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Metamath Proof Explorer

Theorem dalem52

Description: Lemma for dath . Lines G H and P Q intersect at an atom. (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 ) ) ) )
dalem44.m 
|- ./\ = ( meet ` K )
dalem44.o 
|- O = ( LPlanes ` K )
dalem44.y 
|- Y = ( ( P .\/ Q ) .\/ R )
dalem44.z 
|- Z = ( ( S .\/ T ) .\/ U )
dalem44.g 
|- G = ( ( c .\/ P ) ./\ ( d .\/ S ) )
dalem44.h 
|- H = ( ( c .\/ Q ) ./\ ( d .\/ T ) )
dalem44.i 
|- I = ( ( c .\/ R ) ./\ ( d .\/ U ) )
Assertion dalem52 
|- ( ( ph /\ Y = Z /\ ps ) -> ( ( G .\/ H ) ./\ ( P .\/ Q ) ) 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 dalem44.m 
 |-  ./\ = ( meet ` K )
7 dalem44.o 
 |-  O = ( LPlanes ` K )
8 dalem44.y 
 |-  Y = ( ( P .\/ Q ) .\/ R )
9 dalem44.z 
 |-  Z = ( ( S .\/ T ) .\/ U )
10 dalem44.g 
 |-  G = ( ( c .\/ P ) ./\ ( d .\/ S ) )
11 dalem44.h 
 |-  H = ( ( c .\/ Q ) ./\ ( d .\/ T ) )
12 dalem44.i 
 |-  I = ( ( c .\/ R ) ./\ ( d .\/ U ) )
13 1 dalemkehl 
 |-  ( ph -> K e. HL )
14 13 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> K e. HL )
15 5 4 dalemcceb 
 |-  ( ps -> c e. ( Base ` K ) )
16 15 3ad2ant3 
 |-  ( ( ph /\ Y = Z /\ ps ) -> c e. ( Base ` K ) )
17 14 16 jca 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( K e. HL /\ c e. ( Base ` K ) ) )
18 1 2 3 4 5 6 7 8 9 10 dalem23 
 |-  ( ( ph /\ Y = Z /\ ps ) -> G e. A )
19 1 2 3 4 5 6 7 8 9 11 dalem29 
 |-  ( ( ph /\ Y = Z /\ ps ) -> H e. A )
20 1 2 3 4 5 6 7 8 9 12 dalem34 
 |-  ( ( ph /\ Y = Z /\ ps ) -> I e. A )
21 18 19 20 3jca 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( G e. A /\ H e. A /\ I e. A ) )
22 1 dalempea 
 |-  ( ph -> P e. A )
23 1 dalemqea 
 |-  ( ph -> Q e. A )
24 1 dalemrea 
 |-  ( ph -> R e. A )
25 22 23 24 3jca 
 |-  ( ph -> ( P e. A /\ Q e. A /\ R e. A ) )
26 25 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( P e. A /\ Q e. A /\ R e. A ) )
27 1 2 3 4 5 6 7 8 9 10 11 12 dalem42 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( G .\/ H ) .\/ I ) e. O )
28 1 dalemyeo 
 |-  ( ph -> Y e. O )
29 28 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> Y e. O )
30 1 2 3 4 5 6 7 8 9 10 11 12 dalem45 
 |-  ( ( ph /\ Y = Z /\ ps ) -> -. c .<_ ( G .\/ H ) )
31 1 2 3 4 5 6 7 8 9 10 11 12 dalem46 
 |-  ( ( ph /\ Y = Z /\ ps ) -> -. c .<_ ( H .\/ I ) )
32 1 2 3 4 5 6 7 8 9 10 11 12 dalem47 
 |-  ( ( ph /\ Y = Z /\ ps ) -> -. c .<_ ( I .\/ G ) )
33 30 31 32 3jca 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( -. c .<_ ( G .\/ H ) /\ -. c .<_ ( H .\/ I ) /\ -. c .<_ ( I .\/ G ) ) )
