Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
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 4248 | . . . 4 ⊢ (𝑋 ∈ 𝑁 → ¬ 𝑁 = ∅) | |
2 | llnbase.n | . . . . 5 ⊢ 𝑁 = (LLines‘𝐾) | |
3 | 2 | eqeq1i 2742 | . . . 4 ⊢ (𝑁 = ∅ ↔ (LLines‘𝐾) = ∅) |
4 | 1, 3 | sylnib 331 | . . 3 ⊢ (𝑋 ∈ 𝑁 → ¬ (LLines‘𝐾) = ∅) |
5 | fvprc 6709 | . . 3 ⊢ (¬ 𝐾 ∈ V → (LLines‘𝐾) = ∅) | |
6 | 4, 5 | nsyl2 143 | . 2 ⊢ (𝑋 ∈ 𝑁 → 𝐾 ∈ V) |
7 | llnbase.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
8 | eqid 2737 | . . . 4 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾) | |
9 | eqid 2737 | . . . 4 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
10 | 7, 8, 9, 2 | islln 37257 | . . 3 ⊢ (𝐾 ∈ V → (𝑋 ∈ 𝑁 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑝 ∈ (Atoms‘𝐾)𝑝( ⋖ ‘𝐾)𝑋))) |
11 | 10 | simprbda 502 | . 2 ⊢ ((𝐾 ∈ V ∧ 𝑋 ∈ 𝑁) → 𝑋 ∈ 𝐵) |
12 | 6, 11 | mpancom 688 | 1 ⊢ (𝑋 ∈ 𝑁 → 𝑋 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 ∃wrex 3062 Vcvv 3408 ∅c0 4237 class class class wbr 5053 ‘cfv 6380 Basecbs 16760 ⋖ ccvr 37013 Atomscatm 37014 LLinesclln 37242 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-iota 6338 df-fun 6382 df-fv 6388 df-llines 37249 |
This theorem is referenced by: islln2 37262 llnnleat 37264 llnneat 37265 atcvrlln2 37270 llnexatN 37272 llncmp 37273 2llnmat 37275 islpln3 37284 llnmlplnN 37290 lplnle 37291 lplnnle2at 37292 llncvrlpln2 37308 llncvrlpln 37309 2llnmj 37311 lplncmp 37313 lplnexatN 37314 lplnexllnN 37315 2llnm2N 37319 2llnm3N 37320 2llnm4 37321 2llnmeqat 37322 dalem21 37445 dalem54 37477 dalem55 37478 dalem57 37480 dalem60 37483 llnexchb2lem 37619 llnexchb2 37620 llnexch2N 37621 |
Copyright terms: Public domain | W3C validator |