Step |
Hyp |
Ref |
Expression |
1 |
|
ps1.l |
|- .<_ = ( le ` K ) |
2 |
|
ps1.j |
|- .\/ = ( join ` K ) |
3 |
|
ps1.a |
|- A = ( Atoms ` K ) |
4 |
|
oveq1 |
|- ( R = P -> ( R .\/ S ) = ( P .\/ S ) ) |
5 |
4
|
breq2d |
|- ( R = P -> ( ( P .\/ Q ) .<_ ( R .\/ S ) <-> ( P .\/ Q ) .<_ ( P .\/ S ) ) ) |
6 |
4
|
eqeq2d |
|- ( R = P -> ( ( P .\/ Q ) = ( R .\/ S ) <-> ( P .\/ Q ) = ( P .\/ S ) ) ) |
7 |
5 6
|
imbi12d |
|- ( R = P -> ( ( ( P .\/ Q ) .<_ ( R .\/ S ) -> ( P .\/ Q ) = ( R .\/ S ) ) <-> ( ( P .\/ Q ) .<_ ( P .\/ S ) -> ( P .\/ Q ) = ( P .\/ S ) ) ) ) |
8 |
7
|
eqcoms |
|- ( P = R -> ( ( ( P .\/ Q ) .<_ ( R .\/ S ) -> ( P .\/ Q ) = ( R .\/ S ) ) <-> ( ( P .\/ Q ) .<_ ( P .\/ S ) -> ( P .\/ Q ) = ( P .\/ S ) ) ) ) |
9 |
|
simp3 |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A ) ) /\ P =/= R /\ ( P .\/ Q ) .<_ ( R .\/ S ) ) -> ( P .\/ Q ) .<_ ( R .\/ S ) ) |
10 |
|
simp1 |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A ) ) -> K e. HL ) |
11 |
|
simp21 |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A ) ) -> P e. A ) |
12 |
|
simp3l |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A ) ) -> R e. A ) |
13 |
2 3
|
hlatjcom |
|- ( ( K e. HL /\ P e. A /\ R e. A ) -> ( P .\/ R ) = ( R .\/ P ) ) |
14 |
10 11 12 13
|
syl3anc |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A ) ) -> ( P .\/ R ) = ( R .\/ P ) ) |
15 |
14
|
3ad2ant1 |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A ) ) /\ P =/= R /\ ( P .\/ Q ) .<_ ( R .\/ S ) ) -> ( P .\/ R ) = ( R .\/ P ) ) |
16 |
|
hllat |
|- ( K e. HL -> K e. Lat ) |
17 |
16
|
3ad2ant1 |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A ) ) -> K e. Lat ) |
18 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
19 |
18 3
|
atbase |
|- ( P e. A -> P e. ( Base ` K ) ) |
20 |
11 19
|
syl |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A ) ) -> P e. ( Base ` K ) ) |
21 |
|
simp22 |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A ) ) -> Q e. A ) |
22 |
18 3
|
atbase |
|- ( Q e. A -> Q e. ( Base ` K ) ) |
23 |
21 22
|
syl |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A ) ) -> Q e. ( Base ` K ) ) |
24 |
|
simp3r |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A ) ) -> S e. A ) |
25 |
18 2 3
|
hlatjcl |
|- ( ( K e. HL /\ R e. A /\ S e. A ) -> ( R .\/ S ) e. ( Base ` K ) ) |
26 |
10 12 24 25
|
syl3anc |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A ) ) -> ( R .\/ S ) e. ( Base ` K ) ) |
27 |
18 1 2
|
latjle12 |
|- ( ( K e. Lat /\ ( P e. ( Base ` K ) /\ Q e. ( Base ` K ) /\ ( R .\/ S ) e. ( Base ` K ) ) ) -> ( ( P .<_ ( R .\/ S ) /\ Q .<_ ( R .\/ S ) ) <-> ( P .\/ Q ) .<_ ( R .\/ S ) ) ) |
28 |
17 20 23 26 27
|
syl13anc |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A ) ) -> ( ( P .<_ ( R .\/ S ) /\ Q .<_ ( R .\/ S ) ) <-> ( P .\/ Q ) .<_ ( R .\/ S ) ) ) |
29 |
|
simpl |
|- ( ( P .<_ ( R .\/ S ) /\ Q .<_ ( R .\/ S ) ) -> P .<_ ( R .\/ S ) ) |
30 |
28 29
|
syl6bir |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A ) ) -> ( ( P .\/ Q ) .<_ ( R .\/ S ) -> P .<_ ( R .\/ S ) ) ) |
31 |
30
|
adantr |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A ) ) /\ P =/= R ) -> ( ( P .\/ Q ) .<_ ( R .\/ S ) -> P .<_ ( R .\/ S ) ) ) |
