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Metamath Proof Explorer

Theorem ps-2b

Description: Variation of projective geometry axiom ps-2 . (Contributed by NM, 3-Jul-2012)

Ref Expression
Hypotheses ps-2b.l 
|- .<_ = ( le ` K )
ps-2b.j 
|- .\/ = ( join ` K )
ps-2b.m 
|- ./\ = ( meet ` K )
ps-2b.z 
|- .0. = ( 0. ` K )
ps-2b.a 
|- A = ( Atoms ` K )
Assertion ps-2b 
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( -. P .<_ ( Q .\/ R ) /\ S =/= T /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) -> ( ( P .\/ R ) ./\ ( S .\/ T ) ) =/= .0. )

Proof

Step Hyp Ref Expression
1 ps-2b.l 
 |-  .<_ = ( le ` K )
2 ps-2b.j 
 |-  .\/ = ( join ` K )
3 ps-2b.m 
 |-  ./\ = ( meet ` K )
4 ps-2b.z 
 |-  .0. = ( 0. ` K )
5 ps-2b.a 
 |-  A = ( Atoms ` K )
6 simp11 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( -. P .<_ ( Q .\/ R ) /\ S =/= T /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) -> K e. HL )
7 simp12 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( -. P .<_ ( Q .\/ R ) /\ S =/= T /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) -> P e. A )
8 simp13 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( -. P .<_ ( Q .\/ R ) /\ S =/= T /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) -> Q e. A )
9 simp21 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( -. P .<_ ( Q .\/ R ) /\ S =/= T /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) -> R e. A )
10 7 8 9 3jca 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( -. P .<_ ( Q .\/ R ) /\ S =/= T /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) -> ( P e. A /\ Q e. A /\ R e. A ) )
11 simp22 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( -. P .<_ ( Q .\/ R ) /\ S =/= T /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) -> S e. A )
12 simp23 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( -. P .<_ ( Q .\/ R ) /\ S =/= T /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) -> T e. A )
13 11 12 jca 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( -. P .<_ ( Q .\/ R ) /\ S =/= T /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) -> ( S e. A /\ T e. A ) )
14 simp31 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( -. P .<_ ( Q .\/ R ) /\ S =/= T /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) -> -. P .<_ ( Q .\/ R ) )
15 simp32 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( -. P .<_ ( Q .\/ R ) /\ S =/= T /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) -> S =/= T )
16 14 15 jca 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( -. P .<_ ( Q .\/ R ) /\ S =/= T /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) -> ( -. P .<_ ( Q .\/ R ) /\ S =/= T ) )
17 simp33 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( -. P .<_ ( Q .\/ R ) /\ S =/= T /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) -> ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) )
18 1 2 5 ps-2 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A ) ) /\ ( ( -. P .<_ ( Q .\/ R ) /\ S =/= T ) /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) -> E. u e. A ( u .<_ ( P .\/ R ) /\ u .<_ ( S .\/ T ) ) )
19 6 10 13 16 17 18 syl32anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( -. P .<_ ( Q .\/ R ) /\ S =/= T /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) -> E. u e. A ( u .<_ ( P .\/ R ) /\ u .<_ ( S .\/ T ) ) )
20 simp111 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( -. P .<_ ( Q .\/ R ) /\ S =/= T /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) /\ u e. A /\ ( u .<_ ( P .\/ R ) /\ u .<_ ( S .\/ T ) ) ) -> K e. HL )
21 hlatl 
 |-  ( K e. HL -> K e. AtLat )
22 20 21 syl 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( -. P .<_ ( Q .\/ R ) /\ S =/= T /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) /\ u e. A /\ ( u .<_ ( P .\/ R ) /\ u .<_ ( S .\/ T ) ) ) -> K e. AtLat )
23 20 hllatd 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( -. P .<_ ( Q .\/ R ) /\ S =/= T /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) /\ u e. A /\ ( u .<_ ( P .\/ R ) /\ u .<_ ( S .\/ T ) ) ) -> K e. Lat )
24 simp112 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( -. P .<_ ( Q .\/ R ) /\ S =/= T /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) /\ u e. A /\ ( u .<_ ( P .\/ R ) /\ u .<_ ( S .\/ T ) ) ) -> P e. A )
