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 37165 | . 2 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑁 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑋))) |
6 | 5 | baibd 543 | 1 ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐵) → (𝑋 ∈ 𝑁 ↔ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ∃wrex 3054 class class class wbr 5030 ‘cfv 6339 Basecbs 16588 ⋖ ccvr 36921 Atomscatm 36922 LLinesclln 37150 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-sbc 3681 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-iota 6297 df-fun 6341 df-fv 6347 df-llines 37157 |
This theorem is referenced by: islln3 37169 llncmp 37181 |
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