Metamath Proof Explorer

Theorem atexchcvrN

Description: Atom exchange property. Version of hlatexch2 with covers relation. (Contributed by NM, 7-Feb-2012) (New usage is discouraged.)

Ref Expression
Hypotheses atexchcvr.j 
|- .\/ = ( join ` K )
atexchcvr.a 
|- A = ( Atoms ` K )
atexchcvr.c 
|- C = (
Assertion atexchcvrN 
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) -> ( P C ( Q .\/ R ) -> Q C ( P .\/ R ) ) )

Proof

Step Hyp Ref Expression
1 atexchcvr.j 
 |-  .\/ = ( join ` K )
2 atexchcvr.a 
 |-  A = ( Atoms ` K )
3 atexchcvr.c 
 |-  C = (
4 simpl1 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) /\ P C ( Q .\/ R ) ) -> K e. HL )
5 simpl21 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) /\ P C ( Q .\/ R ) ) -> P e. A )
6 eqid 
 |-  ( Base ` K ) = ( Base ` K )
7 6 2 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
8 5 7 syl 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) /\ P C ( Q .\/ R ) ) -> P e. ( Base ` K ) )
9 4 hllatd 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) /\ P C ( Q .\/ R ) ) -> K e. Lat )
10 simpl22 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) /\ P C ( Q .\/ R ) ) -> Q e. A )
11 6 2 atbase 
 |-  ( Q e. A -> Q e. ( Base ` K ) )
12 10 11 syl 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) /\ P C ( Q .\/ R ) ) -> Q e. ( Base ` K ) )
13 simpl23 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) /\ P C ( Q .\/ R ) ) -> R e. A )
14 6 2 atbase 
 |-  ( R e. A -> R e. ( Base ` K ) )
15 13 14 syl 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) /\ P C ( Q .\/ R ) ) -> R e. ( Base ` K ) )
16 6 1 latjcl 
 |-  ( ( K e. Lat /\ Q e. ( Base ` K ) /\ R e. ( Base ` K ) ) -> ( Q .\/ R ) e. ( Base ` K ) )
17 9 12 15 16 syl3anc 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) /\ P C ( Q .\/ R ) ) -> ( Q .\/ R ) e. ( Base ` K ) )
18 4 8 17 3jca 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) /\ P C ( Q .\/ R ) ) -> ( K e. HL /\ P e. ( Base ` K ) /\ ( Q .\/ R ) e. ( Base ` K ) ) )
19 eqid 
 |-  ( le ` K ) = ( le ` K )
20 6 19 3 cvrle 
 |-  ( ( ( K e. HL /\ P e. ( Base ` K ) /\ ( Q .\/ R ) e. ( Base ` K ) ) /\ P C ( Q .\/ R ) ) -> P ( le ` K ) ( Q .\/ R ) )
21 18 20 sylancom 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) /\ P C ( Q .\/ R ) ) -> P ( le ` K ) ( Q .\/ R ) )
22 21 ex 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) -> ( P C ( Q .\/ R ) -> P ( le ` K ) ( Q .\/ R ) ) )
23 19 1 2 hlatexch2 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) -> ( P ( le ` K ) ( Q .\/ R ) -> Q ( le ` K ) ( P .\/ R ) ) )
24 simpl1 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) /\ Q ( le ` K ) ( P .\/ R ) ) -> K e. HL )
25 simpl22 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) /\ Q ( le ` K ) ( P .\/ R ) ) -> Q e. A )
26 simpl21 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) /\ Q ( le ` K ) ( P .\/ R ) ) -> P e. A )
27 simpl23 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) /\ Q ( le ` K ) ( P .\/ R ) ) -> R e. A )
28 simpl3 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) /\ Q ( le ` K ) ( P .\/ R ) ) -> P =/= R )
29 simpr 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) /\ Q ( le ` K ) ( P .\/ R ) ) -> Q ( le ` K ) ( P .\/ R ) )
30 19 1 3 2 atcvrj2 
 |-  ( ( K e. HL /\ ( Q e. A /\ P e. A /\ R e. A ) /\ ( P =/= R /\ Q ( le ` K ) ( P .\/ R ) ) ) -> Q C ( P .\/ R ) )
31 24 25 26 27 28 29 30 syl132anc 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) /\ Q ( le ` K ) ( P .\/ R ) ) -> Q C ( P .\/ R ) )
32 31 ex 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) -> ( Q ( le ` K ) ( P .\/ R ) -> Q C ( P .\/ R ) ) )
33 22 23 32 3syld 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) -> ( P C ( Q .\/ R ) -> Q C ( P .\/ R ) ) )