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Mirrors > Home > MPE Home > Th. List > Mathboxes > islln2 | Structured version Visualization version GIF version |
Description: The predicate "is a lattice line". (Contributed by NM, 23-Jun-2012.) |
Ref | Expression |
---|---|
islln3.b | ⊢ 𝐵 = (Base‘𝐾) |
islln3.j | ⊢ ∨ = (join‘𝐾) |
islln3.a | ⊢ 𝐴 = (Atoms‘𝐾) |
islln3.n | ⊢ 𝑁 = (LLines‘𝐾) |
Ref | Expression |
---|---|
islln2 | ⊢ (𝐾 ∈ HL → (𝑋 ∈ 𝑁 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝 ∨ 𝑞))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | islln3.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
2 | islln3.n | . . . 4 ⊢ 𝑁 = (LLines‘𝐾) | |
3 | 1, 2 | llnbase 37523 | . . 3 ⊢ (𝑋 ∈ 𝑁 → 𝑋 ∈ 𝐵) |
4 | 3 | pm4.71ri 561 | . 2 ⊢ (𝑋 ∈ 𝑁 ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝑁)) |
5 | islln3.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
6 | islln3.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
7 | 1, 5, 6, 2 | islln3 37524 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∈ 𝑁 ↔ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝 ∨ 𝑞)))) |
8 | 7 | pm5.32da 579 | . 2 ⊢ (𝐾 ∈ HL → ((𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝑁) ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝 ∨ 𝑞))))) |
9 | 4, 8 | syl5bb 283 | 1 ⊢ (𝐾 ∈ HL → (𝑋 ∈ 𝑁 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝 ∨ 𝑞))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∃wrex 3065 ‘cfv 6433 (class class class)co 7275 Basecbs 16912 joincjn 18029 Atomscatm 37277 HLchlt 37364 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-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 |
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-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 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-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-proset 18013 df-poset 18031 df-plt 18048 df-lub 18064 df-glb 18065 df-join 18066 df-meet 18067 df-p0 18143 df-lat 18150 df-clat 18217 df-oposet 37190 df-ol 37192 df-oml 37193 df-covers 37280 df-ats 37281 df-atl 37312 df-cvlat 37336 df-hlat 37365 df-llines 37512 |
This theorem is referenced by: islpln5 37549 lplnnlelln 37557 llncvrlpln2 37571 2llnjN 37581 lvolnlelln 37598 |
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