Step |
Hyp |
Ref |
Expression |
1 |
|
4that.l |
|- .<_ = ( le ` K ) |
2 |
|
4that.j |
|- .\/ = ( join ` K ) |
3 |
|
4that.a |
|- A = ( Atoms ` K ) |
4 |
|
4that.h |
|- H = ( LHyp ` K ) |
5 |
|
simp21l |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> P e. A ) |
6 |
5
|
ad2antrr |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ S .<_ ( P .\/ Q ) ) /\ S = P ) -> P e. A ) |
7 |
|
simp21r |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> -. P .<_ W ) |
8 |
7
|
ad2antrr |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ S .<_ ( P .\/ Q ) ) /\ S = P ) -> -. P .<_ W ) |
9 |
|
oveq1 |
|- ( P = S -> ( P .\/ P ) = ( S .\/ P ) ) |
10 |
9
|
eqcoms |
|- ( S = P -> ( P .\/ P ) = ( S .\/ P ) ) |
11 |
10
|
adantl |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ S .<_ ( P .\/ Q ) ) /\ S = P ) -> ( P .\/ P ) = ( S .\/ P ) ) |
12 |
|
breq1 |
|- ( z = P -> ( z .<_ W <-> P .<_ W ) ) |
13 |
12
|
notbid |
|- ( z = P -> ( -. z .<_ W <-> -. P .<_ W ) ) |
14 |
|
oveq2 |
|- ( z = P -> ( P .\/ z ) = ( P .\/ P ) ) |
15 |
|
oveq2 |
|- ( z = P -> ( S .\/ z ) = ( S .\/ P ) ) |
16 |
14 15
|
eqeq12d |
|- ( z = P -> ( ( P .\/ z ) = ( S .\/ z ) <-> ( P .\/ P ) = ( S .\/ P ) ) ) |
17 |
13 16
|
anbi12d |
|- ( z = P -> ( ( -. z .<_ W /\ ( P .\/ z ) = ( S .\/ z ) ) <-> ( -. P .<_ W /\ ( P .\/ P ) = ( S .\/ P ) ) ) ) |
18 |
17
|
rspcev |
|- ( ( P e. A /\ ( -. P .<_ W /\ ( P .\/ P ) = ( S .\/ P ) ) ) -> E. z e. A ( -. z .<_ W /\ ( P .\/ z ) = ( S .\/ z ) ) ) |
19 |
6 8 11 18
|
syl12anc |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ S .<_ ( P .\/ Q ) ) /\ S = P ) -> E. z e. A ( -. z .<_ W /\ ( P .\/ z ) = ( S .\/ z ) ) ) |
20 |
|
simpl3r |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ S .<_ ( P .\/ Q ) ) -> E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) |
21 |
20
|
ad2antrr |
|- ( ( ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ S .<_ ( P .\/ Q ) ) /\ S =/= P ) /\ S = Q ) -> E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) |
22 |
|
breq1 |
|- ( r = z -> ( r .<_ W <-> z .<_ W ) ) |
23 |
22
|
notbid |
|- ( r = z -> ( -. r .<_ W <-> -. z .<_ W ) ) |
24 |
|
oveq2 |
|- ( r = z -> ( P .\/ r ) = ( P .\/ z ) ) |
25 |
|
oveq2 |
|- ( r = z -> ( Q .\/ r ) = ( Q .\/ z ) ) |
26 |
24 25
|
eqeq12d |
|- ( r = z -> ( ( P .\/ r ) = ( Q .\/ r ) <-> ( P .\/ z ) = ( Q .\/ z ) ) ) |
27 |
23 26
|
anbi12d |
|- ( r = z -> ( ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) <-> ( -. z .<_ W /\ ( P .\/ z ) = ( Q .\/ z ) ) ) ) |
28 |
27
|
cbvrexvw |
|- ( E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) <-> E. z e. A ( -. z .<_ W /\ ( P .\/ z ) = ( Q .\/ z ) ) ) |
29 |
|
oveq1 |
|- ( S = Q -> ( S .\/ z ) = ( Q .\/ z ) ) |
30 |
29
|
eqeq2d |
|- ( S = Q -> ( ( P .\/ z ) = ( S .\/ z ) <-> ( P .\/ z ) = ( Q .\/ z ) ) ) |
31 |
30
|
anbi2d |
|- ( S = Q -> ( ( -. z .<_ W /\ ( P .\/ z ) = ( S .\/ z ) ) <-> ( -. z .<_ W /\ ( P .\/ z ) = ( Q .\/ z ) ) ) ) |
32 |
31
|
rexbidv |
|- ( S = Q -> ( E. z e. A ( -. z .<_ W /\ ( P .\/ z ) = ( S .\/ z ) ) <-> E. z e. A ( -. z .<_ W /\ ( P .\/ z ) = ( Q .\/ z ) ) ) ) |
