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 )