Theorem cvlsupr2 39336

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 1194	. . . . 5 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → 𝑃 ≠ 𝑄)
2	1	necomd 2980	. . . 4 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → 𝑄 ≠ 𝑃)
3		simplr 768	. . . . . . . . 9 ⊢ ((((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ 𝑅 = 𝑃) → (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))
4		oveq2 7395	. . . . . . . . . . . 12 ⊢ (𝑅 = 𝑃 → (𝑃 ∨ 𝑅) = (𝑃 ∨ 𝑃))
5		oveq2 7395	. . . . . . . . . . . 12 ⊢ (𝑅 = 𝑃 → (𝑄 ∨ 𝑅) = (𝑄 ∨ 𝑃))
6	4, 5	eqeq12d 2745	. . . . . . . . . . 11 ⊢ (𝑅 = 𝑃 → ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ↔ (𝑃 ∨ 𝑃) = (𝑄 ∨ 𝑃)))
7		eqcom 2736	. . . . . . . . . . 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 1192	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → 𝐾 ∈ CvLat)
12		cvllat 39319	. . . . . . . . . . 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 1252	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → 𝑃 ∈ 𝐴)
15		eqid 2729	. . . . . . . . . . . 12 ⊢ (Base‘𝐾) = (Base‘𝐾)
16		cvlsupr2.a	. . . . . . . . . . . 12 ⊢ 𝐴 = (Atoms‘𝐾)
17	15, 16	atbase 39282	. . . . . . . . . . 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 18421	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾)) → (𝑃 ∨ 𝑃) = 𝑃)
21	13, 18, 20	syl2anc 584	. . . . . . . . 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 2764	. . . . . . 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 1253	. . . . . . . . 9 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → 𝑄 ∈ 𝐴)
26	15, 16	atbase 39282	. . . . . . . . 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 18410	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾)) → (𝑄 ≤ 𝑃 ↔ (𝑄 ∨ 𝑃) = 𝑃))
30	13, 27, 18, 29	syl3anc 1373	. . . . . . 7 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → (𝑄 ≤ 𝑃 ↔ (𝑄 ∨ 𝑃) = 𝑃))
31		cvlatl 39318	. . . . . . . . 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 39304	. . . . . . . 8 ⊢ ((𝐾 ∈ AtLat ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → (𝑄 ≤ 𝑃 ↔ 𝑄 = 𝑃))
34	32, 25, 14, 33	syl3anc 1373	. . . . . . 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 2946	. . . 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 768	. . . . . . . . 9 ⊢ ((((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ 𝑅 = 𝑄) → (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))
40		oveq2 7395	. . . . . . . . . . 11 ⊢ (𝑅 = 𝑄 → (𝑃 ∨ 𝑅) = (𝑃 ∨ 𝑄))
41		oveq2 7395	. . . . . . . . . . 11 ⊢ (𝑅 = 𝑄 → (𝑄 ∨ 𝑅) = (𝑄 ∨ 𝑄))
42	40, 41	eqeq12d 2745	. . . . . . . . . 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 18421	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑄 ∨ 𝑄) = 𝑄)
46	13, 27, 45	syl2anc 584	. . . . . . . . 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 2764	. . . . . . 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 18410	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑃 ≤ 𝑄 ↔ (𝑃 ∨ 𝑄) = 𝑄))
51	13, 18, 27, 50	syl3anc 1373	. . . . . . 7 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → (𝑃 ≤ 𝑄 ↔ (𝑃 ∨ 𝑄) = 𝑄))
52	28, 16	atcmp 39304	. . . . . . . 8 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ≤ 𝑄 ↔ 𝑃 = 𝑄))
53	32, 14, 25, 52	syl3anc 1373	. . . . . . 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 2946	. . . 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 1254	. . . . . . 7 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → 𝑅 ∈ 𝐴)
59	15, 16	atbase 39282	. . . . . . 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 18407	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → 𝑄 ≤ (𝑄 ∨ 𝑅))
62	13, 27, 60, 61	syl3anc 1373	. . . . 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 5135	. . . 4 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → 𝑄 ≤ (𝑃 ∨ 𝑅))
65	28, 19, 16	cvlatexch1 39329	. . . . 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 1385	. . . 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 1128	. 2 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)))
69		simpr3 1197	. . 3 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝑅 ≤ (𝑃 ∨ 𝑄))
70		simpl1 1192	. . . . . . 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 1252	. . . . . . 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 1253	. . . . . . 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 18406	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
77	71, 73, 75, 76	syl3anc 1373	. . . . 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 5119	. . . 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 1254	. . . . . 6 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝑅 ∈ 𝐴)
80		simpr2 1196	. . . . . 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 39329	. . . . . 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 1385	. . . . 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 1195	. . . . . . 7 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝑅 ≠ 𝑃)
84	83	necomd 2980	. . . . . 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 39328	. . . . . 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 1385	. . . . 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 800	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 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925   class class class wbr 5107  ‘cfv 6511  (class class class)co 7387  Basecbs 17179  lecple 17227  joincjn 18272  Latclat 18390  Atomscatm 39256  AtLatcal 39257  CvLatclc 39258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-proset 18255  df-poset 18274  df-plt 18289  df-lub 18305  df-glb 18306  df-join 18307  df-meet 18308  df-p0 18384  df-lat 18391  df-covers 39259  df-ats 39260  df-atl 39291  df-cvlat 39315

This theorem is referenced by:  cvlsupr3  39337  cvlsupr4  39338  cvlsupr5  39339  cvlsupr6  39340  4atexlemex2  40065  4atex  40070  4atex3  40075  cdleme02N  40216  cdleme0ex2N  40218  cdleme0moN  40219  cdleme0nex  40284