Theorem atcvrj2b 39451

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 1229	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) ∧ 𝑃 = 𝑅) → 𝑄 ≠ 𝑅)
2	1	necomd 2987	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) ∧ 𝑃 = 𝑅) → 𝑅 ≠ 𝑄)
3		simpl1 1192	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) ∧ 𝑃 = 𝑅) → 𝐾 ∈ HL)
4		simpl23 1254	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) ∧ 𝑃 = 𝑅) → 𝑅 ∈ 𝐴)
5		simpl22 1253	. . . . . . 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 39437	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑅 ≠ 𝑄 ↔ 𝑅𝐶(𝑄 ∨ 𝑅)))
10	3, 4, 5, 9	syl3anc 1373	. . . . . 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 232	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) ∧ 𝑃 = 𝑅) → 𝑅𝐶(𝑄 ∨ 𝑅))
12		breq1 5122	. . . . . 6 ⊢ (𝑃 = 𝑅 → (𝑃𝐶(𝑄 ∨ 𝑅) ↔ 𝑅𝐶(𝑄 ∨ 𝑅)))
13	12	adantl 481	. . . . 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 257	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) ∧ 𝑃 = 𝑅) → 𝑃𝐶(𝑄 ∨ 𝑅))
15		simpl1 1192	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) ∧ 𝑃 ≠ 𝑅) → 𝐾 ∈ HL)
16		simpl2 1193	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) ∧ 𝑃 ≠ 𝑅) → (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴))
17		simpr 484	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) ∧ 𝑃 ≠ 𝑅) → 𝑃 ≠ 𝑅)
18		simpl3r 1230	. . . . 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 39450	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) → 𝑃𝐶(𝑄 ∨ 𝑅))
21	15, 16, 17, 18, 20	syl112anc 1376	. . . 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 1121	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ((𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅)) → 𝑃𝐶(𝑄 ∨ 𝑅)))
24		hlatl 39378	. . . . . . 7 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
25	24	ad2antrr 726	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝐾 ∈ AtLat)
26		simplr1 1216	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝑃 ∈ 𝐴)
27		eqid 2735	. . . . . . 7 ⊢ (0.‘𝐾) = (0.‘𝐾)
28	27, 8	atn0 39326	. . . . . 6 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → 𝑃 ≠ (0.‘𝐾))
29	25, 26, 28	syl2anc 584	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝑃 ≠ (0.‘𝐾))
30		simpll 766	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝐾 ∈ HL)
31		eqid 2735	. . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾)
32	31, 8	atbase 39307	. . . . . . . 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 1217	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝑄 ∈ 𝐴)
35		simplr3 1218	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝑅 ∈ 𝐴)
36		simpr 484	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝑃𝐶(𝑄 ∨ 𝑅))
37	31, 6, 27, 7, 8	atcvrj0 39447	. . . . . . 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 1385	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → (𝑃 = (0.‘𝐾) ↔ 𝑄 = 𝑅))
39	38	necon3bid 2976	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → (𝑃 ≠ (0.‘𝐾) ↔ 𝑄 ≠ 𝑅))
40	29, 39	mpbid 232	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝑄 ≠ 𝑅)
41		hllat 39381	. . . . . . . 8 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
42	41	ad2antrr 726	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝐾 ∈ Lat)
43	31, 8	atbase 39307	. . . . . . . 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 39307	. . . . . . . 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 18449	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → (𝑄 ∨ 𝑅) ∈ (Base‘𝐾))
48	42, 44, 46, 47	syl3anc 1373	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → (𝑄 ∨ 𝑅) ∈ (Base‘𝐾))
49	30, 33, 48	3jca 1128	. . . . 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 39296	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝑄 ∨ 𝑅) ∈ (Base‘𝐾)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝑃 ≤ (𝑄 ∨ 𝑅))
51	49, 50	sylancom 588	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝑃 ≤ (𝑄 ∨ 𝑅))
52	40, 51	jca 511	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅)))
53	52	ex 412	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (𝑃𝐶(𝑄 ∨ 𝑅) → (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))))
54	23, 53	impbid 212	1 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ((𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅)) ↔ 𝑃𝐶(𝑄 ∨ 𝑅)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108   ≠ wne 2932   class class class wbr 5119  ‘cfv 6531  (class class class)co 7405  Basecbs 17228  lecple 17278  joincjn 18323  0.cp0 18433  Latclat 18441   ⋖ ccvr 39280  Atomscatm 39281  AtLatcal 39282  HLchlt 39368

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-proset 18306  df-poset 18325  df-plt 18340  df-lub 18356  df-glb 18357  df-join 18358  df-meet 18359  df-p0 18435  df-lat 18442  df-clat 18509  df-oposet 39194  df-ol 39196  df-oml 39197  df-covers 39284  df-ats 39285  df-atl 39316  df-cvlat 39340  df-hlat 39369

This theorem is referenced by:  atcvrj2  39452