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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 4339 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → ¬ 𝑉 = ∅) | |
| 2 | lvolbase.v | . . . . 5 ⊢ 𝑉 = (LVols‘𝐾) | |
| 3 | 2 | eqeq1i 2741 | . . . 4 ⊢ (𝑉 = ∅ ↔ (LVols‘𝐾) = ∅) | 
| 4 | 1, 3 | sylnib 328 | . . 3 ⊢ (𝑋 ∈ 𝑉 → ¬ (LVols‘𝐾) = ∅) | 
| 5 | fvprc 6897 | . . 3 ⊢ (¬ 𝐾 ∈ V → (LVols‘𝐾) = ∅) | |
| 6 | 4, 5 | nsyl2 141 | . 2 ⊢ (𝑋 ∈ 𝑉 → 𝐾 ∈ V) | 
| 7 | lvolbase.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
| 8 | eqid 2736 | . . . 4 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾) | |
| 9 | eqid 2736 | . . . 4 ⊢ (LPlanes‘𝐾) = (LPlanes‘𝐾) | |
| 10 | 7, 8, 9, 2 | islvol 39576 | . . 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 1539 ∈ wcel 2107 ∃wrex 3069 Vcvv 3479 ∅c0 4332 class class class wbr 5142 ‘cfv 6560 Basecbs 17248 ⋖ ccvr 39264 LPlanesclpl 39495 LVolsclvol 39496 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-iota 6513 df-fun 6562 df-fv 6568 df-lvols 39503 | 
| This theorem is referenced by: islvol2 39583 lvolnle3at 39585 lvolneatN 39591 lvolnelln 39592 lvolnelpln 39593 lplncvrlvol2 39618 lvolcmp 39620 2lplnja 39622 | 
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