Step |
Hyp |
Ref |
Expression |
1 |
|
linepsub.n |
|- N = ( Lines ` K ) |
2 |
|
linepsub.s |
|- S = ( PSubSp ` K ) |
3 |
|
ssrab2 |
|- { c e. ( Atoms ` K ) | c ( le ` K ) ( a ( join ` K ) b ) } C_ ( Atoms ` K ) |
4 |
|
sseq1 |
|- ( X = { c e. ( Atoms ` K ) | c ( le ` K ) ( a ( join ` K ) b ) } -> ( X C_ ( Atoms ` K ) <-> { c e. ( Atoms ` K ) | c ( le ` K ) ( a ( join ` K ) b ) } C_ ( Atoms ` K ) ) ) |
5 |
3 4
|
mpbiri |
|- ( X = { c e. ( Atoms ` K ) | c ( le ` K ) ( a ( join ` K ) b ) } -> X C_ ( Atoms ` K ) ) |
6 |
5
|
a1i |
|- ( ( K e. Lat /\ ( a e. ( Atoms ` K ) /\ b e. ( Atoms ` K ) ) ) -> ( X = { c e. ( Atoms ` K ) | c ( le ` K ) ( a ( join ` K ) b ) } -> X C_ ( Atoms ` K ) ) ) |
7 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
8 |
|
eqid |
|- ( Atoms ` K ) = ( Atoms ` K ) |
9 |
7 8
|
atbase |
|- ( a e. ( Atoms ` K ) -> a e. ( Base ` K ) ) |
10 |
7 8
|
atbase |
|- ( b e. ( Atoms ` K ) -> b e. ( Base ` K ) ) |
11 |
9 10
|
anim12i |
|- ( ( a e. ( Atoms ` K ) /\ b e. ( Atoms ` K ) ) -> ( a e. ( Base ` K ) /\ b e. ( Base ` K ) ) ) |
12 |
|
eqid |
|- ( join ` K ) = ( join ` K ) |
13 |
7 12
|
latjcl |
|- ( ( K e. Lat /\ a e. ( Base ` K ) /\ b e. ( Base ` K ) ) -> ( a ( join ` K ) b ) e. ( Base ` K ) ) |
14 |
13
|
3expb |
|- ( ( K e. Lat /\ ( a e. ( Base ` K ) /\ b e. ( Base ` K ) ) ) -> ( a ( join ` K ) b ) e. ( Base ` K ) ) |
15 |
11 14
|
sylan2 |
|- ( ( K e. Lat /\ ( a e. ( Atoms ` K ) /\ b e. ( Atoms ` K ) ) ) -> ( a ( join ` K ) b ) e. ( Base ` K ) ) |
16 |
|
eleq2 |
|- ( X = { c e. ( Atoms ` K ) | c ( le ` K ) ( a ( join ` K ) b ) } -> ( p e. X <-> p e. { c e. ( Atoms ` K ) | c ( le ` K ) ( a ( join ` K ) b ) } ) ) |
17 |
|
breq1 |
|- ( c = p -> ( c ( le ` K ) ( a ( join ` K ) b ) <-> p ( le ` K ) ( a ( join ` K ) b ) ) ) |
18 |
17
|
elrab |
|- ( p e. { c e. ( Atoms ` K ) | c ( le ` K ) ( a ( join ` K ) b ) } <-> ( p e. ( Atoms ` K ) /\ p ( le ` K ) ( a ( join ` K ) b ) ) ) |
19 |
7 8
|
atbase |
|- ( p e. ( Atoms ` K ) -> p e. ( Base ` K ) ) |
20 |
19
|
anim1i |
|- ( ( p e. ( Atoms ` K ) /\ p ( le ` K ) ( a ( join ` K ) b ) ) -> ( p e. ( Base ` K ) /\ p ( le ` K ) ( a ( join ` K ) b ) ) ) |
21 |
18 20
|
sylbi |
|- ( p e. { c e. ( Atoms ` K ) | c ( le ` K ) ( a ( join ` K ) b ) } -> ( p e. ( Base ` K ) /\ p ( le ` K ) ( a ( join ` K ) b ) ) ) |
22 |
16 21
|
syl6bi |
