Step |
Hyp |
Ref |
Expression |
1 |
|
cdlemg12.l |
|- .<_ = ( le ` K ) |
2 |
|
cdlemg12.j |
|- .\/ = ( join ` K ) |
3 |
|
cdlemg12.m |
|- ./\ = ( meet ` K ) |
4 |
|
cdlemg12.a |
|- A = ( Atoms ` K ) |
5 |
|
cdlemg12.h |
|- H = ( LHyp ` K ) |
6 |
|
cdlemg12.t |
|- T = ( ( LTrn ` K ) ` W ) |
7 |
|
cdlemg12b.r |
|- R = ( ( trL ` K ) ` W ) |
8 |
|
cdlemg31.n |
|- N = ( ( P .\/ v ) ./\ ( Q .\/ ( R ` F ) ) ) |
9 |
|
simp11 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> ( K e. HL /\ W e. H ) ) |
10 |
|
simp12 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> ( P e. A /\ -. P .<_ W ) ) |
11 |
|
simp13 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> ( Q e. A /\ -. Q .<_ W ) ) |
12 |
|
simp22 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> ( v e. A /\ v .<_ W ) ) |
13 |
|
simp23l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> F e. T ) |
14 |
|
simp31 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> v =/= ( R ` F ) ) |
15 |
1 2 3 4 5 6 7 8
|
cdlemg31b0a |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) ) ) -> ( N e. A \/ N = ( 0. ` K ) ) ) |
16 |
9 10 11 12 13 14 15
|
syl132anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> ( N e. A \/ N = ( 0. ` K ) ) ) |
17 |
|
simp23r |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> z =/= N ) |
18 |
17
|
adantr |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) /\ ( N e. A \/ N = ( 0. ` K ) ) ) -> z =/= N ) |
19 |
|
simp11l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> K e. HL ) |
20 |
19
|
adantr |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) /\ N e. A ) -> K e. HL ) |
21 |
|
hlatl |
|- ( K e. HL -> K e. AtLat ) |
22 |
20 21
|
syl |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) /\ N e. A ) -> K e. AtLat ) |
23 |
|
simpl21 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) /\ N e. A ) -> z e. A ) |
24 |
|
simpr |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) /\ N e. A ) -> N e. A ) |
25 |
1 4
|
atcmp |
|- ( ( K e. AtLat /\ z e. A /\ N e. A ) -> ( z .<_ N <-> z = N ) ) |
26 |
22 23 24 25
|
syl3anc |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) /\ N e. A ) -> ( z .<_ N <-> z = N ) ) |
27 |
26
|
necon3bbid |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) /\ N e. A ) -> ( -. z .<_ N <-> z =/= N ) ) |
28 |
19
|
adantr |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) /\ N = ( 0. ` K ) ) -> K e. HL ) |
29 |
28 21
|
syl |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) /\ N = ( 0. ` K ) ) -> K e. AtLat ) |
30 |
|
simpl21 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) /\ N = ( 0. ` K ) ) -> z e. A ) |
31 |
|
eqid |
|- ( 0. ` K ) = ( 0. ` K ) |
32 |
1 31 4
|
atnle0 |
|- ( ( K e. AtLat /\ z e. A ) -> -. z .<_ ( 0. ` K ) ) |
33 |
29 30 32
|
syl2anc |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) /\ N = ( 0. ` K ) ) -> -. z .<_ ( 0. ` K ) ) |
34 |
|
simpr |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) /\ N = ( 0. ` K ) ) -> N = ( 0. ` K ) ) |
35 |
34
|
breq2d |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) /\ N = ( 0. ` K ) ) -> ( z .<_ N <-> z .<_ ( 0. ` K ) ) ) |
36 |
33 35
|
mtbird |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) /\ N = ( 0. ` K ) ) -> -. z .<_ N ) |
37 |
17
|
adantr |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) /\ N = ( 0. ` K ) ) -> z =/= N ) |
38 |
36 37
|
2thd |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) /\ N = ( 0. ` K ) ) -> ( -. z .<_ N <-> z =/= N ) ) |
39 |
27 38
|
jaodan |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) /\ ( N e. A \/ N = ( 0. ` K ) ) ) -> ( -. z .<_ N <-> z =/= N ) ) |
