Step |
Hyp |
Ref |
Expression |
1 |
|
dalem.ph |
|- ( ph <-> ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( Y e. O /\ Z e. O ) /\ ( ( -. C .<_ ( P .\/ Q ) /\ -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) ) /\ ( -. C .<_ ( S .\/ T ) /\ -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) ) /\ ( C .<_ ( P .\/ S ) /\ C .<_ ( Q .\/ T ) /\ C .<_ ( R .\/ U ) ) ) ) ) |
2 |
|
dalem.l |
|- .<_ = ( le ` K ) |
3 |
|
dalem.j |
|- .\/ = ( join ` K ) |
4 |
|
dalem.a |
|- A = ( Atoms ` K ) |
5 |
|
dalem.ps |
|- ( ps <-> ( ( c e. A /\ d e. A ) /\ -. c .<_ Y /\ ( d =/= c /\ -. d .<_ Y /\ C .<_ ( c .\/ d ) ) ) ) |
6 |
|
dalem54.m |
|- ./\ = ( meet ` K ) |
7 |
|
dalem54.o |
|- O = ( LPlanes ` K ) |
8 |
|
dalem54.y |
|- Y = ( ( P .\/ Q ) .\/ R ) |
9 |
|
dalem54.z |
|- Z = ( ( S .\/ T ) .\/ U ) |
10 |
|
dalem54.g |
|- G = ( ( c .\/ P ) ./\ ( d .\/ S ) ) |
11 |
|
dalem54.h |
|- H = ( ( c .\/ Q ) ./\ ( d .\/ T ) ) |
12 |
|
dalem54.i |
|- I = ( ( c .\/ R ) ./\ ( d .\/ U ) ) |
13 |
|
dalem54.b1 |
|- B = ( ( ( G .\/ H ) .\/ I ) ./\ Y ) |
14 |
1 2 3 4
|
dalemswapyz |
|- ( ph -> ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( S e. A /\ T e. A /\ U e. A ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) /\ ( Z e. O /\ Y e. O ) /\ ( ( -. C .<_ ( S .\/ T ) /\ -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) ) /\ ( -. C .<_ ( P .\/ Q ) /\ -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) ) /\ ( C .<_ ( S .\/ P ) /\ C .<_ ( T .\/ Q ) /\ C .<_ ( U .\/ R ) ) ) ) ) |
15 |
14
|
3ad2ant1 |
|- ( ( ph /\ Y = Z /\ ps ) -> ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( S e. A /\ T e. A /\ U e. A ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) /\ ( Z e. O /\ Y e. O ) /\ ( ( -. C .<_ ( S .\/ T ) /\ -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) ) /\ ( -. C .<_ ( P .\/ Q ) /\ -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) ) /\ ( C .<_ ( S .\/ P ) /\ C .<_ ( T .\/ Q ) /\ C .<_ ( U .\/ R ) ) ) ) ) |
16 |
|
simp2 |
|- ( ( ph /\ Y = Z /\ ps ) -> Y = Z ) |
17 |
16
|
eqcomd |
|- ( ( ph /\ Y = Z /\ ps ) -> Z = Y ) |
18 |
1 2 3 4 5
|
dalemswapyzps |
|- ( ( ph /\ Y = Z /\ ps ) -> ( ( d e. A /\ c e. A ) /\ -. d .<_ Z /\ ( c =/= d /\ -. c .<_ Z /\ C .<_ ( d .\/ c ) ) ) ) |
19 |
|
biid |
|- ( ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( S e. A /\ T e. A /\ U e. A ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) /\ ( Z e. O /\ Y e. O ) /\ ( ( -. C .<_ ( S .\/ T ) /\ -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) ) /\ ( -. C .<_ ( P .\/ Q ) /\ -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) ) /\ ( C .<_ ( S .\/ P ) /\ C .<_ ( T .\/ Q ) /\ C .<_ ( U .\/ R ) ) ) ) <-> ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( S e. A /\ T e. A /\ U e. A ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) /\ ( Z e. O /\ Y e. O ) /\ ( ( -. C .<_ ( S .\/ T ) /\ -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) ) /\ ( -. C .<_ ( P .\/ Q ) /\ -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) ) /\ ( C .<_ ( S .\/ P ) /\ C .<_ ( T .\/ Q ) /\ C .<_ ( U .\/ R ) ) ) ) ) |
20 |
|
biid |
|- ( ( ( d e. A /\ c e. A ) /\ -. d .<_ Z /\ ( c =/= d /\ -. c .<_ Z /\ C .<_ ( d .\/ c ) ) ) <-> ( ( d e. A /\ c e. A ) /\ -. d .<_ Z /\ ( c =/= d /\ -. c .<_ Z /\ C .<_ ( d .\/ c ) ) ) ) |
21 |
|
eqid |
|- ( ( d .\/ S ) ./\ ( c .\/ P ) ) = ( ( d .\/ S ) ./\ ( c .\/ P ) ) |
22 |
|
eqid |
|- ( ( d .\/ T ) ./\ ( c .\/ Q ) ) = ( ( d .\/ T ) ./\ ( c .\/ Q ) ) |
23 |
|
eqid |
|- ( ( d .\/ U ) ./\ ( c .\/ R ) ) = ( ( d .\/ U ) ./\ ( c .\/ R ) ) |
24 |
|
eqid |
|- ( ( ( ( ( d .\/ S ) ./\ ( c .\/ P ) ) .\/ ( ( d .\/ T ) ./\ ( c .\/ Q ) ) ) .\/ ( ( d .\/ U ) ./\ ( c .\/ R ) ) ) ./\ Z ) = ( ( ( ( ( d .\/ S ) ./\ ( c .\/ P ) ) .\/ ( ( d .\/ T ) ./\ ( c .\/ Q ) ) ) .\/ ( ( d .\/ U ) ./\ ( c .\/ R ) ) ) ./\ Z ) |
25 |
19 2 3 4 20 6 7 9 8 21 22 23 24
|
dalem55 |
|- ( ( ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( S e. A /\ T e. A /\ U e. A ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) /\ ( Z e. O /\ Y e. O ) /\ ( ( -. C .<_ ( S .\/ T ) /\ -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) ) /\ ( -. C .<_ ( P .\/ Q ) /\ -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) ) /\ ( C .<_ ( S .\/ P ) /\ C .<_ ( T .\/ Q ) /\ C .<_ ( U .\/ R ) ) ) ) /\ Z = Y /\ ( ( d e. A /\ c e. A ) /\ -. d .<_ Z /\ ( c =/= d /\ -. c .<_ Z /\ C .<_ ( d .\/ c ) ) ) ) -> ( ( ( ( d .\/ S ) ./\ ( c .\/ P ) ) .\/ ( ( d .\/ T ) ./\ ( c .\/ Q ) ) ) ./\ ( S .\/ T ) ) = ( ( ( ( d .\/ S ) ./\ ( c .\/ P ) ) .\/ ( ( d .\/ T ) ./\ ( c .\/ Q ) ) ) ./\ ( ( ( ( ( d .\/ S ) ./\ ( c .\/ P ) ) .\/ ( ( d .\/ T ) ./\ ( c .\/ Q ) ) ) .\/ ( ( d .\/ U ) ./\ ( c .\/ R ) ) ) ./\ Z ) ) ) |
26 |
15 17 18 25
|
syl3anc |
|- ( ( ph /\ Y = Z /\ ps ) -> ( ( ( ( d .\/ S ) ./\ ( c .\/ P ) ) .\/ ( ( d .\/ T ) ./\ ( c .\/ Q ) ) ) ./\ ( S .\/ T ) ) = ( ( ( ( d .\/ S ) ./\ ( c .\/ P ) ) .\/ ( ( d .\/ T ) ./\ ( c .\/ Q ) ) ) ./\ ( ( ( ( ( d .\/ S ) ./\ ( c .\/ P ) ) .\/ ( ( d .\/ T ) ./\ ( c .\/ Q ) ) ) .\/ ( ( d .\/ U ) ./\ ( c .\/ R ) ) ) ./\ Z ) ) ) |
