Metamath Proof Explorer

Theorem dalem4

Description: Lemma for dalemdnee . (Contributed by NM, 10-Aug-2012)

Ref Expression
Hypotheses dalema.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 ) ) ) ) )
dalemc.l 
|- .<_ = ( le ` K )
dalemc.j 
|- .\/ = ( join ` K )
dalemc.a 
|- A = ( Atoms ` K )
dalem3.m 
|- ./\ = ( meet ` K )
dalem3.o 
|- O = ( LPlanes ` K )
dalem3.y 
|- Y = ( ( P .\/ Q ) .\/ R )
dalem3.z 
|- Z = ( ( S .\/ T ) .\/ U )
dalem3.d 
|- D = ( ( P .\/ Q ) ./\ ( S .\/ T ) )
dalem3.e 
|- E = ( ( Q .\/ R ) ./\ ( T .\/ U ) )
Assertion dalem4 
|- ( ( ph /\ D =/= T ) -> D =/= E )

Proof

Step Hyp Ref Expression
1 dalema.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 dalemc.l 
 |-  .<_ = ( le ` K )
3 dalemc.j 
 |-  .\/ = ( join ` K )
4 dalemc.a 
 |-  A = ( Atoms ` K )
5 dalem3.m 
 |-  ./\ = ( meet ` K )
6 dalem3.o 
 |-  O = ( LPlanes ` K )
7 dalem3.y 
 |-  Y = ( ( P .\/ Q ) .\/ R )
8 dalem3.z 
 |-  Z = ( ( S .\/ T ) .\/ U )
9 dalem3.d 
 |-  D = ( ( P .\/ Q ) ./\ ( S .\/ T ) )
10 dalem3.e 
 |-  E = ( ( Q .\/ R ) ./\ ( T .\/ U ) )
11 1 2 3 4 dalemswapyz 
 |-  ( ph -> ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( S e. A /\ T e. A /\ U e. A ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) /\ ( Z e. O /\ Y e. O ) /\ ( ( -. C .<_ ( S .\/ T ) /\ -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) ) /\ ( -. C .<_ ( P .\/ Q ) /\ -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) ) /\ ( C .<_ ( S .\/ P ) /\ C .<_ ( T .\/ Q ) /\ C .<_ ( U .\/ R ) ) ) ) )
12 11 adantr 
 |-  ( ( ph /\ D =/= T ) -> ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( S e. A /\ T e. A /\ U e. A ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) /\ ( Z e. O /\ Y e. O ) /\ ( ( -. C .<_ ( S .\/ T ) /\ -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) ) /\ ( -. C .<_ ( P .\/ Q ) /\ -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) ) /\ ( C .<_ ( S .\/ P ) /\ C .<_ ( T .\/ Q ) /\ C .<_ ( U .\/ R ) ) ) ) )
13 1 dalemkelat 
 |-  ( ph -> K e. Lat )
14 1 3 4 dalempjqeb 
 |-  ( ph -> ( P .\/ Q ) e. ( Base ` K ) )
15 1 3 4 dalemsjteb 
 |-  ( ph -> ( S .\/ T ) e. ( Base ` K ) )
16 eqid 
 |-  ( Base ` K ) = ( Base ` K )
17 16 5 latmcom 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ ( S .\/ T ) e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ ( S .\/ T ) ) = ( ( S .\/ T ) ./\ ( P .\/ Q ) ) )
18 13 14 15 17 syl3anc 
 |-  ( ph -> ( ( P .\/ Q ) ./\ ( S .\/ T ) ) = ( ( S .\/ T ) ./\ ( P .\/ Q ) ) )
19 9 18 syl5eq 
 |-  ( ph -> D = ( ( S .\/ T ) ./\ ( P .\/ Q ) ) )
20 19 neeq1d 
 |-  ( ph -> ( D =/= T <-> ( ( S .\/ T ) ./\ ( P .\/ Q ) ) =/= T ) )
21 20 biimpa 
