Step |
Hyp |
Ref |
Expression |
1 |
|
pmapjat.b |
|- B = ( Base ` K ) |
2 |
|
pmapjat.j |
|- .\/ = ( join ` K ) |
3 |
|
pmapjat.a |
|- A = ( Atoms ` K ) |
4 |
|
pmapjat.m |
|- M = ( pmap ` K ) |
5 |
|
pmapjat.p |
|- .+ = ( +P ` K ) |
6 |
|
simp1 |
|- ( ( K e. HL /\ X e. B /\ Q e. A ) -> K e. HL ) |
7 |
1 3
|
atbase |
|- ( Q e. A -> Q e. B ) |
8 |
7
|
3ad2ant3 |
|- ( ( K e. HL /\ X e. B /\ Q e. A ) -> Q e. B ) |
9 |
1 3 4
|
pmapssat |
|- ( ( K e. HL /\ Q e. B ) -> ( M ` Q ) C_ A ) |
10 |
6 8 9
|
syl2anc |
|- ( ( K e. HL /\ X e. B /\ Q e. A ) -> ( M ` Q ) C_ A ) |
11 |
3 5
|
padd02 |
|- ( ( K e. HL /\ ( M ` Q ) C_ A ) -> ( (/) .+ ( M ` Q ) ) = ( M ` Q ) ) |
12 |
6 10 11
|
syl2anc |
|- ( ( K e. HL /\ X e. B /\ Q e. A ) -> ( (/) .+ ( M ` Q ) ) = ( M ` Q ) ) |
13 |
12
|
adantr |
|- ( ( ( K e. HL /\ X e. B /\ Q e. A ) /\ X = ( 0. ` K ) ) -> ( (/) .+ ( M ` Q ) ) = ( M ` Q ) ) |
14 |
|
fveq2 |
|- ( X = ( 0. ` K ) -> ( M ` X ) = ( M ` ( 0. ` K ) ) ) |
15 |
|
hlatl |
|- ( K e. HL -> K e. AtLat ) |
16 |
15
|
3ad2ant1 |
|- ( ( K e. HL /\ X e. B /\ Q e. A ) -> K e. AtLat ) |
17 |
|
eqid |
|- ( 0. ` K ) = ( 0. ` K ) |
18 |
17 4
|
pmap0 |
|- ( K e. AtLat -> ( M ` ( 0. ` K ) ) = (/) ) |
19 |
16 18
|
syl |
|- ( ( K e. HL /\ X e. B /\ Q e. A ) -> ( M ` ( 0. ` K ) ) = (/) ) |
20 |
14 19
|
sylan9eqr |
|- ( ( ( K e. HL /\ X e. B /\ Q e. A ) /\ X = ( 0. ` K ) ) -> ( M ` X ) = (/) ) |
21 |
20
|
oveq1d |
|- ( ( ( K e. HL /\ X e. B /\ Q e. A ) /\ X = ( 0. ` K ) ) -> ( ( M ` X ) .+ ( M ` Q ) ) = ( (/) .+ ( M ` Q ) ) ) |
22 |
|
oveq1 |
|- ( X = ( 0. ` K ) -> ( X .\/ Q ) = ( ( 0. ` K ) .\/ Q ) ) |
23 |
|
hlol |
|- ( K e. HL -> K e. OL ) |
24 |
23
|
3ad2ant1 |
|- ( ( K e. HL /\ X e. B /\ Q e. A ) -> K e. OL ) |
25 |
1 2 17
|
olj02 |
|- ( ( K e. OL /\ Q e. B ) -> ( ( 0. ` K ) .\/ Q ) = Q ) |
26 |
24 8 25
|
syl2anc |
|- ( ( K e. HL /\ X e. B /\ Q e. A ) -> ( ( 0. ` K ) .\/ Q ) = Q ) |
27 |
22 26
|
sylan9eqr |
|- ( ( ( K e. HL /\ X e. B /\ Q e. A ) /\ X = ( 0. ` K ) ) -> ( X .\/ Q ) = Q ) |
28 |
27
|
fveq2d |
|- ( ( ( K e. HL /\ X e. B /\ Q e. A ) /\ X = ( 0. ` K ) ) -> ( M ` ( X .\/ Q ) ) = ( M ` Q ) ) |
29 |
13 21 28
|
3eqtr4rd |
|- ( ( ( K e. HL /\ X e. B /\ Q e. A ) /\ X = ( 0. ` K ) ) -> ( M ` ( X .\/ Q ) ) = ( ( M ` X ) .+ ( M ` Q ) ) ) |
