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Theorem lvoli3 35181
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 1085 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ ¬ 𝑄 𝑋) → 𝑋𝑃)
2 simpl3 1086 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ ¬ 𝑄 𝑋) → 𝑄𝐴)
3 simpr 476 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ ¬ 𝑄 𝑋) → ¬ 𝑄 𝑋)
4 eqidd 2652 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ ¬ 𝑄 𝑋) → (𝑋 𝑄) = (𝑋 𝑄))
5 breq2 4689 . . . . . 6 (𝑦 = 𝑋 → (𝑟 𝑦𝑟 𝑋))
65notbid 307 . . . . 5 (𝑦 = 𝑋 → (¬ 𝑟 𝑦 ↔ ¬ 𝑟 𝑋))
7 oveq1 6697 . . . . . 6 (𝑦 = 𝑋 → (𝑦 𝑟) = (𝑋 𝑟))
87eqeq2d 2661 . . . . 5 (𝑦 = 𝑋 → ((𝑋 𝑄) = (𝑦 𝑟) ↔ (𝑋 𝑄) = (𝑋 𝑟)))
96, 8anbi12d 747 . . . 4 (𝑦 = 𝑋 → ((¬ 𝑟 𝑦 ∧ (𝑋 𝑄) = (𝑦 𝑟)) ↔ (¬ 𝑟 𝑋 ∧ (𝑋 𝑄) = (𝑋 𝑟))))
10 breq1 4688 . . . . . 6 (𝑟 = 𝑄 → (𝑟 𝑋𝑄 𝑋))
1110notbid 307 . . . . 5 (𝑟 = 𝑄 → (¬ 𝑟 𝑋 ↔ ¬ 𝑄 𝑋))
12 oveq2 6698 . . . . . 6 (𝑟 = 𝑄 → (𝑋 𝑟) = (𝑋 𝑄))
1312eqeq2d 2661 . . . . 5 (𝑟 = 𝑄 → ((𝑋 𝑄) = (𝑋 𝑟) ↔ (𝑋 𝑄) = (𝑋 𝑄)))
1411, 13anbi12d 747 . . . 4 (𝑟 = 𝑄 → ((¬ 𝑟 𝑋 ∧ (𝑋 𝑄) = (𝑋 𝑟)) ↔ (¬ 𝑄 𝑋 ∧ (𝑋 𝑄) = (𝑋 𝑄))))
159, 14rspc2ev 3355 . . 3 ((𝑋𝑃𝑄𝐴 ∧ (¬ 𝑄 𝑋 ∧ (𝑋 𝑄) = (𝑋 𝑄))) → ∃𝑦𝑃𝑟𝐴𝑟 𝑦 ∧ (𝑋 𝑄) = (𝑦 𝑟)))
161, 2, 3, 4, 15syl112anc 1370 . 2 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ ¬ 𝑄 𝑋) → ∃𝑦𝑃𝑟𝐴𝑟 𝑦 ∧ (𝑋 𝑄) = (𝑦 𝑟)))
17 simpl1 1084 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ ¬ 𝑄 𝑋) → 𝐾 ∈ HL)
18 hllat 34968 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ Lat)
1917, 18syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ ¬ 𝑄 𝑋) → 𝐾 ∈ Lat)
20 eqid 2651 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
21 lvoli3.p . . . . . 6 𝑃 = (LPlanes‘𝐾)
2220, 21lplnbase 35138 . . . . 5 (𝑋𝑃𝑋 ∈ (Base‘𝐾))
231, 22syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ ¬ 𝑄 𝑋) → 𝑋 ∈ (Base‘𝐾))
24 lvoli3.a . . . . . 6 𝐴 = (Atoms‘𝐾)
2520, 24atbase 34894 . . . . 5 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
262, 25syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ ¬ 𝑄 𝑋) → 𝑄 ∈ (Base‘𝐾))
27 lvoli3.j . . . . 5 = (join‘𝐾)
2820, 27latjcl 17098 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑋 𝑄) ∈ (Base‘𝐾))
2919, 23, 26, 28syl3anc 1366 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ ¬ 𝑄 𝑋) → (𝑋 𝑄) ∈ (Base‘𝐾))
30 lvoli3.l . . . 4 = (le‘𝐾)
31 lvoli3.v . . . 4 𝑉 = (LVols‘𝐾)
3220, 30, 27, 24, 21, 31islvol3 35180 . . 3 ((𝐾 ∈ HL ∧ (𝑋 𝑄) ∈ (Base‘𝐾)) → ((𝑋 𝑄) ∈ 𝑉 ↔ ∃𝑦𝑃𝑟𝐴𝑟 𝑦 ∧ (𝑋 𝑄) = (𝑦 𝑟))))
3317, 29, 32syl2anc 694 . 2 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ ¬ 𝑄 𝑋) → ((𝑋 𝑄) ∈ 𝑉 ↔ ∃𝑦𝑃𝑟𝐴𝑟 𝑦 ∧ (𝑋 𝑄) = (𝑦 𝑟))))
3416, 33mpbird 247 1 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ ¬ 𝑄 𝑋) → (𝑋 𝑄) ∈ 𝑉)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wrex 2942   class class class wbr 4685  cfv 5926  (class class class)co 6690  Basecbs 15904  lecple 15995  joincjn 16991  Latclat 17092  Atomscatm 34868  HLchlt 34955  LPlanesclpl 35096  LVolsclvol 35097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-preset 16975  df-poset 16993  df-plt 17005  df-lub 17021  df-glb 17022  df-join 17023  df-meet 17024  df-p0 17086  df-lat 17093  df-clat 17155  df-oposet 34781  df-ol 34783  df-oml 34784  df-covers 34871  df-ats 34872  df-atl 34903  df-cvlat 34927  df-hlat 34956  df-lplanes 35103  df-lvols 35104
This theorem is referenced by:  dalem9  35276  dalem39  35315
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