Theorem atcvrj2b 35500

Description: Condition for an atom to be covered by the join of two others. (Contributed by NM, 7-Feb-2012.)

Hypotheses

Ref	Expression
atcvrj1x.l	⊢ ≤ = (le‘𝐾)
atcvrj1x.j	⊢ ∨ = (join‘𝐾)
atcvrj1x.c	⊢ 𝐶 = ( ⋖ ‘𝐾)
atcvrj1x.a	⊢ 𝐴 = (Atoms‘𝐾)

Assertion

Ref	Expression
atcvrj2b	⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ((𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅)) ↔ 𝑃𝐶(𝑄 ∨ 𝑅)))

Proof of Theorem atcvrj2b

Step	Hyp	Ref	Expression
1		simpl3l 1305	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) ∧ 𝑃 = 𝑅) → 𝑄 ≠ 𝑅)
2	1	necomd 3054	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) ∧ 𝑃 = 𝑅) → 𝑅 ≠ 𝑄)
3		simpl1 1246	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) ∧ 𝑃 = 𝑅) → 𝐾 ∈ HL)
4		simpl23 1343	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) ∧ 𝑃 = 𝑅) → 𝑅 ∈ 𝐴)
5		simpl22 1341	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) ∧ 𝑃 = 𝑅) → 𝑄 ∈ 𝐴)
6		atcvrj1x.j	. . . . . . . 8 ⊢ ∨ = (join‘𝐾)
7		atcvrj1x.c	. . . . . . . 8 ⊢ 𝐶 = ( ⋖ ‘𝐾)
8		atcvrj1x.a	. . . . . . . 8 ⊢ 𝐴 = (Atoms‘𝐾)
9	6, 7, 8	atcvr2 35486	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑅 ≠ 𝑄 ↔ 𝑅𝐶(𝑄 ∨ 𝑅)))
10	3, 4, 5, 9	syl3anc 1494	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) ∧ 𝑃 = 𝑅) → (𝑅 ≠ 𝑄 ↔ 𝑅𝐶(𝑄 ∨ 𝑅)))
11	2, 10	mpbid 224	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) ∧ 𝑃 = 𝑅) → 𝑅𝐶(𝑄 ∨ 𝑅))
12		breq1 4876	. . . . . 6 ⊢ (𝑃 = 𝑅 → (𝑃𝐶(𝑄 ∨ 𝑅) ↔ 𝑅𝐶(𝑄 ∨ 𝑅)))
13	12	adantl 475	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) ∧ 𝑃 = 𝑅) → (𝑃𝐶(𝑄 ∨ 𝑅) ↔ 𝑅𝐶(𝑄 ∨ 𝑅)))
14	11, 13	mpbird 249	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) ∧ 𝑃 = 𝑅) → 𝑃𝐶(𝑄 ∨ 𝑅))
15		simpl1 1246	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) ∧ 𝑃 ≠ 𝑅) → 𝐾 ∈ HL)
16		simpl2 1248	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) ∧ 𝑃 ≠ 𝑅) → (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴))
17		simpr 479	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) ∧ 𝑃 ≠ 𝑅) → 𝑃 ≠ 𝑅)
18		simpl3r 1307	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) ∧ 𝑃 ≠ 𝑅) → 𝑃 ≤ (𝑄 ∨ 𝑅))
19		atcvrj1x.l	. . . . . 6 ⊢ ≤ = (le‘𝐾)
20	19, 6, 7, 8	atcvrj1 35499	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) → 𝑃𝐶(𝑄 ∨ 𝑅))
21	15, 16, 17, 18, 20	syl112anc 1497	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) ∧ 𝑃 ≠ 𝑅) → 𝑃𝐶(𝑄 ∨ 𝑅))
22	14, 21	pm2.61dane 3086	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) → 𝑃𝐶(𝑄 ∨ 𝑅))
23	22	3expia 1154	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ((𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅)) → 𝑃𝐶(𝑄 ∨ 𝑅)))
24		hlatl 35428	. . . . . . 7 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
25	24	ad2antrr 717	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝐾 ∈ AtLat)
26		simplr1 1279	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝑃 ∈ 𝐴)
27		eqid 2825	. . . . . . 7 ⊢ (0.‘𝐾) = (0.‘𝐾)
28	27, 8	atn0 35376	. . . . . 6 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → 𝑃 ≠ (0.‘𝐾))
29	25, 26, 28	syl2anc 579	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝑃 ≠ (0.‘𝐾))
30		simpll 783	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝐾 ∈ HL)
