| Mathbox for Norm Megill |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > lplnbase | Structured version Visualization version GIF version | ||
| Description: A lattice plane is a lattice element. (Contributed by NM, 17-Jun-2012.) |
| Ref | Expression |
|---|---|
| lplnbase.b | ⊢ 𝐵 = (Base‘𝐾) |
| lplnbase.p | ⊢ 𝑃 = (LPlanes‘𝐾) |
| Ref | Expression |
|---|---|
| lplnbase | ⊢ (𝑋 ∈ 𝑃 → 𝑋 ∈ 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | n0i 4340 | . . . 4 ⊢ (𝑋 ∈ 𝑃 → ¬ 𝑃 = ∅) | |
| 2 | lplnbase.p | . . . . 5 ⊢ 𝑃 = (LPlanes‘𝐾) | |
| 3 | 2 | eqeq1i 2742 | . . . 4 ⊢ (𝑃 = ∅ ↔ (LPlanes‘𝐾) = ∅) |
| 4 | 1, 3 | sylnib 328 | . . 3 ⊢ (𝑋 ∈ 𝑃 → ¬ (LPlanes‘𝐾) = ∅) |
| 5 | fvprc 6898 | . . 3 ⊢ (¬ 𝐾 ∈ V → (LPlanes‘𝐾) = ∅) | |
| 6 | 4, 5 | nsyl2 141 | . 2 ⊢ (𝑋 ∈ 𝑃 → 𝐾 ∈ V) |
| 7 | lplnbase.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
| 8 | eqid 2737 | . . . 4 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾) | |
| 9 | eqid 2737 | . . . 4 ⊢ (LLines‘𝐾) = (LLines‘𝐾) | |
| 10 | 7, 8, 9, 2 | islpln 39532 | . . 3 ⊢ (𝐾 ∈ V → (𝑋 ∈ 𝑃 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑥 ∈ (LLines‘𝐾)𝑥( ⋖ ‘𝐾)𝑋))) |
| 11 | 10 | simprbda 498 | . 2 ⊢ ((𝐾 ∈ V ∧ 𝑋 ∈ 𝑃) → 𝑋 ∈ 𝐵) |
| 12 | 6, 11 | mpancom 688 | 1 ⊢ (𝑋 ∈ 𝑃 → 𝑋 ∈ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 Vcvv 3480 ∅c0 4333 class class class wbr 5143 ‘cfv 6561 Basecbs 17247 ⋖ ccvr 39263 LLinesclln 39493 LPlanesclpl 39494 |
| 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 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| 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 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-lplanes 39501 |
| This theorem is referenced by: islpln2 39538 llnmlplnN 39541 lplnnle2at 39543 lplnneat 39547 lplnnelln 39548 llncvrlpln2 39559 2lplnmN 39561 lplncmp 39564 lplnexatN 39565 lplnexllnN 39566 2llnjaN 39568 islvol3 39578 lvoli3 39579 lvolnle3at 39584 lplncvrlvol2 39617 lplncvrlvol 39618 lvolcmp 39619 2lplnm2N 39623 2lplnmj 39624 dalemyeb 39651 dalem10 39675 dalem16 39681 dalem44 39718 dalem55 39729 |
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