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Metamath Proof Explorer

Theorem dalem21

Description: Lemma for dath . Show that lines c d and P S intersect at an atom. (Contributed by NM, 2-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 ) ) ) )
dalem21.m 
|- ./\ = ( meet ` K )
dalem21.o 
|- O = ( LPlanes ` K )
dalem21.y 
|- Y = ( ( P .\/ Q ) .\/ R )
dalem21.z 
|- Z = ( ( S .\/ T ) .\/ U )
Assertion dalem21 
|- ( ( ph /\ Y = Z /\ ps ) -> ( ( c .\/ d ) ./\ ( P .\/ S ) ) e. A )

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 dalem21.m 
 |-  ./\ = ( meet ` K )
7 dalem21.o 
 |-  O = ( LPlanes ` K )
8 dalem21.y 
 |-  Y = ( ( P .\/ Q ) .\/ R )
9 dalem21.z 
 |-  Z = ( ( S .\/ T ) .\/ U )
10 1 dalemkehl 
 |-  ( ph -> K e. HL )
11 10 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> K e. HL )
12 1 2 3 4 5 dalemcjden 
 |-  ( ( ph /\ ps ) -> ( c .\/ d ) e. ( LLines ` K ) )
13 12 3adant2 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( c .\/ d ) e. ( LLines ` K ) )
14 1 2 3 4 7 8 dalempjsen 
 |-  ( ph -> ( P .\/ S ) e. ( LLines ` K ) )
15 14 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( P .\/ S ) e. ( LLines ` K ) )
16 1 2 3 4 7 8 dalemply 
 |-  ( ph -> P .<_ Y )
17 16 adantr 
 |-  ( ( ph /\ Y = Z ) -> P .<_ Y )
18 1 2 3 4 9 dalemsly 
 |-  ( ( ph /\ Y = Z ) -> S .<_ Y )
19 1 dalemkelat 
 |-  ( ph -> K e. Lat )
20 1 4 dalempeb 
 |-  ( ph -> P e. ( Base ` K ) )
21 1 4 dalemseb 
 |-  ( ph -> S e. ( Base ` K ) )
22 1 7 dalemyeb 
 |-  ( ph -> Y e. ( Base ` K ) )
23 eqid 
 |-  ( Base ` K ) = ( Base ` K )
24 23 2 3 latjle12 
 |-  ( ( K e. Lat /\ ( P e. ( Base ` K ) /\ S e. ( Base ` K ) /\ Y e. ( Base ` K ) ) ) -> ( ( P .<_ Y /\ S .<_ Y ) <-> ( P .\/ S ) .<_ Y ) )
25 19 20 21 22 24 syl13anc 
 |-  ( ph -> ( ( P .<_ Y /\ S .<_ Y ) <-> ( P .\/ S ) .<_ Y ) )
26 25 adantr 
 |-  ( ( ph /\ Y = Z ) -> ( ( P .<_ Y /\ S .<_ Y ) <-> ( P .\/ S ) .<_ Y ) )
27 17 18 26 mpbi2and 
 |-  ( ( ph /\ Y = Z ) -> ( P .\/ S ) .<_ Y )
28 27 3adant3 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( P .\/ S ) .<_ Y )
29 5 dalem-ccly 
 |-  ( ps -> -. c .<_ Y )
30 29 adantl 
 |-  ( ( ph /\ ps ) -> -. c .<_ Y )
31 19 adantr 
 |-  ( ( ph /\ ps ) -> K e. Lat )
32 5 4 dalemcceb 
 |-  ( ps -> c e. ( Base ` K ) )
33 32 adantl 
 |-  ( ( ph /\ ps ) -> c e. ( Base ` K ) )
34 5 dalemddea 
 |-  ( ps -> d e. A )
35 23 4 atbase 
 |-  ( d e. A -> d e. ( Base ` K ) )
36 34 35 syl 
 |-  ( ps -> d e. ( Base ` K ) )
37 36 adantl 
 |-  ( ( ph /\ ps ) -> d e. ( Base ` K ) )
38 23 2 3 latlej1 
 |-  ( ( K e. Lat /\ c e. ( Base ` K ) /\ d e. ( Base ` K ) ) -> c .<_ ( c .\/ d ) )
39 31 33 37 38 syl3anc 
 |-  ( ( ph /\ ps ) -> c .<_ ( c .\/ d ) )
40 eqid 
 |-  ( LLines ` K ) = ( LLines ` K )
41 23 40 llnbase 
 |-  ( ( c .\/ d ) e. ( LLines ` K ) -> ( c .\/ d ) e. ( Base ` K ) )
42 12 41 syl 
 |-  ( ( ph /\ ps ) -> ( c .\/ d ) e. ( Base ` K ) )
43 22 adantr 
 |-  ( ( ph /\ ps ) -> Y e. ( Base ` K ) )
44 23 2 lattr 
 |-  ( ( K e. Lat /\ ( c e. ( Base ` K ) /\ ( c .\/ d ) e. ( Base ` K ) /\ Y e. ( Base ` K ) ) ) -> ( ( c .<_ ( c .\/ d ) /\ ( c .\/ d ) .<_ Y ) -> c .<_ Y ) )
