Theorem cdlemg3a

Description: Part of proof of Lemma G in Crawley p. 116, line 19. Show p \/ q = p \/ u. TODO: reformat cdleme0cp to match this, then replace with cdleme0cp . (Contributed by NM, 19-Apr-2013)

	Ref	Expression
Hypotheses	cdlemg3.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdlemg3.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdlemg3.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cdlemg3.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdlemg3.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdlemg3.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
Assertion	cdlemg3a	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge Q\in A\right)\to P\underset{\dot{}}{\vee }Q=P\underset{\dot{}}{\vee }U$$

Proof

Step	Hyp	Ref	Expression
1		cdlemg3.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
2		cdlemg3.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		cdlemg3.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
4		cdlemg3.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
5		cdlemg3.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
6		cdlemg3.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
7	1 2 3 4 5 6	cdleme8	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge Q\in A\right)\to P\underset{\dot{}}{\vee }U=P\underset{\dot{}}{\vee }Q$$
8	7	eqcomd	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge Q\in A\right)\to P\underset{\dot{}}{\vee }Q=P\underset{\dot{}}{\vee }U$$