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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 1188 | . 2 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝑃) ∧ 𝑋𝐶𝑌) → 𝑌 ∈ 𝐵) | |
2 | breq1 5069 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝐶𝑌 ↔ 𝑋𝐶𝑌)) | |
3 | 2 | rspcev 3623 | . . 3 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑋𝐶𝑌) → ∃𝑥 ∈ 𝑃 𝑥𝐶𝑌) |
4 | 3 | 3ad2antl3 1183 | . 2 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝑃) ∧ 𝑋𝐶𝑌) → ∃𝑥 ∈ 𝑃 𝑥𝐶𝑌) |
5 | simpl1 1187 | . . 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 36724 | . . 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 711 | 1 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝑃) ∧ 𝑋𝐶𝑌) → 𝑌 ∈ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 class class class wbr 5066 ‘cfv 6355 Basecbs 16483 ⋖ ccvr 36413 LPlanesclpl 36643 LVolsclvol 36644 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-iota 6314 df-fun 6357 df-fv 6363 df-lvols 36651 |
This theorem is referenced by: lplncvrlvol 36767 |
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