Metamath Proof Explorer

Theorem atlelt

Description: Transfer less-than relation from one atom to another. (Contributed by NM, 7-May-2012)

Ref Expression
Hypotheses atlelt.b 
|- B = ( Base ` K )
atlelt.l 
|- .<_ = ( le ` K )
atlelt.s 
|- .< = ( lt ` K )
atlelt.a 
|- A = ( Atoms ` K )
Assertion atlelt 
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P .<_ X /\ Q .< X ) ) -> P .< X )

Proof

Step Hyp Ref Expression
1 atlelt.b 
 |-  B = ( Base ` K )
2 atlelt.l 
 |-  .<_ = ( le ` K )
3 atlelt.s 
 |-  .< = ( lt ` K )
4 atlelt.a 
 |-  A = ( Atoms ` K )
5 simp3r 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P .<_ X /\ Q .< X ) ) -> Q .< X )
6 breq1 
 |-  ( P = Q -> ( P .< X <-> Q .< X ) )
7 5 6 syl5ibrcom 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P .<_ X /\ Q .< X ) ) -> ( P = Q -> P .< X ) )
8 simp1 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P .<_ X /\ Q .< X ) ) -> K e. HL )
9 simp21 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P .<_ X /\ Q .< X ) ) -> P e. A )
10 simp22 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P .<_ X /\ Q .< X ) ) -> Q e. A )
11 eqid 
 |-  ( join ` K ) = ( join ` K )
12 3 11 4 atlt 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .< ( P ( join ` K ) Q ) <-> P =/= Q ) )
13 8 9 10 12 syl3anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P .<_ X /\ Q .< X ) ) -> ( P .< ( P ( join ` K ) Q ) <-> P =/= Q ) )
14 simp3l 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P .<_ X /\ Q .< X ) ) -> P .<_ X )
15 simp23 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P .<_ X /\ Q .< X ) ) -> X e. B )
16 8 10 15 3jca 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P .<_ X /\ Q .< X ) ) -> ( K e. HL /\ Q e. A /\ X e. B ) )
17 2 3 pltle 
 |-  ( ( K e. HL /\ Q e. A /\ X e. B ) -> ( Q .< X -> Q .<_ X ) )
18 16 5 17 sylc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P .<_ X /\ Q .< X ) ) -> Q .<_ X )
19 hllat 
 |-  ( K e. HL -> K e. Lat )
20 19 3ad2ant1 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P .<_ X /\ Q .< X ) ) -> K e. Lat )
21 1 4 atbase 
 |-  ( P e. A -> P e. B )
22 9 21 syl 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P .<_ X /\ Q .< X ) ) -> P e. B )
23 1 4 atbase 
 |-  ( Q e. A -> Q e. B )
24 10 23 syl 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P .<_ X /\ Q .< X ) ) -> Q e. B )
25 1 2 11 latjle12 
 |-  ( ( K e. Lat /\ ( P e. B /\ Q e. B /\ X e. B ) ) -> ( ( P .<_ X /\ Q .<_ X ) <-> ( P ( join ` K ) Q ) .<_ X ) )
26 20 22 24 15 25 syl13anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P .<_ X /\ Q .< X ) ) -> ( ( P .<_ X /\ Q .<_ X ) <-> ( P ( join ` K ) Q ) .<_ X ) )
27 14 18 26 mpbi2and 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P .<_ X /\ Q .< X ) ) -> ( P ( join ` K ) Q ) .<_ X )
28 hlpos 
 |-  ( K e. HL -> K e. Poset )
29 28 3ad2ant1 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P .<_ X /\ Q .< X ) ) -> K e. Poset )
30 1 11 latjcl 
 |-  ( ( K e. Lat /\ P e. B /\ Q e. B ) -> ( P ( join ` K ) Q ) e. B )
31 20 22 24 30 syl3anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P .<_ X /\ Q .< X ) ) -> ( P ( join ` K ) Q ) e. B )
32 1 2 3 pltletr 
 |-  ( ( K e. Poset /\ ( P e. B /\ ( P ( join ` K ) Q ) e. B /\ X e. B ) ) -> ( ( P .< ( P ( join ` K ) Q ) /\ ( P ( join ` K ) Q ) .<_ X ) -> P .< X ) )
33 29 22 31 15 32 syl13anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P .<_ X /\ Q .< X ) ) -> ( ( P .< ( P ( join ` K ) Q ) /\ ( P ( join ` K ) Q ) .<_ X ) -> P .< X ) )
34 27 33 mpan2d 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P .<_ X /\ Q .< X ) ) -> ( P .< ( P ( join ` K ) Q ) -> P .< X ) )
35 13 34 sylbird 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P .<_ X /\ Q .< X ) ) -> ( P =/= Q -> P .< X ) )
36 7 35 pm2.61dne 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P .<_ X /\ Q .< X ) ) -> P .< X )