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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 4296 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → ¬ 𝑉 = ∅) | |
2 | lvolbase.v | . . . . 5 ⊢ 𝑉 = (LVols‘𝐾) | |
3 | 2 | eqeq1i 2823 | . . . 4 ⊢ (𝑉 = ∅ ↔ (LVols‘𝐾) = ∅) |
4 | 1, 3 | sylnib 329 | . . 3 ⊢ (𝑋 ∈ 𝑉 → ¬ (LVols‘𝐾) = ∅) |
5 | fvprc 6656 | . . 3 ⊢ (¬ 𝐾 ∈ V → (LVols‘𝐾) = ∅) | |
6 | 4, 5 | nsyl2 143 | . 2 ⊢ (𝑋 ∈ 𝑉 → 𝐾 ∈ V) |
7 | lvolbase.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
8 | eqid 2818 | . . . 4 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾) | |
9 | eqid 2818 | . . . 4 ⊢ (LPlanes‘𝐾) = (LPlanes‘𝐾) | |
10 | 7, 8, 9, 2 | islvol 36589 | . . 3 ⊢ (𝐾 ∈ V → (𝑋 ∈ 𝑉 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑥 ∈ (LPlanes‘𝐾)𝑥( ⋖ ‘𝐾)𝑋))) |
11 | 10 | simprbda 499 | . 2 ⊢ ((𝐾 ∈ V ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝐵) |
12 | 6, 11 | mpancom 684 | 1 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ∃wrex 3136 Vcvv 3492 ∅c0 4288 class class class wbr 5057 ‘cfv 6348 Basecbs 16471 ⋖ ccvr 36278 LPlanesclpl 36508 LVolsclvol 36509 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-lvols 36516 |
This theorem is referenced by: islvol2 36596 lvolnle3at 36598 lvolneatN 36604 lvolnelln 36605 lvolnelpln 36606 lplncvrlvol2 36631 lvolcmp 36633 2lplnja 36635 |
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