|- ( X = { c e. ( Atoms ` K ) | c ( le ` K ) ( a ( join ` K ) b ) } -> ( p e. X -> ( p e. ( Base ` K ) /\ p ( le ` K ) ( a ( join ` K ) b ) ) ) ) |
23 |
|
eleq2 |
|- ( X = { c e. ( Atoms ` K ) | c ( le ` K ) ( a ( join ` K ) b ) } -> ( q e. X <-> q e. { c e. ( Atoms ` K ) | c ( le ` K ) ( a ( join ` K ) b ) } ) ) |
24 |
|
breq1 |
|- ( c = q -> ( c ( le ` K ) ( a ( join ` K ) b ) <-> q ( le ` K ) ( a ( join ` K ) b ) ) ) |
25 |
24
|
elrab |
|- ( q e. { c e. ( Atoms ` K ) | c ( le ` K ) ( a ( join ` K ) b ) } <-> ( q e. ( Atoms ` K ) /\ q ( le ` K ) ( a ( join ` K ) b ) ) ) |
26 |
7 8
|
atbase |
|- ( q e. ( Atoms ` K ) -> q e. ( Base ` K ) ) |
27 |
26
|
anim1i |
|- ( ( q e. ( Atoms ` K ) /\ q ( le ` K ) ( a ( join ` K ) b ) ) -> ( q e. ( Base ` K ) /\ q ( le ` K ) ( a ( join ` K ) b ) ) ) |
28 |
25 27
|
sylbi |
|- ( q e. { c e. ( Atoms ` K ) | c ( le ` K ) ( a ( join ` K ) b ) } -> ( q e. ( Base ` K ) /\ q ( le ` K ) ( a ( join ` K ) b ) ) ) |
29 |
23 28
|
syl6bi |
|- ( X = { c e. ( Atoms ` K ) | c ( le ` K ) ( a ( join ` K ) b ) } -> ( q e. X -> ( q e. ( Base ` K ) /\ q ( le ` K ) ( a ( join ` K ) b ) ) ) ) |
30 |
22 29
|
anim12d |
|- ( X = { c e. ( Atoms ` K ) | c ( le ` K ) ( a ( join ` K ) b ) } -> ( ( p e. X /\ q e. X ) -> ( ( p e. ( Base ` K ) /\ p ( le ` K ) ( a ( join ` K ) b ) ) /\ ( q e. ( Base ` K ) /\ q ( le ` K ) ( a ( join ` K ) b ) ) ) ) ) |
31 |
|
an4 |
|- ( ( ( p e. ( Base ` K ) /\ p ( le ` K ) ( a ( join ` K ) b ) ) /\ ( q e. ( Base ` K ) /\ q ( le ` K ) ( a ( join ` K ) b ) ) ) <-> ( ( p e. ( Base ` K ) /\ q e. ( Base ` K ) ) /\ ( p ( le ` K ) ( a ( join ` K ) b ) /\ q ( le ` K ) ( a ( join ` K ) b ) ) ) ) |
32 |
30 31
|
syl6ib |
|- ( X = { c e. ( Atoms ` K ) | c ( le ` K ) ( a ( join ` K ) b ) } -> ( ( p e. X /\ q e. X ) -> ( ( p e. ( Base ` K ) /\ q e. ( Base ` K ) ) /\ ( p ( le ` K ) ( a ( join ` K ) b ) /\ q ( le ` K ) ( a ( join ` K ) b ) ) ) ) ) |
33 |
32
|
imp |
|- ( ( X = { c e. ( Atoms ` K ) | c ( le ` K ) ( a ( join ` K ) b ) } /\ ( p e. X /\ q e. X ) ) -> ( ( p e. ( Base ` K ) /\ q e. ( Base ` K ) ) /\ ( p ( le ` K ) ( a ( join ` K ) b ) /\ q ( le ` K ) ( a ( join ` K ) b ) ) ) ) |
34 |
33
|
anim2i |
