Theorem cvlsupr2 40098

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 1212	. . . . 5 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → 𝑃 ≠ 𝑄)
2	1	necomd 3013	. . . 4 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → 𝑄 ≠ 𝑃)
3		simplr 780	. . . . . . . . 9 ⊢ ((((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ 𝑅 = 𝑃) → (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))
4		oveq2 7420	. . . . . . . . . . . 12 ⊢ (𝑅 = 𝑃 → (𝑃 ∨ 𝑅) = (𝑃 ∨ 𝑃))
5		oveq2 7420	. . . . . . . . . . . 12 ⊢ (𝑅 = 𝑃 → (𝑄 ∨ 𝑅) = (𝑄 ∨ 𝑃))
6	4, 5	eqeq12d 2779	. . . . . . . . . . 11 ⊢ (𝑅 = 𝑃 → ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ↔ (𝑃 ∨ 𝑃) = (𝑄 ∨ 𝑃)))
7		eqcom 2770	. . . . . . . . . . 11 ⊢ ((𝑃 ∨ 𝑃) = (𝑄 ∨ 𝑃) ↔ (𝑄 ∨ 𝑃) = (𝑃 ∨ 𝑃))
8	6, 7	bitrdi 290	. . . . . . . . . 10 ⊢ (𝑅 = 𝑃 → ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ↔ (𝑄 ∨ 𝑃) = (𝑃 ∨ 𝑃)))
9	8	adantl 486	. . . . . . . . 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 235	. . . . . . . 8 ⊢ ((((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ 𝑅 = 𝑃) → (𝑄 ∨ 𝑃) = (𝑃 ∨ 𝑃))
11		simpl1 1210	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → 𝐾 ∈ CvLat)
12		cvllat 40081	. . . . . . . . . . 11 ⊢ (𝐾 ∈ CvLat → 𝐾 ∈ Lat)
13	11, 12	syl 18	. . . . . . . . . 10 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → 𝐾 ∈ Lat)
14		simpl21 1270	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → 𝑃 ∈ 𝐴)
15		eqid 2763	. . . . . . . . . . . 12 ⊢ (Base‘𝐾) = (Base‘𝐾)
16		cvlsupr2.a	. . . . . . . . . . . 12 ⊢ 𝐴 = (Atoms‘𝐾)
17	15, 16	atbase 40044	. . . . . . . . . . 11 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
18	14, 17	syl 18	. . . . . . . . . 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 18519	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾)) → (𝑃 ∨ 𝑃) = 𝑃)
21	13, 18, 20	syl2anc 595	. . . . . . . . 9 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → (𝑃 ∨ 𝑃) = 𝑃)
22	21	adantr 485	. . . . . . . 8 ⊢ ((((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ 𝑅 = 𝑃) → (𝑃 ∨ 𝑃) = 𝑃)
23	10, 22	eqtrd 2798	. . . . . . 7 ⊢ ((((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ 𝑅 = 𝑃) → (𝑄 ∨ 𝑃) = 𝑃)
24	23	ex 417	. . . . . 6 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → (𝑅 = 𝑃 → (𝑄 ∨ 𝑃) = 𝑃))
25		simpl22 1271	. . . . . . . . 9 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → 𝑄 ∈ 𝐴)
26	15, 16	atbase 40044	. . . . . . . . 9 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
27	25, 26	syl 18	. . . . . . . 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 18508	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾)) → (𝑄 ≤ 𝑃 ↔ (𝑄 ∨ 𝑃) = 𝑃))
30	13, 27, 18, 29	syl3anc 1398	. . . . . . 7 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → (𝑄 ≤ 𝑃 ↔ (𝑄 ∨ 𝑃) = 𝑃))
31		cvlatl 40080	. . . . . . . . 9 ⊢ (𝐾 ∈ CvLat → 𝐾 ∈ AtLat)
32	11, 31	syl 18	. . . . . . . 8 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → 𝐾 ∈ AtLat)
33	28, 16	atcmp 40066	. . . . . . . 8 ⊢ ((𝐾 ∈ AtLat ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → (𝑄 ≤ 𝑃 ↔ 𝑄 = 𝑃))
34	32, 25, 14, 33	syl3anc 1398	. . . . . . 7 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → (𝑄 ≤ 𝑃 ↔ 𝑄 = 𝑃))
35	30, 34	bitr3d 284	. . . . . 6 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → ((𝑄 ∨ 𝑃) = 𝑃 ↔ 𝑄 = 𝑃))
36	24, 35	sylibd 242	. . . . 5 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → (𝑅 = 𝑃 → 𝑄 = 𝑃))
37	36	necon3d 2979	. . . 4 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → (𝑄 ≠ 𝑃 → 𝑅 ≠ 𝑃))
38	2, 37	mpd 16	. . 3 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → 𝑅 ≠ 𝑃)
39		simplr 780	. . . . . . . . 9 ⊢ ((((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ 𝑅 = 𝑄) → (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))
40		oveq2 7420	. . . . . . . . . . 11 ⊢ (𝑅 = 𝑄 → (𝑃 ∨ 𝑅) = (𝑃 ∨ 𝑄))
41		oveq2 7420	. . . . . . . . . . 11 ⊢ (𝑅 = 𝑄 → (𝑄 ∨ 𝑅) = (𝑄 ∨ 𝑄))
42	40, 41	eqeq12d 2779	. . . . . . . . . 10 ⊢ (𝑅 = 𝑄 → ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ↔ (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑄)))
43	42	adantl 486	. . . . . . . . 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 235	. . . . . . . 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 18519	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑄 ∨ 𝑄) = 𝑄)
46	13, 27, 45	syl2anc 595	. . . . . . . . 9 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → (𝑄 ∨ 𝑄) = 𝑄)
47	46	adantr 485	. . . . . . . 8 ⊢ ((((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ 𝑅 = 𝑄) → (𝑄 ∨ 𝑄) = 𝑄)
