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Theorem lvolnlelln 38759
Description: A lattice line cannot majorize a lattice volume. (Contributed by NM, 14-Jul-2012.)
Hypotheses
Ref Expression
lvolnlelln.l ≀ = (leβ€˜πΎ)
lvolnlelln.n 𝑁 = (LLinesβ€˜πΎ)
lvolnlelln.v 𝑉 = (LVolsβ€˜πΎ)
Assertion
Ref Expression
lvolnlelln ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑁) β†’ Β¬ 𝑋 ≀ π‘Œ)
Proof of Theorem lvolnlelln
Dummy variables π‘ž 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp3 1137 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑁) β†’ π‘Œ ∈ 𝑁)
2 eqid 2731 . . . . 5 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
3 eqid 2731 . . . . 5 (joinβ€˜πΎ) = (joinβ€˜πΎ)
4 eqid 2731 . . . . 5 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
5 lvolnlelln.n . . . . 5 𝑁 = (LLinesβ€˜πΎ)
62, 3, 4, 5islln2 38686 . . . 4 (𝐾 ∈ HL β†’ (π‘Œ ∈ 𝑁 ↔ (π‘Œ ∈ (Baseβ€˜πΎ) ∧ βˆƒπ‘ ∈ (Atomsβ€˜πΎ)βˆƒπ‘ž ∈ (Atomsβ€˜πΎ)(𝑝 β‰  π‘ž ∧ π‘Œ = (𝑝(joinβ€˜πΎ)π‘ž)))))
763ad2ant1 1132 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑁) β†’ (π‘Œ ∈ 𝑁 ↔ (π‘Œ ∈ (Baseβ€˜πΎ) ∧ βˆƒπ‘ ∈ (Atomsβ€˜πΎ)βˆƒπ‘ž ∈ (Atomsβ€˜πΎ)(𝑝 β‰  π‘ž ∧ π‘Œ = (𝑝(joinβ€˜πΎ)π‘ž)))))
81, 7mpbid 231 . 2 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑁) β†’ (π‘Œ ∈ (Baseβ€˜πΎ) ∧ βˆƒπ‘ ∈ (Atomsβ€˜πΎ)βˆƒπ‘ž ∈ (Atomsβ€˜πΎ)(𝑝 β‰  π‘ž ∧ π‘Œ = (𝑝(joinβ€˜πΎ)π‘ž))))
9 simp11 1202 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑁) ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ (𝑝 β‰  π‘ž ∧ π‘Œ = (𝑝(joinβ€˜πΎ)π‘ž))) β†’ 𝐾 ∈ HL)
10 simp12 1203 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑁) ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ (𝑝 β‰  π‘ž ∧ π‘Œ = (𝑝(joinβ€˜πΎ)π‘ž))) β†’ 𝑋 ∈ 𝑉)
11 simp2l 1198 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑁) ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ (𝑝 β‰  π‘ž ∧ π‘Œ = (𝑝(joinβ€˜πΎ)π‘ž))) β†’ 𝑝 ∈ (Atomsβ€˜πΎ))
12 simp2r 1199 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑁) ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ (𝑝 β‰  π‘ž ∧ π‘Œ = (𝑝(joinβ€˜πΎ)π‘ž))) β†’ π‘ž ∈ (Atomsβ€˜πΎ))
13 lvolnlelln.l . . . . . . . 8 ≀ = (leβ€˜πΎ)
14 lvolnlelln.v . . . . . . . 8 𝑉 = (LVolsβ€˜πΎ)
1513, 3, 4, 14lvolnle3at 38757 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ))) β†’ Β¬ 𝑋 ≀ ((𝑝(joinβ€˜πΎ)𝑝)(joinβ€˜πΎ)π‘ž))
169, 10, 11, 11, 12, 15syl23anc 1376 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑁) ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ (𝑝 β‰  π‘ž ∧ π‘Œ = (𝑝(joinβ€˜πΎ)π‘ž))) β†’ Β¬ 𝑋 ≀ ((𝑝(joinβ€˜πΎ)𝑝)(joinβ€˜πΎ)π‘ž))
