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Mirrors > Home > MPE Home > Th. List > Mathboxes > llnbase | Structured version Visualization version GIF version |
Description: A lattice line is a lattice element. (Contributed by NM, 16-Jun-2012.) |
Ref | Expression |
---|---|
llnbase.b | ⊢ 𝐵 = (Base‘𝐾) |
llnbase.n | ⊢ 𝑁 = (LLines‘𝐾) |
Ref | Expression |
---|---|
llnbase | ⊢ (𝑋 ∈ 𝑁 → 𝑋 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0i 3920 | . . . 4 ⊢ (𝑋 ∈ 𝑁 → ¬ 𝑁 = ∅) | |
2 | llnbase.n | . . . . 5 ⊢ 𝑁 = (LLines‘𝐾) | |
3 | 2 | eqeq1i 2627 | . . . 4 ⊢ (𝑁 = ∅ ↔ (LLines‘𝐾) = ∅) |
4 | 1, 3 | sylnib 318 | . . 3 ⊢ (𝑋 ∈ 𝑁 → ¬ (LLines‘𝐾) = ∅) |
5 | fvprc 6185 | . . 3 ⊢ (¬ 𝐾 ∈ V → (LLines‘𝐾) = ∅) | |
6 | 4, 5 | nsyl2 142 | . 2 ⊢ (𝑋 ∈ 𝑁 → 𝐾 ∈ V) |
7 | llnbase.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
8 | eqid 2622 | . . . 4 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾) | |
9 | eqid 2622 | . . . 4 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
10 | 7, 8, 9, 2 | islln 34792 | . . 3 ⊢ (𝐾 ∈ V → (𝑋 ∈ 𝑁 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑝 ∈ (Atoms‘𝐾)𝑝( ⋖ ‘𝐾)𝑋))) |
11 | 10 | simprbda 653 | . 2 ⊢ ((𝐾 ∈ V ∧ 𝑋 ∈ 𝑁) → 𝑋 ∈ 𝐵) |
12 | 6, 11 | mpancom 703 | 1 ⊢ (𝑋 ∈ 𝑁 → 𝑋 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ∃wrex 2913 Vcvv 3200 ∅c0 3915 class class class wbr 4653 ‘cfv 5888 Basecbs 15857 ⋖ ccvr 34549 Atomscatm 34550 LLinesclln 34777 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-llines 34784 |
This theorem is referenced by: islln2 34797 llnnleat 34799 llnneat 34800 atcvrlln2 34805 llnexatN 34807 llncmp 34808 2llnmat 34810 islpln3 34819 llnmlplnN 34825 lplnle 34826 lplnnle2at 34827 llncvrlpln2 34843 llncvrlpln 34844 2llnmj 34846 lplncmp 34848 lplnexatN 34849 lplnexllnN 34850 2llnm2N 34854 2llnm3N 34855 2llnm4 34856 2llnmeqat 34857 dalem21 34980 dalem54 35012 dalem55 35013 dalem57 35015 dalem60 35018 llnexchb2lem 35154 llnexchb2 35155 llnexch2N 35156 |
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