Theorem cvlatexchb1

Description: A version of cvlexchb1 for atoms. (Contributed by NM, 5-Nov-2012)

	Ref	Expression
Hypotheses	cvlatexch.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cvlatexch.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cvlatexch.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
Assertion	cvlatexchb1	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge P\ne R\right)\to \left(P\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }Q\right)\leftrightarrow R\underset{\dot{}}{\vee }P=R\underset{\dot{}}{\vee }Q\right)$$

Proof

Step	Hyp	Ref	Expression
1		cvlatexch.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
2		cvlatexch.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		cvlatexch.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
4		cvlatl	$$⊢K\in \mathrm{CvLat}\to K\in \mathrm{AtLat}$$
5	4	adantr	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\right)\to K\in \mathrm{AtLat}$$
6		simpr1	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\right)\to P\in A$$
7		simpr3	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\right)\to R\in A$$
8	1 3	atncmp	$$⊢\left(K\in \mathrm{AtLat}\wedge P\in A\wedge R\in A\right)\to \left(\neg P\underset{\dot{}}{\le }R\leftrightarrow P\ne R\right)$$
9	5 6 7 8	syl3anc	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\right)\to \left(\neg P\underset{\dot{}}{\le }R\leftrightarrow P\ne R\right)$$
10		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
11	10 3	atbase	$$⊢R\in A\to R\in {\mathrm{Base}}_K$$
12	10 1 2 3	cvlexchb1	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(P\in A\wedge Q\in A\wedge R\in {\mathrm{Base}}_K\right)\wedge \neg P\underset{\dot{}}{\le }R\right)\to \left(P\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }Q\right)\leftrightarrow R\underset{\dot{}}{\vee }P=R\underset{\dot{}}{\vee }Q\right)$$
13	12	3expia	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(P\in A\wedge Q\in A\wedge R\in {\mathrm{Base}}_K\right)\right)\to \left(\neg P\underset{\dot{}}{\le }R\to \left(P\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }Q\right)\leftrightarrow R\underset{\dot{}}{\vee }P=R\underset{\dot{}}{\vee }Q\right)\right)$$
14	11 13	syl3anr3	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\right)\to \left(\neg P\underset{\dot{}}{\le }R\to \left(P\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }Q\right)\leftrightarrow R\underset{\dot{}}{\vee }P=R\underset{\dot{}}{\vee }Q\right)\right)$$
15	9 14	sylbird	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\right)\to \left(P\ne R\to \left(P\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }Q\right)\leftrightarrow R\underset{\dot{}}{\vee }P=R\underset{\dot{}}{\vee }Q\right)\right)$$
16	15	3impia	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge P\ne R\right)\to \left(P\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }Q\right)\leftrightarrow R\underset{\dot{}}{\vee }P=R\underset{\dot{}}{\vee }Q\right)$$