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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > islln | 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 |
---|---|
islln | ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑁 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | llnset.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
2 | llnset.c | . . . 4 ⊢ 𝐶 = ( ⋖ ‘𝐾) | |
3 | llnset.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | llnset.n | . . . 4 ⊢ 𝑁 = (LLines‘𝐾) | |
5 | 1, 2, 3, 4 | llnset 38839 | . . 3 ⊢ (𝐾 ∈ 𝐷 → 𝑁 = {𝑥 ∈ 𝐵 ∣ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑥}) |
6 | 5 | eleq2d 2818 | . 2 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑁 ↔ 𝑋 ∈ {𝑥 ∈ 𝐵 ∣ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑥})) |
7 | breq2 5152 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑝𝐶𝑥 ↔ 𝑝𝐶𝑋)) | |
8 | 7 | rexbidv 3177 | . . 3 ⊢ (𝑥 = 𝑋 → (∃𝑝 ∈ 𝐴 𝑝𝐶𝑥 ↔ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑋)) |
9 | 8 | elrab 3683 | . 2 ⊢ (𝑋 ∈ {𝑥 ∈ 𝐵 ∣ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑥} ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑋)) |
10 | 6, 9 | bitrdi 287 | 1 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑁 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ∃wrex 3069 {crab 3431 class class class wbr 5148 ‘cfv 6543 Basecbs 17151 ⋖ ccvr 38595 Atomscatm 38596 LLinesclln 38825 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-iota 6495 df-fun 6545 df-fv 6551 df-llines 38832 |
This theorem is referenced by: islln4 38841 llni 38842 llnbase 38843 llnnleat 38847 |
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