27 |
1
|
dalemkelat |
|- ( ph -> K e. Lat ) |
28 |
27
|
3ad2ant1 |
|- ( ( ph /\ Y = Z /\ ps ) -> K e. Lat ) |
29 |
1
|
dalemkehl |
|- ( ph -> K e. HL ) |
30 |
29
|
3ad2ant1 |
|- ( ( ph /\ Y = Z /\ ps ) -> K e. HL ) |
31 |
5
|
dalemccea |
|- ( ps -> c e. A ) |
32 |
31
|
3ad2ant3 |
|- ( ( ph /\ Y = Z /\ ps ) -> c e. A ) |
33 |
1
|
dalempea |
|- ( ph -> P e. A ) |
34 |
33
|
3ad2ant1 |
|- ( ( ph /\ Y = Z /\ ps ) -> P e. A ) |
35 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
36 |
35 3 4
|
hlatjcl |
|- ( ( K e. HL /\ c e. A /\ P e. A ) -> ( c .\/ P ) e. ( Base ` K ) ) |
37 |
30 32 34 36
|
syl3anc |
|- ( ( ph /\ Y = Z /\ ps ) -> ( c .\/ P ) e. ( Base ` K ) ) |
38 |
5
|
dalemddea |
|- ( ps -> d e. A ) |
39 |
38
|
3ad2ant3 |
|- ( ( ph /\ Y = Z /\ ps ) -> d e. A ) |
40 |
1
|
dalemsea |
|- ( ph -> S e. A ) |
41 |
40
|
3ad2ant1 |
|- ( ( ph /\ Y = Z /\ ps ) -> S e. A ) |
42 |
35 3 4
|
hlatjcl |
|- ( ( K e. HL /\ d e. A /\ S e. A ) -> ( d .\/ S ) e. ( Base ` K ) ) |
43 |
30 39 41 42
|
syl3anc |
|- ( ( ph /\ Y = Z /\ ps ) -> ( d .\/ S ) e. ( Base ` K ) ) |
44 |
35 6
|
latmcom |
|- ( ( K e. Lat /\ ( c .\/ P ) e. ( Base ` K ) /\ ( d .\/ S ) e. ( Base ` K ) ) -> ( ( c .\/ P ) ./\ ( d .\/ S ) ) = ( ( d .\/ S ) ./\ ( c .\/ P ) ) ) |
45 |
28 37 43 44
|
syl3anc |
|- ( ( ph /\ Y = Z /\ ps ) -> ( ( c .\/ P ) ./\ ( d .\/ S ) ) = ( ( d .\/ S ) ./\ ( c .\/ P ) ) ) |
46 |
10 45
|
syl5eq |
|- ( ( ph /\ Y = Z /\ ps ) -> G = ( ( d .\/ S ) ./\ ( c .\/ P ) ) ) |
47 |
1
|
dalemqea |
|- ( ph -> Q e. A ) |
48 |
47
|
3ad2ant1 |
|- ( ( ph /\ Y = Z /\ ps ) -> Q e. A ) |
49 |
35 3 4
|
hlatjcl |
|- ( ( K e. HL /\ c e. A /\ Q e. A ) -> ( c .\/ Q ) e. ( Base ` K ) ) |
50 |
30 32 48 49
|
syl3anc |
|- ( ( ph /\ Y = Z /\ ps ) -> ( c .\/ Q ) e. ( Base ` K ) ) |
51 |
1
|
dalemtea |
|- ( ph -> T e. A ) |
52 |
51
|
3ad2ant1 |
|- ( ( ph /\ Y = Z /\ ps ) -> T e. A ) |
53 |
35 3 4
|
hlatjcl |
|- ( ( K e. HL /\ d e. A /\ T e. A ) -> ( d .\/ T ) e. ( Base ` K ) ) |
54 |
30 39 52 53
|
syl3anc |
|- ( ( ph /\ Y = Z /\ ps ) -> ( d .\/ T ) e. ( Base ` K ) ) |
55 |
35 6
|
latmcom |
|- ( ( K e. Lat /\ ( c .\/ Q ) e. ( Base ` K ) /\ ( d .\/ T ) e. ( Base ` K ) ) -> ( ( c .\/ Q ) ./\ ( d .\/ T ) ) = ( ( d .\/ T ) ./\ ( c .\/ Q ) ) ) |
56 |
28 50 54 55
|
syl3anc |
