Metamath Proof Explorer

Theorem cdlema2N

Description: A condition for required for proof of Lemma A in Crawley p. 112. (Contributed by NM, 9-May-2012) (New usage is discouraged.)

Ref Expression
Hypotheses cdlema2.b 
|- B = ( Base ` K )
cdlema2.l 
|- .<_ = ( le ` K )
cdlema2.j 
|- .\/ = ( join ` K )
cdlema2.m 
|- ./\ = ( meet ` K )
cdlema2.z 
|- .0. = ( 0. ` K )
cdlema2.a 
|- A = ( Atoms ` K )
Assertion cdlema2N 
|- ( ( ( K e. HL /\ X e. B ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( ( R =/= P /\ R .<_ ( P .\/ Q ) ) /\ ( P .<_ X /\ -. Q .<_ X ) ) ) -> ( R ./\ X ) = .0. )

Proof

Step Hyp Ref Expression
1 cdlema2.b 
 |-  B = ( Base ` K )
2 cdlema2.l 
 |-  .<_ = ( le ` K )
3 cdlema2.j 
 |-  .\/ = ( join ` K )
4 cdlema2.m 
 |-  ./\ = ( meet ` K )
5 cdlema2.z 
 |-  .0. = ( 0. ` K )
6 cdlema2.a 
 |-  A = ( Atoms ` K )
7 simp3ll 
 |-  ( ( ( K e. HL /\ X e. B ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( ( R =/= P /\ R .<_ ( P .\/ Q ) ) /\ ( P .<_ X /\ -. Q .<_ X ) ) ) -> R =/= P )
8 simp3rl 
 |-  ( ( ( K e. HL /\ X e. B ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( ( R =/= P /\ R .<_ ( P .\/ Q ) ) /\ ( P .<_ X /\ -. Q .<_ X ) ) ) -> P .<_ X )
9 simp3rr 
 |-  ( ( ( K e. HL /\ X e. B ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( ( R =/= P /\ R .<_ ( P .\/ Q ) ) /\ ( P .<_ X /\ -. Q .<_ X ) ) ) -> -. Q .<_ X )
10 simp3lr 
 |-  ( ( ( K e. HL /\ X e. B ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( ( R =/= P /\ R .<_ ( P .\/ Q ) ) /\ ( P .<_ X /\ -. Q .<_ X ) ) ) -> R .<_ ( P .\/ Q ) )
11 8 9 10 3jca 
 |-  ( ( ( K e. HL /\ X e. B ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( ( R =/= P /\ R .<_ ( P .\/ Q ) ) /\ ( P .<_ X /\ -. Q .<_ X ) ) ) -> ( P .<_ X /\ -. Q .<_ X /\ R .<_ ( P .\/ Q ) ) )
12 1 2 3 6 exatleN 
 |-  ( ( ( K e. HL /\ X e. B ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P .<_ X /\ -. Q .<_ X /\ R .<_ ( P .\/ Q ) ) ) -> ( R .<_ X <-> R = P ) )
13 11 12 syld3an3 
 |-  ( ( ( K e. HL /\ X e. B ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( ( R =/= P /\ R .<_ ( P .\/ Q ) ) /\ ( P .<_ X /\ -. Q .<_ X ) ) ) -> ( R .<_ X <-> R = P ) )
14 13 necon3bbid 
 |-  ( ( ( K e. HL /\ X e. B ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( ( R =/= P /\ R .<_ ( P .\/ Q ) ) /\ ( P .<_ X /\ -. Q .<_ X ) ) ) -> ( -. R .<_ X <-> R =/= P ) )
15 7 14 mpbird 
 |-  ( ( ( K e. HL /\ X e. B ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( ( R =/= P /\ R .<_ ( P .\/ Q ) ) /\ ( P .<_ X /\ -. Q .<_ X ) ) ) -> -. R .<_ X )
16 simp1l 
 |-  ( ( ( K e. HL /\ X e. B ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( ( R =/= P /\ R .<_ ( P .\/ Q ) ) /\ ( P .<_ X /\ -. Q .<_ X ) ) ) -> K e. HL )
17 hlatl 
 |-  ( K e. HL -> K e. AtLat )
18 16 17 syl 
 |-  ( ( ( K e. HL /\ X e. B ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( ( R =/= P /\ R .<_ ( P .\/ Q ) ) /\ ( P .<_ X /\ -. Q .<_ X ) ) ) -> K e. AtLat )
19 simp23 
 |-  ( ( ( K e. HL /\ X e. B ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( ( R =/= P /\ R .<_ ( P .\/ Q ) ) /\ ( P .<_ X /\ -. Q .<_ X ) ) ) -> R e. A )
20 simp1r 
 |-  ( ( ( K e. HL /\ X e. B ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( ( R =/= P /\ R .<_ ( P .\/ Q ) ) /\ ( P .<_ X /\ -. Q .<_ X ) ) ) -> X e. B )
21 1 2 4 5 6 atnle 
 |-  ( ( K e. AtLat /\ R e. A /\ X e. B ) -> ( -. R .<_ X <-> ( R ./\ X ) = .0. ) )
22 18 19 20 21 syl3anc 
 |-  ( ( ( K e. HL /\ X e. B ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( ( R =/= P /\ R .<_ ( P .\/ Q ) ) /\ ( P .<_ X /\ -. Q .<_ X ) ) ) -> ( -. R .<_ X <-> ( R ./\ X ) = .0. ) )
23 15 22 mpbid 
 |-  ( ( ( K e. HL /\ X e. B ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( ( R =/= P /\ R .<_ ( P .\/ Q ) ) /\ ( P .<_ X /\ -. Q .<_ X ) ) ) -> ( R ./\ X ) = .0. )