Theorem cvlsupr2 39928

Description: Two equivalent ways of expressing that 𝑅 is a superposition of 𝑃 and 𝑄. (Contributed by NM, 5-Nov-2012.)

Hypotheses

Ref	Expression
cvlsupr2.a	⊢ 𝐴 = (Atoms‘𝐾)
cvlsupr2.l	⊢ ≤ = (le‘𝐾)
cvlsupr2.j	⊢ ∨ = (join‘𝐾)

Assertion

Ref	Expression
cvlsupr2	⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ↔ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))))

Proof of Theorem cvlsupr2

Step	Hyp	Ref	Expression
1		simpl3 1206	. . . . 5 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → 𝑃 ≠ 𝑄)
2	1	necomd 3011	. . . 4 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → 𝑄 ≠ 𝑃)
3		simplr 778	. . . . . . . . 9 ⊢ ((((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ 𝑅 = 𝑃) → (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))
4		oveq2 7399	. . . . . . . . . . . 12 ⊢ (𝑅 = 𝑃 → (𝑃 ∨ 𝑅) = (𝑃 ∨ 𝑃))
5		oveq2 7399	. . . . . . . . . . . 12 ⊢ (𝑅 = 𝑃 → (𝑄 ∨ 𝑅) = (𝑄 ∨ 𝑃))
6	4, 5	eqeq12d 2777	. . . . . . . . . . 11 ⊢ (𝑅 = 𝑃 → ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ↔ (𝑃 ∨ 𝑃) = (𝑄 ∨ 𝑃)))
7		eqcom 2768	. . . . . . . . . . 11 ⊢ ((𝑃 ∨ 𝑃) = (𝑄 ∨ 𝑃) ↔ (𝑄 ∨ 𝑃) = (𝑃 ∨ 𝑃))
8	6, 7	bitrdi 289	. . . . . . . . . 10 ⊢ (𝑅 = 𝑃 → ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ↔ (𝑄 ∨ 𝑃) = (𝑃 ∨ 𝑃)))
9	8	adantl 485	. . . . . . . . 9 ⊢ ((((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ 𝑅 = 𝑃) → ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ↔ (𝑄 ∨ 𝑃) = (𝑃 ∨ 𝑃)))
10	3, 9	mpbid 234	. . . . . . . 8 ⊢ ((((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ 𝑅 = 𝑃) → (𝑄 ∨ 𝑃) = (𝑃 ∨ 𝑃))
11		simpl1 1204	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → 𝐾 ∈ CvLat)
12		cvllat 39911	. . . . . . . . . . 11 ⊢ (𝐾 ∈ CvLat → 𝐾 ∈ Lat)
13	11, 12	syl 17	. . . . . . . . . 10 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → 𝐾 ∈ Lat)
14		simpl21 1264	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → 𝑃 ∈ 𝐴)
15		eqid 2761	. . . . . . . . . . . 12 ⊢ (Base‘𝐾) = (Base‘𝐾)
16		cvlsupr2.a	. . . . . . . . . . . 12 ⊢ 𝐴 = (Atoms‘𝐾)
17	15, 16	atbase 39874	. . . . . . . . . . 11 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
18	14, 17	syl 17	. . . . . . . . . 10 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → 𝑃 ∈ (Base‘𝐾))
19		cvlsupr2.j	. . . . . . . . . . 11 ⊢ ∨ = (join‘𝐾)
20	15, 19	latjidm 18485	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾)) → (𝑃 ∨ 𝑃) = 𝑃)
21	13, 18, 20	syl2anc 593	. . . . . . . . 9 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → (𝑃 ∨ 𝑃) = 𝑃)
22	21	adantr 484	. . . . . . . 8 ⊢ ((((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ 𝑅 = 𝑃) → (𝑃 ∨ 𝑃) = 𝑃)
23	10, 22	eqtrd 2796	. . . . . . 7 ⊢ ((((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ 𝑅 = 𝑃) → (𝑄 ∨ 𝑃) = 𝑃)
24	23	ex 416	. . . . . 6 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → (𝑅 = 𝑃 → (𝑄 ∨ 𝑃) = 𝑃))
25		simpl22 1265	. . . . . . . . 9 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → 𝑄 ∈ 𝐴)
26	15, 16	atbase 39874	. . . . . . . . 9 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
27	25, 26	syl 17	. . . . . . . 8 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → 𝑄 ∈ (Base‘𝐾))
28		cvlsupr2.l	. . . . . . . . 9 ⊢ ≤ = (le‘𝐾)
29	15, 28, 19	latleeqj1 18474	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾)) → (𝑄 ≤ 𝑃 ↔ (𝑄 ∨ 𝑃) = 𝑃))
30	13, 27, 18, 29	syl3anc 1389	. . . . . . 7 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → (𝑄 ≤ 𝑃 ↔ (𝑄 ∨ 𝑃) = 𝑃))
31		cvlatl 39910	. . . . . . . . 9 ⊢ (𝐾 ∈ CvLat → 𝐾 ∈ AtLat)
32	11, 31	syl 17	. . . . . . . 8 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → 𝐾 ∈ AtLat)
