Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > islln4 | Structured version Visualization version GIF version |
Description: The predicate "is a lattice line". (Contributed by NM, 16-Jun-2012.) |
Ref | Expression |
---|---|
llnset.b | ⊢ 𝐵 = (Base‘𝐾) |
llnset.c | ⊢ 𝐶 = ( ⋖ ‘𝐾) |
llnset.a | ⊢ 𝐴 = (Atoms‘𝐾) |
llnset.n | ⊢ 𝑁 = (LLines‘𝐾) |
Ref | Expression |
---|---|
islln4 | ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐵) → (𝑋 ∈ 𝑁 ↔ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | llnset.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | llnset.c | . . 3 ⊢ 𝐶 = ( ⋖ ‘𝐾) | |
3 | llnset.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | llnset.n | . . 3 ⊢ 𝑁 = (LLines‘𝐾) | |
5 | 1, 2, 3, 4 | islln 36641 | . 2 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑁 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑋))) |
6 | 5 | baibd 542 | 1 ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐵) → (𝑋 ∈ 𝑁 ↔ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∃wrex 3139 class class class wbr 5065 ‘cfv 6354 Basecbs 16482 ⋖ ccvr 36397 Atomscatm 36398 LLinesclln 36626 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-iota 6313 df-fun 6356 df-fv 6362 df-llines 36633 |
This theorem is referenced by: islln3 36645 llncmp 36657 |
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