Theorem 4atexlemntlpq

Description: Lemma for 4atexlem7 . (Contributed by NM, 24-Nov-2012)

	Ref	Expression
Hypotheses	4thatlem.ph	$$⊢\phi \leftrightarrow \left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(S\in A\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }R=Q\underset{\dot{}}{\vee }R\right)\wedge \left(T\in A\wedge U\underset{\dot{}}{\vee }T=V\underset{\dot{}}{\vee }T\right)\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)$$
	4thatlem0.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	4thatlem0.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	4thatlem0.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	4thatlem0.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	4thatlem0.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	4thatlem0.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
	4thatlem0.v	$$⊢V=\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W$$
Assertion	4atexlemntlpq	$$⊢\phi \to \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$

Proof

Step	Hyp	Ref	Expression
1		4thatlem.ph	$$⊢\phi \leftrightarrow \left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(S\in A\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }R=Q\underset{\dot{}}{\vee }R\right)\wedge \left(T\in A\wedge U\underset{\dot{}}{\vee }T=V\underset{\dot{}}{\vee }T\right)\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)$$
2		4thatlem0.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		4thatlem0.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
4		4thatlem0.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
5		4thatlem0.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
6		4thatlem0.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
7		4thatlem0.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
8		4thatlem0.v	$$⊢V=\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W$$
9	1 2 3 4 5 6 7 8	4atexlemtlw	$$⊢\phi \to T\underset{\dot{}}{\le }W$$
10	1	4atexlemkc	$$⊢\phi \to K\in \mathrm{CvLat}$$
11	1 2 3 4 5 6 7	4atexlemu	$$⊢\phi \to U\in A$$
12	1 2 3 4 5 6 7 8	4atexlemv	$$⊢\phi \to V\in A$$
13	1	4atexlemt	$$⊢\phi \to T\in A$$
14	1 2 3 4 5 6 7 8	4atexlemunv	$$⊢\phi \to U\ne V$$
15	1	4atexlemutvt	$$⊢\phi \to U\underset{\dot{}}{\vee }T=V\underset{\dot{}}{\vee }T$$
16	5 3	cvlsupr5	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(U\in A\wedge V\in A\wedge T\in A\right)\wedge \left(U\ne V\wedge U\underset{\dot{}}{\vee }T=V\underset{\dot{}}{\vee }T\right)\right)\to T\ne U$$
17	10 11 12 13 14 15 16	syl132anc	$$⊢\phi \to T\ne U$$
18	17	adantr	$$⊢\left(\phi \wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to T\ne U$$
19	1	4atexlemk	$$⊢\phi \to K\in \mathrm{HL}$$
20	1	4atexlemw	$$⊢\phi \to W\in H$$
21	19 20	jca	$$⊢\phi \to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
22	21	adantr	$$⊢\left(\phi \wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
23	1	4atexlempw	$$⊢\phi \to \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)$$
24	23	adantr	$$⊢\left(\phi \wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)$$
25	1	4atexlemq	$$⊢\phi \to Q\in A$$
26	25	adantr	$$⊢\left(\phi \wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to Q\in A$$
27	13	adantr	$$⊢\left(\phi \wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to T\in A$$
28	1	4atexlempnq	$$⊢\phi \to P\ne Q$$
29	28	adantr	$$⊢\left(\phi \wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to P\ne Q$$
30		simpr	$$⊢\left(\phi \wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
31	2 3 4 5 6 7	lhpat3	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\right)\wedge \left(Q\in A\wedge T\in A\right)\wedge \left(P\ne Q\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \left(\neg T\underset{\dot{}}{\le }W\leftrightarrow T\ne U\right)$$
32	22 24 26 27 29 30 31	syl222anc	$$⊢\left(\phi \wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to \left(\neg T\underset{\dot{}}{\le }W\leftrightarrow T\ne U\right)$$
33	18 32	mpbird	$$⊢\left(\phi \wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to \neg T\underset{\dot{}}{\le }W$$
34	33	ex	$$⊢\phi \to \left(T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\to \neg T\underset{\dot{}}{\le }W\right)$$
35	9 34	mt2d	$$⊢\phi \to \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$