30 |
|
simpll1 |
|- ( ( ( ( K e. HL /\ X e. B /\ Q e. A ) /\ X =/= ( 0. ` K ) ) /\ p e. A ) -> K e. HL ) |
31 |
30
|
adantr |
|- ( ( ( ( ( K e. HL /\ X e. B /\ Q e. A ) /\ X =/= ( 0. ` K ) ) /\ p e. A ) /\ p ( le ` K ) ( X .\/ Q ) ) -> K e. HL ) |
32 |
|
simpll2 |
|- ( ( ( ( K e. HL /\ X e. B /\ Q e. A ) /\ X =/= ( 0. ` K ) ) /\ p e. A ) -> X e. B ) |
33 |
32
|
adantr |
|- ( ( ( ( ( K e. HL /\ X e. B /\ Q e. A ) /\ X =/= ( 0. ` K ) ) /\ p e. A ) /\ p ( le ` K ) ( X .\/ Q ) ) -> X e. B ) |
34 |
|
simplr |
|- ( ( ( ( ( K e. HL /\ X e. B /\ Q e. A ) /\ X =/= ( 0. ` K ) ) /\ p e. A ) /\ p ( le ` K ) ( X .\/ Q ) ) -> p e. A ) |
35 |
|
simpll3 |
|- ( ( ( ( K e. HL /\ X e. B /\ Q e. A ) /\ X =/= ( 0. ` K ) ) /\ p e. A ) -> Q e. A ) |
36 |
35
|
adantr |
|- ( ( ( ( ( K e. HL /\ X e. B /\ Q e. A ) /\ X =/= ( 0. ` K ) ) /\ p e. A ) /\ p ( le ` K ) ( X .\/ Q ) ) -> Q e. A ) |
37 |
33 34 36
|
3jca |
|- ( ( ( ( ( K e. HL /\ X e. B /\ Q e. A ) /\ X =/= ( 0. ` K ) ) /\ p e. A ) /\ p ( le ` K ) ( X .\/ Q ) ) -> ( X e. B /\ p e. A /\ Q e. A ) ) |
38 |
|
simpllr |
|- ( ( ( ( ( K e. HL /\ X e. B /\ Q e. A ) /\ X =/= ( 0. ` K ) ) /\ p e. A ) /\ p ( le ` K ) ( X .\/ Q ) ) -> X =/= ( 0. ` K ) ) |
39 |
|
simpr |
|- ( ( ( ( ( K e. HL /\ X e. B /\ Q e. A ) /\ X =/= ( 0. ` K ) ) /\ p e. A ) /\ p ( le ` K ) ( X .\/ Q ) ) -> p ( le ` K ) ( X .\/ Q ) ) |
40 |
|
eqid |
|- ( le ` K ) = ( le ` K ) |
41 |
1 40 2 17 3
|
cvrat42 |
|- ( ( K e. HL /\ ( X e. B /\ p e. A /\ Q e. A ) ) -> ( ( X =/= ( 0. ` K ) /\ p ( le ` K ) ( X .\/ Q ) ) -> E. q e. A ( q ( le ` K ) X /\ p ( le ` K ) ( q .\/ Q ) ) ) ) |
42 |
41
|
imp |
|- ( ( ( K e. HL /\ ( X e. B /\ p e. A /\ Q e. A ) ) /\ ( X =/= ( 0. ` K ) /\ p ( le ` K ) ( X .\/ Q ) ) ) -> E. q e. A ( q ( le ` K ) X /\ p ( le ` K ) ( q .\/ Q ) ) ) |
43 |
31 37 38 39 42
|
syl22anc |
|- ( ( ( ( ( K e. HL /\ X e. B /\ Q e. A ) /\ X =/= ( 0. ` K ) ) /\ p e. A ) /\ p ( le ` K ) ( X .\/ Q ) ) -> E. q e. A ( q ( le ` K ) X /\ p ( le ` K ) ( q .\/ Q ) ) ) |
44 |
43
|
ex |
|- ( ( ( ( K e. HL /\ X e. B /\ Q e. A ) /\ X =/= ( 0. ` K ) ) /\ p e. A ) -> ( p ( le ` K ) ( X .\/ Q ) -> E. q e. A ( q ( le ` K ) X /\ p ( le ` K ) ( q .\/ Q ) ) ) ) |
45 |
1 40 3 4
|
elpmap |