32 |
|
simpl1 |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A ) ) /\ P =/= R ) -> K e. HL ) |
33 |
|
simpl21 |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A ) ) /\ P =/= R ) -> P e. A ) |
34 |
|
simpl3r |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A ) ) /\ P =/= R ) -> S e. A ) |
35 |
|
simpl3l |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A ) ) /\ P =/= R ) -> R e. A ) |
36 |
|
simpr |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A ) ) /\ P =/= R ) -> P =/= R ) |
37 |
1 2 3
|
hlatexchb1 |
|- ( ( K e. HL /\ ( P e. A /\ S e. A /\ R e. A ) /\ P =/= R ) -> ( P .<_ ( R .\/ S ) <-> ( R .\/ P ) = ( R .\/ S ) ) ) |
38 |
32 33 34 35 36 37
|
syl131anc |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A ) ) /\ P =/= R ) -> ( P .<_ ( R .\/ S ) <-> ( R .\/ P ) = ( R .\/ S ) ) ) |
39 |
31 38
|
sylibd |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A ) ) /\ P =/= R ) -> ( ( P .\/ Q ) .<_ ( R .\/ S ) -> ( R .\/ P ) = ( R .\/ S ) ) ) |
40 |
39
|
3impia |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A ) ) /\ P =/= R /\ ( P .\/ Q ) .<_ ( R .\/ S ) ) -> ( R .\/ P ) = ( R .\/ S ) ) |
41 |
15 40
|
eqtrd |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A ) ) /\ P =/= R /\ ( P .\/ Q ) .<_ ( R .\/ S ) ) -> ( P .\/ R ) = ( R .\/ S ) ) |
42 |
9 41
|
breqtrrd |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A ) ) /\ P =/= R /\ ( P .\/ Q ) .<_ ( R .\/ S ) ) -> ( P .\/ Q ) .<_ ( P .\/ R ) ) |
43 |
42
|
3expia |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A ) ) /\ P =/= R ) -> ( ( P .\/ Q ) .<_ ( R .\/ S ) -> ( P .\/ Q ) .<_ ( P .\/ R ) ) ) |
44 |
18 2 3
|
hlatjcl |
|- ( ( K e. HL /\ P e. A /\ R e. A ) -> ( P .\/ R ) e. ( Base ` K ) ) |
45 |
10 11 12 44
|
syl3anc |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A ) ) -> ( P .\/ R ) e. ( Base ` K ) ) |
46 |
18 1 2
|
latjle12 |
|- ( ( K e. Lat /\ ( P e. ( Base ` K ) /\ Q e. ( Base ` K ) /\ ( P .\/ R ) e. ( Base ` K ) ) ) -> ( ( P .<_ ( P .\/ R ) /\ Q .<_ ( P .\/ R ) ) <-> ( P .\/ Q ) .<_ ( P .\/ R ) ) ) |
47 |
17 20 23 45 46
|
syl13anc |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A ) ) -> ( ( P .<_ ( P .\/ R ) /\ Q .<_ ( P .\/ R ) ) <-> ( P .\/ Q ) .<_ ( P .\/ R ) ) ) |
48 |
|
simpr |
|- ( ( P .<_ ( P .\/ R ) /\ Q .<_ ( P .\/ R ) ) -> Q .<_ ( P .\/ R ) ) |
49 |
|
simp23 |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A ) ) -> P =/= Q ) |
50 |
49
|
necomd |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A ) ) -> Q =/= P ) |
51 |
1 2 3
|
hlatexchb1 |
|- ( ( K e. HL /\ ( Q e. A /\ R e. A /\ P e. A ) /\ Q =/= P ) -> ( Q .<_ ( P .\/ R ) <-> ( P .\/ Q ) = ( P .\/ R ) ) ) |
52 |
10 21 12 11 50 51
|
syl131anc |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A ) ) -> ( Q .<_ ( P .\/ R ) <-> ( P .\/ Q ) = ( P .\/ R ) ) ) |
53 |
48 52
|
syl5ib |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A ) ) -> ( ( P .<_ ( P .\/ R ) /\ Q .<_ ( P .\/ R ) ) -> ( P .\/ Q ) = ( P .\/ R ) ) ) |
54 |
47 53
|
sylbird |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A ) ) -> ( ( P .\/ Q ) .<_ ( P .\/ R ) -> ( P .\/ Q ) = ( P .\/ R ) ) ) |