25 simp121 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( -. P .<_ ( Q .\/ R ) /\ S =/= T /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) /\ u e. A /\ ( u .<_ ( P .\/ R ) /\ u .<_ ( S .\/ T ) ) ) -> R e. A )
26 eqid 
 |-  ( Base ` K ) = ( Base ` K )
27 26 2 5 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ R e. A ) -> ( P .\/ R ) e. ( Base ` K ) )
28 20 24 25 27 syl3anc 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( -. P .<_ ( Q .\/ R ) /\ S =/= T /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) /\ u e. A /\ ( u .<_ ( P .\/ R ) /\ u .<_ ( S .\/ T ) ) ) -> ( P .\/ R ) e. ( Base ` K ) )
29 simp122 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( -. P .<_ ( Q .\/ R ) /\ S =/= T /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) /\ u e. A /\ ( u .<_ ( P .\/ R ) /\ u .<_ ( S .\/ T ) ) ) -> S e. A )
30 simp123 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( -. P .<_ ( Q .\/ R ) /\ S =/= T /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) /\ u e. A /\ ( u .<_ ( P .\/ R ) /\ u .<_ ( S .\/ T ) ) ) -> T e. A )
31 26 2 5 hlatjcl 
 |-  ( ( K e. HL /\ S e. A /\ T e. A ) -> ( S .\/ T ) e. ( Base ` K ) )
32 20 29 30 31 syl3anc 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( -. P .<_ ( Q .\/ R ) /\ S =/= T /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) /\ u e. A /\ ( u .<_ ( P .\/ R ) /\ u .<_ ( S .\/ T ) ) ) -> ( S .\/ T ) e. ( Base ` K ) )
33 26 3 latmcl 
 |-  ( ( K e. Lat /\ ( P .\/ R ) e. ( Base ` K ) /\ ( S .\/ T ) e. ( Base ` K ) ) -> ( ( P .\/ R ) ./\ ( S .\/ T ) ) e. ( Base ` K ) )
34 23 28 32 33 syl3anc 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( -. P .<_ ( Q .\/ R ) /\ S =/= T /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) /\ u e. A /\ ( u .<_ ( P .\/ R ) /\ u .<_ ( S .\/ T ) ) ) -> ( ( P .\/ R ) ./\ ( S .\/ T ) ) e. ( Base ` K ) )
35 simp2 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( -. P .<_ ( Q .\/ R ) /\ S =/= T /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) /\ u e. A /\ ( u .<_ ( P .\/ R ) /\ u .<_ ( S .\/ T ) ) ) -> u e. A )
36 simp3 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( -. P .<_ ( Q .\/ R ) /\ S =/= T /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) /\ u e. A /\ ( u .<_ ( P .\/ R ) /\ u .<_ ( S .\/ T ) ) ) -> ( u .<_ ( P .\/ R ) /\ u .<_ ( S .\/ T ) ) )
37 26 5 atbase 
 |-  ( u e. A -> u e. ( Base ` K ) )
38 35 37 syl 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( -. P .<_ ( Q .\/ R ) /\ S =/= T /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) /\ u e. A /\ ( u .<_ ( P .\/ R ) /\ u .<_ ( S .\/ T ) ) ) -> u e. ( Base ` K ) )
39 26 1 3 latlem12 
 |-  ( ( K e. Lat /\ ( u e. ( Base ` K ) /\ ( P .\/ R ) e. ( Base ` K ) /\ ( S .\/ T ) e. ( Base ` K ) ) ) -> ( ( u .<_ ( P .\/ R ) /\ u .<_ ( S .\/ T ) ) <-> u .<_ ( ( P .\/ R ) ./\ ( S .\/ T ) ) ) )
40 23 38 28 32 39 syl13anc 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( -. P .<_ ( Q .\/ R ) /\ S =/= T /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) /\ u e. A /\ ( u .<_ ( P .\/ R ) /\ u .<_ ( S .\/ T ) ) ) -> ( ( u .<_ ( P .\/ R ) /\ u .<_ ( S .\/ T ) ) <-> u .<_ ( ( P .\/ R ) ./\ ( S .\/ T ) ) ) )
41 36 40 mpbid 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( -. P .<_ ( Q .\/ R ) /\ S =/= T /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) /\ u e. A /\ ( u .<_ ( P .\/ R ) /\ u .<_ ( S .\/ T ) ) ) -> u .<_ ( ( P .\/ R ) ./\ ( S .\/ T ) ) )
42 26 1 4 5 atlen0 
 |-  ( ( ( K e. AtLat /\ ( ( P .\/ R ) ./\ ( S .\/ T ) ) e. ( Base ` K ) /\ u e. A ) /\ u .<_ ( ( P .\/ R ) ./\ ( S .\/ T ) ) ) -> ( ( P .\/ R ) ./\ ( S .\/ T ) ) =/= .0. )
43 22 34 35 41 42 syl31anc 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( -. P .<_ ( Q .\/ R ) /\ S =/= T /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) /\ u e. A /\ ( u .<_ ( P .\/ R ) /\ u .<_ ( S .\/ T ) ) ) -> ( ( P .\/ R ) ./\ ( S .\/ T ) ) =/= .0. )
44 43 rexlimdv3a 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( -. P .<_ ( Q .\/ R ) /\ S =/= T /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) -> ( E. u e. A ( u .<_ ( P .\/ R ) /\ u .<_ ( S .\/ T ) ) -> ( ( P .\/ R ) ./\ ( S .\/ T ) ) =/= .0. ) )
45 19 44 mpd 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( -. P .<_ ( Q .\/ R ) /\ S =/= T /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) -> ( ( P .\/ R ) ./\ ( S .\/ T ) ) =/= .0. )