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Metamath Proof Explorer

Theorem cvrat

Description: A nonzero Hilbert lattice element less than the join of two atoms is an atom. ( atcvati analog.) (Contributed by NM, 22-Nov-2011)

Ref Expression
Hypotheses cvrat.b 
|- B = ( Base ` K )
cvrat.s 
|- .< = ( lt ` K )
cvrat.j 
|- .\/ = ( join ` K )
cvrat.z 
|- .0. = ( 0. ` K )
cvrat.a 
|- A = ( Atoms ` K )
Assertion cvrat 
|- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> ( ( X =/= .0. /\ X .< ( P .\/ Q ) ) -> X e. A ) )

Proof

Step Hyp Ref Expression
1 cvrat.b 
 |-  B = ( Base ` K )
2 cvrat.s 
 |-  .< = ( lt ` K )
3 cvrat.j 
 |-  .\/ = ( join ` K )
4 cvrat.z 
 |-  .0. = ( 0. ` K )
5 cvrat.a 
 |-  A = ( Atoms ` K )
6 1 2 3 4 5 cvratlem 
 |-  ( ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) /\ ( X =/= .0. /\ X .< ( P .\/ Q ) ) ) -> ( -. P ( le ` K ) X -> X e. A ) )
7 hllat 
 |-  ( K e. HL -> K e. Lat )
8 7 adantr 
 |-  ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> K e. Lat )
9 simpr2 
 |-  ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> P e. A )
10 1 5 atbase 
 |-  ( P e. A -> P e. B )
11 9 10 syl 
 |-  ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> P e. B )
12 simpr3 
 |-  ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> Q e. A )
13 1 5 atbase 
 |-  ( Q e. A -> Q e. B )
14 12 13 syl 
 |-  ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> Q e. B )
15 1 3 latjcom 
 |-  ( ( K e. Lat /\ P e. B /\ Q e. B ) -> ( P .\/ Q ) = ( Q .\/ P ) )
16 8 11 14 15 syl3anc 
 |-  ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> ( P .\/ Q ) = ( Q .\/ P ) )
17 16 breq2d 
 |-  ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> ( X .< ( P .\/ Q ) <-> X .< ( Q .\/ P ) ) )
18 17 anbi2d 
 |-  ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> ( ( X =/= .0. /\ X .< ( P .\/ Q ) ) <-> ( X =/= .0. /\ X .< ( Q .\/ P ) ) ) )
19 simpl 
 |-  ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> K e. HL )
20 simpr1 
 |-  ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> X e. B )
21 1 2 3 4 5 cvratlem 
 |-  ( ( ( K e. HL /\ ( X e. B /\ Q e. A /\ P e. A ) ) /\ ( X =/= .0. /\ X .< ( Q .\/ P ) ) ) -> ( -. Q ( le ` K ) X -> X e. A ) )
22 21 ex 
 |-  ( ( K e. HL /\ ( X e. B /\ Q e. A /\ P e. A ) ) -> ( ( X =/= .0. /\ X .< ( Q .\/ P ) ) -> ( -. Q ( le ` K ) X -> X e. A ) ) )
23 19 20 12 9 22 syl13anc 
 |-  ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> ( ( X =/= .0. /\ X .< ( Q .\/ P ) ) -> ( -. Q ( le ` K ) X -> X e. A ) ) )
24 18 23 sylbid 
 |-  ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> ( ( X =/= .0. /\ X .< ( P .\/ Q ) ) -> ( -. Q ( le ` K ) X -> X e. A ) ) )
25 24 imp 
 |-  ( ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) /\ ( X =/= .0. /\ X .< ( P .\/ Q ) ) ) -> ( -. Q ( le ` K ) X -> X e. A ) )
26 hlpos 
 |-  ( K e. HL -> K e. Poset )
27 26 adantr 
 |-  ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> K e. Poset )
28 1 3 latjcl 
 |-  ( ( K e. Lat /\ P e. B /\ Q e. B ) -> ( P .\/ Q ) e. B )
29 8 11 14 28 syl3anc 
 |-  ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> ( P .\/ Q ) e. B )
30 eqid 
 |-  ( le ` K ) = ( le ` K )
31 1 30 2 pltnle 
 |-  ( ( ( K e. Poset /\ X e. B /\ ( P .\/ Q ) e. B ) /\ X .< ( P .\/ Q ) ) -> -. ( P .\/ Q ) ( le ` K ) X )
32 31 ex 
 |-  ( ( K e. Poset /\ X e. B /\ ( P .\/ Q ) e. B ) -> ( X .< ( P .\/ Q ) -> -. ( P .\/ Q ) ( le ` K ) X ) )
33 27 20 29 32 syl3anc 
 |-  ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> ( X .< ( P .\/ Q ) -> -. ( P .\/ Q ) ( le ` K ) X ) )
34 1 30 3 latjle12 
 |-  ( ( K e. Lat /\ ( P e. B /\ Q e. B /\ X e. B ) ) -> ( ( P ( le ` K ) X /\ Q ( le ` K ) X ) <-> ( P .\/ Q ) ( le ` K ) X ) )
35 8 11 14 20 34 syl13anc 
 |-  ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> ( ( P ( le ` K ) X /\ Q ( le ` K ) X ) <-> ( P .\/ Q ) ( le ` K ) X ) )
36 35 biimpd 
 |-  ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> ( ( P ( le ` K ) X /\ Q ( le ` K ) X ) -> ( P .\/ Q ) ( le ` K ) X ) )
37 33 36 nsyld 
 |-  ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> ( X .< ( P .\/ Q ) -> -. ( P ( le ` K ) X /\ Q ( le ` K ) X ) ) )
38 ianor 
 |-  ( -. ( P ( le ` K ) X /\ Q ( le ` K ) X ) <-> ( -. P ( le ` K ) X \/ -. Q ( le ` K ) X ) )
39 37 38 imbitrdi 
 |-  ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> ( X .< ( P .\/ Q ) -> ( -. P ( le ` K ) X \/ -. Q ( le ` K ) X ) ) )
40 39 imp 
 |-  ( ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) /\ X .< ( P .\/ Q ) ) -> ( -. P ( le ` K ) X \/ -. Q ( le ` K ) X ) )
41 40 adantrl 
 |-  ( ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) /\ ( X =/= .0. /\ X .< ( P .\/ Q ) ) ) -> ( -. P ( le ` K ) X \/ -. Q ( le ` K ) X ) )
42 6 25 41 mpjaod 
 |-  ( ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) /\ ( X =/= .0. /\ X .< ( P .\/ Q ) ) ) -> X e. A )
43 42 ex 
 |-  ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> ( ( X =/= .0. /\ X .< ( P .\/ Q ) ) -> X e. A ) )