|- ( ( ( K e. Lat /\ ( a ( join ` K ) b ) e. ( Base ` K ) ) /\ ( X = { c e. ( Atoms ` K ) | c ( le ` K ) ( a ( join ` K ) b ) } /\ ( p e. X /\ q e. X ) ) ) -> ( ( K e. Lat /\ ( a ( join ` K ) b ) e. ( Base ` K ) ) /\ ( ( p e. ( Base ` K ) /\ q e. ( Base ` K ) ) /\ ( p ( le ` K ) ( a ( join ` K ) b ) /\ q ( le ` K ) ( a ( join ` K ) b ) ) ) ) ) |
35 |
34
|
anassrs |
|- ( ( ( ( K e. Lat /\ ( a ( join ` K ) b ) e. ( Base ` K ) ) /\ X = { c e. ( Atoms ` K ) | c ( le ` K ) ( a ( join ` K ) b ) } ) /\ ( p e. X /\ q e. X ) ) -> ( ( K e. Lat /\ ( a ( join ` K ) b ) e. ( Base ` K ) ) /\ ( ( p e. ( Base ` K ) /\ q e. ( Base ` K ) ) /\ ( p ( le ` K ) ( a ( join ` K ) b ) /\ q ( le ` K ) ( a ( join ` K ) b ) ) ) ) ) |
36 |
7 8
|
atbase |
|- ( r e. ( Atoms ` K ) -> r e. ( Base ` K ) ) |
37 |
|
eqid |
|- ( le ` K ) = ( le ` K ) |
38 |
7 37 12
|
latjle12 |
|- ( ( K e. Lat /\ ( p e. ( Base ` K ) /\ q e. ( Base ` K ) /\ ( a ( join ` K ) b ) e. ( Base ` K ) ) ) -> ( ( p ( le ` K ) ( a ( join ` K ) b ) /\ q ( le ` K ) ( a ( join ` K ) b ) ) <-> ( p ( join ` K ) q ) ( le ` K ) ( a ( join ` K ) b ) ) ) |
39 |
38
|
biimpd |
|- ( ( K e. Lat /\ ( p e. ( Base ` K ) /\ q e. ( Base ` K ) /\ ( a ( join ` K ) b ) e. ( Base ` K ) ) ) -> ( ( p ( le ` K ) ( a ( join ` K ) b ) /\ q ( le ` K ) ( a ( join ` K ) b ) ) -> ( p ( join ` K ) q ) ( le ` K ) ( a ( join ` K ) b ) ) ) |
40 |
39
|
3exp2 |
|- ( K e. Lat -> ( p e. ( Base ` K ) -> ( q e. ( Base ` K ) -> ( ( a ( join ` K ) b ) e. ( Base ` K ) -> ( ( p ( le ` K ) ( a ( join ` K ) b ) /\ q ( le ` K ) ( a ( join ` K ) b ) ) -> ( p ( join ` K ) q ) ( le ` K ) ( a ( join ` K ) b ) ) ) ) ) ) |
41 |
40
|
impd |
|- ( K e. Lat -> ( ( p e. ( Base ` K ) /\ q e. ( Base ` K ) ) -> ( ( a ( join ` K ) b ) e. ( Base ` K ) -> ( ( p ( le ` K ) ( a ( join ` K ) b ) /\ q ( le ` K ) ( a ( join ` K ) b ) ) -> ( p ( join ` K ) q ) ( le ` K ) ( a ( join ` K ) b ) ) ) ) ) |
42 |
41
|
com23 |
|- ( K e. Lat -> ( ( a ( join ` K ) b ) e. ( Base ` K ) -> ( ( p e. ( Base ` K ) /\ q e. ( Base ` K ) ) -> ( ( p ( le ` K ) ( a ( join ` K ) b ) /\ q ( le ` K ) ( a ( join ` K ) b ) ) -> ( p ( join ` K ) q ) ( le ` K ) ( a ( join ` K ) b ) ) ) ) ) |
43 |
42
|
imp43 |
|- ( ( ( K e. Lat /\ ( a ( join ` K ) b ) e. ( Base ` K ) ) /\ ( ( p e. ( Base ` K ) /\ q e. ( Base ` K ) ) /\ ( p ( le ` K ) ( a ( join ` K ) b ) /\ q ( le ` K ) ( a ( join ` K ) b ) ) ) ) -> ( p ( join ` K ) q ) ( le ` K ) ( a ( join ` K ) b ) ) |
44 |
43
|
adantr |
