Theorem atcvrj2b 38961

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 1225	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) ∧ 𝑃 = 𝑅) → 𝑄 ≠ 𝑅)
2	1	necomd 2986	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) ∧ 𝑃 = 𝑅) → 𝑅 ≠ 𝑄)
3		simpl1 1188	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) ∧ 𝑃 = 𝑅) → 𝐾 ∈ HL)
4		simpl23 1250	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) ∧ 𝑃 = 𝑅) → 𝑅 ∈ 𝐴)
5		simpl22 1249	. . . . . . 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 38947	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑅 ≠ 𝑄 ↔ 𝑅𝐶(𝑄 ∨ 𝑅)))
10	3, 4, 5, 9	syl3anc 1368	. . . . . 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 231	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) ∧ 𝑃 = 𝑅) → 𝑅𝐶(𝑄 ∨ 𝑅))
12		breq1 5146	. . . . . 6 ⊢ (𝑃 = 𝑅 → (𝑃𝐶(𝑄 ∨ 𝑅) ↔ 𝑅𝐶(𝑄 ∨ 𝑅)))
13	12	adantl 480	. . . . 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 256	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) ∧ 𝑃 = 𝑅) → 𝑃𝐶(𝑄 ∨ 𝑅))
15		simpl1 1188	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) ∧ 𝑃 ≠ 𝑅) → 𝐾 ∈ HL)
16		simpl2 1189	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) ∧ 𝑃 ≠ 𝑅) → (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴))
17		simpr 483	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) ∧ 𝑃 ≠ 𝑅) → 𝑃 ≠ 𝑅)
18		simpl3r 1226	. . . . 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 38960	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) → 𝑃𝐶(𝑄 ∨ 𝑅))
21	15, 16, 17, 18, 20	syl112anc 1371	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) ∧ 𝑃 ≠ 𝑅) → 𝑃𝐶(𝑄 ∨ 𝑅))
22	14, 21	pm2.61dane 3019	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) → 𝑃𝐶(𝑄 ∨ 𝑅))
23	22	3expia 1118	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ((𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅)) → 𝑃𝐶(𝑄 ∨ 𝑅)))
24		hlatl 38888	. . . . . . 7 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
25	24	ad2antrr 724	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝐾 ∈ AtLat)
26		simplr1 1212	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝑃 ∈ 𝐴)
27		eqid 2725	. . . . . . 7 ⊢ (0.‘𝐾) = (0.‘𝐾)
28	27, 8	atn0 38836	. . . . . 6 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → 𝑃 ≠ (0.‘𝐾))
29	25, 26, 28	syl2anc 582	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝑃 ≠ (0.‘𝐾))
30		simpll 765	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝐾 ∈ HL)
31		eqid 2725	. . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾)
32	31, 8	atbase 38817	. . . . . . . 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 1213	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝑄 ∈ 𝐴)
35		simplr3 1214	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝑅 ∈ 𝐴)
36		simpr 483	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝑃𝐶(𝑄 ∨ 𝑅))
37	31, 6, 27, 7, 8	atcvrj0 38957	. . . . . . 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 1380	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → (𝑃 = (0.‘𝐾) ↔ 𝑄 = 𝑅))
39	38	necon3bid 2975	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → (𝑃 ≠ (0.‘𝐾) ↔ 𝑄 ≠ 𝑅))
40	29, 39	mpbid 231	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝑄 ≠ 𝑅)
41		hllat 38891	. . . . . . . 8 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
42	41	ad2antrr 724	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝐾 ∈ Lat)
43	31, 8	atbase 38817	. . . . . . . 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 38817	. . . . . . . 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 18430	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → (𝑄 ∨ 𝑅) ∈ (Base‘𝐾))
48	42, 44, 46, 47	syl3anc 1368	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → (𝑄 ∨ 𝑅) ∈ (Base‘𝐾))
49	30, 33, 48	3jca 1125	. . . . 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 38806	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝑄 ∨ 𝑅) ∈ (Base‘𝐾)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝑃 ≤ (𝑄 ∨ 𝑅))
51	49, 50	sylancom 586	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝑃 ≤ (𝑄 ∨ 𝑅))
52	40, 51	jca 510	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅)))
53	52	ex 411	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (𝑃𝐶(𝑄 ∨ 𝑅) → (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))))
54	23, 53	impbid 211	1 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ((𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅)) ↔ 𝑃𝐶(𝑄 ∨ 𝑅)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ≠ wne 2930   class class class wbr 5143  ‘cfv 6543  (class class class)co 7416  Basecbs 17179  lecple 17239  joincjn 18302  0.cp0 18414  Latclat 18422   ⋖ ccvr 38790  Atomscatm 38791  AtLatcal 38792  HLchlt 38878

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7372  df-ov 7419  df-oprab 7420  df-proset 18286  df-poset 18304  df-plt 18321  df-lub 18337  df-glb 18338  df-join 18339  df-meet 18340  df-p0 18416  df-lat 18423  df-clat 18490  df-oposet 38704  df-ol 38706  df-oml 38707  df-covers 38794  df-ats 38795  df-atl 38826  df-cvlat 38850  df-hlat 38879

This theorem is referenced by:  atcvrj2  38962