Step |
Hyp |
Ref |
Expression |
1 |
|
cdleme0nex.l |
|- .<_ = ( le ` K ) |
2 |
|
cdleme0nex.j |
|- .\/ = ( join ` K ) |
3 |
|
cdleme0nex.a |
|- A = ( Atoms ` K ) |
4 |
|
simp3r |
|- ( ( ( K e. HL /\ R .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) -> -. R .<_ W ) |
5 |
|
simp12 |
|- ( ( ( K e. HL /\ R .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) -> R .<_ ( P .\/ Q ) ) |
6 |
4 5
|
jca |
|- ( ( ( K e. HL /\ R .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) -> ( -. R .<_ W /\ R .<_ ( P .\/ Q ) ) ) |
7 |
|
simp3l |
|- ( ( ( K e. HL /\ R .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) -> R e. A ) |
8 |
|
simp13 |
|- ( ( ( K e. HL /\ R .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) -> -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) |
9 |
|
ralnex |
|- ( A. r e. A -. ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) <-> -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) |
10 |
8 9
|
sylibr |
|- ( ( ( K e. HL /\ R .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) -> A. r e. A -. ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) |
11 |
|
breq1 |
|- ( r = R -> ( r .<_ W <-> R .<_ W ) ) |
12 |
11
|
notbid |
|- ( r = R -> ( -. r .<_ W <-> -. R .<_ W ) ) |
13 |
|
oveq2 |
|- ( r = R -> ( P .\/ r ) = ( P .\/ R ) ) |
14 |
|
oveq2 |
|- ( r = R -> ( Q .\/ r ) = ( Q .\/ R ) ) |
15 |
13 14
|
eqeq12d |
|- ( r = R -> ( ( P .\/ r ) = ( Q .\/ r ) <-> ( P .\/ R ) = ( Q .\/ R ) ) ) |
16 |
12 15
|
anbi12d |
|- ( r = R -> ( ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) <-> ( -. R .<_ W /\ ( P .\/ R ) = ( Q .\/ R ) ) ) ) |
17 |
16
|
notbid |
|- ( r = R -> ( -. ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) <-> -. ( -. R .<_ W /\ ( P .\/ R ) = ( Q .\/ R ) ) ) ) |
18 |
17
|
rspcva |
|- ( ( R e. A /\ A. r e. A -. ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) -> -. ( -. R .<_ W /\ ( P .\/ R ) = ( Q .\/ R ) ) ) |
19 |
7 10 18
|
syl2anc |
|- ( ( ( K e. HL /\ R .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) -> -. ( -. R .<_ W /\ ( P .\/ R ) = ( Q .\/ R ) ) ) |
20 |
|
simp11 |
|- ( ( ( K e. HL /\ R .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) -> K e. HL ) |
21 |
|
hlcvl |
|- ( K e. HL -> K e. CvLat ) |
22 |
20 21
|
syl |
|- ( ( ( K e. HL /\ R .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) -> K e. CvLat ) |
23 |
|
simp21 |
|- ( ( ( K e. HL /\ R .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) -> P e. A ) |
24 |
|
simp22 |
|- ( ( ( K e. HL /\ R .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) -> Q e. A ) |
25 |
|
simp23 |
|- ( ( ( K e. HL /\ R .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) -> P =/= Q ) |
