Metamath Proof Explorer

Theorem dalem58

Description: Lemma for dath . Analogue of dalem57 for E . (Contributed by NM, 10-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 ) ) ) )
dalem58.m 
|- ./\ = ( meet ` K )
dalem58.o 
|- O = ( LPlanes ` K )
dalem58.y 
|- Y = ( ( P .\/ Q ) .\/ R )
dalem58.z 
|- Z = ( ( S .\/ T ) .\/ U )
dalem58.e 
|- E = ( ( Q .\/ R ) ./\ ( T .\/ U ) )
dalem58.g 
|- G = ( ( c .\/ P ) ./\ ( d .\/ S ) )
dalem58.h 
|- H = ( ( c .\/ Q ) ./\ ( d .\/ T ) )
dalem58.i 
|- I = ( ( c .\/ R ) ./\ ( d .\/ U ) )
dalem58.b1 
|- B = ( ( ( G .\/ H ) .\/ I ) ./\ Y )
Assertion dalem58 
|- ( ( ph /\ Y = Z /\ ps ) -> E .<_ 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 dalem58.m 
 |-  ./\ = ( meet ` K )
7 dalem58.o 
 |-  O = ( LPlanes ` K )
8 dalem58.y 
 |-  Y = ( ( P .\/ Q ) .\/ R )
9 dalem58.z 
 |-  Z = ( ( S .\/ T ) .\/ U )
10 dalem58.e 
 |-  E = ( ( Q .\/ R ) ./\ ( T .\/ U ) )
11 dalem58.g 
 |-  G = ( ( c .\/ P ) ./\ ( d .\/ S ) )
12 dalem58.h 
 |-  H = ( ( c .\/ Q ) ./\ ( d .\/ T ) )
13 dalem58.i 
 |-  I = ( ( c .\/ R ) ./\ ( d .\/ U ) )
14 dalem58.b1 
 |-  B = ( ( ( G .\/ H ) .\/ I ) ./\ Y )
15 1 2 3 4 8 9 dalemrot 
 |-  ( ph -> ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( Q e. A /\ R e. A /\ P e. A ) /\ ( T e. A /\ U e. A /\ S e. A ) ) /\ ( ( ( Q .\/ R ) .\/ P ) e. O /\ ( ( T .\/ U ) .\/ S ) e. O ) /\ ( ( -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) /\ -. C .<_ ( P .\/ Q ) ) /\ ( -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) /\ -. C .<_ ( S .\/ T ) ) /\ ( C .<_ ( Q .\/ T ) /\ C .<_ ( R .\/ U ) /\ C .<_ ( P .\/ S ) ) ) ) )
16 15 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( Q e. A /\ R e. A /\ P e. A ) /\ ( T e. A /\ U e. A /\ S e. A ) ) /\ ( ( ( Q .\/ R ) .\/ P ) e. O /\ ( ( T .\/ U ) .\/ S ) e. O ) /\ ( ( -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) /\ -. C .<_ ( P .\/ Q ) ) /\ ( -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) /\ -. C .<_ ( S .\/ T ) ) /\ ( C .<_ ( Q .\/ T ) /\ C .<_ ( R .\/ U ) /\ C .<_ ( P .\/ S ) ) ) ) )
17 1 2 3 4 8 9 dalemrotyz 
 |-  ( ( ph /\ Y = Z ) -> ( ( Q .\/ R ) .\/ P ) = ( ( T .\/ U ) .\/ S ) )
18 17 3adant3 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( Q .\/ R ) .\/ P ) = ( ( T .\/ U ) .\/ S ) )
19 1 2 3 4 5 8 dalemrotps 
 |-  ( ( ph /\ ps ) -> ( ( c e. A /\ d e. A ) /\ -. c .<_ ( ( Q .\/ R ) .\/ P ) /\ ( d =/= c /\ -. d .<_ ( ( Q .\/ R ) .\/ P ) /\ C .<_ ( c .\/ d ) ) ) )
20 19 3adant2 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( c e. A /\ d e. A ) /\ -. c .<_ ( ( Q .\/ R ) .\/ P ) /\ ( d =/= c /\ -. d .<_ ( ( Q .\/ R ) .\/ P ) /\ C .<_ ( c .\/ d ) ) ) )
21 biid 