|- ( ( ( ( K e. Lat /\ ( a ( join ` K ) b ) e. ( Base ` K ) ) /\ ( ( p e. ( Base ` K ) /\ q e. ( Base ` K ) ) /\ ( p ( le ` K ) ( a ( join ` K ) b ) /\ q ( le ` K ) ( a ( join ` K ) b ) ) ) ) /\ r e. ( Base ` K ) ) -> ( p ( join ` K ) q ) ( le ` K ) ( a ( join ` K ) b ) ) |
45 |
7 12
|
latjcl |
|- ( ( K e. Lat /\ p e. ( Base ` K ) /\ q e. ( Base ` K ) ) -> ( p ( join ` K ) q ) e. ( Base ` K ) ) |
46 |
45
|
3expib |
|- ( K e. Lat -> ( ( p e. ( Base ` K ) /\ q e. ( Base ` K ) ) -> ( p ( join ` K ) q ) e. ( Base ` K ) ) ) |
47 |
7 37
|
lattr |
|- ( ( K e. Lat /\ ( r e. ( Base ` K ) /\ ( p ( join ` K ) q ) e. ( Base ` K ) /\ ( a ( join ` K ) b ) e. ( Base ` K ) ) ) -> ( ( r ( le ` K ) ( p ( join ` K ) q ) /\ ( p ( join ` K ) q ) ( le ` K ) ( a ( join ` K ) b ) ) -> r ( le ` K ) ( a ( join ` K ) b ) ) ) |
48 |
47
|
3exp2 |
|- ( K e. Lat -> ( r e. ( Base ` K ) -> ( ( p ( join ` K ) q ) e. ( Base ` K ) -> ( ( a ( join ` K ) b ) e. ( Base ` K ) -> ( ( r ( le ` K ) ( p ( join ` K ) q ) /\ ( p ( join ` K ) q ) ( le ` K ) ( a ( join ` K ) b ) ) -> r ( le ` K ) ( a ( join ` K ) b ) ) ) ) ) ) |
49 |
48
|
com24 |
|- ( K e. Lat -> ( ( a ( join ` K ) b ) e. ( Base ` K ) -> ( ( p ( join ` K ) q ) e. ( Base ` K ) -> ( r e. ( Base ` K ) -> ( ( r ( le ` K ) ( p ( join ` K ) q ) /\ ( p ( join ` K ) q ) ( le ` K ) ( a ( join ` K ) b ) ) -> r ( le ` K ) ( a ( join ` K ) b ) ) ) ) ) ) |
50 |
46 49
|
syl5d |
|- ( K e. Lat -> ( ( a ( join ` K ) b ) e. ( Base ` K ) -> ( ( p e. ( Base ` K ) /\ q e. ( Base ` K ) ) -> ( r e. ( Base ` K ) -> ( ( r ( le ` K ) ( p ( join ` K ) q ) /\ ( p ( join ` K ) q ) ( le ` K ) ( a ( join ` K ) b ) ) -> r ( le ` K ) ( a ( join ` K ) b ) ) ) ) ) ) |
51 |
50
|
imp41 |
|- ( ( ( ( K e. Lat /\ ( a ( join ` K ) b ) e. ( Base ` K ) ) /\ ( p e. ( Base ` K ) /\ q e. ( Base ` K ) ) ) /\ r e. ( Base ` K ) ) -> ( ( r ( le ` K ) ( p ( join ` K ) q ) /\ ( p ( join ` K ) q ) ( le ` K ) ( a ( join ` K ) b ) ) -> r ( le ` K ) ( a ( join ` K ) b ) ) ) |
52 |
51
|
adantlrr |
|- ( ( ( ( K e. Lat /\ ( a ( join ` K ) b ) e. ( Base ` K ) ) /\ ( ( p e. ( Base ` K ) /\ q e. ( Base ` K ) ) /\ ( p ( le ` K ) ( a ( join ` K ) b ) /\ q ( le ` K ) ( a ( join ` K ) b ) ) ) ) /\ r e. ( Base ` K ) ) -> ( ( r ( le ` K ) ( p ( join ` K ) q ) /\ ( p ( join ` K ) q ) ( le ` K ) ( a ( join ` K ) b ) ) -> r ( le ` K ) ( a ( join ` K ) b ) ) ) |
