Theorem atcvrj2

Description: Condition for an atom to be covered by the join of two others. (Contributed by NM, 7-Feb-2012)

	Ref	Expression
Hypotheses	atcvrj1x.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	atcvrj1x.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	atcvrj1x.c	$$⊢C={⋖}_K$$
	atcvrj1x.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
Assertion	atcvrj2	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(Q\ne R\wedge P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\right)\right)\to PC\left(Q\underset{\dot{}}{\vee }R\right)$$

Proof

Step	Hyp	Ref	Expression
1		atcvrj1x.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
2		atcvrj1x.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		atcvrj1x.c	$$⊢C={⋖}_K$$
4		atcvrj1x.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
5	1 2 3 4	atcvrj2b	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\right)\to \left(\left(Q\ne R\wedge P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\right)\leftrightarrow PC\left(Q\underset{\dot{}}{\vee }R\right)\right)$$
6	5	biimp3a	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(Q\ne R\wedge P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\right)\right)\to PC\left(Q\underset{\dot{}}{\vee }R\right)$$