 |-  ( ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( Q e. A /\ R e. A /\ P e. A ) /\ ( T e. A /\ U e. A /\ S e. A ) ) /\ ( ( ( Q .\/ R ) .\/ P ) e. O /\ ( ( T .\/ U ) .\/ S ) e. O ) /\ ( ( -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) /\ -. C .<_ ( P .\/ Q ) ) /\ ( -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) /\ -. C .<_ ( S .\/ T ) ) /\ ( C .<_ ( Q .\/ T ) /\ C .<_ ( R .\/ U ) /\ C .<_ ( P .\/ S ) ) ) ) <-> ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( Q e. A /\ R e. A /\ P e. A ) /\ ( T e. A /\ U e. A /\ S e. A ) ) /\ ( ( ( Q .\/ R ) .\/ P ) e. O /\ ( ( T .\/ U ) .\/ S ) e. O ) /\ ( ( -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) /\ -. C .<_ ( P .\/ Q ) ) /\ ( -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) /\ -. C .<_ ( S .\/ T ) ) /\ ( C .<_ ( Q .\/ T ) /\ C .<_ ( R .\/ U ) /\ C .<_ ( P .\/ S ) ) ) ) )
22 biid 
 |-  ( ( ( c e. A /\ d e. A ) /\ -. c .<_ ( ( Q .\/ R ) .\/ P ) /\ ( d =/= c /\ -. d .<_ ( ( Q .\/ R ) .\/ P ) /\ C .<_ ( c .\/ d ) ) ) <-> ( ( c e. A /\ d e. A ) /\ -. c .<_ ( ( Q .\/ R ) .\/ P ) /\ ( d =/= c /\ -. d .<_ ( ( Q .\/ R ) .\/ P ) /\ C .<_ ( c .\/ d ) ) ) )
23 eqid 
 |-  ( ( Q .\/ R ) .\/ P ) = ( ( Q .\/ R ) .\/ P )
24 eqid 
 |-  ( ( T .\/ U ) .\/ S ) = ( ( T .\/ U ) .\/ S )
25 eqid 
 |-  ( ( ( H .\/ I ) .\/ G ) ./\ ( ( Q .\/ R ) .\/ P ) ) = ( ( ( H .\/ I ) .\/ G ) ./\ ( ( Q .\/ R ) .\/ P ) )
26 21 2 3 4 22 6 7 23 24 10 12 13 11 25 dalem57 
 |-  ( ( ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( Q e. A /\ R e. A /\ P e. A ) /\ ( T e. A /\ U e. A /\ S e. A ) ) /\ ( ( ( Q .\/ R ) .\/ P ) e. O /\ ( ( T .\/ U ) .\/ S ) e. O ) /\ ( ( -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) /\ -. C .<_ ( P .\/ Q ) ) /\ ( -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) /\ -. C .<_ ( S .\/ T ) ) /\ ( C .<_ ( Q .\/ T ) /\ C .<_ ( R .\/ U ) /\ C .<_ ( P .\/ S ) ) ) ) /\ ( ( Q .\/ R ) .\/ P ) = ( ( T .\/ U ) .\/ S ) /\ ( ( c e. A /\ d e. A ) /\ -. c .<_ ( ( Q .\/ R ) .\/ P ) /\ ( d =/= c /\ -. d .<_ ( ( Q .\/ R ) .\/ P ) /\ C .<_ ( c .\/ d ) ) ) ) -> E .<_ ( ( ( H .\/ I ) .\/ G ) ./\ ( ( Q .\/ R ) .\/ P ) ) )
27 16 18 20 26 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> E .<_ ( ( ( H .\/ I ) .\/ G ) ./\ ( ( Q .\/ R ) .\/ P ) ) )
28 1 dalemkehl 
 |-  ( ph -> K e. HL )
29 28 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> K e. HL )
30 1 2 3 4 5 6 7 8 9 12 dalem29 
 |-  ( ( ph /\ Y = Z /\ ps ) -> H e. A )
31 1 2 3 4 5 6 7 8 9 13 dalem34 
 |-  ( ( ph /\ Y = Z /\ ps ) -> I e. A )
32 1 2 3 4 5 6 7 8 9 11 dalem23 
 |-  ( ( ph /\ Y = Z /\ ps ) -> G e. A )
33 3 4 hlatjrot 
 |-  ( ( K e. HL /\ ( H e. A /\ I e. A /\ G e. A ) ) -> ( ( H .\/ I ) .\/ G ) = ( ( G .\/ H ) .\/ I ) )
34 29 30 31 32 33 syl13anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( H .\/ I ) .\/ G ) = ( ( G .\/ H ) .\/ I ) )
35 1 3 4 dalemqrprot 
 |-  ( ph -> ( ( Q .\/ R ) .\/ P ) = ( ( P .\/ Q ) .\/ R ) )
36 35 8 eqtr4di 
 |-  ( ph -> ( ( Q .\/ R ) .\/ P ) = Y )
37 36 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( Q .\/ R ) .\/ P ) = Y )
38 34 37 oveq12d 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( ( H .\/ I ) .\/ G ) ./\ ( ( Q .\/ R ) .\/ P ) ) = ( ( ( G .\/ H ) .\/ I ) ./\ Y ) )
39 38 14 eqtr4di 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( ( H .\/ I ) .\/ G ) ./\ ( ( Q .\/ R ) .\/ P ) ) = B )
40 27 39 breqtrd 
 |-  ( ( ph /\ Y = Z /\ ps ) -> E .<_ B )