Theorem cdleme9tN

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

	Ref	Expression
Hypotheses	cdleme9t.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdleme9t.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdleme9t.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cdleme9t.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdleme9t.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdleme9t.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
	cdleme9t.g	$$⊢F=\left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme9t.x	$$⊢X=\left(P\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W$$
Assertion	cdleme9tN	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge Q\in A\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to F\underset{\dot{}}{\vee }X=Q\underset{\dot{}}{\vee }X$$

Proof

Step	Hyp	Ref	Expression
1		cdleme9t.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
2		cdleme9t.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		cdleme9t.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
4		cdleme9t.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
5		cdleme9t.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
6		cdleme9t.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
7		cdleme9t.g	$$⊢F=\left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W\right)\right)$$
8		cdleme9t.x	$$⊢X=\left(P\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W$$
9	1 2 3 4 5 6 7 8	cdleme9	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge Q\in A\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to F\underset{\dot{}}{\vee }X=Q\underset{\dot{}}{\vee }X$$