53 |
44 52
|
mpan2d |
|- ( ( ( ( K e. Lat /\ ( a ( join ` K ) b ) e. ( Base ` K ) ) /\ ( ( p e. ( Base ` K ) /\ q e. ( Base ` K ) ) /\ ( p ( le ` K ) ( a ( join ` K ) b ) /\ q ( le ` K ) ( a ( join ` K ) b ) ) ) ) /\ r e. ( Base ` K ) ) -> ( r ( le ` K ) ( p ( join ` K ) q ) -> r ( le ` K ) ( a ( join ` K ) b ) ) ) |
54 |
35 36 53
|
syl2an |
|- ( ( ( ( ( K e. Lat /\ ( a ( join ` K ) b ) e. ( Base ` K ) ) /\ X = { c e. ( Atoms ` K ) | c ( le ` K ) ( a ( join ` K ) b ) } ) /\ ( p e. X /\ q e. X ) ) /\ r e. ( Atoms ` K ) ) -> ( r ( le ` K ) ( p ( join ` K ) q ) -> r ( le ` K ) ( a ( join ` K ) b ) ) ) |
55 |
|
simpr |
|- ( ( ( ( ( K e. Lat /\ ( a ( join ` K ) b ) e. ( Base ` K ) ) /\ X = { c e. ( Atoms ` K ) | c ( le ` K ) ( a ( join ` K ) b ) } ) /\ ( p e. X /\ q e. X ) ) /\ r e. ( Atoms ` K ) ) -> r e. ( Atoms ` K ) ) |
56 |
54 55
|
jctild |
|- ( ( ( ( ( K e. Lat /\ ( a ( join ` K ) b ) e. ( Base ` K ) ) /\ X = { c e. ( Atoms ` K ) | c ( le ` K ) ( a ( join ` K ) b ) } ) /\ ( p e. X /\ q e. X ) ) /\ r e. ( Atoms ` K ) ) -> ( r ( le ` K ) ( p ( join ` K ) q ) -> ( r e. ( Atoms ` K ) /\ r ( le ` K ) ( a ( join ` K ) b ) ) ) ) |
57 |
|
eleq2 |
|- ( X = { c e. ( Atoms ` K ) | c ( le ` K ) ( a ( join ` K ) b ) } -> ( r e. X <-> r e. { c e. ( Atoms ` K ) | c ( le ` K ) ( a ( join ` K ) b ) } ) ) |
58 |
|
breq1 |
|- ( c = r -> ( c ( le ` K ) ( a ( join ` K ) b ) <-> r ( le ` K ) ( a ( join ` K ) b ) ) ) |
59 |
58
|
elrab |
|- ( r e. { c e. ( Atoms ` K ) | c ( le ` K ) ( a ( join ` K ) b ) } <-> ( r e. ( Atoms ` K ) /\ r ( le ` K ) ( a ( join ` K ) b ) ) ) |
60 |
57 59
|
bitrdi |
|- ( X = { c e. ( Atoms ` K ) | c ( le ` K ) ( a ( join ` K ) b ) } -> ( r e. X <-> ( r e. ( Atoms ` K ) /\ r ( le ` K ) ( a ( join ` K ) b ) ) ) ) |
61 |
60
|
ad3antlr |
|- ( ( ( ( ( K e. Lat /\ ( a ( join ` K ) b ) e. ( Base ` K ) ) /\ X = { c e. ( Atoms ` K ) | c ( le ` K ) ( a ( join ` K ) b ) } ) /\ ( p e. X /\ q e. X ) ) /\ r e. ( Atoms ` K ) ) -> ( r e. X <-> ( r e. ( Atoms ` K ) /\ r ( le ` K ) ( a ( join ` K ) b ) ) ) ) |
62 |
56 61
|
sylibrd |
|- ( ( ( ( ( K e. Lat /\ ( a ( join ` K ) b ) e. ( Base ` K ) ) /\ X = { c e. ( Atoms ` K ) | c ( le ` K ) ( a ( join ` K ) b ) } ) /\ ( p e. X /\ q e. X ) ) /\ r e. ( Atoms ` K ) ) -> ( r ( le ` K ) ( p ( join ` K ) q ) -> r e. X ) ) |
63 |
62
|
ralrimiva |
