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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > islvol2 | Structured version Visualization version GIF version |
Description: The predicate "is a 3-dim lattice volume" in terms of atoms. (Contributed by NM, 1-Jul-2012.) |
Ref | Expression |
---|---|
islvol5.b | ⊢ 𝐵 = (Base‘𝐾) |
islvol5.l | ⊢ ≤ = (le‘𝐾) |
islvol5.j | ⊢ ∨ = (join‘𝐾) |
islvol5.a | ⊢ 𝐴 = (Atoms‘𝐾) |
islvol5.v | ⊢ 𝑉 = (LVols‘𝐾) |
Ref | Expression |
---|---|
islvol2 | ⊢ (𝐾 ∈ HL → (𝑋 ∈ 𝑉 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 ((𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑝 ∨ 𝑞) ∧ ¬ 𝑠 ≤ ((𝑝 ∨ 𝑞) ∨ 𝑟)) ∧ 𝑋 = (((𝑝 ∨ 𝑞) ∨ 𝑟) ∨ 𝑠))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | islvol5.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
2 | islvol5.v | . . . 4 ⊢ 𝑉 = (LVols‘𝐾) | |
3 | 1, 2 | lvolbase 36874 | . . 3 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ 𝐵) |
4 | 3 | pm4.71ri 564 | . 2 ⊢ (𝑋 ∈ 𝑉 ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉)) |
5 | islvol5.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
6 | islvol5.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
7 | islvol5.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
8 | 1, 5, 6, 7, 2 | islvol5 36875 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∈ 𝑉 ↔ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 ((𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑝 ∨ 𝑞) ∧ ¬ 𝑠 ≤ ((𝑝 ∨ 𝑞) ∨ 𝑟)) ∧ 𝑋 = (((𝑝 ∨ 𝑞) ∨ 𝑟) ∨ 𝑠)))) |
9 | 8 | pm5.32da 582 | . 2 ⊢ (𝐾 ∈ HL → ((𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉) ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 ((𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑝 ∨ 𝑞) ∧ ¬ 𝑠 ≤ ((𝑝 ∨ 𝑞) ∨ 𝑟)) ∧ 𝑋 = (((𝑝 ∨ 𝑞) ∨ 𝑟) ∨ 𝑠))))) |
10 | 4, 9 | syl5bb 286 | 1 ⊢ (𝐾 ∈ HL → (𝑋 ∈ 𝑉 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 ((𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑝 ∨ 𝑞) ∧ ¬ 𝑠 ≤ ((𝑝 ∨ 𝑞) ∨ 𝑟)) ∧ 𝑋 = (((𝑝 ∨ 𝑞) ∨ 𝑟) ∨ 𝑠))))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∃wrex 3107 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 lecple 16564 joincjn 17546 Atomscatm 36559 HLchlt 36646 LVolsclvol 36789 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-proset 17530 df-poset 17548 df-plt 17560 df-lub 17576 df-glb 17577 df-join 17578 df-meet 17579 df-p0 17641 df-lat 17648 df-clat 17710 df-oposet 36472 df-ol 36474 df-oml 36475 df-covers 36562 df-ats 36563 df-atl 36594 df-cvlat 36618 df-hlat 36647 df-llines 36794 df-lplanes 36795 df-lvols 36796 |
This theorem is referenced by: lplncvrlvol2 36911 |
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