Theorem 2atnelvolN

Description: The join of two atoms is not a lattice volume. (Contributed by NM, 17-Jul-2012) (New usage is discouraged.)

	Ref	Expression
Hypotheses	3atnelvol.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	3atnelvol.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	3atnelvol.v	$$⊢V=\mathrm{LVols}\left(K\right)$$
Assertion	2atnelvolN	$$⊢\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\to \neg P\underset{\dot{}}{\vee }Q\in V$$

Proof

Step	Hyp	Ref	Expression
1		3atnelvol.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
2		3atnelvol.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
3		3atnelvol.v	$$⊢V=\mathrm{LVols}\left(K\right)$$
4	1 2	hlatjidm	$$⊢\left(K\in \mathrm{HL}\wedge P\in A\right)\to P\underset{\dot{}}{\vee }P=P$$
5	4	3adant3	$$⊢\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\to P\underset{\dot{}}{\vee }P=P$$
6	5	oveq1d	$$⊢\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\to \left(P\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\vee }Q=P\underset{\dot{}}{\vee }Q$$
7		simp1	$$⊢\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\to K\in \mathrm{HL}$$
8		simp2	$$⊢\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\to P\in A$$
9		simp3	$$⊢\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\to Q\in A$$
10	1 2 3	3atnelvolN	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge P\in A\wedge Q\in A\right)\right)\to \neg \left(P\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\vee }Q\in V$$
11	7 8 8 9 10	syl13anc	$$⊢\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\to \neg \left(P\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\vee }Q\in V$$
12	6 11	eqneltrrd	$$⊢\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\to \neg P\underset{\dot{}}{\vee }Q\in V$$