Theorem lvolex3N 40398

Description: There is an atom outside of a lattice plane i.e. a 3-dimensional lattice volume exists. (Contributed by NM, 28-Jul-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
lvolex3.l	⊢ ≤ = (le‘𝐾)
lvolex3.a	⊢ 𝐴 = (Atoms‘𝐾)
lvolex3.p	⊢ 𝑃 = (LPlanes‘𝐾)

Assertion

Ref	Expression
lvolex3N	⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) → ∃𝑞 ∈ 𝐴 ¬ 𝑞 ≤ 𝑋)

Distinct variable groups:   𝐴,𝑞   𝐾,𝑞   ≤ ,𝑞   𝑋,𝑞

Allowed substitution hint:   𝑃(𝑞)

Proof of Theorem lvolex3N

Dummy variables 𝑟 𝑠 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2762	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		lvolex3.l	. . . 4 ⊢ ≤ = (le‘𝐾)
3		eqid 2762	. . . 4 ⊢ (join‘𝐾) = (join‘𝐾)
4		lvolex3.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
5		lvolex3.p	. . . 4 ⊢ 𝑃 = (LPlanes‘𝐾)
6	1, 2, 3, 4, 5	islpln2 40396	. . 3 ⊢ (𝐾 ∈ HL → (𝑋 ∈ 𝑃 ↔ (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 ∃𝑡 ∈ 𝐴 (𝑟 ≠ 𝑠 ∧ ¬ 𝑡 ≤ (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡)))))
7		simp1l 1216	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑡 ∈ 𝐴 ∧ (𝑟 ≠ 𝑠 ∧ ¬ 𝑡 ≤ (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡))) → 𝐾 ∈ HL)
8		simp1rl 1257	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑡 ∈ 𝐴 ∧ (𝑟 ≠ 𝑠 ∧ ¬ 𝑡 ≤ (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡))) → 𝑟 ∈ 𝐴)
9		simp1rr 1258	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑡 ∈ 𝐴 ∧ (𝑟 ≠ 𝑠 ∧ ¬ 𝑡 ≤ (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡))) → 𝑠 ∈ 𝐴)
10		simp2 1155	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑡 ∈ 𝐴 ∧ (𝑟 ≠ 𝑠 ∧ ¬ 𝑡 ≤ (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡))) → 𝑡 ∈ 𝐴)
11	3, 2, 4	3dim3 40329	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴 ∧ 𝑡 ∈ 𝐴)) → ∃𝑞 ∈ 𝐴 ¬ 𝑞 ≤ ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡))
12	7, 8, 9, 10, 11	syl13anc 1399	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑡 ∈ 𝐴 ∧ (𝑟 ≠ 𝑠 ∧ ¬ 𝑡 ≤ (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡))) → ∃𝑞 ∈ 𝐴 ¬ 𝑞 ≤ ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡))
13		simp33 1230	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑡 ∈ 𝐴 ∧ (𝑟 ≠ 𝑠 ∧ ¬ 𝑡 ≤ (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡))) → 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡))
14		breq2 5111	. . . . . . . . . 10 ⊢ (𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡) → (𝑞 ≤ 𝑋 ↔ 𝑞 ≤ ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡)))
15	14	notbid 321	. . . . . . . . 9 ⊢ (𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡) → (¬ 𝑞 ≤ 𝑋 ↔ ¬ 𝑞 ≤ ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡)))
16	15	rexbidv 3188	. . . . . . . 8 ⊢ (𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡) → (∃𝑞 ∈ 𝐴 ¬ 𝑞 ≤ 𝑋 ↔ ∃𝑞 ∈ 𝐴 ¬ 𝑞 ≤ ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡)))
17	13, 16	syl 18	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑡 ∈ 𝐴 ∧ (𝑟 ≠ 𝑠 ∧ ¬ 𝑡 ≤ (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡))) → (∃𝑞 ∈ 𝐴 ¬ 𝑞 ≤ 𝑋 ↔ ∃𝑞 ∈ 𝐴 ¬ 𝑞 ≤ ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡)))
18	12, 17	mpbird 260	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑡 ∈ 𝐴 ∧ (𝑟 ≠ 𝑠 ∧ ¬ 𝑡 ≤ (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡))) → ∃𝑞 ∈ 𝐴 ¬ 𝑞 ≤ 𝑋)
19	18	rexlimdv3a 3169	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) → (∃𝑡 ∈ 𝐴 (𝑟 ≠ 𝑠 ∧ ¬ 𝑡 ≤ (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡)) → ∃𝑞 ∈ 𝐴 ¬ 𝑞 ≤ 𝑋))
20	19	rexlimdvva 3221	. . . 4 ⊢ (𝐾 ∈ HL → (∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 ∃𝑡 ∈ 𝐴 (𝑟 ≠ 𝑠 ∧ ¬ 𝑡 ≤ (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡)) → ∃𝑞 ∈ 𝐴 ¬ 𝑞 ≤ 𝑋))
21	20	adantld 496	. . 3 ⊢ (𝐾 ∈ HL → ((𝑋 ∈ (Base‘𝐾) ∧ ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 ∃𝑡 ∈ 𝐴 (𝑟 ≠ 𝑠 ∧ ¬ 𝑡 ≤ (𝑟(join‘𝐾)𝑠) ∧ 𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡))) → ∃𝑞 ∈ 𝐴 ¬ 𝑞 ≤ 𝑋))
22	6, 21	sylbid 243	. 2 ⊢ (𝐾 ∈ HL → (𝑋 ∈ 𝑃 → ∃𝑞 ∈ 𝐴 ¬ 𝑞 ≤ 𝑋))
23	22	imp 412	1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) → ∃𝑞 ∈ 𝐴 ¬ 𝑞 ≤ 𝑋)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2957  ∃wrex 3088   class class class wbr 5107  ‘cfv 6537  (class class class)co 7416  Basecbs 17305  lecple 17353  joincjn 18403  Atomscatm 40123  HLchlt 40210  LPlanesclpl 40352

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-ov 7419  df-oprab 7420  df-proset 18386  df-poset 18405  df-plt 18420  df-lub 18436  df-glb 18437  df-join 18438  df-meet 18439  df-p0 18515  df-p1 18516  df-lat 18524  df-clat 18591  df-oposet 40036  df-ol 40038  df-oml 40039  df-covers 40126  df-ats 40127  df-atl 40158  df-cvlat 40182  df-hlat 40211  df-llines 40358  df-lplanes 40359

This theorem is used by: (None)