|- ( ( ( ( K e. Lat /\ ( a ( join ` K ) b ) e. ( Base ` K ) ) /\ X = { c e. ( Atoms ` K ) | c ( le ` K ) ( a ( join ` K ) b ) } ) /\ ( p e. X /\ q e. X ) ) -> A. r e. ( Atoms ` K ) ( r ( le ` K ) ( p ( join ` K ) q ) -> r e. X ) ) |
64 |
63
|
ralrimivva |
|- ( ( ( K e. Lat /\ ( a ( join ` K ) b ) e. ( Base ` K ) ) /\ X = { c e. ( Atoms ` K ) | c ( le ` K ) ( a ( join ` K ) b ) } ) -> A. p e. X A. q e. X A. r e. ( Atoms ` K ) ( r ( le ` K ) ( p ( join ` K ) q ) -> r e. X ) ) |
65 |
64
|
ex |
|- ( ( K e. Lat /\ ( a ( join ` K ) b ) e. ( Base ` K ) ) -> ( X = { c e. ( Atoms ` K ) | c ( le ` K ) ( a ( join ` K ) b ) } -> A. p e. X A. q e. X A. r e. ( Atoms ` K ) ( r ( le ` K ) ( p ( join ` K ) q ) -> r e. X ) ) ) |
66 |
15 65
|
syldan |
|- ( ( K e. Lat /\ ( a e. ( Atoms ` K ) /\ b e. ( Atoms ` K ) ) ) -> ( X = { c e. ( Atoms ` K ) | c ( le ` K ) ( a ( join ` K ) b ) } -> A. p e. X A. q e. X A. r e. ( Atoms ` K ) ( r ( le ` K ) ( p ( join ` K ) q ) -> r e. X ) ) ) |
67 |
6 66
|
jcad |
|- ( ( K e. Lat /\ ( a e. ( Atoms ` K ) /\ b e. ( Atoms ` K ) ) ) -> ( X = { c e. ( Atoms ` K ) | c ( le ` K ) ( a ( join ` K ) b ) } -> ( X C_ ( Atoms ` K ) /\ A. p e. X A. q e. X A. r e. ( Atoms ` K ) ( r ( le ` K ) ( p ( join ` K ) q ) -> r e. X ) ) ) ) |
68 |
67
|
adantld |
|- ( ( K e. Lat /\ ( a e. ( Atoms ` K ) /\ b e. ( Atoms ` K ) ) ) -> ( ( a =/= b /\ X = { c e. ( Atoms ` K ) | c ( le ` K ) ( a ( join ` K ) b ) } ) -> ( X C_ ( Atoms ` K ) /\ A. p e. X A. q e. X A. r e. ( Atoms ` K ) ( r ( le ` K ) ( p ( join ` K ) q ) -> r e. X ) ) ) ) |
69 |
68
|
rexlimdvva |
|- ( K e. Lat -> ( E. a e. ( Atoms ` K ) E. b e. ( Atoms ` K ) ( a =/= b /\ X = { c e. ( Atoms ` K ) | c ( le ` K ) ( a ( join ` K ) b ) } ) -> ( X C_ ( Atoms ` K ) /\ A. p e. X A. q e. X A. r e. ( Atoms ` K ) ( r ( le ` K ) ( p ( join ` K ) q ) -> r e. X ) ) ) ) |
70 |
37 12 8 1
|
isline |
|- ( K e. Lat -> ( X e. N <-> E. a e. ( Atoms ` K ) E. b e. ( Atoms ` K ) ( a =/= b /\ X = { c e. ( Atoms ` K ) | c ( le ` K ) ( a ( join ` K ) b ) } ) ) ) |
71 |
37 12 8 2
|
ispsubsp |
|- ( K e. Lat -> ( X e. S <-> ( X C_ ( Atoms ` K ) /\ A. p e. X A. q e. X A. r e. ( Atoms ` K ) ( r ( le ` K ) ( p ( join ` K ) q ) -> r e. X ) ) ) ) |
72 |
69 70 71
|
3imtr4d |
|- ( K e. Lat -> ( X e. N -> X e. S ) ) |
73 |
72
|
imp |
|- ( ( K e. Lat /\ X e. N ) -> X e. S ) |