34 1 2 3 4 5 6 7 8 9 10 11 12 dalem48 
 |-  ( ( ph /\ ps ) -> -. c .<_ ( P .\/ Q ) )
35 1 2 3 4 5 6 7 8 9 10 11 12 dalem49 
 |-  ( ( ph /\ ps ) -> -. c .<_ ( Q .\/ R ) )
36 1 2 3 4 5 6 7 8 9 10 11 12 dalem50 
 |-  ( ( ph /\ ps ) -> -. c .<_ ( R .\/ P ) )
37 34 35 36 3jca 
 |-  ( ( ph /\ ps ) -> ( -. c .<_ ( P .\/ Q ) /\ -. c .<_ ( Q .\/ R ) /\ -. c .<_ ( R .\/ P ) ) )
38 37 3adant2 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( -. c .<_ ( P .\/ Q ) /\ -. c .<_ ( Q .\/ R ) /\ -. c .<_ ( R .\/ P ) ) )
39 1 2 3 4 5 6 7 8 9 10 dalem27 
 |-  ( ( ph /\ Y = Z /\ ps ) -> c .<_ ( G .\/ P ) )
40 1 2 3 4 5 6 7 8 9 11 dalem32 
 |-  ( ( ph /\ Y = Z /\ ps ) -> c .<_ ( H .\/ Q ) )
41 1 2 3 4 5 6 7 8 9 12 dalem36 
 |-  ( ( ph /\ Y = Z /\ ps ) -> c .<_ ( I .\/ R ) )
42 39 40 41 3jca 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( c .<_ ( G .\/ P ) /\ c .<_ ( H .\/ Q ) /\ c .<_ ( I .\/ R ) ) )
43 biid 
 |-  ( ( ( ( K e. HL /\ c e. ( Base ` K ) ) /\ ( G e. A /\ H e. A /\ I e. A ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) /\ ( ( ( G .\/ H ) .\/ I ) e. O /\ Y e. O ) /\ ( ( -. c .<_ ( G .\/ H ) /\ -. c .<_ ( H .\/ I ) /\ -. c .<_ ( I .\/ G ) ) /\ ( -. c .<_ ( P .\/ Q ) /\ -. c .<_ ( Q .\/ R ) /\ -. c .<_ ( R .\/ P ) ) /\ ( c .<_ ( G .\/ P ) /\ c .<_ ( H .\/ Q ) /\ c .<_ ( I .\/ R ) ) ) ) <-> ( ( ( K e. HL /\ c e. ( Base ` K ) ) /\ ( G e. A /\ H e. A /\ I e. A ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) /\ ( ( ( G .\/ H ) .\/ I ) e. O /\ Y e. O ) /\ ( ( -. c .<_ ( G .\/ H ) /\ -. c .<_ ( H .\/ I ) /\ -. c .<_ ( I .\/ G ) ) /\ ( -. c .<_ ( P .\/ Q ) /\ -. c .<_ ( Q .\/ R ) /\ -. c .<_ ( R .\/ P ) ) /\ ( c .<_ ( G .\/ P ) /\ c .<_ ( H .\/ Q ) /\ c .<_ ( I .\/ R ) ) ) ) )
44 eqid 
 |-  ( ( G .\/ H ) .\/ I ) = ( ( G .\/ H ) .\/ I )
45 eqid 
 |-  ( ( G .\/ H ) ./\ ( P .\/ Q ) ) = ( ( G .\/ H ) ./\ ( P .\/ Q ) )
46 43 2 3 4 6 7 44 8 45 dalemdea 
 |-  ( ( ( ( K e. HL /\ c e. ( Base ` K ) ) /\ ( G e. A /\ H e. A /\ I e. A ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) /\ ( ( ( G .\/ H ) .\/ I ) e. O /\ Y e. O ) /\ ( ( -. c .<_ ( G .\/ H ) /\ -. c .<_ ( H .\/ I ) /\ -. c .<_ ( I .\/ G ) ) /\ ( -. c .<_ ( P .\/ Q ) /\ -. c .<_ ( Q .\/ R ) /\ -. c .<_ ( R .\/ P ) ) /\ ( c .<_ ( G .\/ P ) /\ c .<_ ( H .\/ Q ) /\ c .<_ ( I .\/ R ) ) ) ) -> ( ( G .\/ H ) ./\ ( P .\/ Q ) ) e. A )
47 17 21 26 27 29 33 38 42 46 syl323anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( G .\/ H ) ./\ ( P .\/ Q ) ) e. A )