| Mathbox for Norm Megill |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > islvol4 | Structured version Visualization version GIF version | ||
| Description: The predicate "is a 3-dim lattice volume". (Contributed by NM, 1-Jul-2012.) |
| Ref | Expression |
|---|---|
| lvolset.b | ⊢ 𝐵 = (Base‘𝐾) |
| lvolset.c | ⊢ 𝐶 = ( ⋖ ‘𝐾) |
| lvolset.p | ⊢ 𝑃 = (LPlanes‘𝐾) |
| lvolset.v | ⊢ 𝑉 = (LVols‘𝐾) |
| Ref | Expression |
|---|---|
| islvol4 | ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (𝑋 ∈ 𝑉 ↔ ∃𝑦 ∈ 𝑃 𝑦𝐶𝑋)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lvolset.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
| 2 | lvolset.c | . . 3 ⊢ 𝐶 = ( ⋖ ‘𝐾) | |
| 3 | lvolset.p | . . 3 ⊢ 𝑃 = (LPlanes‘𝐾) | |
| 4 | lvolset.v | . . 3 ⊢ 𝑉 = (LVols‘𝐾) | |
| 5 | 1, 2, 3, 4 | islvol 39553 | . 2 ⊢ (𝐾 ∈ 𝐴 → (𝑋 ∈ 𝑉 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑦 ∈ 𝑃 𝑦𝐶𝑋))) |
| 6 | 5 | baibd 539 | 1 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (𝑋 ∈ 𝑉 ↔ ∃𝑦 ∈ 𝑃 𝑦𝐶𝑋)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃wrex 3069 class class class wbr 5141 ‘cfv 6559 Basecbs 17243 ⋖ ccvr 39241 LPlanesclpl 39472 LVolsclvol 39473 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5294 ax-nul 5304 ax-pr 5430 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 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 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-br 5142 df-opab 5204 df-mpt 5224 df-id 5576 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-iota 6512 df-fun 6561 df-fv 6567 df-lvols 39480 |
| This theorem is referenced by: islvol3 39556 lvolcmp 39597 |
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