Theorem cvlsupr2 39713

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 1195	. . . . 5 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → 𝑃 ≠ 𝑄)
2	1	necomd 2988	. . . 4 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → 𝑄 ≠ 𝑃)
3		simplr 769	. . . . . . . . 9 ⊢ ((((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ 𝑅 = 𝑃) → (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))
4		oveq2 7376	. . . . . . . . . . . 12 ⊢ (𝑅 = 𝑃 → (𝑃 ∨ 𝑅) = (𝑃 ∨ 𝑃))
5		oveq2 7376	. . . . . . . . . . . 12 ⊢ (𝑅 = 𝑃 → (𝑄 ∨ 𝑅) = (𝑄 ∨ 𝑃))
6	4, 5	eqeq12d 2753	. . . . . . . . . . 11 ⊢ (𝑅 = 𝑃 → ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ↔ (𝑃 ∨ 𝑃) = (𝑄 ∨ 𝑃)))
7		eqcom 2744	. . . . . . . . . . 11 ⊢ ((𝑃 ∨ 𝑃) = (𝑄 ∨ 𝑃) ↔ (𝑄 ∨ 𝑃) = (𝑃 ∨ 𝑃))
8	6, 7	bitrdi 287	. . . . . . . . . 10 ⊢ (𝑅 = 𝑃 → ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ↔ (𝑄 ∨ 𝑃) = (𝑃 ∨ 𝑃)))
9	8	adantl 481	. . . . . . . . 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 232	. . . . . . . 8 ⊢ ((((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ 𝑅 = 𝑃) → (𝑄 ∨ 𝑃) = (𝑃 ∨ 𝑃))
11		simpl1 1193	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → 𝐾 ∈ CvLat)
12		cvllat 39696	. . . . . . . . . . 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 1253	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → 𝑃 ∈ 𝐴)
15		eqid 2737	. . . . . . . . . . . 12 ⊢ (Base‘𝐾) = (Base‘𝐾)
16		cvlsupr2.a	. . . . . . . . . . . 12 ⊢ 𝐴 = (Atoms‘𝐾)
17	15, 16	atbase 39659	. . . . . . . . . . 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 18397	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾)) → (𝑃 ∨ 𝑃) = 𝑃)
21	13, 18, 20	syl2anc 585	. . . . . . . . 9 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → (𝑃 ∨ 𝑃) = 𝑃)
22	21	adantr 480	. . . . . . . 8 ⊢ ((((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ 𝑅 = 𝑃) → (𝑃 ∨ 𝑃) = 𝑃)
23	10, 22	eqtrd 2772	. . . . . . 7 ⊢ ((((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ 𝑅 = 𝑃) → (𝑄 ∨ 𝑃) = 𝑃)
24	23	ex 412	. . . . . 6 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → (𝑅 = 𝑃 → (𝑄 ∨ 𝑃) = 𝑃))
25		simpl22 1254	. . . . . . . . 9 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → 𝑄 ∈ 𝐴)
26	15, 16	atbase 39659	. . . . . . . . 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 18386	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾)) → (𝑄 ≤ 𝑃 ↔ (𝑄 ∨ 𝑃) = 𝑃))
30	13, 27, 18, 29	syl3anc 1374	. . . . . . 7 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → (𝑄 ≤ 𝑃 ↔ (𝑄 ∨ 𝑃) = 𝑃))
31		cvlatl 39695	. . . . . . . . 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 39681	. . . . . . . 8 ⊢ ((𝐾 ∈ AtLat ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → (𝑄 ≤ 𝑃 ↔ 𝑄 = 𝑃))
34	32, 25, 14, 33	syl3anc 1374	. . . . . . 7 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → (𝑄 ≤ 𝑃 ↔ 𝑄 = 𝑃))
35	30, 34	bitr3d 281	. . . . . 6 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → ((𝑄 ∨ 𝑃) = 𝑃 ↔ 𝑄 = 𝑃))
36	24, 35	sylibd 239	. . . . 5 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → (𝑅 = 𝑃 → 𝑄 = 𝑃))
37	36	necon3d 2954	. . . 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 769	. . . . . . . . 9 ⊢ ((((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ 𝑅 = 𝑄) → (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))
40		oveq2 7376	. . . . . . . . . . 11 ⊢ (𝑅 = 𝑄 → (𝑃 ∨ 𝑅) = (𝑃 ∨ 𝑄))
41		oveq2 7376	. . . . . . . . . . 11 ⊢ (𝑅 = 𝑄 → (𝑄 ∨ 𝑅) = (𝑄 ∨ 𝑄))
42	40, 41	eqeq12d 2753	. . . . . . . . . 10 ⊢ (𝑅 = 𝑄 → ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ↔ (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑄)))
43	42	adantl 481	. . . . . . . . 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 232	. . . . . . . 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 18397	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑄 ∨ 𝑄) = 𝑄)
46	13, 27, 45	syl2anc 585	. . . . . . . . 9 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → (𝑄 ∨ 𝑄) = 𝑄)
