Metamath Proof Explorer


Theorem dalem59

Description: Lemma for dath . Analogue of dalem57 for F . (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 ) ) ) )
dalem59.m
|- ./\ = ( meet ` K )
dalem59.o
|- O = ( LPlanes ` K )
dalem59.y
|- Y = ( ( P .\/ Q ) .\/ R )
dalem59.z
|- Z = ( ( S .\/ T ) .\/ U )
dalem59.f
|- F = ( ( R .\/ P ) ./\ ( U .\/ S ) )
dalem59.g
|- G = ( ( c .\/ P ) ./\ ( d .\/ S ) )
dalem59.h
|- H = ( ( c .\/ Q ) ./\ ( d .\/ T ) )
dalem59.i
|- I = ( ( c .\/ R ) ./\ ( d .\/ U ) )
dalem59.b1
|- B = ( ( ( G .\/ H ) .\/ I ) ./\ Y )
Assertion dalem59
|- ( ( ph /\ Y = Z /\ ps ) -> F .<_ 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 dalem59.m
 |-  ./\ = ( meet ` K )
7 dalem59.o
 |-  O = ( LPlanes ` K )
8 dalem59.y
 |-  Y = ( ( P .\/ Q ) .\/ R )
9 dalem59.z
 |-  Z = ( ( S .\/ T ) .\/ U )
10 dalem59.f
 |-  F = ( ( R .\/ P ) ./\ ( U .\/ S ) )
11 dalem59.g
 |-  G = ( ( c .\/ P ) ./\ ( d .\/ S ) )
12 dalem59.h
 |-  H = ( ( c .\/ Q ) ./\ ( d .\/ T ) )
13 dalem59.i
 |-  I = ( ( c .\/ R ) ./\ ( d .\/ U ) )
14 dalem59.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 dalem58
 |-  ( ( ( ( ( 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 ) ) ) ) -> F .<_ ( ( ( H .\/ I ) .\/ G ) ./\ ( ( Q .\/ R ) .\/ P ) ) )
27 16 18 20 26 syl3anc
 |-  ( ( ph /\ Y = Z /\ ps ) -> F .<_ ( ( ( 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 ) -> F .<_ B )