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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 37169 | . . 3 ⊢ (𝑋 ∈ 𝑁 → 𝑋 ∈ 𝐵) |
4 | 3 | pm4.71ri 564 | . 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 37170 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∈ 𝑁 ↔ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝 ∨ 𝑞)))) |
8 | 7 | pm5.32da 582 | . 2 ⊢ (𝐾 ∈ HL → ((𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝑁) ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝 ∨ 𝑞))))) |
9 | 4, 8 | syl5bb 286 | 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 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ≠ wne 2935 ∃wrex 3055 ‘cfv 6340 (class class class)co 7173 Basecbs 16589 joincjn 17673 Atomscatm 36923 HLchlt 37010 LLinesclln 37151 |
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 2711 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 |
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 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-op 4524 df-uni 4798 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-id 5430 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7130 df-ov 7176 df-oprab 7177 df-proset 17657 df-poset 17675 df-plt 17687 df-lub 17703 df-glb 17704 df-join 17705 df-meet 17706 df-p0 17768 df-lat 17775 df-clat 17837 df-oposet 36836 df-ol 36838 df-oml 36839 df-covers 36926 df-ats 36927 df-atl 36958 df-cvlat 36982 df-hlat 37011 df-llines 37158 |
This theorem is referenced by: islpln5 37195 lplnnlelln 37203 llncvrlpln2 37217 2llnjN 37227 lvolnlelln 37244 |
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