|- ( ( K e. HL /\ X e. B ) -> ( q e. ( M ` X ) <-> ( q e. A /\ q ( le ` K ) X ) ) ) |
46 |
45
|
3adant3 |
|- ( ( K e. HL /\ X e. B /\ Q e. A ) -> ( q e. ( M ` X ) <-> ( q e. A /\ q ( le ` K ) X ) ) ) |
47 |
|
df-rex |
|- ( E. r e. ( M ` Q ) p ( le ` K ) ( q .\/ r ) <-> E. r ( r e. ( M ` Q ) /\ p ( le ` K ) ( q .\/ r ) ) ) |
48 |
3 4
|
elpmapat |
|- ( ( K e. HL /\ Q e. A ) -> ( r e. ( M ` Q ) <-> r = Q ) ) |
49 |
48
|
3adant2 |
|- ( ( K e. HL /\ X e. B /\ Q e. A ) -> ( r e. ( M ` Q ) <-> r = Q ) ) |
50 |
49
|
anbi1d |
|- ( ( K e. HL /\ X e. B /\ Q e. A ) -> ( ( r e. ( M ` Q ) /\ p ( le ` K ) ( q .\/ r ) ) <-> ( r = Q /\ p ( le ` K ) ( q .\/ r ) ) ) ) |
51 |
50
|
exbidv |
|- ( ( K e. HL /\ X e. B /\ Q e. A ) -> ( E. r ( r e. ( M ` Q ) /\ p ( le ` K ) ( q .\/ r ) ) <-> E. r ( r = Q /\ p ( le ` K ) ( q .\/ r ) ) ) ) |
52 |
47 51
|
bitr2id |
|- ( ( K e. HL /\ X e. B /\ Q e. A ) -> ( E. r ( r = Q /\ p ( le ` K ) ( q .\/ r ) ) <-> E. r e. ( M ` Q ) p ( le ` K ) ( q .\/ r ) ) ) |
53 |
|
oveq2 |
|- ( r = Q -> ( q .\/ r ) = ( q .\/ Q ) ) |
54 |
53
|
breq2d |
|- ( r = Q -> ( p ( le ` K ) ( q .\/ r ) <-> p ( le ` K ) ( q .\/ Q ) ) ) |
55 |
54
|
ceqsexgv |
|- ( Q e. A -> ( E. r ( r = Q /\ p ( le ` K ) ( q .\/ r ) ) <-> p ( le ` K ) ( q .\/ Q ) ) ) |
56 |
55
|
3ad2ant3 |
|- ( ( K e. HL /\ X e. B /\ Q e. A ) -> ( E. r ( r = Q /\ p ( le ` K ) ( q .\/ r ) ) <-> p ( le ` K ) ( q .\/ Q ) ) ) |
57 |
52 56
|
bitr3d |
|- ( ( K e. HL /\ X e. B /\ Q e. A ) -> ( E. r e. ( M ` Q ) p ( le ` K ) ( q .\/ r ) <-> p ( le ` K ) ( q .\/ Q ) ) ) |
58 |
46 57
|
anbi12d |
|- ( ( K e. HL /\ X e. B /\ Q e. A ) -> ( ( q e. ( M ` X ) /\ E. r e. ( M ` Q ) p ( le ` K ) ( q .\/ r ) ) <-> ( ( q e. A /\ q ( le ` K ) X ) /\ p ( le ` K ) ( q .\/ Q ) ) ) ) |
59 |
|
anass |
|- ( ( ( q e. A /\ q ( le ` K ) X ) /\ p ( le ` K ) ( q .\/ Q ) ) <-> ( q e. A /\ ( q ( le ` K ) X /\ p ( le ` K ) ( q .\/ Q ) ) ) ) |
60 |
58 59
|
bitrdi |
|- ( ( K e. HL /\ X e. B /\ Q e. A ) -> ( ( q e. ( M ` X ) /\ E. r e. ( M ` Q ) p ( le ` K ) ( q .\/ r ) ) <-> ( q e. A /\ ( q ( le ` K ) X /\ p ( le ` K ) ( q .\/ Q ) ) ) ) ) |
61 |
60
|
rexbidv2 |
|- ( ( K e. HL /\ X e. B /\ Q e. A ) -> ( E. q e. ( M ` X ) E. r e. ( M ` Q ) p ( le ` K ) ( q .\/ r ) <-> E. q e. A ( q ( le ` K ) X /\ p ( le ` K ) ( q .\/ Q ) ) ) ) |
