt ` K ) = ( lt ` K ) |
48 |
1 47 33 2 3
|
lhp2lt |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) -> ( X ( join ` K ) Y ) ( lt ` K ) W ) |
49 |
44 24 45 28 46 48
|
syl122anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) -> ( X ( join ` K ) Y ) ( lt ` K ) W ) |
50 |
21 33 2
|
hlatjcl |
|- ( ( K e. HL /\ X e. A /\ Y e. A ) -> ( X ( join ` K ) Y ) e. ( Base ` K ) ) |
51 |
18 24 28 50
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) -> ( X ( join ` K ) Y ) e. ( Base ` K ) ) |
52 |
|
simp11r |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) -> W e. H ) |
53 |
21 3
|
lhpbase |
|- ( W e. H -> W e. ( Base ` K ) ) |
54 |
52 53
|
syl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) -> W e. ( Base ` K ) ) |
55 |
21 1 47 2
|
hlrelat1 |
|- ( ( K e. HL /\ ( X ( join ` K ) Y ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( X ( join ` K ) Y ) ( lt ` K ) W -> E. p e. A ( -. p .<_ ( X ( join ` K ) Y ) /\ p .<_ W ) ) ) |
56 |
18 51 54 55
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) -> ( ( X ( join ` K ) Y ) ( lt ` K ) W -> E. p e. A ( -. p .<_ ( X ( join ` K ) Y ) /\ p .<_ W ) ) ) |
57 |
49 56
|
mpd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) -> E. p e. A ( -. p .<_ ( X ( join ` K ) Y ) /\ p .<_ W ) ) |
58 |
43 57
|
reximddv |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) -> E. p e. A ( p .<_ W /\ ( p =/= X /\ p =/= Y /\ p =/= Z ) ) ) |
59 |
58
|
3expa |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y ) /\ Z .<_ ( X ( join ` K ) Y ) ) -> E. p e. A ( p .<_ W /\ ( p =/= X /\ p =/= Y /\ p =/= Z ) ) ) |
60 |
|
simp11l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) -> K e. HL ) |
61 |
60
|
adantr |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> K e. HL ) |
62 |
61
|
hllatd |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> K e. Lat ) |
63 |
22
|
ad2antrl |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> p e. ( Base ` K ) ) |
64 |
|
simp121 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) -> X e. A ) |
65 |
64
|
adantr |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> X e. A ) |
66 |
|
simp122 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) -> Y e. A ) |
67 |
66
|
adantr |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> Y e. A ) |
68 |
61 65 67 50
|
syl3anc |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> ( X ( join ` K ) Y ) e. ( Base ` K ) ) |
69 |
|
simp11r |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) -> W e. H ) |
70 |
69
|
adantr |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> W e. H ) |
71 |
70 53
|
syl |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> W e. ( Base ` K ) ) |
72 |
|
simprr3 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> p .<_ ( X ( join ` K ) Y ) ) |
73 |
|
simp131 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) -> X .<_ W ) |
74 |
73
|
adantr |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> X .<_ W ) |
75 |
|
simp132 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) -> Y .<_ W ) |
76 |
75
|
adantr |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> Y .<_ W ) |
77 |
65 26
|
syl |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> X e. ( Base ` K ) ) |
78 |
67 30
|
syl |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> Y e. ( Base ` K ) ) |
79 |
21 1 33
|
latjle12 |
|- ( ( K e. Lat /\ ( X e. ( Base ` K ) /\ Y e. ( Base ` K ) /\ W e. ( Base ` K ) ) ) -> ( ( X .<_ W /\ Y .<_ W ) <-> ( X ( join ` K ) Y ) .<_ W ) ) |
80 |
62 77 78 71 79
|
syl13anc |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> ( ( X .<_ W /\ Y .<_ W ) <-> ( X ( join ` K ) Y ) .<_ W ) ) |
81 |
74 76 80
|
mpbi2and |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> ( X ( join ` K ) Y ) .<_ W ) |
82 |
21 1 62 63 68 71 72 81
|
lattrd |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> p .<_ W ) |
83 |
|
simprr1 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> p =/= X ) |
84 |
|
simprr2 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> p =/= Y ) |
85 |
|
simpl3 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> -. Z .<_ ( X ( join ` K ) Y ) ) |
86 |
|
nbrne2 |
|- ( ( p .<_ ( X ( join ` K ) Y ) /\ -. Z .<_ ( X ( join ` K ) Y ) ) -> p =/= Z ) |
87 |
72 85 86
|
syl2anc |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> p =/= Z ) |
88 |
83 84 87
|
3jca |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> ( p =/= X /\ p =/= Y /\ p =/= Z ) ) |
89 |
82 88
|
jca |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> ( p .<_ W /\ ( p =/= X /\ p =/= Y /\ p =/= Z ) ) ) |
90 |
|
simp2 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) -> X =/= Y ) |
91 |
1 33 2
|
hlsupr |
|- ( ( ( K e. HL /\ X e. A /\ Y e. A ) /\ X =/= Y ) -> E. p e. A ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) |
92 |
60 64 66 90 91
|
syl31anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) -> E. p e. A ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) |
93 |
89 92
|
reximddv |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) -> E. p e. A ( p .<_ W /\ ( p =/= X /\ p =/= Y /\ p =/= Z ) ) ) |
94 |
93
|
3expa |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y ) /\ -. Z .<_ ( X ( join ` K ) Y ) ) -> E. p e. A ( p .<_ W /\ ( p =/= X /\ p =/= Y /\ p =/= Z ) ) ) |
95 |
59 94
|
pm2.61dan |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y ) -> E. p e. A ( p .<_ W /\ ( p =/= X /\ p =/= Y /\ p =/= Z ) ) ) |
96 |
16 95
|
pm2.61dane |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) -> E. p e. A ( p .<_ W /\ ( p =/= X /\ p =/= Y /\ p =/= Z ) ) ) |