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Theorem lvolnlelln 35371
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 ∧ 𝑋𝑉𝑌𝑁) → ¬ 𝑋 𝑌)

Proof of Theorem lvolnlelln
Dummy variables 𝑞 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp3 1133 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑁) → 𝑌𝑁)
2 eqid 2758 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
3 eqid 2758 . . . . 5 (join‘𝐾) = (join‘𝐾)
4 eqid 2758 . . . . 5 (Atoms‘𝐾) = (Atoms‘𝐾)
5 lvolnlelln.n . . . . 5 𝑁 = (LLines‘𝐾)
62, 3, 4, 5islln2 35298 . . . 4 (𝐾 ∈ HL → (𝑌𝑁 ↔ (𝑌 ∈ (Base‘𝐾) ∧ ∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝𝑞𝑌 = (𝑝(join‘𝐾)𝑞)))))
763ad2ant1 1128 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑁) → (𝑌𝑁 ↔ (𝑌 ∈ (Base‘𝐾) ∧ ∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝𝑞𝑌 = (𝑝(join‘𝐾)𝑞)))))
81, 7mpbid 222 . 2 ((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑁) → (𝑌 ∈ (Base‘𝐾) ∧ ∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝𝑞𝑌 = (𝑝(join‘𝐾)𝑞))))
9 simp11 1246 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑌 = (𝑝(join‘𝐾)𝑞))) → 𝐾 ∈ HL)
10 simp12 1247 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑌 = (𝑝(join‘𝐾)𝑞))) → 𝑋𝑉)
11 simp2l 1242 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑌 = (𝑝(join‘𝐾)𝑞))) → 𝑝 ∈ (Atoms‘𝐾))
12 simp2r 1243 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑌 = (𝑝(join‘𝐾)𝑞))) → 𝑞 ∈ (Atoms‘𝐾))
13 lvolnlelln.l . . . . . . . 8 = (le‘𝐾)
14 lvolnlelln.v . . . . . . . 8 𝑉 = (LVols‘𝐾)
1513, 3, 4, 14lvolnle3at 35369 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑉) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → ¬ 𝑋 ((𝑝(join‘𝐾)𝑝)(join‘𝐾)𝑞))
169, 10, 11, 11, 12, 15syl23anc 1484 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑌 = (𝑝(join‘𝐾)𝑞))) → ¬ 𝑋 ((𝑝(join‘𝐾)𝑝)(join‘𝐾)𝑞))
17 simp3r 1245 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑌 = (𝑝(join‘𝐾)𝑞))) → 𝑌 = (𝑝(join‘𝐾)𝑞))
183, 4hlatjidm 35156 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾)) → (𝑝(join‘𝐾)𝑝) = 𝑝)
199, 11, 18syl2anc 696 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑌 = (𝑝(join‘𝐾)𝑞))) → (𝑝(join‘𝐾)𝑝) = 𝑝)
2019oveq1d 6826 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑌 = (𝑝(join‘𝐾)𝑞))) → ((𝑝(join‘𝐾)𝑝)(join‘𝐾)𝑞) = (𝑝(join‘𝐾)𝑞))
2117, 20eqtr4d 2795 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑌 = (𝑝(join‘𝐾)𝑞))) → 𝑌 = ((𝑝(join‘𝐾)𝑝)(join‘𝐾)𝑞))
2221breq2d 4814 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑌 = (𝑝(join‘𝐾)𝑞))) → (𝑋 𝑌𝑋 ((𝑝(join‘𝐾)𝑝)(join‘𝐾)𝑞)))
2316, 22mtbird 314 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑌 = (𝑝(join‘𝐾)𝑞))) → ¬ 𝑋 𝑌)
24233exp 1113 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑁) → ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) → ((𝑝𝑞𝑌 = (𝑝(join‘𝐾)𝑞)) → ¬ 𝑋 𝑌)))
2524rexlimdvv 3173 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑁) → (∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝𝑞𝑌 = (𝑝(join‘𝐾)𝑞)) → ¬ 𝑋 𝑌))
2625adantld 484 . 2 ((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑁) → ((𝑌 ∈ (Base‘𝐾) ∧ ∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝𝑞𝑌 = (𝑝(join‘𝐾)𝑞))) → ¬ 𝑋 𝑌))
278, 26mpd 15 1 ((𝐾 ∈ HL ∧ 𝑋𝑉𝑌𝑁) → ¬ 𝑋 𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1072   = wceq 1630  wcel 2137  wne 2930  wrex 3049   class class class wbr 4802  cfv 6047  (class class class)co 6811  Basecbs 16057  lecple 16148  joincjn 17143  Atomscatm 35051  HLchlt 35138  LLinesclln 35278  LVolsclvol 35280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-rep 4921  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-reu 3055  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-op 4326  df-uni 4587  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-id 5172  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-riota 6772  df-ov 6814  df-oprab 6815  df-preset 17127  df-poset 17145  df-plt 17157  df-lub 17173  df-glb 17174  df-join 17175  df-meet 17176  df-p0 17238  df-lat 17245  df-clat 17307  df-oposet 34964  df-ol 34966  df-oml 34967  df-covers 35054  df-ats 35055  df-atl 35086  df-cvlat 35110  df-hlat 35139  df-llines 35285  df-lplanes 35286  df-lvols 35287
This theorem is referenced by:  lvolnelln  35376
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