Theorem 4atexlemtlw

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	4atexlemtlw	$$⊢\phi \to T\underset{\dot{}}{\le }W$$

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		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
10	1	4atexlemkl	$$⊢\phi \to K\in \mathrm{Lat}$$
11	1	4atexlemt	$$⊢\phi \to T\in A$$
12	9 5	atbase	$$⊢T\in A\to T\in {\mathrm{Base}}_K$$
13	11 12	syl	$$⊢\phi \to T\in {\mathrm{Base}}_K$$
14	1	4atexlemk	$$⊢\phi \to K\in \mathrm{HL}$$
15	1 2 3 4 5 6 7	4atexlemu	$$⊢\phi \to U\in A$$
16	1 2 3 4 5 6 7 8	4atexlemv	$$⊢\phi \to V\in A$$
17	9 3 5	hlatjcl	$$⊢\left(K\in \mathrm{HL}\wedge U\in A\wedge V\in A\right)\to U\underset{\dot{}}{\vee }V\in {\mathrm{Base}}_K$$
18	14 15 16 17	syl3anc	$$⊢\phi \to U\underset{\dot{}}{\vee }V\in {\mathrm{Base}}_K$$
19	1 6	4atexlemwb	$$⊢\phi \to W\in {\mathrm{Base}}_K$$
20	1	4atexlemkc	$$⊢\phi \to K\in \mathrm{CvLat}$$
21	1 2 3 4 5 6 7 8	4atexlemunv	$$⊢\phi \to U\ne V$$
22	1	4atexlemutvt	$$⊢\phi \to U\underset{\dot{}}{\vee }T=V\underset{\dot{}}{\vee }T$$
23	5 2 3	cvlsupr4	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(U\in A\wedge V\in A\wedge T\in A\right)\wedge \left(U\ne V\wedge U\underset{\dot{}}{\vee }T=V\underset{\dot{}}{\vee }T\right)\right)\to T\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }V\right)$$
24	20 15 16 11 21 22 23	syl132anc	$$⊢\phi \to T\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }V\right)$$
25	1	4atexlemp	$$⊢\phi \to P\in A$$
26	1	4atexlemq	$$⊢\phi \to Q\in A$$
27	9 3 5	hlatjcl	$$⊢\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\to P\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K$$
28	14 25 26 27	syl3anc	$$⊢\phi \to P\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K$$
29	9 2 4	latmle2	$$⊢\left(K\in \mathrm{Lat}\wedge P\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K\wedge W\in {\mathrm{Base}}_K\right)\to \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\le }W$$
30	10 28 19 29	syl3anc	$$⊢\phi \to \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\le }W$$
31	7 30	eqbrtrid	$$⊢\phi \to U\underset{\dot{}}{\le }W$$
32	1 3 5	4atexlempsb	$$⊢\phi \to P\underset{\dot{}}{\vee }S\in {\mathrm{Base}}_K$$
33	9 2 4	latmle2	$$⊢\left(K\in \mathrm{Lat}\wedge P\underset{\dot{}}{\vee }S\in {\mathrm{Base}}_K\wedge W\in {\mathrm{Base}}_K\right)\to \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\le }W$$
34	10 32 19 33	syl3anc	$$⊢\phi \to \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\le }W$$
35	8 34	eqbrtrid	$$⊢\phi \to V\underset{\dot{}}{\le }W$$
36	9 5	atbase	$$⊢U\in A\to U\in {\mathrm{Base}}_K$$
37	15 36	syl	$$⊢\phi \to U\in {\mathrm{Base}}_K$$
38	9 5	atbase	$$⊢V\in A\to V\in {\mathrm{Base}}_K$$
39	16 38	syl	$$⊢\phi \to V\in {\mathrm{Base}}_K$$
40	9 2 3	latjle12	$$⊢\left(K\in \mathrm{Lat}\wedge \left(U\in {\mathrm{Base}}_K\wedge V\in {\mathrm{Base}}_K\wedge W\in {\mathrm{Base}}_K\right)\right)\to \left(\left(U\underset{\dot{}}{\le }W\wedge V\underset{\dot{}}{\le }W\right)\leftrightarrow \left(U\underset{\dot{}}{\vee }V\right)\underset{\dot{}}{\le }W\right)$$
41	10 37 39 19 40	syl13anc	$$⊢\phi \to \left(\left(U\underset{\dot{}}{\le }W\wedge V\underset{\dot{}}{\le }W\right)\leftrightarrow \left(U\underset{\dot{}}{\vee }V\right)\underset{\dot{}}{\le }W\right)$$
42	31 35 41	mpbi2and	$$⊢\phi \to \left(U\underset{\dot{}}{\vee }V\right)\underset{\dot{}}{\le }W$$
43	9 2 10 13 18 19 24 42	lattrd	$$⊢\phi \to T\underset{\dot{}}{\le }W$$