Theorem 4atexlemcnd

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	$$⊢V=\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W$$
	4thatlem0.c	$$⊢C=\left(Q\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }S\right)$$
	4thatlem0.d	$$⊢D=\left(R\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }S\right)$$
Assertion	4atexlemcnd	$$⊢\phi \to C\ne D$$

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		4thatlem0.c	$$⊢C=\left(Q\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }S\right)$$
10		4thatlem0.d	$$⊢D=\left(R\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }S\right)$$
11	1 2 3 4 5 6 7 8	4atexlemtlw	$$⊢\phi \to T\underset{\dot{}}{\le }W$$
12	1 2 3 4 5 6 7 8 9	4atexlemnclw	$$⊢\phi \to \neg C\underset{\dot{}}{\le }W$$
13		nbrne2	$$⊢\left(T\underset{\dot{}}{\le }W\wedge \neg C\underset{\dot{}}{\le }W\right)\to T\ne C$$
14	11 12 13	syl2anc	$$⊢\phi \to T\ne C$$
15	1	4atexlemk	$$⊢\phi \to K\in \mathrm{HL}$$
16	1	4atexlemq	$$⊢\phi \to Q\in A$$
17	1	4atexlemt	$$⊢\phi \to T\in A$$
18	3 5	hlatjcom	$$⊢\left(K\in \mathrm{HL}\wedge Q\in A\wedge T\in A\right)\to Q\underset{\dot{}}{\vee }T=T\underset{\dot{}}{\vee }Q$$
19	15 16 17 18	syl3anc	$$⊢\phi \to Q\underset{\dot{}}{\vee }T=T\underset{\dot{}}{\vee }Q$$
20		simp221	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(S\in A\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }R=Q\underset{\dot{}}{\vee }R\right)\wedge \left(T\in A\wedge U\underset{\dot{}}{\vee }T=V\underset{\dot{}}{\vee }T\right)\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to R\in A$$
21	1 20	sylbi	$$⊢\phi \to R\in A$$
22	3 5	hlatjcom	$$⊢\left(K\in \mathrm{HL}\wedge R\in A\wedge T\in A\right)\to R\underset{\dot{}}{\vee }T=T\underset{\dot{}}{\vee }R$$
23	15 21 17 22	syl3anc	$$⊢\phi \to R\underset{\dot{}}{\vee }T=T\underset{\dot{}}{\vee }R$$
24	19 23	oveq12d	$$⊢\phi \to \left(Q\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(R\underset{\dot{}}{\vee }T\right)=\left(T\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }R\right)$$
25	1	4atexlemkc	$$⊢\phi \to K\in \mathrm{CvLat}$$
26	1	4atexlemp	$$⊢\phi \to P\in A$$
27	1	4atexlempnq	$$⊢\phi \to P\ne Q$$
28		simp223	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(S\in A\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }R=Q\underset{\dot{}}{\vee }R\right)\wedge \left(T\in A\wedge U\underset{\dot{}}{\vee }T=V\underset{\dot{}}{\vee }T\right)\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to P\underset{\dot{}}{\vee }R=Q\underset{\dot{}}{\vee }R$$
29	1 28	sylbi	$$⊢\phi \to P\underset{\dot{}}{\vee }R=Q\underset{\dot{}}{\vee }R$$
30	5 3	cvlsupr6	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge P\underset{\dot{}}{\vee }R=Q\underset{\dot{}}{\vee }R\right)\right)\to R\ne Q$$
31	30	necomd	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge P\underset{\dot{}}{\vee }R=Q\underset{\dot{}}{\vee }R\right)\right)\to Q\ne R$$
32	25 26 16 21 27 29 31	syl132anc	$$⊢\phi \to Q\ne R$$
33	1 2 3 4 5 6 7 8	4atexlemntlpq	$$⊢\phi \to \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
34	5 3	cvlsupr7	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge P\underset{\dot{}}{\vee }R=Q\underset{\dot{}}{\vee }R\right)\right)\to P\underset{\dot{}}{\vee }Q=R\underset{\dot{}}{\vee }Q$$
35	25 26 16 21 27 29 34	syl132anc	$$⊢\phi \to P\underset{\dot{}}{\vee }Q=R\underset{\dot{}}{\vee }Q$$
36	3 5	hlatjcom	$$⊢\left(K\in \mathrm{HL}\wedge Q\in A\wedge R\in A\right)\to Q\underset{\dot{}}{\vee }R=R\underset{\dot{}}{\vee }Q$$
37	15 16 21 36	syl3anc	$$⊢\phi \to Q\underset{\dot{}}{\vee }R=R\underset{\dot{}}{\vee }Q$$
38	35 37	eqtr4d	$$⊢\phi \to P\underset{\dot{}}{\vee }Q=Q\underset{\dot{}}{\vee }R$$
39	38	breq2d	$$⊢\phi \to \left(T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\leftrightarrow T\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\right)$$
40	33 39	mtbid	$$⊢\phi \to \neg T\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)$$
41	2 3 4 5	2llnma2	$$⊢\left(K\in \mathrm{HL}\wedge \left(Q\in A\wedge R\in A\wedge T\in A\right)\wedge \left(Q\ne R\wedge \neg T\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\right)\right)\to \left(T\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }R\right)=T$$
42	15 16 21 17 32 40 41	syl132anc	$$⊢\phi \to \left(T\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }R\right)=T$$
43	24 42	eqtr2d	$$⊢\phi \to T=\left(Q\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(R\underset{\dot{}}{\vee }T\right)$$
44	43	adantr	$$⊢\left(\phi \wedge C=D\right)\to T=\left(Q\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(R\underset{\dot{}}{\vee }T\right)$$
45	1	4atexlemkl	$$⊢\phi \to K\in \mathrm{Lat}$$
46	1 3 5	4atexlemqtb	$$⊢\phi \to Q\underset{\dot{}}{\vee }T\in {\mathrm{Base}}_K$$