62 |
61
|
ad2antrr |
|- ( ( ( ( K e. HL /\ X e. B /\ Q e. A ) /\ X =/= ( 0. ` K ) ) /\ p e. A ) -> ( E. q e. ( M ` X ) E. r e. ( M ` Q ) p ( le ` K ) ( q .\/ r ) <-> E. q e. A ( q ( le ` K ) X /\ p ( le ` K ) ( q .\/ Q ) ) ) ) |
63 |
44 62
|
sylibrd |
|- ( ( ( ( K e. HL /\ X e. B /\ Q e. A ) /\ X =/= ( 0. ` K ) ) /\ p e. A ) -> ( p ( le ` K ) ( X .\/ Q ) -> E. q e. ( M ` X ) E. r e. ( M ` Q ) p ( le ` K ) ( q .\/ r ) ) ) |
64 |
63
|
imdistanda |
|- ( ( ( K e. HL /\ X e. B /\ Q e. A ) /\ X =/= ( 0. ` K ) ) -> ( ( p e. A /\ p ( le ` K ) ( X .\/ Q ) ) -> ( p e. A /\ E. q e. ( M ` X ) E. r e. ( M ` Q ) p ( le ` K ) ( q .\/ r ) ) ) ) |
65 |
|
hllat |
|- ( K e. HL -> K e. Lat ) |
66 |
65
|
3ad2ant1 |
|- ( ( K e. HL /\ X e. B /\ Q e. A ) -> K e. Lat ) |
67 |
|
simp2 |
|- ( ( K e. HL /\ X e. B /\ Q e. A ) -> X e. B ) |
68 |
1 2
|
latjcl |
|- ( ( K e. Lat /\ X e. B /\ Q e. B ) -> ( X .\/ Q ) e. B ) |
69 |
66 67 8 68
|
syl3anc |
|- ( ( K e. HL /\ X e. B /\ Q e. A ) -> ( X .\/ Q ) e. B ) |
70 |
1 40 3 4
|
elpmap |
|- ( ( K e. HL /\ ( X .\/ Q ) e. B ) -> ( p e. ( M ` ( X .\/ Q ) ) <-> ( p e. A /\ p ( le ` K ) ( X .\/ Q ) ) ) ) |
71 |
6 69 70
|
syl2anc |
|- ( ( K e. HL /\ X e. B /\ Q e. A ) -> ( p e. ( M ` ( X .\/ Q ) ) <-> ( p e. A /\ p ( le ` K ) ( X .\/ Q ) ) ) ) |
72 |
71
|
adantr |
|- ( ( ( K e. HL /\ X e. B /\ Q e. A ) /\ X =/= ( 0. ` K ) ) -> ( p e. ( M ` ( X .\/ Q ) ) <-> ( p e. A /\ p ( le ` K ) ( X .\/ Q ) ) ) ) |
73 |
1 3 4
|
pmapssat |
|- ( ( K e. HL /\ X e. B ) -> ( M ` X ) C_ A ) |
74 |
73
|
3adant3 |
|- ( ( K e. HL /\ X e. B /\ Q e. A ) -> ( M ` X ) C_ A ) |
75 |
66 74 10
|
3jca |
|- ( ( K e. HL /\ X e. B /\ Q e. A ) -> ( K e. Lat /\ ( M ` X ) C_ A /\ ( M ` Q ) C_ A ) ) |
76 |
75
|
adantr |
|- ( ( ( K e. HL /\ X e. B /\ Q e. A ) /\ X =/= ( 0. ` K ) ) -> ( K e. Lat /\ ( M ` X ) C_ A /\ ( M ` Q ) C_ A ) ) |
77 |
1 17 4
|
pmapeq0 |
|- ( ( K e. HL /\ X e. B ) -> ( ( M ` X ) = (/) <-> X = ( 0. ` K ) ) ) |
78 |
77
|
3adant3 |
|- ( ( K e. HL /\ X e. B /\ Q e. A ) -> ( ( M ` X ) = (/) <-> X = ( 0. ` K ) ) ) |
79 |
78
|
necon3bid |
|- ( ( K e. HL /\ X e. B /\ Q e. A ) -> ( ( M ` X ) =/= (/) <-> X =/= ( 0. ` K ) ) ) |
80 |
79