31		eqid 2825	. . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾)
32	31, 8	atbase 35357	. . . . . . . 8 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
33	26, 32	syl 17	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝑃 ∈ (Base‘𝐾))
34		simplr2 1281	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝑄 ∈ 𝐴)
35		simplr3 1283	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝑅 ∈ 𝐴)
36		simpr 479	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝑃𝐶(𝑄 ∨ 𝑅))
37	31, 6, 27, 7, 8	atcvrj0 35496	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → (𝑃 = (0.‘𝐾) ↔ 𝑄 = 𝑅))
38	30, 33, 34, 35, 36, 37	syl131anc 1506	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → (𝑃 = (0.‘𝐾) ↔ 𝑄 = 𝑅))
39	38	necon3bid 3043	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → (𝑃 ≠ (0.‘𝐾) ↔ 𝑄 ≠ 𝑅))
40	29, 39	mpbid 224	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝑄 ≠ 𝑅)
41		hllat 35431	. . . . . . . 8 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
42	41	ad2antrr 717	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝐾 ∈ Lat)
43	31, 8	atbase 35357	. . . . . . . 8 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
44	34, 43	syl 17	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝑄 ∈ (Base‘𝐾))
45	31, 8	atbase 35357	. . . . . . . 8 ⊢ (𝑅 ∈ 𝐴 → 𝑅 ∈ (Base‘𝐾))
46	35, 45	syl 17	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝑅 ∈ (Base‘𝐾))
47	31, 6	latjcl 17404	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → (𝑄 ∨ 𝑅) ∈ (Base‘𝐾))
48	42, 44, 46, 47	syl3anc 1494	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → (𝑄 ∨ 𝑅) ∈ (Base‘𝐾))
49	30, 33, 48	3jca 1162	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → (𝐾 ∈ HL ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝑄 ∨ 𝑅) ∈ (Base‘𝐾)))
50	31, 19, 7	cvrle 35346	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝑄 ∨ 𝑅) ∈ (Base‘𝐾)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝑃 ≤ (𝑄 ∨ 𝑅))
51	49, 50	sylancom 582	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝑃 ≤ (𝑄 ∨ 𝑅))
52	40, 51	jca 507	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅)))
53	52	ex 403	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (𝑃𝐶(𝑄 ∨ 𝑅) → (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))))
54	23, 53	impbid 204	1 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ((𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅)) ↔ 𝑃𝐶(𝑄 ∨ 𝑅)))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1111   = wceq 1656   ∈ wcel 2164   ≠ wne 2999   class class class wbr 4873  ‘cfv 6123  (class class class)co 6905  Basecbs 16222  lecple 16312  joincjn 17297  0.cp0 17390  Latclat 17398   ⋖ ccvr 35330  Atomscatm 35331  AtLatcal 35332  HLchlt 35418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-proset 17281  df-poset 17299  df-plt 17311  df-lub 17327  df-glb 17328  df-join 17329  df-meet 17330  df-p0 17392  df-lat 17399  df-clat 17461  df-oposet 35244  df-ol 35246  df-oml 35247  df-covers 35334  df-ats 35335  df-atl 35366  df-cvlat 35390  df-hlat 35419

This theorem is referenced by:  atcvrj2  35501