Metamath Proof Explorer

Theorem dalem62

Description: Lemma for dath . Eliminate the condition ps containing dummy variables c and d . (Contributed by NM, 11-Aug-2012)

Ref Expression
Hypotheses dalem62.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 ) ) ) ) )
dalem62.l 
|- .<_ = ( le ` K )
dalem62.j 
|- .\/ = ( join ` K )
dalem62.a 
|- A = ( Atoms ` K )
dalem62.m 
|- ./\ = ( meet ` K )
dalem62.o 
|- O = ( LPlanes ` K )
dalem62.y 
|- Y = ( ( P .\/ Q ) .\/ R )
dalem62.z 
|- Z = ( ( S .\/ T ) .\/ U )
dalem62.d 
|- D = ( ( P .\/ Q ) ./\ ( S .\/ T ) )
dalem62.e 
|- E = ( ( Q .\/ R ) ./\ ( T .\/ U ) )
dalem62.f 
|- F = ( ( R .\/ P ) ./\ ( U .\/ S ) )
Assertion dalem62 
|- ( ( ph /\ Y = Z ) -> F .<_ ( D .\/ E ) )

Proof

Step Hyp Ref Expression
1 dalem62.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 dalem62.l 
 |-  .<_ = ( le ` K )
3 dalem62.j 
 |-  .\/ = ( join ` K )
4 dalem62.a 
 |-  A = ( Atoms ` K )
5 dalem62.m 
 |-  ./\ = ( meet ` K )
6 dalem62.o 
 |-  O = ( LPlanes ` K )
7 dalem62.y 
 |-  Y = ( ( P .\/ Q ) .\/ R )
8 dalem62.z 
 |-  Z = ( ( S .\/ T ) .\/ U )
9 dalem62.d 
 |-  D = ( ( P .\/ Q ) ./\ ( S .\/ T ) )
10 dalem62.e 
 |-  E = ( ( Q .\/ R ) ./\ ( T .\/ U ) )
11 dalem62.f 
 |-  F = ( ( R .\/ P ) ./\ ( U .\/ S ) )
12 biid 
 |-  ( ( ( c e. A /\ d e. A ) /\ -. c .<_ Y /\ ( d =/= c /\ -. d .<_ Y /\ C .<_ ( c .\/ d ) ) ) <-> ( ( c e. A /\ d e. A ) /\ -. c .<_ Y /\ ( d =/= c /\ -. d .<_ Y /\ C .<_ ( c .\/ d ) ) ) )
13 1 2 3 4 12 6 7 8 dalem20 
 |-  ( ( ph /\ Y = Z ) -> E. c E. d ( ( c e. A /\ d e. A ) /\ -. c .<_ Y /\ ( d =/= c /\ -. d .<_ Y /\ C .<_ ( c .\/ d ) ) ) )
14 1 2 3 4 12 5 6 7 8 9 10 11 dalem61 
 |-  ( ( ph /\ Y = Z /\ ( ( c e. A /\ d e. A ) /\ -. c .<_ Y /\ ( d =/= c /\ -. d .<_ Y /\ C .<_ ( c .\/ d ) ) ) ) -> F .<_ ( D .\/ E ) )
15 14 3expia 
 |-  ( ( ph /\ Y = Z ) -> ( ( ( c e. A /\ d e. A ) /\ -. c .<_ Y /\ ( d =/= c /\ -. d .<_ Y /\ C .<_ ( c .\/ d ) ) ) -> F .<_ ( D .\/ E ) ) )
16 15 exlimdvv 
 |-  ( ( ph /\ Y = Z ) -> ( E. c E. d ( ( c e. A /\ d e. A ) /\ -. c .<_ Y /\ ( d =/= c /\ -. d .<_ Y /\ C .<_ ( c .\/ d ) ) ) -> F .<_ ( D .\/ E ) ) )
17 13 16 mpd 
 |-  ( ( ph /\ Y = Z ) -> F .<_ ( D .\/ E ) )