Theorem lvolnlelpln 39584

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 1138	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑃) → 𝑌 ∈ 𝑃)
2		eqid 2729	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
3		lvolnlelpln.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
4		eqid 2729	. . . . 5 ⊢ (join‘𝐾) = (join‘𝐾)
5		eqid 2729	. . . . 5 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
6		lvolnlelpln.p	. . . . 5 ⊢ 𝑃 = (LPlanes‘𝐾)
7	2, 3, 4, 5, 6	islpln2 39535	. . . 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 1133	. . 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 232	. 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 1267	. . . . . . . 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 1268	. . . . . . . 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 1199	. . . . . . . 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 1200	. . . . . . . 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 1201	. . . . . . . 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 39581	. . . . . . . 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 1379	. . . . . . 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 1212	. . . . . . . 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 5107	. . . . . . 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 325	. . . . . 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 1119	. . . . 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 3185	. . . 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 3130	. . 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 490	. 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 15	1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑃) → ¬ 𝑋 ≤ 𝑌)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∃wrex 3053   class class class wbr 5095  ‘cfv 6486  (class class class)co 7353  Basecbs 17139  lecple 17187  joincjn 18236  Atomscatm 39261  HLchlt 39348  LPlanesclpl 39491  LVolsclvol 39492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  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 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  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-lat 18357  df-clat 18424  df-oposet 39174  df-ol 39176  df-oml 39177  df-covers 39264  df-ats 39265  df-atl 39296  df-cvlat 39320  df-hlat 39349  df-llines 39497  df-lplanes 39498  df-lvols 39499

This theorem is referenced by:  lvolnelpln  39589