33 |
28 32
|
bitr4id |
|- ( S = Q -> ( E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) <-> E. z e. A ( -. z .<_ W /\ ( P .\/ z ) = ( S .\/ z ) ) ) ) |
34 |
33
|
adantl |
|- ( ( ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ S .<_ ( P .\/ Q ) ) /\ S =/= P ) /\ S = Q ) -> ( E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) <-> E. z e. A ( -. z .<_ W /\ ( P .\/ z ) = ( S .\/ z ) ) ) ) |
35 |
21 34
|
mpbid |
|- ( ( ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ S .<_ ( P .\/ Q ) ) /\ S =/= P ) /\ S = Q ) -> E. z e. A ( -. z .<_ W /\ ( P .\/ z ) = ( S .\/ z ) ) ) |
36 |
|
simp22l |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> Q e. A ) |
37 |
36
|
ad3antrrr |
|- ( ( ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ S .<_ ( P .\/ Q ) ) /\ S =/= P ) /\ S =/= Q ) -> Q e. A ) |
38 |
|
simp22r |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> -. Q .<_ W ) |
39 |
38
|
ad3antrrr |
|- ( ( ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ S .<_ ( P .\/ Q ) ) /\ S =/= P ) /\ S =/= Q ) -> -. Q .<_ W ) |
40 |
|
simp3l |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> P =/= Q ) |
41 |
40
|
necomd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> Q =/= P ) |
42 |
41
|
ad3antrrr |
|- ( ( ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ S .<_ ( P .\/ Q ) ) /\ S =/= P ) /\ S =/= Q ) -> Q =/= P ) |
43 |
|
simpr |
|- ( ( ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ S .<_ ( P .\/ Q ) ) /\ S =/= P ) /\ S =/= Q ) -> S =/= Q ) |
44 |
43
|
necomd |
|- ( ( ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ S .<_ ( P .\/ Q ) ) /\ S =/= P ) /\ S =/= Q ) -> Q =/= S ) |
45 |
|
simpllr |
|- ( ( ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ S .<_ ( P .\/ Q ) ) /\ S =/= P ) /\ S =/= Q ) -> S .<_ ( P .\/ Q ) ) |
46 |
|
simp1l |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> K e. HL ) |
47 |
|
hlcvl |
|- ( K e. HL -> K e. CvLat ) |
48 |
46 47
|
syl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> K e. CvLat ) |
49 |
48
|
ad3antrrr |
|- ( ( ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ S .<_ ( P .\/ Q ) ) /\ S =/= P ) /\ S =/= Q ) -> K e. CvLat ) |
50 |
|
simp23 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> S e. A ) |
51 |
50
|
ad3antrrr |
|- ( ( ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ S .<_ ( P .\/ Q ) ) /\ S =/= P ) /\ S =/= Q ) -> S e. A ) |
52 |
5
|
ad3antrrr |
|- ( ( ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ S .<_ ( P .\/ Q ) ) /\ S =/= P ) /\ S =/= Q ) -> P e. A ) |
53 |
|
simplr |
|- ( ( ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ S .<_ ( P .\/ Q ) ) /\ S =/= P ) /\ S =/= Q ) -> S =/= P ) |
54 |
1 2 3
|
cvlatexch1 |
|- ( ( K e. CvLat /\ ( S e. A /\ Q e. A /\ P e. A ) /\ S =/= P ) -> ( S .<_ ( P .\/ Q ) -> Q .<_ ( P .\/ S ) ) ) |
55 |
49 51 37 52 53 54
|
syl131anc |
|- ( ( ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ S .<_ ( P .\/ Q ) ) /\ S =/= P ) /\ S =/= Q ) -> ( S .<_ ( P .\/ Q ) -> Q .<_ ( P .\/ S ) ) ) |
