Theorem cdleme21at

Description: Part of proof of Lemma E in Crawley p. 115. (Contributed by NM, 29-Nov-2012)

	Ref	Expression
Hypotheses	cdleme21.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdleme21.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdleme21.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cdleme21.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdleme21.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdleme21.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
Assertion	cdleme21at	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge Q\in A\right)\wedge \left(\left(S\in A\wedge P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\wedge \left(z\in A\wedge P\underset{\dot{}}{\vee }z=S\underset{\dot{}}{\vee }z\right)\right)\to T\ne z$$

Proof

Step	Hyp	Ref	Expression
1		cdleme21.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
2		cdleme21.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		cdleme21.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
4		cdleme21.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
5		cdleme21.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
6		cdleme21.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
7	1 2 3 4 5 6	cdleme21c	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge Q\in A\right)\wedge \left(S\in A\wedge P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge P\underset{\dot{}}{\vee }z=S\underset{\dot{}}{\vee }z\right)\right)\to \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }z\right)$$
8	7	3adant2r	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge Q\in A\right)\wedge \left(\left(S\in A\wedge P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\wedge \left(z\in A\wedge P\underset{\dot{}}{\vee }z=S\underset{\dot{}}{\vee }z\right)\right)\to \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }z\right)$$
9		simp2r	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge Q\in A\right)\wedge \left(\left(S\in A\wedge P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\wedge \left(z\in A\wedge P\underset{\dot{}}{\vee }z=S\underset{\dot{}}{\vee }z\right)\right)\to U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)$$
10		oveq2	$$⊢T=z\to S\underset{\dot{}}{\vee }T=S\underset{\dot{}}{\vee }z$$
11	10	breq2d	$$⊢T=z\to \left(U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\leftrightarrow U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }z\right)\right)$$
12	9 11	syl5ibcom	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge Q\in A\right)\wedge \left(\left(S\in A\wedge P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\wedge \left(z\in A\wedge P\underset{\dot{}}{\vee }z=S\underset{\dot{}}{\vee }z\right)\right)\to \left(T=z\to U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }z\right)\right)$$
13	12	necon3bd	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge Q\in A\right)\wedge \left(\left(S\in A\wedge P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\wedge \left(z\in A\wedge P\underset{\dot{}}{\vee }z=S\underset{\dot{}}{\vee }z\right)\right)\to \left(\neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }z\right)\to T\ne z\right)$$
14	8 13	mpd	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge Q\in A\right)\wedge \left(\left(S\in A\wedge P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\wedge \left(z\in A\wedge P\underset{\dot{}}{\vee }z=S\underset{\dot{}}{\vee }z\right)\right)\to T\ne z$$