47	1 3 5	4atexlempsb	$$⊢\phi \to P\underset{\dot{}}{\vee }S\in {\mathrm{Base}}_K$$
48		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
49	48 2 4	latmle1	$$⊢\left(K\in \mathrm{Lat}\wedge Q\underset{\dot{}}{\vee }T\in {\mathrm{Base}}_K\wedge P\underset{\dot{}}{\vee }S\in {\mathrm{Base}}_K\right)\to \left(\left(Q\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }S\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)$$
50	45 46 47 49	syl3anc	$$⊢\phi \to \left(\left(Q\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }S\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)$$
51	9 50	eqbrtrid	$$⊢\phi \to C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)$$
52	51	adantr	$$⊢\left(\phi \wedge C=D\right)\to C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)$$
53		simpr	$$⊢\left(\phi \wedge C=D\right)\to C=D$$
54	48 3 5	hlatjcl	$$⊢\left(K\in \mathrm{HL}\wedge R\in A\wedge T\in A\right)\to R\underset{\dot{}}{\vee }T\in {\mathrm{Base}}_K$$
55	15 21 17 54	syl3anc	$$⊢\phi \to R\underset{\dot{}}{\vee }T\in {\mathrm{Base}}_K$$
56	48 2 4	latmle1	$$⊢\left(K\in \mathrm{Lat}\wedge R\underset{\dot{}}{\vee }T\in {\mathrm{Base}}_K\wedge P\underset{\dot{}}{\vee }S\in {\mathrm{Base}}_K\right)\to \left(\left(R\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }S\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }T\right)$$
57	45 55 47 56	syl3anc	$$⊢\phi \to \left(\left(R\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }S\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }T\right)$$
58	10 57	eqbrtrid	$$⊢\phi \to D\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }T\right)$$
59	58	adantr	$$⊢\left(\phi \wedge C=D\right)\to D\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }T\right)$$
60	53 59	eqbrtrd	$$⊢\left(\phi \wedge C=D\right)\to C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }T\right)$$
61	1 2 3 4 5 6 7 8 9	4atexlemc	$$⊢\phi \to C\in A$$
62	48 5	atbase	$$⊢C\in A\to C\in {\mathrm{Base}}_K$$
63	61 62	syl	$$⊢\phi \to C\in {\mathrm{Base}}_K$$
64	48 2 4	latlem12	$$⊢\left(K\in \mathrm{Lat}\wedge \left(C\in {\mathrm{Base}}_K\wedge Q\underset{\dot{}}{\vee }T\in {\mathrm{Base}}_K\wedge R\underset{\dot{}}{\vee }T\in {\mathrm{Base}}_K\right)\right)\to \left(\left(C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)\wedge C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }T\right)\right)\leftrightarrow C\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(R\underset{\dot{}}{\vee }T\right)\right)\right)$$
65	45 63 46 55 64	syl13anc	$$⊢\phi \to \left(\left(C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)\wedge C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }T\right)\right)\leftrightarrow C\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(R\underset{\dot{}}{\vee }T\right)\right)\right)$$
66	65	adantr	$$⊢\left(\phi \wedge C=D\right)\to \left(\left(C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)\wedge C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }T\right)\right)\leftrightarrow C\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(R\underset{\dot{}}{\vee }T\right)\right)\right)$$
67	52 60 66	mpbi2and	$$⊢\left(\phi \wedge C=D\right)\to C\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(R\underset{\dot{}}{\vee }T\right)\right)$$
68		hlatl	$$⊢K\in \mathrm{HL}\to K\in \mathrm{AtLat}$$
69	15 68	syl	$$⊢\phi \to K\in \mathrm{AtLat}$$
70	43 17	eqeltrrd	$$⊢\phi \to \left(Q\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(R\underset{\dot{}}{\vee }T\right)\in A$$
71	2 5	atcmp	$$⊢\left(K\in \mathrm{AtLat}\wedge C\in A\wedge \left(Q\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(R\underset{\dot{}}{\vee }T\right)\in A\right)\to \left(C\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(R\underset{\dot{}}{\vee }T\right)\right)\leftrightarrow C=\left(Q\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(R\underset{\dot{}}{\vee }T\right)\right)$$
72	69 61 70 71	syl3anc	$$⊢\phi \to \left(C\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(R\underset{\dot{}}{\vee }T\right)\right)\leftrightarrow C=\left(Q\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(R\underset{\dot{}}{\vee }T\right)\right)$$
73	72	adantr	$$⊢\left(\phi \wedge C=D\right)\to \left(C\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(R\underset{\dot{}}{\vee }T\right)\right)\leftrightarrow C=\left(Q\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(R\underset{\dot{}}{\vee }T\right)\right)$$
74	67 73	mpbid	$$⊢\left(\phi \wedge C=D\right)\to C=\left(Q\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(R\underset{\dot{}}{\vee }T\right)$$
75	44 74	eqtr4d	$$⊢\left(\phi \wedge C=D\right)\to T=C$$
76	75	ex	$$⊢\phi \to \left(C=D\to T=C\right)$$
77	76	necon3d	$$⊢\phi \to \left(T\ne C\to C\ne D\right)$$
78	14 77	mpd	$$⊢\phi \to C\ne D$$