Theorem cdleme8tN

Description: Part of proof of Lemma E in Crawley p. 113, 2nd paragraph on p. 114. X represents t_1. In their notation, we prove p \/ t_1 = p \/ t. (Contributed by NM, 8-Oct-2012) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cdleme8t.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdleme8t.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdleme8t.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cdleme8t.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdleme8t.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdleme8t.x	$$⊢X=\left(P\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W$$
Assertion	cdleme8tN	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge T\in A\right)\to P\underset{\dot{}}{\vee }X=P\underset{\dot{}}{\vee }T$$

Proof

Step	Hyp	Ref	Expression
1		cdleme8t.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
2		cdleme8t.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		cdleme8t.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
4		cdleme8t.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
5		cdleme8t.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
6		cdleme8t.x	$$⊢X=\left(P\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W$$
7	1 2 3 4 5 6	cdleme8	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge T\in A\right)\to P\underset{\dot{}}{\vee }X=P\underset{\dot{}}{\vee }T$$