Step |
Hyp |
Ref |
Expression |
1 |
|
lplnexat.l |
|- .<_ = ( le ` K ) |
2 |
|
lplnexat.j |
|- .\/ = ( join ` K ) |
3 |
|
lplnexat.a |
|- A = ( Atoms ` K ) |
4 |
|
lplnexat.n |
|- N = ( LLines ` K ) |
5 |
|
lplnexat.p |
|- P = ( LPlanes ` K ) |
6 |
|
simpl2 |
|- ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) -> X e. P ) |
7 |
|
simpl1 |
|- ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) -> K e. HL ) |
8 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
9 |
8 5
|
lplnbase |
|- ( X e. P -> X e. ( Base ` K ) ) |
10 |
6 9
|
syl |
|- ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) -> X e. ( Base ` K ) ) |
11 |
8 1 2 3 4 5
|
islpln3 |
|- ( ( K e. HL /\ X e. ( Base ` K ) ) -> ( X e. P <-> E. z e. N E. r e. A ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) |
12 |
7 10 11
|
syl2anc |
|- ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) -> ( X e. P <-> E. z e. N E. r e. A ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) |
13 |
6 12
|
mpbid |
|- ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) -> E. z e. N E. r e. A ( -. r .<_ z /\ X = ( z .\/ r ) ) ) |
14 |
|
simpll1 |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> K e. HL ) |
15 |
|
simpr2l |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> z e. N ) |
16 |
|
simpll3 |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> Q e. A ) |
17 |
|
simpr1 |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> Q .<_ z ) |
18 |
1 2 3 4
|
llnexatN |
|- ( ( ( K e. HL /\ z e. N /\ Q e. A ) /\ Q .<_ z ) -> E. s e. A ( Q =/= s /\ z = ( Q .\/ s ) ) ) |
19 |
14 15 16 17 18
|
syl31anc |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> E. s e. A ( Q =/= s /\ z = ( Q .\/ s ) ) ) |
20 |
|
simp1l1 |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) /\ ( s e. A /\ ( Q =/= s /\ z = ( Q .\/ s ) ) ) ) -> K e. HL ) |
21 |
|
simp22r |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) /\ ( s e. A /\ ( Q =/= s /\ z = ( Q .\/ s ) ) ) ) -> r e. A ) |
22 |
|
simp3l |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) /\ ( s e. A /\ ( Q =/= s /\ z = ( Q .\/ s ) ) ) ) -> s e. A ) |
23 |
|
simp1l3 |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) /\ ( s e. A /\ ( Q =/= s /\ z = ( Q .\/ s ) ) ) ) -> Q e. A ) |
24 |
|
simp23l |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) /\ ( s e. A /\ ( Q =/= s /\ z = ( Q .\/ s ) ) ) ) -> -. r .<_ z ) |
25 |
|
simp3rr |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) /\ ( s e. A /\ ( Q =/= s /\ z = ( Q .\/ s ) ) ) ) -> z = ( Q .\/ s ) ) |
26 |
25
|
breq2d |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) /\ ( s e. A /\ ( Q =/= s /\ z = ( Q .\/ s ) ) ) ) -> ( r .<_ z <-> r .<_ ( Q .\/ s ) ) ) |
27 |
24 26
|
mtbid |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) /\ ( s e. A /\ ( Q =/= s /\ z = ( Q .\/ s ) ) ) ) -> -. r .<_ ( Q .\/ s ) ) |
28 |
1 2 3
|
atnlej2 |
|- ( ( K e. HL /\ ( r e. A /\ Q e. A /\ s e. A ) /\ -. r .<_ ( Q .\/ s ) ) -> r =/= s ) |
