Theorem lvolex3N 39833

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 2735	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		lvolex3.l	. . . 4 ⊢ ≤ = (le‘𝐾)
3		eqid 2735	. . . 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 39831	. . 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 1199	. . . . . . . 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 1240	. . . . . . . 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 1241	. . . . . . . 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 1138	. . . . . . . 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 39764	. . . . . . . 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 1375	. . . . . . 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 1213	. . . . . . . 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 5101	. . . . . . . . . 10 ⊢ (𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡) → (𝑞 ≤ 𝑋 ↔ 𝑞 ≤ ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡)))
15	14	notbid 318	. . . . . . . . 9 ⊢ (𝑋 = ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡) → (¬ 𝑞 ≤ 𝑋 ↔ ¬ 𝑞 ≤ ((𝑟(join‘𝐾)𝑠)(join‘𝐾)𝑡)))
16	15	rexbidv 3159	. . . . . . . 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 17	. . . . . . 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 257	. . . . . 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 3140	. . . . 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 3192	. . . 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 490	. . 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 240	. 2 ⊢ (𝐾 ∈ HL → (𝑋 ∈ 𝑃 → ∃𝑞 ∈ 𝐴 ¬ 𝑞 ≤ 𝑋))
23	22	imp 406	1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) → ∃𝑞 ∈ 𝐴 ¬ 𝑞 ≤ 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2931  ∃wrex 3059   class class class wbr 5097  ‘cfv 6491  (class class class)co 7358  Basecbs 17138  lecple 17186  joincjn 18236  Atomscatm 39558  HLchlt 39645  LPlanesclpl 39787

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 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680

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 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3349  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-riota 7315  df-ov 7361  df-oprab 7362  df-proset 18219  df-poset 18238  df-plt 18253  df-lub 18269  df-glb 18270  df-join 18271  df-meet 18272  df-p0 18348  df-p1 18349  df-lat 18357  df-clat 18424  df-oposet 39471  df-ol 39473  df-oml 39474  df-covers 39561  df-ats 39562  df-atl 39593  df-cvlat 39617  df-hlat 39646  df-llines 39793  df-lplanes 39794

This theorem is referenced by: (None)