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Metamath Proof Explorer

Theorem dalem56

Description: Lemma for dath . Analogue of dalem55 for line S T . (Contributed by NM, 8-Aug-2012)

Ref Expression
Hypotheses 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 ) ) ) ) )
dalem.l 
|- .<_ = ( le ` K )
dalem.j 
|- .\/ = ( join ` K )
dalem.a 
|- A = ( Atoms ` K )
dalem.ps 
|- ( ps <-> ( ( c e. A /\ d e. A ) /\ -. c .<_ Y /\ ( d =/= c /\ -. d .<_ Y /\ C .<_ ( c .\/ d ) ) ) )
dalem54.m 
|- ./\ = ( meet ` K )
dalem54.o 
|- O = ( LPlanes ` K )
dalem54.y 
|- Y = ( ( P .\/ Q ) .\/ R )
dalem54.z 
|- Z = ( ( S .\/ T ) .\/ U )
dalem54.g 
|- G = ( ( c .\/ P ) ./\ ( d .\/ S ) )
dalem54.h 
|- H = ( ( c .\/ Q ) ./\ ( d .\/ T ) )
dalem54.i 
|- I = ( ( c .\/ R ) ./\ ( d .\/ U ) )
dalem54.b1 
|- B = ( ( ( G .\/ H ) .\/ I ) ./\ Y )
Assertion dalem56 
|- ( ( ph /\ Y = Z /\ ps ) -> ( ( G .\/ H ) ./\ ( S .\/ T ) ) = ( ( G .\/ H ) ./\ B ) )

Proof

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 eqtrid 
 |-  ( ( 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 eqtrid 
 |-  ( ( 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 eqtrid 
 |-  ( ( 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 eqtrid 
 |-  ( ( 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 ) )