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Mirrors > Home > MPE Home > Th. List > Mathboxes > lvoli | Structured version Visualization version GIF version |
Description: Condition implying 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 |
---|---|
lvoli | ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝑃) ∧ 𝑋𝐶𝑌) → 𝑌 ∈ 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl2 1194 | . 2 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝑃) ∧ 𝑋𝐶𝑌) → 𝑌 ∈ 𝐵) | |
2 | breq1 5042 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝐶𝑌 ↔ 𝑋𝐶𝑌)) | |
3 | 2 | rspcev 3527 | . . 3 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑋𝐶𝑌) → ∃𝑥 ∈ 𝑃 𝑥𝐶𝑌) |
4 | 3 | 3ad2antl3 1189 | . 2 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝑃) ∧ 𝑋𝐶𝑌) → ∃𝑥 ∈ 𝑃 𝑥𝐶𝑌) |
5 | simpl1 1193 | . . 3 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝑃) ∧ 𝑋𝐶𝑌) → 𝐾 ∈ 𝐷) | |
6 | lvolset.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
7 | lvolset.c | . . . 4 ⊢ 𝐶 = ( ⋖ ‘𝐾) | |
8 | lvolset.p | . . . 4 ⊢ 𝑃 = (LPlanes‘𝐾) | |
9 | lvolset.v | . . . 4 ⊢ 𝑉 = (LVols‘𝐾) | |
10 | 6, 7, 8, 9 | islvol 37273 | . . 3 ⊢ (𝐾 ∈ 𝐷 → (𝑌 ∈ 𝑉 ↔ (𝑌 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝑃 𝑥𝐶𝑌))) |
11 | 5, 10 | syl 17 | . 2 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝑃) ∧ 𝑋𝐶𝑌) → (𝑌 ∈ 𝑉 ↔ (𝑌 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝑃 𝑥𝐶𝑌))) |
12 | 1, 4, 11 | mpbir2and 713 | 1 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝑃) ∧ 𝑋𝐶𝑌) → 𝑌 ∈ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ∃wrex 3052 class class class wbr 5039 ‘cfv 6358 Basecbs 16666 ⋖ ccvr 36962 LPlanesclpl 37192 LVolsclvol 37193 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-iota 6316 df-fun 6360 df-fv 6366 df-lvols 37200 |
This theorem is referenced by: lplncvrlvol 37316 |
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