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Metamath Proof Explorer

Theorem dalem23

Description: Lemma for dath . Show that auxiliary atom G is 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 ) ) ) )
dalem23.m 
|- ./\ = ( meet ` K )
dalem23.o 
|- O = ( LPlanes ` K )
dalem23.y 
|- Y = ( ( P .\/ Q ) .\/ R )
dalem23.z 
|- Z = ( ( S .\/ T ) .\/ U )
dalem23.g 
|- G = ( ( c .\/ P ) ./\ ( d .\/ S ) )
Assertion dalem23 
|- ( ( ph /\ Y = Z /\ ps ) -> G 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 dalem23.m 
 |-  ./\ = ( meet ` K )
7 dalem23.o 
 |-  O = ( LPlanes ` K )
8 dalem23.y 
 |-  Y = ( ( P .\/ Q ) .\/ R )
9 dalem23.z 
 |-  Z = ( ( S .\/ T ) .\/ U )
10 dalem23.g 
 |-  G = ( ( c .\/ P ) ./\ ( d .\/ S ) )
11 1 dalemkehl 
 |-  ( ph -> K e. HL )
12 11 adantr 
 |-  ( ( ph /\ ps ) -> K e. HL )
13 5 dalemccea 
 |-  ( ps -> c e. A )
14 13 adantl 
 |-  ( ( ph /\ ps ) -> c e. A )
15 1 dalempea 
 |-  ( ph -> P e. A )
16 15 adantr 
 |-  ( ( ph /\ ps ) -> P e. A )
17 5 dalemddea 
 |-  ( ps -> d e. A )
18 17 adantl 
 |-  ( ( ph /\ ps ) -> d e. A )
19 1 dalemsea 
 |-  ( ph -> S e. A )
20 19 adantr 
 |-  ( ( ph /\ ps ) -> S e. A )
21 3 4 hlatj4 
 |-  ( ( K e. HL /\ ( c e. A /\ P e. A ) /\ ( d e. A /\ S e. A ) ) -> ( ( c .\/ P ) .\/ ( d .\/ S ) ) = ( ( c .\/ d ) .\/ ( P .\/ S ) ) )
22 12 14 16 18 20 21 syl122anc 
 |-  ( ( ph /\ ps ) -> ( ( c .\/ P ) .\/ ( d .\/ S ) ) = ( ( c .\/ d ) .\/ ( P .\/ S ) ) )
23 22 3adant2 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( c .\/ P ) .\/ ( d .\/ S ) ) = ( ( c .\/ d ) .\/ ( P .\/ S ) ) )
24 1 2 3 4 5 7 8 9 dalem22 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( c .\/ d ) .\/ ( P .\/ S ) ) e. O )
25 23 24 eqeltrd 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( c .\/ P ) .\/ ( d .\/ S ) ) e. O )
26 11 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> K e. HL )
27 1 2 3 4 7 8 dalemply 
 |-  ( ph -> P .<_ Y )
28 5 dalem-ccly 
 |-  ( ps -> -. c .<_ Y )
29 nbrne2 
 |-  ( ( P .<_ Y /\ -. c .<_ Y ) -> P =/= c )
30 27 28 29 syl2an 
 |-  ( ( ph /\ ps ) -> P =/= c )
31 30 necomd 
 |-  ( ( ph /\ ps ) -> c =/= P )
32 eqid 
 |-  ( LLines ` K ) = ( LLines ` K )
33 3 4 32 llni2 
 |-  ( ( ( K e. HL /\ c e. A /\ P e. A ) /\ c =/= P ) -> ( c .\/ P ) e. ( LLines ` K ) )
34 12 14 16 31 33 syl31anc 
 |-  ( ( ph /\ ps ) -> ( c .\/ P ) e. ( LLines ` K ) )
35 34 3adant2 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( c .\/ P ) e. ( LLines ` K ) )
36 17 3ad2ant3 
 |-  ( ( ph /\ Y = Z /\ ps ) -> d e. A )
37 19 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> S e. A )
38 1 2 3 4 9 dalemsly 
 |-  ( ( ph /\ Y = Z ) -> S .<_ Y )
39 38 3adant3 
 |-  ( ( ph /\ Y = Z /\ ps ) -> S .<_ Y )
40 5 dalem-ddly 
 |-  ( ps -> -. d .<_ Y )
41 40 3ad2ant3 
 |-  ( ( ph /\ Y = Z /\ ps ) -> -. d .<_ Y )
42 nbrne2 
 |-  ( ( S .<_ Y /\ -. d .<_ Y ) -> S =/= d )
43 39 41 42 syl2anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> S =/= d )
44 43 necomd 
 |-  ( ( ph /\ Y = Z /\ ps ) -> d =/= S )
45 3 4 32 llni2 
 |-  ( ( ( K e. HL /\ d e. A /\ S e. A ) /\ d =/= S ) -> ( d .\/ S ) e. ( LLines ` K ) )
46 26 36 37 44 45 syl31anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( d .\/ S ) e. ( LLines ` K ) )
47 3 6 4 32 7 2llnmj 
 |-  ( ( K e. HL /\ ( c .\/ P ) e. ( LLines ` K ) /\ ( d .\/ S ) e. ( LLines ` K ) ) -> ( ( ( c .\/ P ) ./\ ( d .\/ S ) ) e. A <-> ( ( c .\/ P ) .\/ ( d .\/ S ) ) e. O ) )
48 26 35 46 47 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( ( c .\/ P ) ./\ ( d .\/ S ) ) e. A <-> ( ( c .\/ P ) .\/ ( d .\/ S ) ) e. O ) )
49 25 48 mpbird 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( c .\/ P ) ./\ ( d .\/ S ) ) e. A )
50 10 49 eqeltrid 
 |-  ( ( ph /\ Y = Z /\ ps ) -> G e. A )