Theorem cdleme10tN

Description: Part of proof of Lemma E in Crawley p. 113, 2nd paragraph on p. 114. Y represents t_2. In their notation, we prove t \/ t_2 = t \/ r. (Contributed by NM, 8-Oct-2012) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cdleme10t.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdleme10t.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdleme10t.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cdleme10t.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdleme10t.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdleme10t.y	$$⊢Y=\left(R\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W$$
Assertion	cdleme10tN	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge R\in A\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\to T\underset{\dot{}}{\vee }Y=T\underset{\dot{}}{\vee }R$$

Proof

Step	Hyp	Ref	Expression
1		cdleme10t.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
2		cdleme10t.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		cdleme10t.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
4		cdleme10t.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
5		cdleme10t.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
6		cdleme10t.y	$$⊢Y=\left(R\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W$$
7	1 2 3 4 5 6	cdleme10	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge R\in A\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\to T\underset{\dot{}}{\vee }Y=T\underset{\dot{}}{\vee }R$$