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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 4249 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → ¬ 𝑉 = ∅) | |
2 | lvolbase.v | . . . . 5 ⊢ 𝑉 = (LVols‘𝐾) | |
3 | 2 | eqeq1i 2803 | . . . 4 ⊢ (𝑉 = ∅ ↔ (LVols‘𝐾) = ∅) |
4 | 1, 3 | sylnib 331 | . . 3 ⊢ (𝑋 ∈ 𝑉 → ¬ (LVols‘𝐾) = ∅) |
5 | fvprc 6638 | . . 3 ⊢ (¬ 𝐾 ∈ V → (LVols‘𝐾) = ∅) | |
6 | 4, 5 | nsyl2 143 | . 2 ⊢ (𝑋 ∈ 𝑉 → 𝐾 ∈ V) |
7 | lvolbase.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
8 | eqid 2798 | . . . 4 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾) | |
9 | eqid 2798 | . . . 4 ⊢ (LPlanes‘𝐾) = (LPlanes‘𝐾) | |
10 | 7, 8, 9, 2 | islvol 36869 | . . 3 ⊢ (𝐾 ∈ V → (𝑋 ∈ 𝑉 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑥 ∈ (LPlanes‘𝐾)𝑥( ⋖ ‘𝐾)𝑋))) |
11 | 10 | simprbda 502 | . 2 ⊢ ((𝐾 ∈ V ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝐵) |
12 | 6, 11 | mpancom 687 | 1 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 Vcvv 3441 ∅c0 4243 class class class wbr 5030 ‘cfv 6324 Basecbs 16475 ⋖ ccvr 36558 LPlanesclpl 36788 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-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
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-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 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-iota 6283 df-fun 6326 df-fv 6332 df-lvols 36796 |
This theorem is referenced by: islvol2 36876 lvolnle3at 36878 lvolneatN 36884 lvolnelln 36885 lvolnelpln 36886 lplncvrlvol2 36911 lvolcmp 36913 2lplnja 36915 |
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