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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 4363 | . . . 4 ⊢ (𝑋 ∈ 𝑃 → ¬ 𝑃 = ∅) | |
2 | lplnbase.p | . . . . 5 ⊢ 𝑃 = (LPlanes‘𝐾) | |
3 | 2 | eqeq1i 2745 | . . . 4 ⊢ (𝑃 = ∅ ↔ (LPlanes‘𝐾) = ∅) |
4 | 1, 3 | sylnib 328 | . . 3 ⊢ (𝑋 ∈ 𝑃 → ¬ (LPlanes‘𝐾) = ∅) |
5 | fvprc 6912 | . . 3 ⊢ (¬ 𝐾 ∈ V → (LPlanes‘𝐾) = ∅) | |
6 | 4, 5 | nsyl2 141 | . 2 ⊢ (𝑋 ∈ 𝑃 → 𝐾 ∈ V) |
7 | lplnbase.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
8 | eqid 2740 | . . . 4 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾) | |
9 | eqid 2740 | . . . 4 ⊢ (LLines‘𝐾) = (LLines‘𝐾) | |
10 | 7, 8, 9, 2 | islpln 39487 | . . 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 687 | 1 ⊢ (𝑋 ∈ 𝑃 → 𝑋 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2108 ∃wrex 3076 Vcvv 3488 ∅c0 4352 class class class wbr 5166 ‘cfv 6573 Basecbs 17258 ⋖ ccvr 39218 LLinesclln 39448 LPlanesclpl 39449 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-iota 6525 df-fun 6575 df-fv 6581 df-lplanes 39456 |
This theorem is referenced by: islpln2 39493 llnmlplnN 39496 lplnnle2at 39498 lplnneat 39502 lplnnelln 39503 llncvrlpln2 39514 2lplnmN 39516 lplncmp 39519 lplnexatN 39520 lplnexllnN 39521 2llnjaN 39523 islvol3 39533 lvoli3 39534 lvolnle3at 39539 lplncvrlvol2 39572 lplncvrlvol 39573 lvolcmp 39574 2lplnm2N 39578 2lplnmj 39579 dalemyeb 39606 dalem10 39630 dalem16 39636 dalem44 39673 dalem55 39684 |
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