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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 4249 | . . . 4 ⊢ (𝑋 ∈ 𝑃 → ¬ 𝑃 = ∅) | |
2 | lplnbase.p | . . . . 5 ⊢ 𝑃 = (LPlanes‘𝐾) | |
3 | 2 | eqeq1i 2803 | . . . 4 ⊢ (𝑃 = ∅ ↔ (LPlanes‘𝐾) = ∅) |
4 | 1, 3 | sylnib 331 | . . 3 ⊢ (𝑋 ∈ 𝑃 → ¬ (LPlanes‘𝐾) = ∅) |
5 | fvprc 6638 | . . 3 ⊢ (¬ 𝐾 ∈ V → (LPlanes‘𝐾) = ∅) | |
6 | 4, 5 | nsyl2 143 | . 2 ⊢ (𝑋 ∈ 𝑃 → 𝐾 ∈ V) |
7 | lplnbase.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
8 | eqid 2798 | . . . 4 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾) | |
9 | eqid 2798 | . . . 4 ⊢ (LLines‘𝐾) = (LLines‘𝐾) | |
10 | 7, 8, 9, 2 | islpln 36826 | . . 3 ⊢ (𝐾 ∈ V → (𝑋 ∈ 𝑃 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑥 ∈ (LLines‘𝐾)𝑥( ⋖ ‘𝐾)𝑋))) |
11 | 10 | simprbda 502 | . 2 ⊢ ((𝐾 ∈ V ∧ 𝑋 ∈ 𝑃) → 𝑋 ∈ 𝐵) |
12 | 6, 11 | mpancom 687 | 1 ⊢ (𝑋 ∈ 𝑃 → 𝑋 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 Vcvv 3441 ∅c0 4243 class class class wbr 5030 ‘cfv 6324 Basecbs 16475 ⋖ ccvr 36558 LLinesclln 36787 LPlanesclpl 36788 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-lplanes 36795 |
This theorem is referenced by: islpln2 36832 llnmlplnN 36835 lplnnle2at 36837 lplnneat 36841 lplnnelln 36842 llncvrlpln2 36853 2lplnmN 36855 lplncmp 36858 lplnexatN 36859 lplnexllnN 36860 2llnjaN 36862 islvol3 36872 lvoli3 36873 lvolnle3at 36878 lplncvrlvol2 36911 lplncvrlvol 36912 lvolcmp 36913 2lplnm2N 36917 2lplnmj 36918 dalemyeb 36945 dalem10 36969 dalem16 36975 dalem44 37012 dalem55 37023 |
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