Theorem atbtwnex 39911

Description: Given atoms 𝑃 in 𝑋 and 𝑄 not in 𝑋, there exists an atom 𝑟 not in 𝑋 such that the line 𝑄 ∨ 𝑟 intersects 𝑋 at 𝑃. (Contributed by NM, 1-Aug-2012.)

Hypotheses

Ref	Expression
atbtwn.b	⊢ 𝐵 = (Base‘𝐾)
atbtwn.l	⊢ ≤ = (le‘𝐾)
atbtwn.j	⊢ ∨ = (join‘𝐾)
atbtwn.a	⊢ 𝐴 = (Atoms‘𝐾)

Assertion

Ref	Expression
atbtwnex	⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → ∃𝑟 ∈ 𝐴 (𝑟 ≠ 𝑄 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟)))

Distinct variable groups:   𝐴,𝑟   𝐵,𝑟   𝐾,𝑟   ≤ ,𝑟   𝑃,𝑟   𝑄,𝑟   𝑋,𝑟

Allowed substitution hint:   ∨ (𝑟)

Proof of Theorem atbtwnex

Step	Hyp	Ref	Expression
1		simpr2 1197	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → 𝑃 ≤ 𝑋)
2		simpr3 1198	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → ¬ 𝑄 ≤ 𝑋)
3		nbrne2 5106	. . . 4 ⊢ ((𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋) → 𝑃 ≠ 𝑄)
4	1, 2, 3	syl2anc 585	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → 𝑃 ≠ 𝑄)
5		atbtwn.l	. . . 4 ⊢ ≤ = (le‘𝐾)
6		atbtwn.j	. . . 4 ⊢ ∨ = (join‘𝐾)
7		atbtwn.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
8	5, 6, 7	hlsupr 39849	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → ∃𝑟 ∈ 𝐴 (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄)))
9	4, 8	syldan 592	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → ∃𝑟 ∈ 𝐴 (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄)))
10		simp32 1212	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ 𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄))) → 𝑟 ≠ 𝑄)
11		simp31 1211	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ 𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄))) → 𝑟 ≠ 𝑃)
12		simp1l 1199	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ 𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄))) → (𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴))
13		simp2 1138	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ 𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄))) → 𝑟 ∈ 𝐴)
14		simp1r1 1271	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ 𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄))) → 𝑋 ∈ 𝐵)
15		simp1r2 1272	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ 𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄))) → 𝑃 ≤ 𝑋)
16		simp1r3 1273	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ 𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄))) → ¬ 𝑄 ≤ 𝑋)
17		simp33 1213	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ 𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄))) → 𝑟 ≤ (𝑃 ∨ 𝑄))
18		atbtwn.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐾)
19	18, 5, 6, 7	atbtwn 39909	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑟 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄))) → (𝑟 ≠ 𝑃 ↔ ¬ 𝑟 ≤ 𝑋))
20	12, 13, 14, 15, 16, 17, 19	syl123anc 1390	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ 𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄))) → (𝑟 ≠ 𝑃 ↔ ¬ 𝑟 ≤ 𝑋))
21	11, 20	mpbid 232	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ 𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄))) → ¬ 𝑟 ≤ 𝑋)
22		simp1l1 1268	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ 𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄))) → 𝐾 ∈ HL)
23		simp1l2 1269	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ 𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄))) → 𝑃 ∈ 𝐴)
24		simp1l3 1270	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ 𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄))) → 𝑄 ∈ 𝐴)
25	5, 6, 7	hlatexch2 39859	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑟 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑟 ≠ 𝑄) → (𝑟 ≤ (𝑃 ∨ 𝑄) → 𝑃 ≤ (𝑟 ∨ 𝑄)))
26	22, 13, 23, 24, 10, 25	syl131anc 1386	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ 𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄))) → (𝑟 ≤ (𝑃 ∨ 𝑄) → 𝑃 ≤ (𝑟 ∨ 𝑄)))
27	17, 26	mpd 15	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ 𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄))) → 𝑃 ≤ (𝑟 ∨ 𝑄))
28	6, 7	hlatjcom 39831	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) → (𝑄 ∨ 𝑟) = (𝑟 ∨ 𝑄))
29	22, 24, 13, 28	syl3anc 1374	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ 𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄))) → (𝑄 ∨ 𝑟) = (𝑟 ∨ 𝑄))
30	27, 29	breqtrrd 5114	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ 𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄))) → 𝑃 ≤ (𝑄 ∨ 𝑟))
31	10, 21, 30	3jca 1129	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ 𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄))) → (𝑟 ≠ 𝑄 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟)))
32	31	3exp 1120	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → (𝑟 ∈ 𝐴 → ((𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄)) → (𝑟 ≠ 𝑄 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟)))))
33	32	reximdvai 3149	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → (∃𝑟 ∈ 𝐴 (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄)) → ∃𝑟 ∈ 𝐴 (𝑟 ≠ 𝑄 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟))))
34	9, 33	mpd 15	1 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → ∃𝑟 ∈ 𝐴 (𝑟 ≠ 𝑄 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∃wrex 3062   class class class wbr 5086  ‘cfv 6493  (class class class)co 7361  Basecbs 17173  lecple 17221  joincjn 18271  Atomscatm 39726  HLchlt 39813

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 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

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 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7318  df-ov 7364  df-oprab 7365  df-proset 18254  df-poset 18273  df-plt 18288  df-lub 18304  df-glb 18305  df-join 18306  df-meet 18307  df-p0 18383  df-lat 18392  df-clat 18459  df-oposet 39639  df-ol 39641  df-oml 39642  df-covers 39729  df-ats 39730  df-atl 39761  df-cvlat 39785  df-hlat 39814

This theorem is referenced by:  dalem19  40145