Theorem lvolnlelln 39957

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

Hypotheses

Ref	Expression
lvolnlelln.l	⊢ ≤ = (le‘𝐾)
lvolnlelln.n	⊢ 𝑁 = (LLines‘𝐾)
lvolnlelln.v	⊢ 𝑉 = (LVols‘𝐾)

Assertion

Ref	Expression
lvolnlelln	⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑁) → ¬ 𝑋 ≤ 𝑌)

Proof of Theorem lvolnlelln

Dummy variables 𝑞 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp3 1139	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑁) → 𝑌 ∈ 𝑁)
2		eqid 2737	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
3		eqid 2737	. . . . 5 ⊢ (join‘𝐾) = (join‘𝐾)
4		eqid 2737	. . . . 5 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
5		lvolnlelln.n	. . . . 5 ⊢ 𝑁 = (LLines‘𝐾)
6	2, 3, 4, 5	islln2 39884	. . . 4 ⊢ (𝐾 ∈ HL → (𝑌 ∈ 𝑁 ↔ (𝑌 ∈ (Base‘𝐾) ∧ ∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝 ≠ 𝑞 ∧ 𝑌 = (𝑝(join‘𝐾)𝑞)))))
7	6	3ad2ant1 1134	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑁) → (𝑌 ∈ 𝑁 ↔ (𝑌 ∈ (Base‘𝐾) ∧ ∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝 ≠ 𝑞 ∧ 𝑌 = (𝑝(join‘𝐾)𝑞)))))
8	1, 7	mpbid 232	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑁) → (𝑌 ∈ (Base‘𝐾) ∧ ∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝 ≠ 𝑞 ∧ 𝑌 = (𝑝(join‘𝐾)𝑞))))
9		simp11 1205	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑌 = (𝑝(join‘𝐾)𝑞))) → 𝐾 ∈ HL)
10		simp12 1206	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑌 = (𝑝(join‘𝐾)𝑞))) → 𝑋 ∈ 𝑉)
11		simp2l 1201	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑌 = (𝑝(join‘𝐾)𝑞))) → 𝑝 ∈ (Atoms‘𝐾))
12		simp2r 1202	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑌 = (𝑝(join‘𝐾)𝑞))) → 𝑞 ∈ (Atoms‘𝐾))
13		lvolnlelln.l	. . . . . . . 8 ⊢ ≤ = (le‘𝐾)
14		lvolnlelln.v	. . . . . . . 8 ⊢ 𝑉 = (LVols‘𝐾)
15	13, 3, 4, 14	lvolnle3at 39955	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → ¬ 𝑋 ≤ ((𝑝(join‘𝐾)𝑝)(join‘𝐾)𝑞))
16	9, 10, 11, 11, 12, 15	syl23anc 1380	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑌 = (𝑝(join‘𝐾)𝑞))) → ¬ 𝑋 ≤ ((𝑝(join‘𝐾)𝑝)(join‘𝐾)𝑞))
17		simp3r 1204	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑌 = (𝑝(join‘𝐾)𝑞))) → 𝑌 = (𝑝(join‘𝐾)𝑞))
18	3, 4	hlatjidm 39742	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾)) → (𝑝(join‘𝐾)𝑝) = 𝑝)
19	9, 11, 18	syl2anc 585	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑌 = (𝑝(join‘𝐾)𝑞))) → (𝑝(join‘𝐾)𝑝) = 𝑝)
20	19	oveq1d 7383	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑌 = (𝑝(join‘𝐾)𝑞))) → ((𝑝(join‘𝐾)𝑝)(join‘𝐾)𝑞) = (𝑝(join‘𝐾)𝑞))
21	17, 20	eqtr4d 2775	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑌 = (𝑝(join‘𝐾)𝑞))) → 𝑌 = ((𝑝(join‘𝐾)𝑝)(join‘𝐾)𝑞))
22	21	breq2d 5112	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑌 = (𝑝(join‘𝐾)𝑞))) → (𝑋 ≤ 𝑌 ↔ 𝑋 ≤ ((𝑝(join‘𝐾)𝑝)(join‘𝐾)𝑞)))
23	16, 22	mtbird 325	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑌 = (𝑝(join‘𝐾)𝑞))) → ¬ 𝑋 ≤ 𝑌)
24	23	3exp 1120	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑁) → ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) → ((𝑝 ≠ 𝑞 ∧ 𝑌 = (𝑝(join‘𝐾)𝑞)) → ¬ 𝑋 ≤ 𝑌)))
25	24	rexlimdvv 3194	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑁) → (∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝 ≠ 𝑞 ∧ 𝑌 = (𝑝(join‘𝐾)𝑞)) → ¬ 𝑋 ≤ 𝑌))
26	25	adantld 490	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑁) → ((𝑌 ∈ (Base‘𝐾) ∧ ∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝 ≠ 𝑞 ∧ 𝑌 = (𝑝(join‘𝐾)𝑞))) → ¬ 𝑋 ≤ 𝑌))
27	8, 26	mpd 15	1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑁) → ¬ 𝑋 ≤ 𝑌)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∃wrex 3062   class class class wbr 5100  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  lecple 17196  joincjn 18246  Atomscatm 39636  HLchlt 39723  LLinesclln 39864  LVolsclvol 39866

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 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-proset 18229  df-poset 18248  df-plt 18263  df-lub 18279  df-glb 18280  df-join 18281  df-meet 18282  df-p0 18358  df-lat 18367  df-clat 18434  df-oposet 39549  df-ol 39551  df-oml 39552  df-covers 39639  df-ats 39640  df-atl 39671  df-cvlat 39695  df-hlat 39724  df-llines 39871  df-lplanes 39872  df-lvols 39873

This theorem is referenced by:  lvolnelln  39962