|- ( ( ph /\ Y = Z /\ ps ) -> ( ( c .\/ Q ) ./\ ( d .\/ T ) ) = ( ( d .\/ T ) ./\ ( c .\/ Q ) ) ) |
57 |
11 56
|
syl5eq |
|- ( ( ph /\ Y = Z /\ ps ) -> H = ( ( d .\/ T ) ./\ ( c .\/ Q ) ) ) |
58 |
46 57
|
oveq12d |
|- ( ( ph /\ Y = Z /\ ps ) -> ( G .\/ H ) = ( ( ( d .\/ S ) ./\ ( c .\/ P ) ) .\/ ( ( d .\/ T ) ./\ ( c .\/ Q ) ) ) ) |
59 |
58
|
oveq1d |
|- ( ( ph /\ Y = Z /\ ps ) -> ( ( G .\/ H ) ./\ ( S .\/ T ) ) = ( ( ( ( d .\/ S ) ./\ ( c .\/ P ) ) .\/ ( ( d .\/ T ) ./\ ( c .\/ Q ) ) ) ./\ ( S .\/ T ) ) ) |
60 |
1
|
dalemrea |
|- ( ph -> R e. A ) |
61 |
60
|
3ad2ant1 |
|- ( ( ph /\ Y = Z /\ ps ) -> R e. A ) |
62 |
35 3 4
|
hlatjcl |
|- ( ( K e. HL /\ c e. A /\ R e. A ) -> ( c .\/ R ) e. ( Base ` K ) ) |
63 |
30 32 61 62
|
syl3anc |
|- ( ( ph /\ Y = Z /\ ps ) -> ( c .\/ R ) e. ( Base ` K ) ) |
64 |
1
|
dalemuea |
|- ( ph -> U e. A ) |
65 |
64
|
3ad2ant1 |
|- ( ( ph /\ Y = Z /\ ps ) -> U e. A ) |
66 |
35 3 4
|
hlatjcl |
|- ( ( K e. HL /\ d e. A /\ U e. A ) -> ( d .\/ U ) e. ( Base ` K ) ) |
67 |
30 39 65 66
|
syl3anc |
|- ( ( ph /\ Y = Z /\ ps ) -> ( d .\/ U ) e. ( Base ` K ) ) |
68 |
35 6
|
latmcom |
|- ( ( K e. Lat /\ ( c .\/ R ) e. ( Base ` K ) /\ ( d .\/ U ) e. ( Base ` K ) ) -> ( ( c .\/ R ) ./\ ( d .\/ U ) ) = ( ( d .\/ U ) ./\ ( c .\/ R ) ) ) |
69 |
28 63 67 68
|
syl3anc |
|- ( ( ph /\ Y = Z /\ ps ) -> ( ( c .\/ R ) ./\ ( d .\/ U ) ) = ( ( d .\/ U ) ./\ ( c .\/ R ) ) ) |
70 |
12 69
|
syl5eq |
|- ( ( ph /\ Y = Z /\ ps ) -> I = ( ( d .\/ U ) ./\ ( c .\/ R ) ) ) |
71 |
58 70
|
oveq12d |
|- ( ( ph /\ Y = Z /\ ps ) -> ( ( G .\/ H ) .\/ I ) = ( ( ( ( d .\/ S ) ./\ ( c .\/ P ) ) .\/ ( ( d .\/ T ) ./\ ( c .\/ Q ) ) ) .\/ ( ( d .\/ U ) ./\ ( c .\/ R ) ) ) ) |
72 |
71 16
|
oveq12d |
|- ( ( ph /\ Y = Z /\ ps ) -> ( ( ( G .\/ H ) .\/ I ) ./\ Y ) = ( ( ( ( ( d .\/ S ) ./\ ( c .\/ P ) ) .\/ ( ( d .\/ T ) ./\ ( c .\/ Q ) ) ) .\/ ( ( d .\/ U ) ./\ ( c .\/ R ) ) ) ./\ Z ) ) |
73 |
13 72
|
syl5eq |
|- ( ( ph /\ Y = Z /\ ps ) -> B = ( ( ( ( ( d .\/ S ) ./\ ( c .\/ P ) ) .\/ ( ( d .\/ T ) ./\ ( c .\/ Q ) ) ) .\/ ( ( d .\/ U ) ./\ ( c .\/ R ) ) ) ./\ Z ) ) |
74 |
58 73
|
oveq12d |
|- ( ( ph /\ Y = Z /\ ps ) -> ( ( G .\/ H ) ./\ B ) = ( ( ( ( d .\/ S ) ./\ ( c .\/ P ) ) .\/ ( ( d .\/ T ) ./\ ( c .\/ Q ) ) ) ./\ ( ( ( ( ( d .\/ S ) ./\ ( c .\/ P ) ) .\/ ( ( d .\/ T ) ./\ ( c .\/ Q ) ) ) .\/ ( ( d .\/ U ) ./\ ( c .\/ R ) ) ) ./\ Z ) ) ) |
75 |
26 59 74
|
3eqtr4d |
|- ( ( ph /\ Y = Z /\ ps ) -> ( ( G .\/ H ) ./\ ( S .\/ T ) ) = ( ( G .\/ H ) ./\ B ) ) |