26 |
3 1 2
|
cvlsupr2 |
|- ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= Q ) -> ( ( P .\/ R ) = ( Q .\/ R ) <-> ( R =/= P /\ R =/= Q /\ R .<_ ( P .\/ Q ) ) ) ) |
27 |
22 23 24 7 25 26
|
syl131anc |
|- ( ( ( K e. HL /\ R .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) -> ( ( P .\/ R ) = ( Q .\/ R ) <-> ( R =/= P /\ R =/= Q /\ R .<_ ( P .\/ Q ) ) ) ) |
28 |
27
|
anbi2d |
|- ( ( ( K e. HL /\ R .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) -> ( ( -. R .<_ W /\ ( P .\/ R ) = ( Q .\/ R ) ) <-> ( -. R .<_ W /\ ( R =/= P /\ R =/= Q /\ R .<_ ( P .\/ Q ) ) ) ) ) |
29 |
19 28
|
mtbid |
|- ( ( ( K e. HL /\ R .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) -> -. ( -. R .<_ W /\ ( R =/= P /\ R =/= Q /\ R .<_ ( P .\/ Q ) ) ) ) |
30 |
|
ianor |
|- ( -. ( ( R =/= P /\ R =/= Q ) /\ ( -. R .<_ W /\ R .<_ ( P .\/ Q ) ) ) <-> ( -. ( R =/= P /\ R =/= Q ) \/ -. ( -. R .<_ W /\ R .<_ ( P .\/ Q ) ) ) ) |
31 |
|
df-3an |
|- ( ( R =/= P /\ R =/= Q /\ R .<_ ( P .\/ Q ) ) <-> ( ( R =/= P /\ R =/= Q ) /\ R .<_ ( P .\/ Q ) ) ) |
32 |
31
|
anbi2i |
|- ( ( -. R .<_ W /\ ( R =/= P /\ R =/= Q /\ R .<_ ( P .\/ Q ) ) ) <-> ( -. R .<_ W /\ ( ( R =/= P /\ R =/= Q ) /\ R .<_ ( P .\/ Q ) ) ) ) |
33 |
|
an12 |
|- ( ( -. R .<_ W /\ ( ( R =/= P /\ R =/= Q ) /\ R .<_ ( P .\/ Q ) ) ) <-> ( ( R =/= P /\ R =/= Q ) /\ ( -. R .<_ W /\ R .<_ ( P .\/ Q ) ) ) ) |
34 |
32 33
|
bitri |
|- ( ( -. R .<_ W /\ ( R =/= P /\ R =/= Q /\ R .<_ ( P .\/ Q ) ) ) <-> ( ( R =/= P /\ R =/= Q ) /\ ( -. R .<_ W /\ R .<_ ( P .\/ Q ) ) ) ) |
35 |
34
|
notbii |
|- ( -. ( -. R .<_ W /\ ( R =/= P /\ R =/= Q /\ R .<_ ( P .\/ Q ) ) ) <-> -. ( ( R =/= P /\ R =/= Q ) /\ ( -. R .<_ W /\ R .<_ ( P .\/ Q ) ) ) ) |
36 |
|
pm4.62 |
|- ( ( ( R =/= P /\ R =/= Q ) -> -. ( -. R .<_ W /\ R .<_ ( P .\/ Q ) ) ) <-> ( -. ( R =/= P /\ R =/= Q ) \/ -. ( -. R .<_ W /\ R .<_ ( P .\/ Q ) ) ) ) |
37 |
30 35 36
|
3bitr4ri |
|- ( ( ( R =/= P /\ R =/= Q ) -> -. ( -. R .<_ W /\ R .<_ ( P .\/ Q ) ) ) <-> -. ( -. R .<_ W /\ ( R =/= P /\ R =/= Q /\ R .<_ ( P .\/ Q ) ) ) ) |
38 |
29 37
|
sylibr |
|- ( ( ( K e. HL /\ R .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) -> ( ( R =/= P /\ R =/= Q ) -> -. ( -. R .<_ W /\ R .<_ ( P .\/ Q ) ) ) ) |
39 |
6 38
|
mt2d |
|- ( ( ( K e. HL /\ R .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) -> -. ( R =/= P /\ R =/= Q ) ) |
40 |
|
neanior |
|- ( ( R =/= P /\ R =/= Q ) <-> -. ( R = P \/ R = Q ) ) |
41 |
40
|
con2bii |
|- ( ( R = P \/ R = Q ) <-> -. ( R =/= P /\ R =/= Q ) ) |
42 |
39 41
|
sylibr |
|- ( ( ( K e. HL /\ R .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) -> ( R = P \/ R = Q ) ) |