Theorem lvolnlelpln 40392

Description: A lattice plane cannot majorize a lattice volume. (Contributed by NM, 14-Jul-2012.)

Hypotheses

Ref	Expression
lvolnlelpln.l	⊢ ≤ = (le‘𝐾)
lvolnlelpln.p	⊢ 𝑃 = (LPlanes‘𝐾)
lvolnlelpln.v	⊢ 𝑉 = (LVols‘𝐾)

Assertion

Ref	Expression
lvolnlelpln	⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑃) → ¬ 𝑋 ≤ 𝑌)

Proof of Theorem lvolnlelpln

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

Step	Hyp	Ref	Expression
1		simp3 1156	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑃) → 𝑌 ∈ 𝑃)
2		eqid 2765	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
3		lvolnlelpln.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
4		eqid 2765	. . . . 5 ⊢ (join‘𝐾) = (join‘𝐾)
5		eqid 2765	. . . . 5 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
6		lvolnlelpln.p	. . . . 5 ⊢ 𝑃 = (LPlanes‘𝐾)
7	2, 3, 4, 5, 6	islpln2 40343	. . . 4 ⊢ (𝐾 ∈ HL → (𝑌 ∈ 𝑃 ↔ (𝑌 ∈ (Base‘𝐾) ∧ ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)∃𝑠 ∈ (Atoms‘𝐾)(𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞(join‘𝐾)𝑟) ∧ 𝑌 = ((𝑞(join‘𝐾)𝑟)(join‘𝐾)𝑠)))))
8	7	3ad2ant1 1151	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑃) → (𝑌 ∈ 𝑃 ↔ (𝑌 ∈ (Base‘𝐾) ∧ ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)∃𝑠 ∈ (Atoms‘𝐾)(𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞(join‘𝐾)𝑟) ∧ 𝑌 = ((𝑞(join‘𝐾)𝑟)(join‘𝐾)𝑠)))))
9	1, 8	mpbid 235	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑃) → (𝑌 ∈ (Base‘𝐾) ∧ ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)∃𝑠 ∈ (Atoms‘𝐾)(𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞(join‘𝐾)𝑟) ∧ 𝑌 = ((𝑞(join‘𝐾)𝑟)(join‘𝐾)𝑠))))
10		simp1l1 1285	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑃) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾)) ∧ (𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞(join‘𝐾)𝑟) ∧ 𝑌 = ((𝑞(join‘𝐾)𝑟)(join‘𝐾)𝑠))) → 𝐾 ∈ HL)
11		simp1l2 1286	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑃) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾)) ∧ (𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞(join‘𝐾)𝑟) ∧ 𝑌 = ((𝑞(join‘𝐾)𝑟)(join‘𝐾)𝑠))) → 𝑋 ∈ 𝑉)
12		simp1r 1217	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑃) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾)) ∧ (𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞(join‘𝐾)𝑟) ∧ 𝑌 = ((𝑞(join‘𝐾)𝑟)(join‘𝐾)𝑠))) → 𝑞 ∈ (Atoms‘𝐾))
13		simp2l 1218	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑃) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾)) ∧ (𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞(join‘𝐾)𝑟) ∧ 𝑌 = ((𝑞(join‘𝐾)𝑟)(join‘𝐾)𝑠))) → 𝑟 ∈ (Atoms‘𝐾))
14		simp2r 1219	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑃) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾)) ∧ (𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞(join‘𝐾)𝑟) ∧ 𝑌 = ((𝑞(join‘𝐾)𝑟)(join‘𝐾)𝑠))) → 𝑠 ∈ (Atoms‘𝐾))
15		lvolnlelpln.v	. . . . . . . . 9 ⊢ 𝑉 = (LVols‘𝐾)
16	3, 4, 5, 15	lvolnle3at 40389	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) → ¬ 𝑋 ≤ ((𝑞(join‘𝐾)𝑟)(join‘𝐾)𝑠))
17	10, 11, 12, 13, 14, 16	syl23anc 1404	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑃) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾)) ∧ (𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞(join‘𝐾)𝑟) ∧ 𝑌 = ((𝑞(join‘𝐾)𝑟)(join‘𝐾)𝑠))) → ¬ 𝑋 ≤ ((𝑞(join‘𝐾)𝑟)(join‘𝐾)𝑠))
18		simp33 1230	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑃) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾)) ∧ (𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞(join‘𝐾)𝑟) ∧ 𝑌 = ((𝑞(join‘𝐾)𝑟)(join‘𝐾)𝑠))) → 𝑌 = ((𝑞(join‘𝐾)𝑟)(join‘𝐾)𝑠))
19	18	breq2d 5123	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑃) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾)) ∧ (𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞(join‘𝐾)𝑟) ∧ 𝑌 = ((𝑞(join‘𝐾)𝑟)(join‘𝐾)𝑠))) → (𝑋 ≤ 𝑌 ↔ 𝑋 ≤ ((𝑞(join‘𝐾)𝑟)(join‘𝐾)𝑠)))
20	17, 19	mtbird 328	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑃) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾)) ∧ (𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞(join‘𝐾)𝑟) ∧ 𝑌 = ((𝑞(join‘𝐾)𝑟)(join‘𝐾)𝑠))) → ¬ 𝑋 ≤ 𝑌)
21	20	3exp 1137	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑃) ∧ 𝑞 ∈ (Atoms‘𝐾)) → ((𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾)) → ((𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞(join‘𝐾)𝑟) ∧ 𝑌 = ((𝑞(join‘𝐾)𝑟)(join‘𝐾)𝑠)) → ¬ 𝑋 ≤ 𝑌)))
22	21	rexlimdvv 3223	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑃) ∧ 𝑞 ∈ (Atoms‘𝐾)) → (∃𝑟 ∈ (Atoms‘𝐾)∃𝑠 ∈ (Atoms‘𝐾)(𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞(join‘𝐾)𝑟) ∧ 𝑌 = ((𝑞(join‘𝐾)𝑟)(join‘𝐾)𝑠)) → ¬ 𝑋 ≤ 𝑌))
23	22	rexlimdva 3168	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑃) → (∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)∃𝑠 ∈ (Atoms‘𝐾)(𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞(join‘𝐾)𝑟) ∧ 𝑌 = ((𝑞(join‘𝐾)𝑟)(join‘𝐾)𝑠)) → ¬ 𝑋 ≤ 𝑌))
24	23	adantld 496	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑃) → ((𝑌 ∈ (Base‘𝐾) ∧ ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)∃𝑠 ∈ (Atoms‘𝐾)(𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞(join‘𝐾)𝑟) ∧ 𝑌 = ((𝑞(join‘𝐾)𝑟)(join‘𝐾)𝑠))) → ¬ 𝑋 ≤ 𝑌))
25	9, 24	mpd 16	1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑃) → ¬ 𝑋 ≤ 𝑌)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  ∃wrex 3091   class class class wbr 5111  ‘cfv 6540  (class class class)co 7416  Basecbs 17287  lecple 17335  joincjn 18385  Atomscatm 40070  HLchlt 40157  LPlanesclpl 40299  LVolsclvol 40300

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7373  df-ov 7419  df-oprab 7420  df-proset 18368  df-poset 18387  df-plt 18402  df-lub 18418  df-glb 18419  df-join 18420  df-meet 18421  df-p0 18497  df-lat 18506  df-clat 18573  df-oposet 39983  df-ol 39985  df-oml 39986  df-covers 40073  df-ats 40074  df-atl 40105  df-cvlat 40129  df-hlat 40158  df-llines 40305  df-lplanes 40306  df-lvols 40307

This theorem is used by:  lvolnelpln  40397