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Mirrors > Home > MPE Home > Th. List > Mathboxes > lvolbase | Structured version Visualization version GIF version |
Description: A 3-dim lattice volume is a lattice element. (Contributed by NM, 1-Jul-2012.) |
Ref | Expression |
---|---|
lvolbase.b | ⊢ 𝐵 = (Base‘𝐾) |
lvolbase.v | ⊢ 𝑉 = (LVols‘𝐾) |
Ref | Expression |
---|---|
lvolbase | ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0i 4346 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → ¬ 𝑉 = ∅) | |
2 | lvolbase.v | . . . . 5 ⊢ 𝑉 = (LVols‘𝐾) | |
3 | 2 | eqeq1i 2740 | . . . 4 ⊢ (𝑉 = ∅ ↔ (LVols‘𝐾) = ∅) |
4 | 1, 3 | sylnib 328 | . . 3 ⊢ (𝑋 ∈ 𝑉 → ¬ (LVols‘𝐾) = ∅) |
5 | fvprc 6899 | . . 3 ⊢ (¬ 𝐾 ∈ V → (LVols‘𝐾) = ∅) | |
6 | 4, 5 | nsyl2 141 | . 2 ⊢ (𝑋 ∈ 𝑉 → 𝐾 ∈ V) |
7 | lvolbase.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
8 | eqid 2735 | . . . 4 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾) | |
9 | eqid 2735 | . . . 4 ⊢ (LPlanes‘𝐾) = (LPlanes‘𝐾) | |
10 | 7, 8, 9, 2 | islvol 39556 | . . 3 ⊢ (𝐾 ∈ V → (𝑋 ∈ 𝑉 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑥 ∈ (LPlanes‘𝐾)𝑥( ⋖ ‘𝐾)𝑋))) |
11 | 10 | simprbda 498 | . 2 ⊢ ((𝐾 ∈ V ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝐵) |
12 | 6, 11 | mpancom 688 | 1 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 Vcvv 3478 ∅c0 4339 class class class wbr 5148 ‘cfv 6563 Basecbs 17245 ⋖ ccvr 39244 LPlanesclpl 39475 LVolsclvol 39476 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-lvols 39483 |
This theorem is referenced by: islvol2 39563 lvolnle3at 39565 lvolneatN 39571 lvolnelln 39572 lvolnelpln 39573 lplncvrlvol2 39598 lvolcmp 39600 2lplnja 39602 |
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