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Metamath Proof Explorer

Theorem atcvrneN

Description: Inequality derived from atom condition. (Contributed by NM, 7-Feb-2012) (New usage is discouraged.)

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

Proof

Step Hyp Ref Expression
1 atcvrne.j 
 |-  .\/ = ( join ` K )
2 atcvrne.c 
 |-  C = (
3 atcvrne.a 
 |-  A = ( Atoms ` K )
4 hlatl 
 |-  ( K e. HL -> K e. AtLat )
5 4 3ad2ant1 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P C ( Q .\/ R ) ) -> K e. AtLat )
6 simp21 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P C ( Q .\/ R ) ) -> P e. A )
7 eqid 
 |-  ( 0. ` K ) = ( 0. ` K )
8 7 3 atn0 
 |-  ( ( K e. AtLat /\ P e. A ) -> P =/= ( 0. ` K ) )
9 5 6 8 syl2anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P C ( Q .\/ R ) ) -> P =/= ( 0. ` K ) )
10 simp1 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P C ( Q .\/ R ) ) -> K e. HL )
11 eqid 
 |-  ( Base ` K ) = ( Base ` K )
12 11 3 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
13 6 12 syl 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P C ( Q .\/ R ) ) -> P e. ( Base ` K ) )
14 simp22 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P C ( Q .\/ R ) ) -> Q e. A )
15 simp23 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P C ( Q .\/ R ) ) -> R e. A )
16 simp3 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P C ( Q .\/ R ) ) -> P C ( Q .\/ R ) )
17 11 1 7 2 3 atcvrj0 
 |-  ( ( K e. HL /\ ( P e. ( Base ` K ) /\ Q e. A /\ R e. A ) /\ P C ( Q .\/ R ) ) -> ( P = ( 0. ` K ) <-> Q = R ) )
18 10 13 14 15 16 17 syl131anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P C ( Q .\/ R ) ) -> ( P = ( 0. ` K ) <-> Q = R ) )
19 18 necon3bid 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P C ( Q .\/ R ) ) -> ( P =/= ( 0. ` K ) <-> Q =/= R ) )
20 9 19 mpbid 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P C ( Q .\/ R ) ) -> Q =/= R )