40 |
18 39
|
mpbird |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) /\ ( N e. A \/ N = ( 0. ` K ) ) ) -> -. z .<_ N ) |
41 |
16 40
|
mpdan |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> -. z .<_ N ) |
42 |
|
simp32 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> z .<_ ( P .\/ v ) ) |
43 |
19
|
hllatd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> K e. Lat ) |
44 |
|
simp21 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> z e. A ) |
45 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
46 |
45 4
|
atbase |
|- ( z e. A -> z e. ( Base ` K ) ) |
47 |
44 46
|
syl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> z e. ( Base ` K ) ) |
48 |
|
simp12l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> P e. A ) |
49 |
|
simp22l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> v e. A ) |
50 |
45 2 4
|
hlatjcl |
|- ( ( K e. HL /\ P e. A /\ v e. A ) -> ( P .\/ v ) e. ( Base ` K ) ) |
51 |
19 48 49 50
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> ( P .\/ v ) e. ( Base ` K ) ) |
52 |
|
simp13l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> Q e. A ) |
53 |
|
simp33 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> ( F ` P ) =/= P ) |
54 |
1 4 5 6 7
|
trlat |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) =/= P ) ) -> ( R ` F ) e. A ) |
55 |
9 10 13 53 54
|
syl112anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> ( R ` F ) e. A ) |
56 |
45 2 4
|
hlatjcl |
|- ( ( K e. HL /\ Q e. A /\ ( R ` F ) e. A ) -> ( Q .\/ ( R ` F ) ) e. ( Base ` K ) ) |
57 |
19 52 55 56
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> ( Q .\/ ( R ` F ) ) e. ( Base ` K ) ) |
58 |
45 1 3
|
latlem12 |
|- ( ( K e. Lat /\ ( z e. ( Base ` K ) /\ ( P .\/ v ) e. ( Base ` K ) /\ ( Q .\/ ( R ` F ) ) e. ( Base ` K ) ) ) -> ( ( z .<_ ( P .\/ v ) /\ z .<_ ( Q .\/ ( R ` F ) ) ) <-> z .<_ ( ( P .\/ v ) ./\ ( Q .\/ ( R ` F ) ) ) ) ) |
59 |
43 47 51 57 58
|
syl13anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> ( ( z .<_ ( P .\/ v ) /\ z .<_ ( Q .\/ ( R ` F ) ) ) <-> z .<_ ( ( P .\/ v ) ./\ ( Q .\/ ( R ` F ) ) ) ) ) |
60 |
8
|
breq2i |
|- ( z .<_ N <-> z .<_ ( ( P .\/ v ) ./\ ( Q .\/ ( R ` F ) ) ) ) |
61 |
59 60
|
bitr4di |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> ( ( z .<_ ( P .\/ v ) /\ z .<_ ( Q .\/ ( R ` F ) ) ) <-> z .<_ N ) ) |
62 |
61
|
biimpd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> ( ( z .<_ ( P .\/ v ) /\ z .<_ ( Q .\/ ( R ` F ) ) ) -> z .<_ N ) ) |
63 |
42 62
|
mpand |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> ( z .<_ ( Q .\/ ( R ` F ) ) -> z .<_ N ) ) |
64 |
41 63
|
mtod |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> -. z .<_ ( Q .\/ ( R ` F ) ) ) |
65 |
1 5 6 7
|
trlle |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( R ` F ) .<_ W ) |
66 |
9 13 65
|
syl2anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> ( R ` F ) .<_ W ) |
67 |
|
simp13r |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> -. Q .<_ W ) |
68 |
|
nbrne2 |
|- ( ( ( R ` F ) .<_ W /\ -. Q .<_ W ) -> ( R ` F ) =/= Q ) |
69 |
66 67 68
|
syl2anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> ( R ` F ) =/= Q ) |
70 |
1 2 4
|
hlatexch1 |
|- ( ( K e. HL /\ ( ( R ` F ) e. A /\ z e. A /\ Q e. A ) /\ ( R ` F ) =/= Q ) -> ( ( R ` F ) .<_ ( Q .\/ z ) -> z .<_ ( Q .\/ ( R ` F ) ) ) ) |
71 |
19 55 44 52 69 70
|
syl131anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> ( ( R ` F ) .<_ ( Q .\/ z ) -> z .<_ ( Q .\/ ( R ` F ) ) ) ) |
72 |
64 71
|
mtod |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> -. ( R ` F ) .<_ ( Q .\/ z ) ) |