29 |
20 21 23 22 27 28
|
syl131anc |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) /\ ( s e. A /\ ( Q =/= s /\ z = ( Q .\/ s ) ) ) ) -> r =/= s ) |
30 |
2 3 4
|
llni2 |
|- ( ( ( K e. HL /\ r e. A /\ s e. A ) /\ r =/= s ) -> ( r .\/ s ) e. N ) |
31 |
20 21 22 29 30
|
syl31anc |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) /\ ( s e. A /\ ( Q =/= s /\ z = ( Q .\/ s ) ) ) ) -> ( r .\/ s ) e. N ) |
32 |
|
simp3rl |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) /\ ( s e. A /\ ( Q =/= s /\ z = ( Q .\/ s ) ) ) ) -> Q =/= s ) |
33 |
1 2 3
|
hlatcon2 |
|- ( ( K e. HL /\ ( Q e. A /\ s e. A /\ r e. A ) /\ ( Q =/= s /\ -. r .<_ ( Q .\/ s ) ) ) -> -. Q .<_ ( r .\/ s ) ) |
34 |
20 23 22 21 32 27 33
|
syl132anc |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) /\ ( s e. A /\ ( Q =/= s /\ z = ( Q .\/ s ) ) ) ) -> -. Q .<_ ( r .\/ s ) ) |
35 |
|
simp23r |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) /\ ( s e. A /\ ( Q =/= s /\ z = ( Q .\/ s ) ) ) ) -> X = ( z .\/ r ) ) |
36 |
25
|
oveq1d |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) /\ ( s e. A /\ ( Q =/= s /\ z = ( Q .\/ s ) ) ) ) -> ( z .\/ r ) = ( ( Q .\/ s ) .\/ r ) ) |
37 |
20
|
hllatd |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) /\ ( s e. A /\ ( Q =/= s /\ z = ( Q .\/ s ) ) ) ) -> K e. Lat ) |
38 |
8 3
|
atbase |
|- ( Q e. A -> Q e. ( Base ` K ) ) |
39 |
23 38
|
syl |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) /\ ( s e. A /\ ( Q =/= s /\ z = ( Q .\/ s ) ) ) ) -> Q e. ( Base ` K ) ) |
40 |
8 3
|
atbase |
|- ( s e. A -> s e. ( Base ` K ) ) |
41 |
22 40
|
syl |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) /\ ( s e. A /\ ( Q =/= s /\ z = ( Q .\/ s ) ) ) ) -> s e. ( Base ` K ) ) |
42 |
8 3
|
atbase |
|- ( r e. A -> r e. ( Base ` K ) ) |
43 |
21 42
|
syl |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) /\ ( s e. A /\ ( Q =/= s /\ z = ( Q .\/ s ) ) ) ) -> r e. ( Base ` K ) ) |
44 |
8 2
|
latj31 |
|- ( ( K e. Lat /\ ( Q e. ( Base ` K ) /\ s e. ( Base ` K ) /\ r e. ( Base ` K ) ) ) -> ( ( Q .\/ s ) .\/ r ) = ( ( r .\/ s ) .\/ Q ) ) |
45 |
37 39 41 43 44
|
syl13anc |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) /\ ( s e. A /\ ( Q =/= s /\ z = ( Q .\/ s ) ) ) ) -> ( ( Q .\/ s ) .\/ r ) = ( ( r .\/ s ) .\/ Q ) ) |
46 |
35 36 45
|
3eqtrd |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) /\ ( s e. A /\ ( Q =/= s /\ z = ( Q .\/ s ) ) ) ) -> X = ( ( r .\/ s ) .\/ Q ) ) |
47 |
|
breq2 |
|- ( y = ( r .\/ s ) -> ( Q .<_ y <-> Q .<_ ( r .\/ s ) ) ) |
48 |
47
|
notbid |
|- ( y = ( r .\/ s ) -> ( -. Q .<_ y <-> -. Q .<_ ( r .\/ s ) ) ) |
49 |
|
oveq1 |
|- ( y = ( r .\/ s ) -> ( y .\/ Q ) = ( ( r .\/ s ) .\/ Q ) ) |
50 |
49
|
eqeq2d |
|- ( y = ( r .\/ s ) -> ( X = ( y .\/ Q ) <-> X = ( ( r .\/ s ) .\/ Q ) ) ) |
51 |
48 50
|
anbi12d |
