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Theorem lvoli3 36191
Description: Condition implying a 3-dim lattice volume. (Contributed by NM, 2-Aug-2012.)
Hypotheses
Ref Expression
lvoli3.l = (le‘𝐾)
lvoli3.j = (join‘𝐾)
lvoli3.a 𝐴 = (Atoms‘𝐾)
lvoli3.p 𝑃 = (LPlanes‘𝐾)
lvoli3.v 𝑉 = (LVols‘𝐾)
Assertion
Ref Expression
lvoli3 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ ¬ 𝑄 𝑋) → (𝑋 𝑄) ∈ 𝑉)

Proof of Theorem lvoli3
Dummy variables 𝑦 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl2 1173 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ ¬ 𝑄 𝑋) → 𝑋𝑃)
2 simpl3 1174 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ ¬ 𝑄 𝑋) → 𝑄𝐴)
3 simpr 477 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ ¬ 𝑄 𝑋) → ¬ 𝑄 𝑋)
4 eqidd 2774 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ ¬ 𝑄 𝑋) → (𝑋 𝑄) = (𝑋 𝑄))
5 breq2 4930 . . . . . 6 (𝑦 = 𝑋 → (𝑟 𝑦𝑟 𝑋))
65notbid 310 . . . . 5 (𝑦 = 𝑋 → (¬ 𝑟 𝑦 ↔ ¬ 𝑟 𝑋))
7 oveq1 6982 . . . . . 6 (𝑦 = 𝑋 → (𝑦 𝑟) = (𝑋 𝑟))
87eqeq2d 2783 . . . . 5 (𝑦 = 𝑋 → ((𝑋 𝑄) = (𝑦 𝑟) ↔ (𝑋 𝑄) = (𝑋 𝑟)))
96, 8anbi12d 622 . . . 4 (𝑦 = 𝑋 → ((¬ 𝑟 𝑦 ∧ (𝑋 𝑄) = (𝑦 𝑟)) ↔ (¬ 𝑟 𝑋 ∧ (𝑋 𝑄) = (𝑋 𝑟))))
10 breq1 4929 . . . . . 6 (𝑟 = 𝑄 → (𝑟 𝑋𝑄 𝑋))
1110notbid 310 . . . . 5 (𝑟 = 𝑄 → (¬ 𝑟 𝑋 ↔ ¬ 𝑄 𝑋))
12 oveq2 6983 . . . . . 6 (𝑟 = 𝑄 → (𝑋 𝑟) = (𝑋 𝑄))
1312eqeq2d 2783 . . . . 5 (𝑟 = 𝑄 → ((𝑋 𝑄) = (𝑋 𝑟) ↔ (𝑋 𝑄) = (𝑋 𝑄)))
1411, 13anbi12d 622 . . . 4 (𝑟 = 𝑄 → ((¬ 𝑟 𝑋 ∧ (𝑋 𝑄) = (𝑋 𝑟)) ↔ (¬ 𝑄 𝑋 ∧ (𝑋 𝑄) = (𝑋 𝑄))))
159, 14rspc2ev 3545 . . 3 ((𝑋𝑃𝑄𝐴 ∧ (¬ 𝑄 𝑋 ∧ (𝑋 𝑄) = (𝑋 𝑄))) → ∃𝑦𝑃𝑟𝐴𝑟 𝑦 ∧ (𝑋 𝑄) = (𝑦 𝑟)))
161, 2, 3, 4, 15syl112anc 1355 . 2 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ ¬ 𝑄 𝑋) → ∃𝑦𝑃𝑟𝐴𝑟 𝑦 ∧ (𝑋 𝑄) = (𝑦 𝑟)))
17 simpl1 1172 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ ¬ 𝑄 𝑋) → 𝐾 ∈ HL)
1817hllatd 35978 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ ¬ 𝑄 𝑋) → 𝐾 ∈ Lat)
19 eqid 2773 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
20 lvoli3.p . . . . . 6 𝑃 = (LPlanes‘𝐾)
2119, 20lplnbase 36148 . . . . 5 (𝑋𝑃𝑋 ∈ (Base‘𝐾))
221, 21syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ ¬ 𝑄 𝑋) → 𝑋 ∈ (Base‘𝐾))
23 lvoli3.a . . . . . 6 𝐴 = (Atoms‘𝐾)
2419, 23atbase 35903 . . . . 5 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
252, 24syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ ¬ 𝑄 𝑋) → 𝑄 ∈ (Base‘𝐾))
26 lvoli3.j . . . . 5 = (join‘𝐾)
2719, 26latjcl 17532 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑋 𝑄) ∈ (Base‘𝐾))
2818, 22, 25, 27syl3anc 1352 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ ¬ 𝑄 𝑋) → (𝑋 𝑄) ∈ (Base‘𝐾))
29 lvoli3.l . . . 4 = (le‘𝐾)
30 lvoli3.v . . . 4 𝑉 = (LVols‘𝐾)
3119, 29, 26, 23, 20, 30islvol3 36190 . . 3 ((𝐾 ∈ HL ∧ (𝑋 𝑄) ∈ (Base‘𝐾)) → ((𝑋 𝑄) ∈ 𝑉 ↔ ∃𝑦𝑃𝑟𝐴𝑟 𝑦 ∧ (𝑋 𝑄) = (𝑦 𝑟))))
3217, 28, 31syl2anc 576 . 2 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ ¬ 𝑄 𝑋) → ((𝑋 𝑄) ∈ 𝑉 ↔ ∃𝑦𝑃𝑟𝐴𝑟 𝑦 ∧ (𝑋 𝑄) = (𝑦 𝑟))))
3316, 32mpbird 249 1 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ ¬ 𝑄 𝑋) → (𝑋 𝑄) ∈ 𝑉)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 387  w3a 1069   = wceq 1508  wcel 2051  wrex 3084   class class class wbr 4926  cfv 6186  (class class class)co 6975  Basecbs 16338  lecple 16427  joincjn 17425  Latclat 17526  Atomscatm 35877  HLchlt 35964  LPlanesclpl 36106  LVolsclvol 36107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-rep 5046  ax-sep 5057  ax-nul 5064  ax-pow 5116  ax-pr 5183  ax-un 7278
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ne 2963  df-ral 3088  df-rex 3089  df-reu 3090  df-rab 3092  df-v 3412  df-sbc 3677  df-csb 3782  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-nul 4174  df-if 4346  df-pw 4419  df-sn 4437  df-pr 4439  df-op 4443  df-uni 4710  df-iun 4791  df-br 4927  df-opab 4989  df-mpt 5006  df-id 5309  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-res 5416  df-ima 5417  df-iota 6150  df-fun 6188  df-fn 6189  df-f 6190  df-f1 6191  df-fo 6192  df-f1o 6193  df-fv 6194  df-riota 6936  df-ov 6978  df-oprab 6979  df-proset 17409  df-poset 17427  df-plt 17439  df-lub 17455  df-glb 17456  df-join 17457  df-meet 17458  df-p0 17520  df-lat 17527  df-clat 17589  df-oposet 35790  df-ol 35792  df-oml 35793  df-covers 35880  df-ats 35881  df-atl 35912  df-cvlat 35936  df-hlat 35965  df-lplanes 36113  df-lvols 36114
This theorem is referenced by:  dalem9  36286  dalem39  36325
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