47	46	adantr 480	. . . . . . . 8 ⊢ ((((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ 𝑅 = 𝑄) → (𝑄 ∨ 𝑄) = 𝑄)
48	44, 47	eqtrd 2772	. . . . . . 7 ⊢ ((((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ 𝑅 = 𝑄) → (𝑃 ∨ 𝑄) = 𝑄)
49	48	ex 412	. . . . . 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 18386	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑃 ≤ 𝑄 ↔ (𝑃 ∨ 𝑄) = 𝑄))
51	13, 18, 27, 50	syl3anc 1374	. . . . . . 7 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → (𝑃 ≤ 𝑄 ↔ (𝑃 ∨ 𝑄) = 𝑄))
52	28, 16	atcmp 39681	. . . . . . . 8 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ≤ 𝑄 ↔ 𝑃 = 𝑄))
53	32, 14, 25, 52	syl3anc 1374	. . . . . . 7 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → (𝑃 ≤ 𝑄 ↔ 𝑃 = 𝑄))
54	51, 53	bitr3d 281	. . . . . 6 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → ((𝑃 ∨ 𝑄) = 𝑄 ↔ 𝑃 = 𝑄))
55	49, 54	sylibd 239	. . . . 5 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → (𝑅 = 𝑄 → 𝑃 = 𝑄))
56	55	necon3d 2954	. . . 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 1255	. . . . . . 7 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → 𝑅 ∈ 𝐴)
59	15, 16	atbase 39659	. . . . . . 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 18383	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → 𝑄 ≤ (𝑄 ∨ 𝑅))
62	13, 27, 60, 61	syl3anc 1374	. . . . 5 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → 𝑄 ≤ (𝑄 ∨ 𝑅))
63		simpr 484	. . . . 5 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))
64	62, 63	breqtrrd 5128	. . . 4 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → 𝑄 ≤ (𝑃 ∨ 𝑅))
65	28, 19, 16	cvlatexch1 39706	. . . . 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 1386	. . . 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 1129	. 2 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)))
69		simpr3 1198	. . 3 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝑅 ≤ (𝑃 ∨ 𝑄))
70		simpl1 1193	. . . . . . 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 1253	. . . . . . 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 1254	. . . . . . 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 18382	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
77	71, 73, 75, 76	syl3anc 1374	. . . . 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 5112	. . . 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 1255	. . . . . 6 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝑅 ∈ 𝐴)
80		simpr2 1197	. . . . . 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 39706	. . . . . 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 1386	. . . . 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 1196	. . . . . . 7 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝑅 ≠ 𝑃)
84	83	necomd 2988	. . . . . 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 39705	. . . . . 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 1386	. . . . 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 239	. . . 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 240	. . 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 801	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 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933   class class class wbr 5100  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  lecple 17196  joincjn 18246  Latclat 18366  Atomscatm 39633  AtLatcal 39634  CvLatclc 39635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-proset 18229  df-poset 18248  df-plt 18263  df-lub 18279  df-glb 18280  df-join 18281  df-meet 18282  df-p0 18358  df-lat 18367  df-covers 39636  df-ats 39637  df-atl 39668  df-cvlat 39692

This theorem is referenced by:  cvlsupr3  39714  cvlsupr4  39715  cvlsupr5  39716  cvlsupr6  39717  4atexlemex2  40441  4atex  40446  4atex3  40451  cdleme02N  40592  cdleme0ex2N  40594  cdleme0moN  40595  cdleme0nex  40660