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Theorem lvoli3 38040
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 1192 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ ¬ 𝑄 𝑋) → 𝑋𝑃)
2 simpl3 1193 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ ¬ 𝑄 𝑋) → 𝑄𝐴)
3 simpr 485 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ ¬ 𝑄 𝑋) → ¬ 𝑄 𝑋)
4 eqidd 2737 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ ¬ 𝑄 𝑋) → (𝑋 𝑄) = (𝑋 𝑄))
5 breq2 5109 . . . . . 6 (𝑦 = 𝑋 → (𝑟 𝑦𝑟 𝑋))
65notbid 317 . . . . 5 (𝑦 = 𝑋 → (¬ 𝑟 𝑦 ↔ ¬ 𝑟 𝑋))
7 oveq1 7364 . . . . . 6 (𝑦 = 𝑋 → (𝑦 𝑟) = (𝑋 𝑟))
87eqeq2d 2747 . . . . 5 (𝑦 = 𝑋 → ((𝑋 𝑄) = (𝑦 𝑟) ↔ (𝑋 𝑄) = (𝑋 𝑟)))
96, 8anbi12d 631 . . . 4 (𝑦 = 𝑋 → ((¬ 𝑟 𝑦 ∧ (𝑋 𝑄) = (𝑦 𝑟)) ↔ (¬ 𝑟 𝑋 ∧ (𝑋 𝑄) = (𝑋 𝑟))))
10 breq1 5108 . . . . . 6 (𝑟 = 𝑄 → (𝑟 𝑋𝑄 𝑋))
1110notbid 317 . . . . 5 (𝑟 = 𝑄 → (¬ 𝑟 𝑋 ↔ ¬ 𝑄 𝑋))
12 oveq2 7365 . . . . . 6 (𝑟 = 𝑄 → (𝑋 𝑟) = (𝑋 𝑄))
1312eqeq2d 2747 . . . . 5 (𝑟 = 𝑄 → ((𝑋 𝑄) = (𝑋 𝑟) ↔ (𝑋 𝑄) = (𝑋 𝑄)))
1411, 13anbi12d 631 . . . 4 (𝑟 = 𝑄 → ((¬ 𝑟 𝑋 ∧ (𝑋 𝑄) = (𝑋 𝑟)) ↔ (¬ 𝑄 𝑋 ∧ (𝑋 𝑄) = (𝑋 𝑄))))
159, 14rspc2ev 3592 . . 3 ((𝑋𝑃𝑄𝐴 ∧ (¬ 𝑄 𝑋 ∧ (𝑋 𝑄) = (𝑋 𝑄))) → ∃𝑦𝑃𝑟𝐴𝑟 𝑦 ∧ (𝑋 𝑄) = (𝑦 𝑟)))
161, 2, 3, 4, 15syl112anc 1374 . 2 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ ¬ 𝑄 𝑋) → ∃𝑦𝑃𝑟𝐴𝑟 𝑦 ∧ (𝑋 𝑄) = (𝑦 𝑟)))
17 simpl1 1191 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ ¬ 𝑄 𝑋) → 𝐾 ∈ HL)
1817hllatd 37826 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ ¬ 𝑄 𝑋) → 𝐾 ∈ Lat)
19 eqid 2736 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
20 lvoli3.p . . . . . 6 𝑃 = (LPlanes‘𝐾)
2119, 20lplnbase 37997 . . . . 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 37751 . . . . 5 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
252, 24syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ ¬ 𝑄 𝑋) → 𝑄 ∈ (Base‘𝐾))
26 lvoli3.j . . . . 5 = (join‘𝐾)
2719, 26latjcl 18328 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑋 𝑄) ∈ (Base‘𝐾))
2818, 22, 25, 27syl3anc 1371 . . 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 38039 . . 3 ((𝐾 ∈ HL ∧ (𝑋 𝑄) ∈ (Base‘𝐾)) → ((𝑋 𝑄) ∈ 𝑉 ↔ ∃𝑦𝑃𝑟𝐴𝑟 𝑦 ∧ (𝑋 𝑄) = (𝑦 𝑟))))
3217, 28, 31syl2anc 584 . 2 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ ¬ 𝑄 𝑋) → ((𝑋 𝑄) ∈ 𝑉 ↔ ∃𝑦𝑃𝑟𝐴𝑟 𝑦 ∧ (𝑋 𝑄) = (𝑦 𝑟))))
3316, 32mpbird 256 1 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ ¬ 𝑄 𝑋) → (𝑋 𝑄) ∈ 𝑉)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wrex 3073   class class class wbr 5105  cfv 6496  (class class class)co 7357  Basecbs 17083  lecple 17140  joincjn 18200  Latclat 18320  Atomscatm 37725  HLchlt 37812  LPlanesclpl 37955  LVolsclvol 37956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-proset 18184  df-poset 18202  df-plt 18219  df-lub 18235  df-glb 18236  df-join 18237  df-meet 18238  df-p0 18314  df-lat 18321  df-clat 18388  df-oposet 37638  df-ol 37640  df-oml 37641  df-covers 37728  df-ats 37729  df-atl 37760  df-cvlat 37784  df-hlat 37813  df-lplanes 37962  df-lvols 37963
This theorem is referenced by:  dalem9  38135  dalem39  38174
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