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Theorem lvoli3 35358
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 1227 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ ¬ 𝑄 𝑋) → 𝑋𝑃)
2 simpl3 1229 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ ¬ 𝑄 𝑋) → 𝑄𝐴)
3 simpr 479 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ ¬ 𝑄 𝑋) → ¬ 𝑄 𝑋)
4 eqidd 2753 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ ¬ 𝑄 𝑋) → (𝑋 𝑄) = (𝑋 𝑄))
5 breq2 4800 . . . . . 6 (𝑦 = 𝑋 → (𝑟 𝑦𝑟 𝑋))
65notbid 307 . . . . 5 (𝑦 = 𝑋 → (¬ 𝑟 𝑦 ↔ ¬ 𝑟 𝑋))
7 oveq1 6812 . . . . . 6 (𝑦 = 𝑋 → (𝑦 𝑟) = (𝑋 𝑟))
87eqeq2d 2762 . . . . 5 (𝑦 = 𝑋 → ((𝑋 𝑄) = (𝑦 𝑟) ↔ (𝑋 𝑄) = (𝑋 𝑟)))
96, 8anbi12d 749 . . . 4 (𝑦 = 𝑋 → ((¬ 𝑟 𝑦 ∧ (𝑋 𝑄) = (𝑦 𝑟)) ↔ (¬ 𝑟 𝑋 ∧ (𝑋 𝑄) = (𝑋 𝑟))))
10 breq1 4799 . . . . . 6 (𝑟 = 𝑄 → (𝑟 𝑋𝑄 𝑋))
1110notbid 307 . . . . 5 (𝑟 = 𝑄 → (¬ 𝑟 𝑋 ↔ ¬ 𝑄 𝑋))
12 oveq2 6813 . . . . . 6 (𝑟 = 𝑄 → (𝑋 𝑟) = (𝑋 𝑄))
1312eqeq2d 2762 . . . . 5 (𝑟 = 𝑄 → ((𝑋 𝑄) = (𝑋 𝑟) ↔ (𝑋 𝑄) = (𝑋 𝑄)))
1411, 13anbi12d 749 . . . 4 (𝑟 = 𝑄 → ((¬ 𝑟 𝑋 ∧ (𝑋 𝑄) = (𝑋 𝑟)) ↔ (¬ 𝑄 𝑋 ∧ (𝑋 𝑄) = (𝑋 𝑄))))
159, 14rspc2ev 3455 . . 3 ((𝑋𝑃𝑄𝐴 ∧ (¬ 𝑄 𝑋 ∧ (𝑋 𝑄) = (𝑋 𝑄))) → ∃𝑦𝑃𝑟𝐴𝑟 𝑦 ∧ (𝑋 𝑄) = (𝑦 𝑟)))
161, 2, 3, 4, 15syl112anc 1477 . 2 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ ¬ 𝑄 𝑋) → ∃𝑦𝑃𝑟𝐴𝑟 𝑦 ∧ (𝑋 𝑄) = (𝑦 𝑟)))
17 simpl1 1225 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ ¬ 𝑄 𝑋) → 𝐾 ∈ HL)
18 hllat 35145 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ Lat)
1917, 18syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ ¬ 𝑄 𝑋) → 𝐾 ∈ Lat)
20 eqid 2752 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
21 lvoli3.p . . . . . 6 𝑃 = (LPlanes‘𝐾)
2220, 21lplnbase 35315 . . . . 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 35071 . . . . 5 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
262, 25syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ ¬ 𝑄 𝑋) → 𝑄 ∈ (Base‘𝐾))
27 lvoli3.j . . . . 5 = (join‘𝐾)
2820, 27latjcl 17244 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑋 𝑄) ∈ (Base‘𝐾))
2919, 23, 26, 28syl3anc 1473 . . 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 35357 . . 3 ((𝐾 ∈ HL ∧ (𝑋 𝑄) ∈ (Base‘𝐾)) → ((𝑋 𝑄) ∈ 𝑉 ↔ ∃𝑦𝑃𝑟𝐴𝑟 𝑦 ∧ (𝑋 𝑄) = (𝑦 𝑟))))
3317, 29, 32syl2anc 696 . 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 1072   = wceq 1624  wcel 2131  wrex 3043   class class class wbr 4796  cfv 6041  (class class class)co 6805  Basecbs 16051  lecple 16142  joincjn 17137  Latclat 17238  Atomscatm 35045  HLchlt 35132  LPlanesclpl 35273  LVolsclvol 35274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-riota 6766  df-ov 6808  df-oprab 6809  df-preset 17121  df-poset 17139  df-plt 17151  df-lub 17167  df-glb 17168  df-join 17169  df-meet 17170  df-p0 17232  df-lat 17239  df-clat 17301  df-oposet 34958  df-ol 34960  df-oml 34961  df-covers 35048  df-ats 35049  df-atl 35080  df-cvlat 35104  df-hlat 35133  df-lplanes 35280  df-lvols 35281
This theorem is referenced by:  dalem9  35453  dalem39  35492
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