|- ( y = ( r .\/ s ) -> ( ( -. Q .<_ y /\ X = ( y .\/ Q ) ) <-> ( -. Q .<_ ( r .\/ s ) /\ X = ( ( r .\/ s ) .\/ Q ) ) ) ) |
52 |
51
|
rspcev |
|- ( ( ( r .\/ s ) e. N /\ ( -. Q .<_ ( r .\/ s ) /\ X = ( ( r .\/ s ) .\/ Q ) ) ) -> E. y e. N ( -. Q .<_ y /\ X = ( y .\/ Q ) ) ) |
53 |
31 34 46 52
|
syl12anc |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) /\ ( s e. A /\ ( Q =/= s /\ z = ( Q .\/ s ) ) ) ) -> E. y e. N ( -. Q .<_ y /\ X = ( y .\/ Q ) ) ) |
54 |
53
|
3expia |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> ( ( s e. A /\ ( Q =/= s /\ z = ( Q .\/ s ) ) ) -> E. y e. N ( -. Q .<_ y /\ X = ( y .\/ Q ) ) ) ) |
55 |
54
|
expd |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> ( s e. A -> ( ( Q =/= s /\ z = ( Q .\/ s ) ) -> E. y e. N ( -. Q .<_ y /\ X = ( y .\/ Q ) ) ) ) ) |
56 |
55
|
rexlimdv |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> ( E. s e. A ( Q =/= s /\ z = ( Q .\/ s ) ) -> E. y e. N ( -. Q .<_ y /\ X = ( y .\/ Q ) ) ) ) |
57 |
19 56
|
mpd |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> E. y e. N ( -. Q .<_ y /\ X = ( y .\/ Q ) ) ) |
58 |
57
|
3exp2 |
|- ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) -> ( Q .<_ z -> ( ( z e. N /\ r e. A ) -> ( ( -. r .<_ z /\ X = ( z .\/ r ) ) -> E. y e. N ( -. Q .<_ y /\ X = ( y .\/ Q ) ) ) ) ) ) |
59 |
|
simpr2l |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> z e. N ) |
60 |
|
simpr1 |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> -. Q .<_ z ) |
61 |
|
simpll1 |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> K e. HL ) |
62 |
61
|
hllatd |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> K e. Lat ) |
63 |
8 4
|
llnbase |
|- ( z e. N -> z e. ( Base ` K ) ) |
64 |
59 63
|
syl |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> z e. ( Base ` K ) ) |
65 |
|
simpr2r |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> r e. A ) |
66 |
65 42
|
syl |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> r e. ( Base ` K ) ) |
67 |
8 1 2
|
latlej1 |
|- ( ( K e. Lat /\ z e. ( Base ` K ) /\ r e. ( Base ` K ) ) -> z .<_ ( z .\/ r ) ) |
68 |
62 64 66 67
|
syl3anc |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> z .<_ ( z .\/ r ) ) |
69 |
|
simpr3r |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> X = ( z .\/ r ) ) |
70 |
68 69
|
breqtrrd |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> z .<_ X ) |
71 |
|
simplr |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> Q .<_ X ) |
72 |
|
simpll3 |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> Q e. A ) |
73 |
72 38
|
syl |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> Q e. ( Base ` K ) ) |
74 |
|
simpll2 |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> X e. P ) |
75 |
74 9
|
syl |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> X e. ( Base ` K ) ) |
76 |
8 1 2
|
latjle12 |
|- ( ( K e. Lat /\ ( z e. ( Base ` K ) /\ Q e. ( Base ` K ) /\ X e. ( Base ` K ) ) ) -> ( ( z .<_ X /\ Q .<_ X ) <-> ( z .\/ Q ) .<_ X ) ) |
