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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 35594 | . 2 ⊢ (𝐾 ∈ 𝐴 → (𝑋 ∈ 𝑉 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑦 ∈ 𝑃 𝑦𝐶𝑋))) |
6 | 5 | baibd 536 | 1 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (𝑋 ∈ 𝑉 ↔ ∃𝑦 ∈ 𝑃 𝑦𝐶𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ∃wrex 3090 class class class wbr 4843 ‘cfv 6101 Basecbs 16184 ⋖ ccvr 35283 LPlanesclpl 35513 LVolsclvol 35514 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pr 5097 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-sbc 3634 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-iota 6064 df-fun 6103 df-fv 6109 df-lvols 35521 |
This theorem is referenced by: islvol3 35597 lvolcmp 35638 |
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