17 simp3r 1201 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑁) ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ (𝑝 β‰  π‘ž ∧ π‘Œ = (𝑝(joinβ€˜πΎ)π‘ž))) β†’ π‘Œ = (𝑝(joinβ€˜πΎ)π‘ž))
183, 4hlatjidm 38543 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) β†’ (𝑝(joinβ€˜πΎ)𝑝) = 𝑝)
199, 11, 18syl2anc 583 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑁) ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ (𝑝 β‰  π‘ž ∧ π‘Œ = (𝑝(joinβ€˜πΎ)π‘ž))) β†’ (𝑝(joinβ€˜πΎ)𝑝) = 𝑝)
2019oveq1d 7427 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑁) ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ (𝑝 β‰  π‘ž ∧ π‘Œ = (𝑝(joinβ€˜πΎ)π‘ž))) β†’ ((𝑝(joinβ€˜πΎ)𝑝)(joinβ€˜πΎ)π‘ž) = (𝑝(joinβ€˜πΎ)π‘ž))
2117, 20eqtr4d 2774 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑁) ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ (𝑝 β‰  π‘ž ∧ π‘Œ = (𝑝(joinβ€˜πΎ)π‘ž))) β†’ π‘Œ = ((𝑝(joinβ€˜πΎ)𝑝)(joinβ€˜πΎ)π‘ž))
2221breq2d 5161 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑁) ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ (𝑝 β‰  π‘ž ∧ π‘Œ = (𝑝(joinβ€˜πΎ)π‘ž))) β†’ (𝑋 ≀ π‘Œ ↔ 𝑋 ≀ ((𝑝(joinβ€˜πΎ)𝑝)(joinβ€˜πΎ)π‘ž)))
2316, 22mtbird 324 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑁) ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) ∧ (𝑝 β‰  π‘ž ∧ π‘Œ = (𝑝(joinβ€˜πΎ)π‘ž))) β†’ Β¬ 𝑋 ≀ π‘Œ)
24233exp 1118 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑁) β†’ ((𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ)) β†’ ((𝑝 β‰  π‘ž ∧ π‘Œ = (𝑝(joinβ€˜πΎ)π‘ž)) β†’ Β¬ 𝑋 ≀ π‘Œ)))
2524rexlimdvv 3209 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑁) β†’ (βˆƒπ‘ ∈ (Atomsβ€˜πΎ)βˆƒπ‘ž ∈ (Atomsβ€˜πΎ)(𝑝 β‰  π‘ž ∧ π‘Œ = (𝑝(joinβ€˜πΎ)π‘ž)) β†’ Β¬ 𝑋 ≀ π‘Œ))
2625adantld 490 . 2 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑁) β†’ ((π‘Œ ∈ (Baseβ€˜πΎ) ∧ βˆƒπ‘ ∈ (Atomsβ€˜πΎ)βˆƒπ‘ž ∈ (Atomsβ€˜πΎ)(𝑝 β‰  π‘ž ∧ π‘Œ = (𝑝(joinβ€˜πΎ)π‘ž))) β†’ Β¬ 𝑋 ≀ π‘Œ))
278, 26mpd 15 1 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑁) β†’ Β¬ 𝑋 ≀ π‘Œ)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   β‰  wne 2939  βˆƒwrex 3069   class class class wbr 5149  β€˜cfv 6544  (class class class)co 7412  Basecbs 17149  lecple 17209  joincjn 18269  Atomscatm 38437  HLchlt 38524  LLinesclln 38666  LVolsclvol 38668
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-proset 18253  df-poset 18271  df-plt 18288  df-lub 18304  df-glb 18305  df-join 18306  df-meet 18307  df-p0 18383  df-lat 18390  df-clat 18457  df-oposet 38350  df-ol 38352  df-oml 38353  df-covers 38440  df-ats 38441  df-atl 38472  df-cvlat 38496  df-hlat 38525  df-llines 38673  df-lplanes 38674  df-lvols 38675
This theorem is referenced by:  lvolnelln  38764
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