Metamath Proof Explorer

Theorem dalem1

Description: Lemma for dath . Show the lines P S and Q T are different. (Contributed by NM, 9-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 )
dalem1.o 
|- O = ( LPlanes ` K )
dalem1.y 
|- Y = ( ( P .\/ Q ) .\/ R )
Assertion dalem1 
|- ( ph -> ( P .\/ S ) =/= ( Q .\/ T ) )

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 dalem1.o 
 |-  O = ( LPlanes ` K )
6 dalem1.y 
 |-  Y = ( ( P .\/ Q ) .\/ R )
7 1 dalemclpjs 
 |-  ( ph -> C .<_ ( P .\/ S ) )
8 1 dalem-clpjq 
 |-  ( ph -> -. C .<_ ( P .\/ Q ) )
9 8 adantr 
 |-  ( ( ph /\ ( P .\/ S ) = ( Q .\/ T ) ) -> -. C .<_ ( P .\/ Q ) )
10 1 dalemkehl 
 |-  ( ph -> K e. HL )
11 1 dalempea 
 |-  ( ph -> P e. A )
12 1 dalemsea 
 |-  ( ph -> S e. A )
13 2 3 4 hlatlej1 
 |-  ( ( K e. HL /\ P e. A /\ S e. A ) -> P .<_ ( P .\/ S ) )
14 10 11 12 13 syl3anc 
 |-  ( ph -> P .<_ ( P .\/ S ) )
15 14 adantr 
 |-  ( ( ph /\ ( P .\/ S ) = ( Q .\/ T ) ) -> P .<_ ( P .\/ S ) )
16 1 dalemqea 
 |-  ( ph -> Q e. A )
17 1 dalemtea 
 |-  ( ph -> T e. A )
18 2 3 4 hlatlej1 
 |-  ( ( K e. HL /\ Q e. A /\ T e. A ) -> Q .<_ ( Q .\/ T ) )
19 10 16 17 18 syl3anc 
 |-  ( ph -> Q .<_ ( Q .\/ T ) )
20 19 adantr 
 |-  ( ( ph /\ ( P .\/ S ) = ( Q .\/ T ) ) -> Q .<_ ( Q .\/ T ) )
21 simpr 
 |-  ( ( ph /\ ( P .\/ S ) = ( Q .\/ T ) ) -> ( P .\/ S ) = ( Q .\/ T ) )
22 20 21 breqtrrd 
 |-  ( ( ph /\ ( P .\/ S ) = ( Q .\/ T ) ) -> Q .<_ ( P .\/ S ) )
23 1 dalemkelat 
 |-  ( ph -> K e. Lat )
24 1 4 dalempeb 
 |-  ( ph -> P e. ( Base ` K ) )
25 1 4 dalemqeb 
 |-  ( ph -> Q e. ( Base ` K ) )
26 eqid 
 |-  ( Base ` K ) = ( Base ` K )
27 26 3 4 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ S e. A ) -> ( P .\/ S ) e. ( Base ` K ) )
28 10 11 12 27 syl3anc 
 |-  ( ph -> ( P .\/ S ) e. ( Base ` K ) )
29 26 2 3 latjle12 
 |-  ( ( K e. Lat /\ ( P e. ( Base ` K ) /\ Q e. ( Base ` K ) /\ ( P .\/ S ) e. ( Base ` K ) ) ) -> ( ( P .<_ ( P .\/ S ) /\ Q .<_ ( P .\/ S ) ) <-> ( P .\/ Q ) .<_ ( P .\/ S ) ) )
30 23 24 25 28 29 syl13anc 
 |-  ( ph -> ( ( P .<_ ( P .\/ S ) /\ Q .<_ ( P .\/ S ) ) <-> ( P .\/ Q ) .<_ ( P .\/ S ) ) )
31 30 adantr 
 |-  ( ( ph /\ ( P .\/ S ) = ( Q .\/ T ) ) -> ( ( P .<_ ( P .\/ S ) /\ Q .<_ ( P .\/ S ) ) <-> ( P .\/ Q ) .<_ ( P .\/ S ) ) )
32 15 22 31 mpbi2and 
 |-  ( ( ph /\ ( P .\/ S ) = ( Q .\/ T ) ) -> ( P .\/ Q ) .<_ ( P .\/ S ) )
33 1 dalemrea 
 |-  ( ph -> R e. A )
34 1 dalemyeo 
 |-  ( ph -> Y e. O )
35 3 4 5 6 lplnri1 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ Y e. O ) -> P =/= Q )
36 10 11 16 33 34 35 syl131anc 
 |-  ( ph -> P =/= Q )
37 2 3 4 ps-1 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( P e. A /\ S e. A ) ) -> ( ( P .\/ Q ) .<_ ( P .\/ S ) <-> ( P .\/ Q ) = ( P .\/ S ) ) )
38 10 11 16 36 11 12 37 syl132anc 
 |-  ( ph -> ( ( P .\/ Q ) .<_ ( P .\/ S ) <-> ( P .\/ Q ) = ( P .\/ S ) ) )
39 38 adantr 
 |-  ( ( ph /\ ( P .\/ S ) = ( Q .\/ T ) ) -> ( ( P .\/ Q ) .<_ ( P .\/ S ) <-> ( P .\/ Q ) = ( P .\/ S ) ) )
40 32 39 mpbid 
 |-  ( ( ph /\ ( P .\/ S ) = ( Q .\/ T ) ) -> ( P .\/ Q ) = ( P .\/ S ) )
41 40 breq2d 
 |-  ( ( ph /\ ( P .\/ S ) = ( Q .\/ T ) ) -> ( C .<_ ( P .\/ Q ) <-> C .<_ ( P .\/ S ) ) )
42 9 41 mtbid 
 |-  ( ( ph /\ ( P .\/ S ) = ( Q .\/ T ) ) -> -. C .<_ ( P .\/ S ) )
43 42 ex 
 |-  ( ph -> ( ( P .\/ S ) = ( Q .\/ T ) -> -. C .<_ ( P .\/ S ) ) )
44 43 necon2ad 
 |-  ( ph -> ( C .<_ ( P .\/ S ) -> ( P .\/ S ) =/= ( Q .\/ T ) ) )
45 7 44 mpd 
 |-  ( ph -> ( P .\/ S ) =/= ( Q .\/ T ) )