48	44, 47	eqtrd 2798	. . . . . . 7 ⊢ ((((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ 𝑅 = 𝑄) → (𝑃 ∨ 𝑄) = 𝑄)
49	48	ex 417	. . . . . 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 18508	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑃 ≤ 𝑄 ↔ (𝑃 ∨ 𝑄) = 𝑄))
51	13, 18, 27, 50	syl3anc 1398	. . . . . . 7 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → (𝑃 ≤ 𝑄 ↔ (𝑃 ∨ 𝑄) = 𝑄))
52	28, 16	atcmp 40066	. . . . . . . 8 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ≤ 𝑄 ↔ 𝑃 = 𝑄))
53	32, 14, 25, 52	syl3anc 1398	. . . . . . 7 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → (𝑃 ≤ 𝑄 ↔ 𝑃 = 𝑄))
54	51, 53	bitr3d 284	. . . . . 6 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → ((𝑃 ∨ 𝑄) = 𝑄 ↔ 𝑃 = 𝑄))
55	49, 54	sylibd 242	. . . . 5 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → (𝑅 = 𝑄 → 𝑃 = 𝑄))
56	55	necon3d 2979	. . . 4 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → (𝑃 ≠ 𝑄 → 𝑅 ≠ 𝑄))
57	1, 56	mpd 16	. . 3 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → 𝑅 ≠ 𝑄)
58		simpl23 1272	. . . . . . 7 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → 𝑅 ∈ 𝐴)
59	15, 16	atbase 40044	. . . . . . 7 ⊢ (𝑅 ∈ 𝐴 → 𝑅 ∈ (Base‘𝐾))
60	58, 59	syl 18	. . . . . 6 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → 𝑅 ∈ (Base‘𝐾))
61	15, 28, 19	latlej1 18505	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → 𝑄 ≤ (𝑄 ∨ 𝑅))
62	13, 27, 60, 61	syl3anc 1398	. . . . 5 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → 𝑄 ≤ (𝑄 ∨ 𝑅))
63		simpr 489	. . . . 5 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))
64	62, 63	breqtrrd 5140	. . . 4 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → 𝑄 ≤ (𝑃 ∨ 𝑅))
65	28, 19, 16	cvlatexch1 40091	. . . . 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 1410	. . . 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 16	. . 3 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → 𝑅 ≤ (𝑃 ∨ 𝑄))
68	38, 57, 67	3jca 1146	. 2 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)))
69		simpr3 1215	. . 3 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝑅 ≤ (𝑃 ∨ 𝑄))
70		simpl1 1210	. . . . . . 7 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝐾 ∈ CvLat)
71	70, 12	syl 18	. . . . . 6 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝐾 ∈ Lat)
72		simpl21 1270	. . . . . . 7 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝑃 ∈ 𝐴)
73	72, 17	syl 18	. . . . . 6 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝑃 ∈ (Base‘𝐾))
74		simpl22 1271	. . . . . . 7 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝑄 ∈ 𝐴)
75	74, 26	syl 18	. . . . . 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 18504	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
77	71, 73, 75, 76	syl3anc 1398	. . . . 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 5122	. . . 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 1272	. . . . . 6 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝑅 ∈ 𝐴)
80		simpr2 1214	. . . . . 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 40091	. . . . . 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 1410	. . . . 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 1213	. . . . . . 7 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝑅 ≠ 𝑃)
84	83	necomd 3013	. . . . . 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 40090	. . . . . 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 1410	. . . . 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 242	. . . 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 243	. . 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 16	. 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 812	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 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ≠ wne 2958   class class class wbr 5110  ‘cfv 6538  (class class class)co 7412  Basecbs 17270  lecple 17318  joincjn 18368  Latclat 18488  Atomscatm 40018  AtLatcal 40019  CvLatclc 40020

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-proset 18351  df-poset 18370  df-plt 18385  df-lub 18401  df-glb 18402  df-join 18403  df-meet 18404  df-p0 18480  df-lat 18489  df-covers 40021  df-ats 40022  df-atl 40053  df-cvlat 40077

This theorem is referenced by:  cvlsupr3  40099  cvlsupr4  40100  cvlsupr5  40101  cvlsupr6  40102  4atexlemex2  40826  4atex  40831  4atex3  40836  cdleme02N  40977  cdleme0ex2N  40979  cdleme0moN  40980  cdleme0nex  41045