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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 4267 | . . . 4 ⊢ (𝑋 ∈ 𝑁 → ¬ 𝑁 = ∅) | |
2 | llnbase.n | . . . . 5 ⊢ 𝑁 = (LLines‘𝐾) | |
3 | 2 | eqeq1i 2743 | . . . 4 ⊢ (𝑁 = ∅ ↔ (LLines‘𝐾) = ∅) |
4 | 1, 3 | sylnib 328 | . . 3 ⊢ (𝑋 ∈ 𝑁 → ¬ (LLines‘𝐾) = ∅) |
5 | fvprc 6766 | . . 3 ⊢ (¬ 𝐾 ∈ V → (LLines‘𝐾) = ∅) | |
6 | 4, 5 | nsyl2 141 | . 2 ⊢ (𝑋 ∈ 𝑁 → 𝐾 ∈ V) |
7 | llnbase.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
8 | eqid 2738 | . . . 4 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾) | |
9 | eqid 2738 | . . . 4 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
10 | 7, 8, 9, 2 | islln 37520 | . . 3 ⊢ (𝐾 ∈ V → (𝑋 ∈ 𝑁 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑝 ∈ (Atoms‘𝐾)𝑝( ⋖ ‘𝐾)𝑋))) |
11 | 10 | simprbda 499 | . 2 ⊢ ((𝐾 ∈ V ∧ 𝑋 ∈ 𝑁) → 𝑋 ∈ 𝐵) |
12 | 6, 11 | mpancom 685 | 1 ⊢ (𝑋 ∈ 𝑁 → 𝑋 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 ∃wrex 3065 Vcvv 3432 ∅c0 4256 class class class wbr 5074 ‘cfv 6433 Basecbs 16912 ⋖ ccvr 37276 Atomscatm 37277 LLinesclln 37505 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-iota 6391 df-fun 6435 df-fv 6441 df-llines 37512 |
This theorem is referenced by: islln2 37525 llnnleat 37527 llnneat 37528 atcvrlln2 37533 llnexatN 37535 llncmp 37536 2llnmat 37538 islpln3 37547 llnmlplnN 37553 lplnle 37554 lplnnle2at 37555 llncvrlpln2 37571 llncvrlpln 37572 2llnmj 37574 lplncmp 37576 lplnexatN 37577 lplnexllnN 37578 2llnm2N 37582 2llnm3N 37583 2llnm4 37584 2llnmeqat 37585 dalem21 37708 dalem54 37740 dalem55 37741 dalem57 37743 dalem60 37746 llnexchb2lem 37882 llnexchb2 37883 llnexch2N 37884 |
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