Theorem llni2

Description: The join of two different atoms is a lattice line. (Contributed by NM, 26-Jun-2012)

	Ref	Expression
Hypotheses	llni2.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	llni2.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	llni2.n	$$⊢N=\mathrm{LLines}\left(K\right)$$
Assertion	llni2	$$⊢\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge P\ne Q\right)\to P\underset{\dot{}}{\vee }Q\in N$$

Proof

Step	Hyp	Ref	Expression
1		llni2.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
2		llni2.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
3		llni2.n	$$⊢N=\mathrm{LLines}\left(K\right)$$
4		simpl2	$$⊢\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge P\ne Q\right)\to P\in A$$
5		simpl3	$$⊢\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge P\ne Q\right)\to Q\in A$$
6		simpr	$$⊢\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge P\ne Q\right)\to P\ne Q$$
7		eqidd	$$⊢\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge P\ne Q\right)\to P\underset{\dot{}}{\vee }Q=P\underset{\dot{}}{\vee }Q$$
8		neeq1	$$⊢r=P\to \left(r\ne s\leftrightarrow P\ne s\right)$$
9		oveq1	$$⊢r=P\to r\underset{\dot{}}{\vee }s=P\underset{\dot{}}{\vee }s$$
10	9	eqeq2d	$$⊢r=P\to \left(P\underset{\dot{}}{\vee }Q=r\underset{\dot{}}{\vee }s\leftrightarrow P\underset{\dot{}}{\vee }Q=P\underset{\dot{}}{\vee }s\right)$$
11	8 10	anbi12d	$$⊢r=P\to \left(\left(r\ne s\wedge P\underset{\dot{}}{\vee }Q=r\underset{\dot{}}{\vee }s\right)\leftrightarrow \left(P\ne s\wedge P\underset{\dot{}}{\vee }Q=P\underset{\dot{}}{\vee }s\right)\right)$$
12		neeq2	$$⊢s=Q\to \left(P\ne s\leftrightarrow P\ne Q\right)$$
13		oveq2	$$⊢s=Q\to P\underset{\dot{}}{\vee }s=P\underset{\dot{}}{\vee }Q$$
14	13	eqeq2d	$$⊢s=Q\to \left(P\underset{\dot{}}{\vee }Q=P\underset{\dot{}}{\vee }s\leftrightarrow P\underset{\dot{}}{\vee }Q=P\underset{\dot{}}{\vee }Q\right)$$
15	12 14	anbi12d	$$⊢s=Q\to \left(\left(P\ne s\wedge P\underset{\dot{}}{\vee }Q=P\underset{\dot{}}{\vee }s\right)\leftrightarrow \left(P\ne Q\wedge P\underset{\dot{}}{\vee }Q=P\underset{\dot{}}{\vee }Q\right)\right)$$
16	11 15	rspc2ev	$$⊢\left(P\in A\wedge Q\in A\wedge \left(P\ne Q\wedge P\underset{\dot{}}{\vee }Q=P\underset{\dot{}}{\vee }Q\right)\right)\to \exists r\in A\exists s\in A\left(r\ne s\wedge P\underset{\dot{}}{\vee }Q=r\underset{\dot{}}{\vee }s\right)$$
17	4 5 6 7 16	syl112anc	$$⊢\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge P\ne Q\right)\to \exists r\in A\exists s\in A\left(r\ne s\wedge P\underset{\dot{}}{\vee }Q=r\underset{\dot{}}{\vee }s\right)$$
18		simpl1	$$⊢\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge P\ne Q\right)\to K\in \mathrm{HL}$$
19		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
20	19 1 2	hlatjcl	$$⊢\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\to P\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K$$
21	20	adantr	$$⊢\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge P\ne Q\right)\to P\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K$$
22	19 1 2 3	islln3	$$⊢\left(K\in \mathrm{HL}\wedge P\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K\right)\to \left(P\underset{\dot{}}{\vee }Q\in N\leftrightarrow \exists r\in A\exists s\in A\left(r\ne s\wedge P\underset{\dot{}}{\vee }Q=r\underset{\dot{}}{\vee }s\right)\right)$$
23	18 21 22	syl2anc	$$⊢\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge P\ne Q\right)\to \left(P\underset{\dot{}}{\vee }Q\in N\leftrightarrow \exists r\in A\exists s\in A\left(r\ne s\wedge P\underset{\dot{}}{\vee }Q=r\underset{\dot{}}{\vee }s\right)\right)$$
24	17 23	mpbird	$$⊢\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge P\ne Q\right)\to P\underset{\dot{}}{\vee }Q\in N$$