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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 3920 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → ¬ 𝑉 = ∅) | |
2 | lvolbase.v | . . . . 5 ⊢ 𝑉 = (LVols‘𝐾) | |
3 | 2 | eqeq1i 2627 | . . . 4 ⊢ (𝑉 = ∅ ↔ (LVols‘𝐾) = ∅) |
4 | 1, 3 | sylnib 318 | . . 3 ⊢ (𝑋 ∈ 𝑉 → ¬ (LVols‘𝐾) = ∅) |
5 | fvprc 6185 | . . 3 ⊢ (¬ 𝐾 ∈ V → (LVols‘𝐾) = ∅) | |
6 | 4, 5 | nsyl2 142 | . 2 ⊢ (𝑋 ∈ 𝑉 → 𝐾 ∈ V) |
7 | lvolbase.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
8 | eqid 2622 | . . . 4 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾) | |
9 | eqid 2622 | . . . 4 ⊢ (LPlanes‘𝐾) = (LPlanes‘𝐾) | |
10 | 7, 8, 9, 2 | islvol 34859 | . . 3 ⊢ (𝐾 ∈ V → (𝑋 ∈ 𝑉 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑥 ∈ (LPlanes‘𝐾)𝑥( ⋖ ‘𝐾)𝑋))) |
11 | 10 | simprbda 653 | . 2 ⊢ ((𝐾 ∈ V ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝐵) |
12 | 6, 11 | mpancom 703 | 1 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ∃wrex 2913 Vcvv 3200 ∅c0 3915 class class class wbr 4653 ‘cfv 5888 Basecbs 15857 ⋖ ccvr 34549 LPlanesclpl 34778 LVolsclvol 34779 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-lvols 34786 |
This theorem is referenced by: islvol2 34866 lvolnle3at 34868 lvolneatN 34874 lvolnelln 34875 lvolnelpln 34876 lplncvrlvol2 34901 lvolcmp 34903 2lplnja 34905 |
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