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Metamath Proof Explorer

Theorem ps-2c

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

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

Proof

Step Hyp Ref Expression
1 2atm.l 
 |-  .<_ = ( le ` K )
2 2atm.j 
 |-  .\/ = ( join ` K )
3 2atm.m 
 |-  ./\ = ( meet ` K )
4 2atm.a 
 |-  A = ( Atoms ` K )
5 simp11 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( ( -. P .<_ ( Q .\/ R ) /\ S =/= T ) /\ ( P .\/ R ) =/= ( S .\/ T ) /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) -> K e. HL )
6 simp12 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( ( -. P .<_ ( Q .\/ R ) /\ S =/= T ) /\ ( P .\/ R ) =/= ( S .\/ T ) /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) -> P e. A )
7 simp21 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( ( -. P .<_ ( Q .\/ R ) /\ S =/= T ) /\ ( P .\/ R ) =/= ( S .\/ T ) /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) -> R e. A )
8 5 hllatd 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( ( -. P .<_ ( Q .\/ R ) /\ S =/= T ) /\ ( P .\/ R ) =/= ( S .\/ T ) /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) -> K e. Lat )
9 eqid 
 |-  ( Base ` K ) = ( Base ` K )
10 9 4 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
11 6 10 syl 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( ( -. P .<_ ( Q .\/ R ) /\ S =/= T ) /\ ( P .\/ R ) =/= ( S .\/ T ) /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) -> P e. ( Base ` K ) )
12 simp13 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( ( -. P .<_ ( Q .\/ R ) /\ S =/= T ) /\ ( P .\/ R ) =/= ( S .\/ T ) /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) -> Q e. A )
13 9 4 atbase 
 |-  ( Q e. A -> Q e. ( Base ` K ) )
14 12 13 syl 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( ( -. P .<_ ( Q .\/ R ) /\ S =/= T ) /\ ( P .\/ R ) =/= ( S .\/ T ) /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) -> Q e. ( Base ` K ) )
15 9 4 atbase 
 |-  ( R e. A -> R e. ( Base ` K ) )
16 7 15 syl 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( ( -. P .<_ ( Q .\/ R ) /\ S =/= T ) /\ ( P .\/ R ) =/= ( S .\/ T ) /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) -> R e. ( Base ` K ) )
17 simp31l 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( ( -. P .<_ ( Q .\/ R ) /\ S =/= T ) /\ ( P .\/ R ) =/= ( S .\/ T ) /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) -> -. P .<_ ( Q .\/ R ) )
18 9 1 2 latnlej1r 
 |-  ( ( K e. Lat /\ ( P e. ( Base ` K ) /\ Q e. ( Base ` K ) /\ R e. ( Base ` K ) ) /\ -. P .<_ ( Q .\/ R ) ) -> P =/= R )
19 8 11 14 16 17 18 syl131anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( ( -. P .<_ ( Q .\/ R ) /\ S =/= T ) /\ ( P .\/ R ) =/= ( S .\/ T ) /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) -> P =/= R )
20 eqid 
 |-  ( LLines ` K ) = ( LLines ` K )
21 2 4 20 llni2 
 |-  ( ( ( K e. HL /\ P e. A /\ R e. A ) /\ P =/= R ) -> ( P .\/ R ) e. ( LLines ` K ) )
22 5 6 7 19 21 syl31anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( ( -. P .<_ ( Q .\/ R ) /\ S =/= T ) /\ ( P .\/ R ) =/= ( S .\/ T ) /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) -> ( P .\/ R ) e. ( LLines ` K ) )
23 simp22 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( ( -. P .<_ ( Q .\/ R ) /\ S =/= T ) /\ ( P .\/ R ) =/= ( S .\/ T ) /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) -> S e. A )
24 simp23 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( ( -. P .<_ ( Q .\/ R ) /\ S =/= T ) /\ ( P .\/ R ) =/= ( S .\/ T ) /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) -> T e. A )
25 simp31r 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( ( -. P .<_ ( Q .\/ R ) /\ S =/= T ) /\ ( P .\/ R ) =/= ( S .\/ T ) /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) -> S =/= T )
26 2 4 20 llni2 
 |-  ( ( ( K e. HL /\ S e. A /\ T e. A ) /\ S =/= T ) -> ( S .\/ T ) e. ( LLines ` K ) )
27 5 23 24 25 26 syl31anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( ( -. P .<_ ( Q .\/ R ) /\ S =/= T ) /\ ( P .\/ R ) =/= ( S .\/ T ) /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) -> ( S .\/ T ) e. ( LLines ` K ) )
28 simp32 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( ( -. P .<_ ( Q .\/ R ) /\ S =/= T ) /\ ( P .\/ R ) =/= ( S .\/ T ) /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) -> ( P .\/ R ) =/= ( S .\/ T ) )
29 simp33 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( ( -. P .<_ ( Q .\/ R ) /\ S =/= T ) /\ ( P .\/ R ) =/= ( S .\/ T ) /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) -> ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) )
30 eqid 
 |-  ( 0. ` K ) = ( 0. ` K )
31 1 2 3 30 4 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. ` K ) )
32 5 6 12 7 23 24 17 25 29 31 syl333anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( ( -. P .<_ ( Q .\/ R ) /\ S =/= T ) /\ ( P .\/ R ) =/= ( S .\/ T ) /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) -> ( ( P .\/ R ) ./\ ( S .\/ T ) ) =/= ( 0. ` K ) )
33 3 30 4 20 2llnmat 
 |-  ( ( ( K e. HL /\ ( P .\/ R ) e. ( LLines ` K ) /\ ( S .\/ T ) e. ( LLines ` K ) ) /\ ( ( P .\/ R ) =/= ( S .\/ T ) /\ ( ( P .\/ R ) ./\ ( S .\/ T ) ) =/= ( 0. ` K ) ) ) -> ( ( P .\/ R ) ./\ ( S .\/ T ) ) e. A )
34 5 22 27 28 32 33 syl32anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( ( -. P .<_ ( Q .\/ R ) /\ S =/= T ) /\ ( P .\/ R ) =/= ( S .\/ T ) /\ ( S .<_ ( P .\/ Q ) /\ T .<_ ( Q .\/ R ) ) ) ) -> ( ( P .\/ R ) ./\ ( S .\/ T ) ) e. A )