33	28, 16	atcmp 39896	. . . . . . . 8 ⊢ ((𝐾 ∈ AtLat ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → (𝑄 ≤ 𝑃 ↔ 𝑄 = 𝑃))
34	32, 25, 14, 33	syl3anc 1389	. . . . . . 7 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → (𝑄 ≤ 𝑃 ↔ 𝑄 = 𝑃))
35	30, 34	bitr3d 283	. . . . . 6 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → ((𝑄 ∨ 𝑃) = 𝑃 ↔ 𝑄 = 𝑃))
36	24, 35	sylibd 241	. . . . 5 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → (𝑅 = 𝑃 → 𝑄 = 𝑃))
37	36	necon3d 2977	. . . 4 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → (𝑄 ≠ 𝑃 → 𝑅 ≠ 𝑃))
38	2, 37	mpd 15	. . 3 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → 𝑅 ≠ 𝑃)
39		simplr 778	. . . . . . . . 9 ⊢ ((((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ 𝑅 = 𝑄) → (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))
40		oveq2 7399	. . . . . . . . . . 11 ⊢ (𝑅 = 𝑄 → (𝑃 ∨ 𝑅) = (𝑃 ∨ 𝑄))
41		oveq2 7399	. . . . . . . . . . 11 ⊢ (𝑅 = 𝑄 → (𝑄 ∨ 𝑅) = (𝑄 ∨ 𝑄))
42	40, 41	eqeq12d 2777	. . . . . . . . . 10 ⊢ (𝑅 = 𝑄 → ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ↔ (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑄)))
43	42	adantl 485	. . . . . . . . 9 ⊢ ((((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ 𝑅 = 𝑄) → ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ↔ (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑄)))
44	39, 43	mpbid 234	. . . . . . . 8 ⊢ ((((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ 𝑅 = 𝑄) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑄))
45	15, 19	latjidm 18485	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑄 ∨ 𝑄) = 𝑄)
46	13, 27, 45	syl2anc 593	. . . . . . . . 9 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → (𝑄 ∨ 𝑄) = 𝑄)
47	46	adantr 484	. . . . . . . 8 ⊢ ((((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ 𝑅 = 𝑄) → (𝑄 ∨ 𝑄) = 𝑄)
48	44, 47	eqtrd 2796	. . . . . . 7 ⊢ ((((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ 𝑅 = 𝑄) → (𝑃 ∨ 𝑄) = 𝑄)
49	48	ex 416	. . . . . 6 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → (𝑅 = 𝑄 → (𝑃 ∨ 𝑄) = 𝑄))
50	15, 28, 19	latleeqj1 18474	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑃 ≤ 𝑄 ↔ (𝑃 ∨ 𝑄) = 𝑄))
51	13, 18, 27, 50	syl3anc 1389	. . . . . . 7 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → (𝑃 ≤ 𝑄 ↔ (𝑃 ∨ 𝑄) = 𝑄))
52	28, 16	atcmp 39896	. . . . . . . 8 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ≤ 𝑄 ↔ 𝑃 = 𝑄))
53	32, 14, 25, 52	syl3anc 1389	. . . . . . 7 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → (𝑃 ≤ 𝑄 ↔ 𝑃 = 𝑄))
54	51, 53	bitr3d 283	. . . . . 6 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → ((𝑃 ∨ 𝑄) = 𝑄 ↔ 𝑃 = 𝑄))
55	49, 54	sylibd 241	. . . . 5 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → (𝑅 = 𝑄 → 𝑃 = 𝑄))
56	55	necon3d 2977	. . . 4 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → (𝑃 ≠ 𝑄 → 𝑅 ≠ 𝑄))
57	1, 56	mpd 15	. . 3 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → 𝑅 ≠ 𝑄)
58		simpl23 1266	. . . . . . 7 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → 𝑅 ∈ 𝐴)
59	15, 16	atbase 39874	. . . . . . 7 ⊢ (𝑅 ∈ 𝐴 → 𝑅 ∈ (Base‘𝐾))
60	58, 59	syl 17	. . . . . 6 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → 𝑅 ∈ (Base‘𝐾))
61	15, 28, 19	latlej1 18471	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → 𝑄 ≤ (𝑄 ∨ 𝑅))
62	13, 27, 60, 61	syl3anc 1389	. . . . 5 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → 𝑄 ≤ (𝑄 ∨ 𝑅))
63		simpr 488	. . . . 5 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))
64	62, 63	breqtrrd 5125	. . . 4 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → 𝑄 ≤ (𝑃 ∨ 𝑅))