77 |
62 64 73 75 76
|
syl13anc |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> ( ( z .<_ X /\ Q .<_ X ) <-> ( z .\/ Q ) .<_ X ) ) |
78 |
70 71 77
|
mpbi2and |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> ( z .\/ Q ) .<_ X ) |
79 |
8 2
|
latjcl |
|- ( ( K e. Lat /\ z e. ( Base ` K ) /\ Q e. ( Base ` K ) ) -> ( z .\/ Q ) e. ( Base ` K ) ) |
80 |
62 64 73 79
|
syl3anc |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> ( z .\/ Q ) e. ( Base ` K ) ) |
81 |
|
eqid |
|- ( |
82 |
8 1 2 81 3
|
cvr1 |
|- ( ( K e. HL /\ z e. ( Base ` K ) /\ Q e. A ) -> ( -. Q .<_ z <-> z ( |
83 |
61 64 72 82
|
syl3anc |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> ( -. Q .<_ z <-> z ( |
84 |
60 83
|
mpbid |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> z ( |
85 |
8 81 4 5
|
lplni |
|- ( ( ( K e. HL /\ ( z .\/ Q ) e. ( Base ` K ) /\ z e. N ) /\ z ( ( z .\/ Q ) e. P ) |
86 |
61 80 59 84 85
|
syl31anc |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> ( z .\/ Q ) e. P ) |
87 |
1 5
|
lplncmp |
|- ( ( K e. HL /\ ( z .\/ Q ) e. P /\ X e. P ) -> ( ( z .\/ Q ) .<_ X <-> ( z .\/ Q ) = X ) ) |
88 |
61 86 74 87
|
syl3anc |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> ( ( z .\/ Q ) .<_ X <-> ( z .\/ Q ) = X ) ) |
89 |
78 88
|
mpbid |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> ( z .\/ Q ) = X ) |
90 |
89
|
eqcomd |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> X = ( z .\/ Q ) ) |
91 |
|
breq2 |
|- ( y = z -> ( Q .<_ y <-> Q .<_ z ) ) |
92 |
91
|
notbid |
|- ( y = z -> ( -. Q .<_ y <-> -. Q .<_ z ) ) |
93 |
|
oveq1 |
|- ( y = z -> ( y .\/ Q ) = ( z .\/ Q ) ) |
94 |
93
|
eqeq2d |
|- ( y = z -> ( X = ( y .\/ Q ) <-> X = ( z .\/ Q ) ) ) |
95 |
92 94
|
anbi12d |
|- ( y = z -> ( ( -. Q .<_ y /\ X = ( y .\/ Q ) ) <-> ( -. Q .<_ z /\ X = ( z .\/ Q ) ) ) ) |
96 |
95
|
rspcev |
|- ( ( z e. N /\ ( -. Q .<_ z /\ X = ( z .\/ Q ) ) ) -> E. y e. N ( -. Q .<_ y /\ X = ( y .\/ Q ) ) ) |
97 |
59 60 90 96
|
syl12anc |
|- ( ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) /\ ( -. Q .<_ z /\ ( z e. N /\ r e. A ) /\ ( -. r .<_ z /\ X = ( z .\/ r ) ) ) ) -> E. y e. N ( -. Q .<_ y /\ X = ( y .\/ Q ) ) ) |
98 |
97
|
3exp2 |
|- ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) -> ( -. Q .<_ z -> ( ( z e. N /\ r e. A ) -> ( ( -. r .<_ z /\ X = ( z .\/ r ) ) -> E. y e. N ( -. Q .<_ y /\ X = ( y .\/ Q ) ) ) ) ) ) |
99 |
58 98
|
pm2.61d |
|- ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) -> ( ( z e. N /\ r e. A ) -> ( ( -. r .<_ z /\ X = ( z .\/ r ) ) -> E. y e. N ( -. Q .<_ y /\ X = ( y .\/ Q ) ) ) ) ) |
100 |
99
|
rexlimdvv |
|- ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) -> ( E. z e. N E. r e. A ( -. r .<_ z /\ X = ( z .\/ r ) ) -> E. y e. N ( -. Q .<_ y /\ X = ( y .\/ Q ) ) ) ) |
101 |
13 100
|
mpd |
|- ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ Q .<_ X ) -> E. y e. N ( -. Q .<_ y /\ X = ( y .\/ Q ) ) ) |