45 31 33 42 43 44 syl13anc 
 |-  ( ( ph /\ ps ) -> ( ( c .<_ ( c .\/ d ) /\ ( c .\/ d ) .<_ Y ) -> c .<_ Y ) )
46 39 45 mpand 
 |-  ( ( ph /\ ps ) -> ( ( c .\/ d ) .<_ Y -> c .<_ Y ) )
47 30 46 mtod 
 |-  ( ( ph /\ ps ) -> -. ( c .\/ d ) .<_ Y )
48 47 3adant2 
 |-  ( ( ph /\ Y = Z /\ ps ) -> -. ( c .\/ d ) .<_ Y )
49 nbrne2 
 |-  ( ( ( P .\/ S ) .<_ Y /\ -. ( c .\/ d ) .<_ Y ) -> ( P .\/ S ) =/= ( c .\/ d ) )
50 28 48 49 syl2anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( P .\/ S ) =/= ( c .\/ d ) )
51 50 necomd 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( c .\/ d ) =/= ( P .\/ S ) )
52 hlatl 
 |-  ( K e. HL -> K e. AtLat )
53 10 52 syl 
 |-  ( ph -> K e. AtLat )
54 53 adantr 
 |-  ( ( ph /\ ps ) -> K e. AtLat )
55 1 dalempea 
 |-  ( ph -> P e. A )
56 1 dalemsea 
 |-  ( ph -> S e. A )
57 23 3 4 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ S e. A ) -> ( P .\/ S ) e. ( Base ` K ) )
58 10 55 56 57 syl3anc 
 |-  ( ph -> ( P .\/ S ) e. ( Base ` K ) )
59 58 adantr 
 |-  ( ( ph /\ ps ) -> ( P .\/ S ) e. ( Base ` K ) )
60 23 6 latmcl 
 |-  ( ( K e. Lat /\ ( c .\/ d ) e. ( Base ` K ) /\ ( P .\/ S ) e. ( Base ` K ) ) -> ( ( c .\/ d ) ./\ ( P .\/ S ) ) e. ( Base ` K ) )
61 31 42 59 60 syl3anc 
 |-  ( ( ph /\ ps ) -> ( ( c .\/ d ) ./\ ( P .\/ S ) ) e. ( Base ` K ) )
62 1 2 3 4 7 8 dalemcea 
 |-  ( ph -> C e. A )
63 62 adantr 
 |-  ( ( ph /\ ps ) -> C e. A )
64 5 dalemclccjdd 
 |-  ( ps -> C .<_ ( c .\/ d ) )
65 64 adantl 
 |-  ( ( ph /\ ps ) -> C .<_ ( c .\/ d ) )
66 1 dalemclpjs 
 |-  ( ph -> C .<_ ( P .\/ S ) )
67 66 adantr 
 |-  ( ( ph /\ ps ) -> C .<_ ( P .\/ S ) )
68 1 4 dalemceb 
 |-  ( ph -> C e. ( Base ` K ) )
69 68 adantr 
 |-  ( ( ph /\ ps ) -> C e. ( Base ` K ) )
70 23 2 6 latlem12 
 |-  ( ( K e. Lat /\ ( C e. ( Base ` K ) /\ ( c .\/ d ) e. ( Base ` K ) /\ ( P .\/ S ) e. ( Base ` K ) ) ) -> ( ( C .<_ ( c .\/ d ) /\ C .<_ ( P .\/ S ) ) <-> C .<_ ( ( c .\/ d ) ./\ ( P .\/ S ) ) ) )
71 31 69 42 59 70 syl13anc 
 |-  ( ( ph /\ ps ) -> ( ( C .<_ ( c .\/ d ) /\ C .<_ ( P .\/ S ) ) <-> C .<_ ( ( c .\/ d ) ./\ ( P .\/ S ) ) ) )
72 65 67 71 mpbi2and 
 |-  ( ( ph /\ ps ) -> C .<_ ( ( c .\/ d ) ./\ ( P .\/ S ) ) )
73 eqid 
 |-  ( 0. ` K ) = ( 0. ` K )
74 23 2 73 4 atlen0 
 |-  ( ( ( K e. AtLat /\ ( ( c .\/ d ) ./\ ( P .\/ S ) ) e. ( Base ` K ) /\ C e. A ) /\ C .<_ ( ( c .\/ d ) ./\ ( P .\/ S ) ) ) -> ( ( c .\/ d ) ./\ ( P .\/ S ) ) =/= ( 0. ` K ) )
75 54 61 63 72 74 syl31anc 
 |-  ( ( ph /\ ps ) -> ( ( c .\/ d ) ./\ ( P .\/ S ) ) =/= ( 0. ` K ) )
76 75 3adant2 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( c .\/ d ) ./\ ( P .\/ S ) ) =/= ( 0. ` K ) )
77 6 73 4 40 2llnmat 
 |-  ( ( ( K e. HL /\ ( c .\/ d ) e. ( LLines ` K ) /\ ( P .\/ S ) e. ( LLines ` K ) ) /\ ( ( c .\/ d ) =/= ( P .\/ S ) /\ ( ( c .\/ d ) ./\ ( P .\/ S ) ) =/= ( 0. ` K ) ) ) -> ( ( c .\/ d ) ./\ ( P .\/ S ) ) e. A )
78 11 13 15 51 76 77 syl32anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( c .\/ d ) ./\ ( P .\/ S ) ) e. A )