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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 4346 | . . . 4 ⊢ (𝑋 ∈ 𝑃 → ¬ 𝑃 = ∅) | |
2 | lplnbase.p | . . . . 5 ⊢ 𝑃 = (LPlanes‘𝐾) | |
3 | 2 | eqeq1i 2740 | . . . 4 ⊢ (𝑃 = ∅ ↔ (LPlanes‘𝐾) = ∅) |
4 | 1, 3 | sylnib 328 | . . 3 ⊢ (𝑋 ∈ 𝑃 → ¬ (LPlanes‘𝐾) = ∅) |
5 | fvprc 6899 | . . 3 ⊢ (¬ 𝐾 ∈ V → (LPlanes‘𝐾) = ∅) | |
6 | 4, 5 | nsyl2 141 | . 2 ⊢ (𝑋 ∈ 𝑃 → 𝐾 ∈ V) |
7 | lplnbase.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
8 | eqid 2735 | . . . 4 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾) | |
9 | eqid 2735 | . . . 4 ⊢ (LLines‘𝐾) = (LLines‘𝐾) | |
10 | 7, 8, 9, 2 | islpln 39513 | . . 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 1537 ∈ wcel 2106 ∃wrex 3068 Vcvv 3478 ∅c0 4339 class class class wbr 5148 ‘cfv 6563 Basecbs 17245 ⋖ ccvr 39244 LLinesclln 39474 LPlanesclpl 39475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-lplanes 39482 |
This theorem is referenced by: islpln2 39519 llnmlplnN 39522 lplnnle2at 39524 lplnneat 39528 lplnnelln 39529 llncvrlpln2 39540 2lplnmN 39542 lplncmp 39545 lplnexatN 39546 lplnexllnN 39547 2llnjaN 39549 islvol3 39559 lvoli3 39560 lvolnle3at 39565 lplncvrlvol2 39598 lplncvrlvol 39599 lvolcmp 39600 2lplnm2N 39604 2lplnmj 39605 dalemyeb 39632 dalem10 39656 dalem16 39662 dalem44 39699 dalem55 39710 |
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