56 |
45 55
|
mpd |
|- ( ( ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ S .<_ ( P .\/ Q ) ) /\ S =/= P ) /\ S =/= Q ) -> Q .<_ ( P .\/ S ) ) |
57 |
53
|
necomd |
|- ( ( ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ S .<_ ( P .\/ Q ) ) /\ S =/= P ) /\ S =/= Q ) -> P =/= S ) |
58 |
3 1 2
|
cvlsupr2 |
|- ( ( K e. CvLat /\ ( P e. A /\ S e. A /\ Q e. A ) /\ P =/= S ) -> ( ( P .\/ Q ) = ( S .\/ Q ) <-> ( Q =/= P /\ Q =/= S /\ Q .<_ ( P .\/ S ) ) ) ) |
59 |
49 52 51 37 57 58
|
syl131anc |
|- ( ( ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ S .<_ ( P .\/ Q ) ) /\ S =/= P ) /\ S =/= Q ) -> ( ( P .\/ Q ) = ( S .\/ Q ) <-> ( Q =/= P /\ Q =/= S /\ Q .<_ ( P .\/ S ) ) ) ) |
60 |
42 44 56 59
|
mpbir3and |
|- ( ( ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ S .<_ ( P .\/ Q ) ) /\ S =/= P ) /\ S =/= Q ) -> ( P .\/ Q ) = ( S .\/ Q ) ) |
61 |
|
breq1 |
|- ( z = Q -> ( z .<_ W <-> Q .<_ W ) ) |
62 |
61
|
notbid |
|- ( z = Q -> ( -. z .<_ W <-> -. Q .<_ W ) ) |
63 |
|
oveq2 |
|- ( z = Q -> ( P .\/ z ) = ( P .\/ Q ) ) |
64 |
|
oveq2 |
|- ( z = Q -> ( S .\/ z ) = ( S .\/ Q ) ) |
65 |
63 64
|
eqeq12d |
|- ( z = Q -> ( ( P .\/ z ) = ( S .\/ z ) <-> ( P .\/ Q ) = ( S .\/ Q ) ) ) |
66 |
62 65
|
anbi12d |
|- ( z = Q -> ( ( -. z .<_ W /\ ( P .\/ z ) = ( S .\/ z ) ) <-> ( -. Q .<_ W /\ ( P .\/ Q ) = ( S .\/ Q ) ) ) ) |
67 |
66
|
rspcev |
|- ( ( Q e. A /\ ( -. Q .<_ W /\ ( P .\/ Q ) = ( S .\/ Q ) ) ) -> E. z e. A ( -. z .<_ W /\ ( P .\/ z ) = ( S .\/ z ) ) ) |
68 |
37 39 60 67
|
syl12anc |
|- ( ( ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ S .<_ ( P .\/ Q ) ) /\ S =/= P ) /\ S =/= Q ) -> E. z e. A ( -. z .<_ W /\ ( P .\/ z ) = ( S .\/ z ) ) ) |
69 |
35 68
|
pm2.61dane |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ S .<_ ( P .\/ Q ) ) /\ S =/= P ) -> E. z e. A ( -. z .<_ W /\ ( P .\/ z ) = ( S .\/ z ) ) ) |
70 |
19 69
|
pm2.61dane |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ S .<_ ( P .\/ Q ) ) -> E. z e. A ( -. z .<_ W /\ ( P .\/ z ) = ( S .\/ z ) ) ) |
71 |
|
simpl1 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( K e. HL /\ W e. H ) ) |
72 |
|
simpl2 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) ) |
73 |
|
simpl3l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ -. S .<_ ( P .\/ Q ) ) -> P =/= Q ) |
74 |
|
simpr |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ -. S .<_ ( P .\/ Q ) ) -> -. S .<_ ( P .\/ Q ) ) |
75 |
|
simpl3r |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ -. S .<_ ( P .\/ Q ) ) -> E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) |
76 |
1 2 3 4
|
4atexlem7 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> E. z e. A ( -. z .<_ W /\ ( P .\/ z ) = ( S .\/ z ) ) ) |
77 |
71 72 73 74 75 76
|
syl113anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ -. S .<_ ( P .\/ Q ) ) -> E. z e. A ( -. z .<_ W /\ ( P .\/ z ) = ( S .\/ z ) ) ) |
78 |
70 77
|
pm2.61dan |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> E. z e. A ( -. z .<_ W /\ ( P .\/ z ) = ( S .\/ z ) ) ) |