55 |
54
|
adantr |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A ) ) /\ P =/= R ) -> ( ( P .\/ Q ) .<_ ( P .\/ R ) -> ( P .\/ Q ) = ( P .\/ R ) ) ) |
56 |
43 55
|
syld |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A ) ) /\ P =/= R ) -> ( ( P .\/ Q ) .<_ ( R .\/ S ) -> ( P .\/ Q ) = ( P .\/ R ) ) ) |
57 |
56
|
3impia |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A ) ) /\ P =/= R /\ ( P .\/ Q ) .<_ ( R .\/ S ) ) -> ( P .\/ Q ) = ( P .\/ R ) ) |
58 |
57 41
|
eqtrd |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A ) ) /\ P =/= R /\ ( P .\/ Q ) .<_ ( R .\/ S ) ) -> ( P .\/ Q ) = ( R .\/ S ) ) |
59 |
58
|
3expia |
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A ) ) /\ P =/= R ) -> ( ( P .\/ Q ) .<_ ( R .\/ S ) -> ( P .\/ Q ) = ( R .\/ S ) ) ) |
60 |
18 2 3
|
hlatjcl |
|- ( ( K e. HL /\ P e. A /\ S e. A ) -> ( P .\/ S ) e. ( Base ` K ) ) |
61 |
10 11 24 60
|
syl3anc |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A ) ) -> ( P .\/ S ) e. ( Base ` K ) ) |
62 |
18 1 2
|
latjle12 |
|- ( ( K e. Lat /\ ( P e. ( Base ` K ) /\ Q e. ( Base ` K ) /\ ( P .\/ S ) e. ( Base ` K ) ) ) -> ( ( P .<_ ( P .\/ S ) /\ Q .<_ ( P .\/ S ) ) <-> ( P .\/ Q ) .<_ ( P .\/ S ) ) ) |
63 |
17 20 23 61 62
|
syl13anc |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A ) ) -> ( ( P .<_ ( P .\/ S ) /\ Q .<_ ( P .\/ S ) ) <-> ( P .\/ Q ) .<_ ( P .\/ S ) ) ) |
64 |
|
simpr |
|- ( ( P .<_ ( P .\/ S ) /\ Q .<_ ( P .\/ S ) ) -> Q .<_ ( P .\/ S ) ) |
65 |
63 64
|
syl6bir |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A ) ) -> ( ( P .\/ Q ) .<_ ( P .\/ S ) -> Q .<_ ( P .\/ S ) ) ) |
66 |
1 2 3
|
hlatexchb1 |
|- ( ( K e. HL /\ ( Q e. A /\ S e. A /\ P e. A ) /\ Q =/= P ) -> ( Q .<_ ( P .\/ S ) <-> ( P .\/ Q ) = ( P .\/ S ) ) ) |
67 |
10 21 24 11 50 66
|
syl131anc |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A ) ) -> ( Q .<_ ( P .\/ S ) <-> ( P .\/ Q ) = ( P .\/ S ) ) ) |
68 |
65 67
|
sylibd |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A ) ) -> ( ( P .\/ Q ) .<_ ( P .\/ S ) -> ( P .\/ Q ) = ( P .\/ S ) ) ) |
69 |
8 59 68
|
pm2.61ne |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A ) ) -> ( ( P .\/ Q ) .<_ ( R .\/ S ) -> ( P .\/ Q ) = ( R .\/ S ) ) ) |
70 |
18 2 3
|
hlatjcl |
|- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) ) |
71 |
10 11 21 70
|
syl3anc |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A ) ) -> ( P .\/ Q ) e. ( Base ` K ) ) |
72 |
18 1
|
latref |
|- ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) ) -> ( P .\/ Q ) .<_ ( P .\/ Q ) ) |
73 |
17 71 72
|
syl2anc |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A ) ) -> ( P .\/ Q ) .<_ ( P .\/ Q ) ) |
74 |
|
breq2 |
|- ( ( P .\/ Q ) = ( R .\/ S ) -> ( ( P .\/ Q ) .<_ ( P .\/ Q ) <-> ( P .\/ Q ) .<_ ( R .\/ S ) ) ) |
75 |
73 74
|
syl5ibcom |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A ) ) -> ( ( P .\/ Q ) = ( R .\/ S ) -> ( P .\/ Q ) .<_ ( R .\/ S ) ) ) |
76 |
69 75
|
impbid |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A ) ) -> ( ( P .\/ Q ) .<_ ( R .\/ S ) <-> ( P .\/ Q ) = ( R .\/ S ) ) ) |