Theorem 4atexlemunv

Description: Lemma for 4atexlem7 . (Contributed by NM, 21-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	4atexlemunv	$$⊢\phi \to U\ne V$$

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	4atexlemnslpq	$$⊢\phi \to \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
10	1	4atexlemk	$$⊢\phi \to K\in \mathrm{HL}$$
11	1	4atexlemp	$$⊢\phi \to P\in A$$
12	1	4atexlems	$$⊢\phi \to S\in A$$
13	2 3 5	hlatlej2	$$⊢\left(K\in \mathrm{HL}\wedge P\in A\wedge S\in A\right)\to S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)$$
14	10 11 12 13	syl3anc	$$⊢\phi \to S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)$$
15	14	adantr	$$⊢\left(\phi \wedge U=V\right)\to S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)$$
16	1	4atexlemkl	$$⊢\phi \to K\in \mathrm{Lat}$$
17	1 3 5	4atexlempsb	$$⊢\phi \to P\underset{\dot{}}{\vee }S\in {\mathrm{Base}}_K$$
18	1 6	4atexlemwb	$$⊢\phi \to W\in {\mathrm{Base}}_K$$
19		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
20	19 2 4	latmle1	$$⊢\left(K\in \mathrm{Lat}\wedge P\underset{\dot{}}{\vee }S\in {\mathrm{Base}}_K\wedge W\in {\mathrm{Base}}_K\right)\to \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)$$
21	16 17 18 20	syl3anc	$$⊢\phi \to \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)$$
22	8 21	eqbrtrid	$$⊢\phi \to V\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)$$
23	1	4atexlemkc	$$⊢\phi \to K\in \mathrm{CvLat}$$
24	1 2 3 4 5 6 7 8	4atexlemv	$$⊢\phi \to V\in A$$
25	19 2 4	latmle2	$$⊢\left(K\in \mathrm{Lat}\wedge P\underset{\dot{}}{\vee }S\in {\mathrm{Base}}_K\wedge W\in {\mathrm{Base}}_K\right)\to \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\le }W$$
26	16 17 18 25	syl3anc	$$⊢\phi \to \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\le }W$$
27	8 26	eqbrtrid	$$⊢\phi \to V\underset{\dot{}}{\le }W$$
28	1	4atexlempw	$$⊢\phi \to \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)$$
29	28	simprd	$$⊢\phi \to \neg P\underset{\dot{}}{\le }W$$
30		nbrne2	$$⊢\left(V\underset{\dot{}}{\le }W\wedge \neg P\underset{\dot{}}{\le }W\right)\to V\ne P$$
31	27 29 30	syl2anc	$$⊢\phi \to V\ne P$$
32	2 3 5	cvlatexchb1	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(V\in A\wedge S\in A\wedge P\in A\right)\wedge V\ne P\right)\to \left(V\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\leftrightarrow P\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }S\right)$$
33	23 24 12 11 31 32	syl131anc	$$⊢\phi \to \left(V\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\leftrightarrow P\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }S\right)$$
34	22 33	mpbid	$$⊢\phi \to P\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }S$$
35	34	adantr	$$⊢\left(\phi \wedge U=V\right)\to P\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }S$$
36		oveq2	$$⊢U=V\to P\underset{\dot{}}{\vee }U=P\underset{\dot{}}{\vee }V$$
37	36	eqcomd	$$⊢U=V\to P\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }U$$
38	1	4atexlemq	$$⊢\phi \to Q\in A$$
39	19 3 5	hlatjcl	$$⊢\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\to P\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K$$
40	10 11 38 39	syl3anc	$$⊢\phi \to P\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K$$
41	19 2 4	latmle1	$$⊢\left(K\in \mathrm{Lat}\wedge P\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K\wedge W\in {\mathrm{Base}}_K\right)\to \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
42	16 40 18 41	syl3anc	$$⊢\phi \to \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
43	7 42	eqbrtrid	$$⊢\phi \to U\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
44	1 2 3 4 5 6 7	4atexlemu	$$⊢\phi \to U\in A$$
45	19 2 4	latmle2	$$⊢\left(K\in \mathrm{Lat}\wedge P\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K\wedge W\in {\mathrm{Base}}_K\right)\to \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\le }W$$
46	16 40 18 45	syl3anc	$$⊢\phi \to \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\le }W$$
47	7 46	eqbrtrid	$$⊢\phi \to U\underset{\dot{}}{\le }W$$
48		nbrne2	$$⊢\left(U\underset{\dot{}}{\le }W\wedge \neg P\underset{\dot{}}{\le }W\right)\to U\ne P$$
49	47 29 48	syl2anc	$$⊢\phi \to U\ne P$$
50	2 3 5	cvlatexchb1	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(U\in A\wedge Q\in A\wedge P\in A\right)\wedge U\ne P\right)\to \left(U\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\leftrightarrow P\underset{\dot{}}{\vee }U=P\underset{\dot{}}{\vee }Q\right)$$
51	23 44 38 11 49 50	syl131anc	$$⊢\phi \to \left(U\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\leftrightarrow P\underset{\dot{}}{\vee }U=P\underset{\dot{}}{\vee }Q\right)$$
52	43 51	mpbid	$$⊢\phi \to P\underset{\dot{}}{\vee }U=P\underset{\dot{}}{\vee }Q$$
53	37 52	sylan9eqr	$$⊢\left(\phi \wedge U=V\right)\to P\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q$$
54	35 53	eqtr3d	$$⊢\left(\phi \wedge U=V\right)\to P\underset{\dot{}}{\vee }S=P\underset{\dot{}}{\vee }Q$$
55	15 54	breqtrd	$$⊢\left(\phi \wedge U=V\right)\to S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
56	55	ex	$$⊢\phi \to \left(U=V\to S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)$$
57	56	necon3bd	$$⊢\phi \to \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\to U\ne V\right)$$
58	9 57	mpd	$$⊢\phi \to U\ne V$$