Theorem lvolnlelpln 40751

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 2760	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
3		lvolnlelpln.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
4		eqid 2760	. . . . 5 ⊢ (join‘𝐾) = (join‘𝐾)
5		eqid 2760	. . . . 5 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
6		lvolnlelpln.p	. . . . 5 ⊢ 𝑃 = (LPlanes‘𝐾)
7	2, 3, 4, 5, 6	islpln2 40702	. . . 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 40748	. . . . . . . 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 5114	. . . . . . 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 3218	. . . 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 3163	. . 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 1569   ∈ wcel 2144   ≠ wne 2955  ∃wrex 3086   class class class wbr 5102  ‘cfv 6543  (class class class)co 7424  Basecbs 17396  lecple 17444  joincjn 18494  Atomscatm 40429  HLchlt 40516  LPlanesclpl 40658  LVolsclvol 40659

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7381  df-ov 7427  df-oprab 7428  df-proset 18477  df-poset 18496  df-plt 18511  df-lub 18527  df-glb 18528  df-join 18529  df-meet 18530  df-p0 18606  df-lat 18615  df-clat 18682  df-oposet 40342  df-ol 40344  df-oml 40345  df-covers 40432  df-ats 40433  df-atl 40464  df-cvlat 40488  df-hlat 40517  df-llines 40664  df-lplanes 40665  df-lvols 40666

This theorem is used by:  lvolnelpln  40756