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Metamath Proof Explorer

Theorem dalem3

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 dalem3 
|- ( ( ph /\ D =/= Q ) -> 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 dalemkehl 
 |-  ( ph -> K e. HL )
12 1 dalempea 
 |-  ( ph -> P e. A )
13 1 dalemqea 
 |-  ( ph -> Q e. A )
14 1 dalemrea 
 |-  ( ph -> R e. A )
15 1 dalemyeo 
 |-  ( ph -> Y e. O )
16 2 3 4 6 7 lplnric 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ Y e. O ) -> -. R .<_ ( P .\/ Q ) )
17 11 12 13 14 15 16 syl131anc 
 |-  ( ph -> -. R .<_ ( P .\/ Q ) )
18 17 adantr 
 |-  ( ( ph /\ D =/= Q ) -> -. R .<_ ( P .\/ Q ) )
19 1 dalemkelat 
 |-  ( ph -> K e. Lat )
20 eqid 
 |-  ( Base ` K ) = ( Base ` K )
21 20 3 4 hlatjcl 
 |-  ( ( K e. HL /\ Q e. A /\ R e. A ) -> ( Q .\/ R ) e. ( Base ` K ) )
22 11 13 14 21 syl3anc 
 |-  ( ph -> ( Q .\/ R ) e. ( Base ` K ) )
23 1 3 4 dalemtjueb 
 |-  ( ph -> ( T .\/ U ) e. ( Base ` K ) )
24 20 2 5 latmle1 
 |-  ( ( K e. Lat /\ ( Q .\/ R ) e. ( Base ` K ) /\ ( T .\/ U ) e. ( Base ` K ) ) -> ( ( Q .\/ R ) ./\ ( T .\/ U ) ) .<_ ( Q .\/ R ) )
25 19 22 23 24 syl3anc 
 |-  ( ph -> ( ( Q .\/ R ) ./\ ( T .\/ U ) ) .<_ ( Q .\/ R ) )
26 10 25 eqbrtrid 
 |-  ( ph -> E .<_ ( Q .\/ R ) )
27 breq1 
 |-  ( D = E -> ( D .<_ ( Q .\/ R ) <-> E .<_ ( Q .\/ R ) ) )
28 26 27 syl5ibrcom 
 |-  ( ph -> ( D = E -> D .<_ ( Q .\/ R ) ) )
29 28 adantr 
 |-  ( ( ph /\ D =/= Q ) -> ( D = E -> D .<_ ( Q .\/ R ) ) )
30 11 adantr 
 |-  ( ( ph /\ D =/= Q ) -> K e. HL )
31 1 2 3 4 5 6 7 8 9 dalemdea 
 |-  ( ph -> D e. A )
32 31 adantr 
 |-  ( ( ph /\ D =/= Q ) -> D e. A )
33 14 adantr 
 |-  ( ( ph /\ D =/= Q ) -> R e. A )
34 13 adantr 
 |-  ( ( ph /\ D =/= Q ) -> Q e. A )
35 simpr 
 |-  ( ( ph /\ D =/= Q ) -> D =/= Q )
36 2 3 4 hlatexch1 
 |-  ( ( K e. HL /\ ( D e. A /\ R e. A /\ Q e. A ) /\ D =/= Q ) -> ( D .<_ ( Q .\/ R ) -> R .<_ ( Q .\/ D ) ) )
37 30 32 33 34 35 36 syl131anc 
 |-  ( ( ph /\ D =/= Q ) -> ( D .<_ ( Q .\/ R ) -> R .<_ ( Q .\/ D ) ) )
38 2 3 4 hlatlej2 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> Q .<_ ( P .\/ Q ) )
39 11 12 13 38 syl3anc 
 |-  ( ph -> Q .<_ ( P .\/ Q ) )
40 1 3 4 dalempjqeb 
 |-  ( ph -> ( P .\/ Q ) e. ( Base ` K ) )
41 1 3 4 dalemsjteb 
 |-  ( ph -> ( S .\/ T ) e. ( Base ` K ) )
42 20 2 5 latmle1 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ ( S .\/ T ) e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ ( S .\/ T ) ) .<_ ( P .\/ Q ) )
43 19 40 41 42 syl3anc 
 |-  ( ph -> ( ( P .\/ Q ) ./\ ( S .\/ T ) ) .<_ ( P .\/ Q ) )
44 9 43 eqbrtrid 
 |-  ( ph -> D .<_ ( P .\/ Q ) )
45 1 4 dalemqeb 
 |-  ( ph -> Q e. ( Base ` K ) )
46 20 4 atbase 
 |-  ( D e. A -> D e. ( Base ` K ) )
47 31 46 syl 
 |-  ( ph -> D e. ( Base ` K ) )
48 20 2 3 latjle12 
 |-  ( ( K e. Lat /\ ( Q e. ( Base ` K ) /\ D e. ( Base ` K ) /\ ( P .\/ Q ) e. ( Base ` K ) ) ) -> ( ( Q .<_ ( P .\/ Q ) /\ D .<_ ( P .\/ Q ) ) <-> ( Q .\/ D ) .<_ ( P .\/ Q ) ) )
49 19 45 47 40 48 syl13anc 
 |-  ( ph -> ( ( Q .<_ ( P .\/ Q ) /\ D .<_ ( P .\/ Q ) ) <-> ( Q .\/ D ) .<_ ( P .\/ Q ) ) )
50 39 44 49 mpbi2and 
 |-  ( ph -> ( Q .\/ D ) .<_ ( P .\/ Q ) )
51 1 4 dalemreb 
 |-  ( ph -> R e. ( Base ` K ) )
52 20 3 4 hlatjcl 
 |-  ( ( K e. HL /\ Q e. A /\ D e. A ) -> ( Q .\/ D ) e. ( Base ` K ) )
53 11 13 31 52 syl3anc 
 |-  ( ph -> ( Q .\/ D ) e. ( Base ` K ) )
54 20 2 lattr 
 |-  ( ( K e. Lat /\ ( R e. ( Base ` K ) /\ ( Q .\/ D ) e. ( Base ` K ) /\ ( P .\/ Q ) e. ( Base ` K ) ) ) -> ( ( R .<_ ( Q .\/ D ) /\ ( Q .\/ D ) .<_ ( P .\/ Q ) ) -> R .<_ ( P .\/ Q ) ) )
55 19 51 53 40 54 syl13anc 
 |-  ( ph -> ( ( R .<_ ( Q .\/ D ) /\ ( Q .\/ D ) .<_ ( P .\/ Q ) ) -> R .<_ ( P .\/ Q ) ) )
56 50 55 mpan2d 
 |-  ( ph -> ( R .<_ ( Q .\/ D ) -> R .<_ ( P .\/ Q ) ) )
57 56 adantr 
 |-  ( ( ph /\ D =/= Q ) -> ( R .<_ ( Q .\/ D ) -> R .<_ ( P .\/ Q ) ) )
58 29 37 57 3syld 
 |-  ( ( ph /\ D =/= Q ) -> ( D = E -> R .<_ ( P .\/ Q ) ) )
59 58 necon3bd 
 |-  ( ( ph /\ D =/= Q ) -> ( -. R .<_ ( P .\/ Q ) -> D =/= E ) )
60 18 59 mpd 
 |-  ( ( ph /\ D =/= Q ) -> D =/= E )