 |-  ( ( ph /\ D =/= T ) -> ( ( S .\/ T ) ./\ ( P .\/ Q ) ) =/= T )
22 biid 
 |-  ( ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( S e. A /\ T e. A /\ U e. A ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) /\ ( Z e. O /\ Y e. O ) /\ ( ( -. C .<_ ( S .\/ T ) /\ -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) ) /\ ( -. C .<_ ( P .\/ Q ) /\ -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) ) /\ ( C .<_ ( S .\/ P ) /\ C .<_ ( T .\/ Q ) /\ C .<_ ( U .\/ R ) ) ) ) <-> ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( S e. A /\ T e. A /\ U e. A ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) /\ ( Z e. O /\ Y e. O ) /\ ( ( -. C .<_ ( S .\/ T ) /\ -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) ) /\ ( -. C .<_ ( P .\/ Q ) /\ -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) ) /\ ( C .<_ ( S .\/ P ) /\ C .<_ ( T .\/ Q ) /\ C .<_ ( U .\/ R ) ) ) ) )
23 eqid 
 |-  ( ( S .\/ T ) ./\ ( P .\/ Q ) ) = ( ( S .\/ T ) ./\ ( P .\/ Q ) )
24 eqid 
 |-  ( ( T .\/ U ) ./\ ( Q .\/ R ) ) = ( ( T .\/ U ) ./\ ( Q .\/ R ) )
25 22 2 3 4 5 6 8 7 23 24 dalem3 
 |-  ( ( ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( S e. A /\ T e. A /\ U e. A ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) /\ ( Z e. O /\ Y e. O ) /\ ( ( -. C .<_ ( S .\/ T ) /\ -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) ) /\ ( -. C .<_ ( P .\/ Q ) /\ -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) ) /\ ( C .<_ ( S .\/ P ) /\ C .<_ ( T .\/ Q ) /\ C .<_ ( U .\/ R ) ) ) ) /\ ( ( S .\/ T ) ./\ ( P .\/ Q ) ) =/= T ) -> ( ( S .\/ T ) ./\ ( P .\/ Q ) ) =/= ( ( T .\/ U ) ./\ ( Q .\/ R ) ) )
26 12 21 25 syl2anc 
 |-  ( ( ph /\ D =/= T ) -> ( ( S .\/ T ) ./\ ( P .\/ Q ) ) =/= ( ( T .\/ U ) ./\ ( Q .\/ R ) ) )
27 19 adantr 
 |-  ( ( ph /\ D =/= T ) -> D = ( ( S .\/ T ) ./\ ( P .\/ Q ) ) )
28 1 dalemkehl 
 |-  ( ph -> K e. HL )
29 1 dalemqea 
 |-  ( ph -> Q e. A )
30 1 dalemrea 
 |-  ( ph -> R e. A )
31 16 3 4 hlatjcl 
 |-  ( ( K e. HL /\ Q e. A /\ R e. A ) -> ( Q .\/ R ) e. ( Base ` K ) )
32 28 29 30 31 syl3anc 
 |-  ( ph -> ( Q .\/ R ) e. ( Base ` K ) )
33 1 3 4 dalemtjueb 
 |-  ( ph -> ( T .\/ U ) e. ( Base ` K ) )
34 16 5 latmcom 
 |-  ( ( K e. Lat /\ ( Q .\/ R ) e. ( Base ` K ) /\ ( T .\/ U ) e. ( Base ` K ) ) -> ( ( Q .\/ R ) ./\ ( T .\/ U ) ) = ( ( T .\/ U ) ./\ ( Q .\/ R ) ) )
35 13 32 33 34 syl3anc 
 |-  ( ph -> ( ( Q .\/ R ) ./\ ( T .\/ U ) ) = ( ( T .\/ U ) ./\ ( Q .\/ R ) ) )
36 10 35 syl5eq 
 |-  ( ph -> E = ( ( T .\/ U ) ./\ ( Q .\/ R ) ) )
37 36 adantr 
 |-  ( ( ph /\ D =/= T ) -> E = ( ( T .\/ U ) ./\ ( Q .\/ R ) ) )
38 26 27 37 3netr4d 
 |-  ( ( ph /\ D =/= T ) -> D =/= E )