|
biimpar |
|- ( ( ( K e. HL /\ X e. B /\ Q e. A ) /\ X =/= ( 0. ` K ) ) -> ( M ` X ) =/= (/) ) |
81 |
|
simp3 |
|- ( ( K e. HL /\ X e. B /\ Q e. A ) -> Q e. A ) |
82 |
17 3
|
atn0 |
|- ( ( K e. AtLat /\ Q e. A ) -> Q =/= ( 0. ` K ) ) |
83 |
16 81 82
|
syl2anc |
|- ( ( K e. HL /\ X e. B /\ Q e. A ) -> Q =/= ( 0. ` K ) ) |
84 |
1 17 4
|
pmapeq0 |
|- ( ( K e. HL /\ Q e. B ) -> ( ( M ` Q ) = (/) <-> Q = ( 0. ` K ) ) ) |
85 |
6 8 84
|
syl2anc |
|- ( ( K e. HL /\ X e. B /\ Q e. A ) -> ( ( M ` Q ) = (/) <-> Q = ( 0. ` K ) ) ) |
86 |
85
|
necon3bid |
|- ( ( K e. HL /\ X e. B /\ Q e. A ) -> ( ( M ` Q ) =/= (/) <-> Q =/= ( 0. ` K ) ) ) |
87 |
83 86
|
mpbird |
|- ( ( K e. HL /\ X e. B /\ Q e. A ) -> ( M ` Q ) =/= (/) ) |
88 |
87
|
adantr |
|- ( ( ( K e. HL /\ X e. B /\ Q e. A ) /\ X =/= ( 0. ` K ) ) -> ( M ` Q ) =/= (/) ) |
89 |
40 2 3 5
|
elpaddn0 |
|- ( ( ( K e. Lat /\ ( M ` X ) C_ A /\ ( M ` Q ) C_ A ) /\ ( ( M ` X ) =/= (/) /\ ( M ` Q ) =/= (/) ) ) -> ( p e. ( ( M ` X ) .+ ( M ` Q ) ) <-> ( p e. A /\ E. q e. ( M ` X ) E. r e. ( M ` Q ) p ( le ` K ) ( q .\/ r ) ) ) ) |
90 |
76 80 88 89
|
syl12anc |
|- ( ( ( K e. HL /\ X e. B /\ Q e. A ) /\ X =/= ( 0. ` K ) ) -> ( p e. ( ( M ` X ) .+ ( M ` Q ) ) <-> ( p e. A /\ E. q e. ( M ` X ) E. r e. ( M ` Q ) p ( le ` K ) ( q .\/ r ) ) ) ) |
91 |
64 72 90
|
3imtr4d |
|- ( ( ( K e. HL /\ X e. B /\ Q e. A ) /\ X =/= ( 0. ` K ) ) -> ( p e. ( M ` ( X .\/ Q ) ) -> p e. ( ( M ` X ) .+ ( M ` Q ) ) ) ) |
92 |
91
|
ssrdv |
|- ( ( ( K e. HL /\ X e. B /\ Q e. A ) /\ X =/= ( 0. ` K ) ) -> ( M ` ( X .\/ Q ) ) C_ ( ( M ` X ) .+ ( M ` Q ) ) ) |
93 |
1 2 4 5
|
pmapjoin |
|- ( ( K e. Lat /\ X e. B /\ Q e. B ) -> ( ( M ` X ) .+ ( M ` Q ) ) C_ ( M ` ( X .\/ Q ) ) ) |
94 |
66 67 8 93
|
syl3anc |
|- ( ( K e. HL /\ X e. B /\ Q e. A ) -> ( ( M ` X ) .+ ( M ` Q ) ) C_ ( M ` ( X .\/ Q ) ) ) |
95 |
94
|
adantr |
|- ( ( ( K e. HL /\ X e. B /\ Q e. A ) /\ X =/= ( 0. ` K ) ) -> ( ( M ` X ) .+ ( M ` Q ) ) C_ ( M ` ( X .\/ Q ) ) ) |
96 |
92 95
|
eqssd |
|- ( ( ( K e. HL /\ X e. B /\ Q e. A ) /\ X =/= ( 0. ` K ) ) -> ( M ` ( X .\/ Q ) ) = ( ( M ` X ) .+ ( M ` Q ) ) ) |
97 |
29 96
|
pm2.61dane |
|- ( ( K e. HL /\ X e. B /\ Q e. A ) -> ( M ` ( X .\/ Q ) ) = ( ( M ` X ) .+ ( M ` Q ) ) ) |