65	28, 19, 16	cvlatexch1 39921	. . . . 5 ⊢ ((𝐾 ∈ CvLat ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) ∧ 𝑄 ≠ 𝑃) → (𝑄 ≤ (𝑃 ∨ 𝑅) → 𝑅 ≤ (𝑃 ∨ 𝑄)))
66	11, 25, 58, 14, 2, 65	syl131anc 1401	. . . 4 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → (𝑄 ≤ (𝑃 ∨ 𝑅) → 𝑅 ≤ (𝑃 ∨ 𝑄)))
67	64, 66	mpd 15	. . 3 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → 𝑅 ≤ (𝑃 ∨ 𝑄))
68	38, 57, 67	3jca 1140	. 2 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)))
69		simpr3 1209	. . 3 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝑅 ≤ (𝑃 ∨ 𝑄))
70		simpl1 1204	. . . . . . 7 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝐾 ∈ CvLat)
71	70, 12	syl 17	. . . . . 6 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝐾 ∈ Lat)
72		simpl21 1264	. . . . . . 7 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝑃 ∈ 𝐴)
73	72, 17	syl 17	. . . . . 6 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝑃 ∈ (Base‘𝐾))
74		simpl22 1265	. . . . . . 7 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝑄 ∈ 𝐴)
75	74, 26	syl 17	. . . . . 6 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝑄 ∈ (Base‘𝐾))
76	15, 19	latjcom 18470	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
77	71, 73, 75, 76	syl3anc 1389	. . . . 5 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
78	77	breq2d 5109	. . . 4 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝑅 ≤ (𝑃 ∨ 𝑄) ↔ 𝑅 ≤ (𝑄 ∨ 𝑃)))
79		simpl23 1266	. . . . . 6 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝑅 ∈ 𝐴)
80		simpr2 1208	. . . . . 6 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝑅 ≠ 𝑄)
81	28, 19, 16	cvlatexch1 39921	. . . . . 6 ⊢ ((𝐾 ∈ CvLat ∧ (𝑅 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑅 ≠ 𝑄) → (𝑅 ≤ (𝑄 ∨ 𝑃) → 𝑃 ≤ (𝑄 ∨ 𝑅)))
82	70, 79, 72, 74, 80, 81	syl131anc 1401	. . . . 5 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝑅 ≤ (𝑄 ∨ 𝑃) → 𝑃 ≤ (𝑄 ∨ 𝑅)))
83		simpr1 1207	. . . . . . 7 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝑅 ≠ 𝑃)
84	83	necomd 3011	. . . . . 6 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝑃 ≠ 𝑅)
85	28, 19, 16	cvlatexchb2 39920	. . . . . 6 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑅) → (𝑃 ≤ (𝑄 ∨ 𝑅) ↔ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)))
86	70, 72, 74, 79, 84, 85	syl131anc 1401	. . . . 5 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝑃 ≤ (𝑄 ∨ 𝑅) ↔ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)))
87	82, 86	sylibd 241	. . . 4 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝑅 ≤ (𝑄 ∨ 𝑃) → (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)))
88	78, 87	sylbid 242	. . 3 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝑅 ≤ (𝑃 ∨ 𝑄) → (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)))
89	69, 88	mpd 15	. 2 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))
90	68, 89	impbida 810	1 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ↔ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141   ≠ wne 2956   class class class wbr 5097  ‘cfv 6516  (class class class)co 7391  Basecbs 17236  lecple 17284  joincjn 18334  Latclat 18454  Atomscatm 39848  AtLatcal 39849  CvLatclc 39850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-riota 7348  df-ov 7394  df-oprab 7395  df-proset 18317  df-poset 18336  df-plt 18351  df-lub 18367  df-glb 18368  df-join 18369  df-meet 18370  df-p0 18446  df-lat 18455  df-covers 39851  df-ats 39852  df-atl 39883  df-cvlat 39907

This theorem is referenced by:  cvlsupr3  39929  cvlsupr4  39930  cvlsupr5  39931  cvlsupr6  39932  4atexlemex2  40656  4atex  40661  4atex3  40666  cdleme02N  40807  cdleme0ex2N  40809  cdleme0moN  40810  cdleme0nex  40875