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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > islln3 | Structured version Visualization version GIF version |
Description: The predicate "is a lattice line". (Contributed by NM, 17-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 |
---|---|
islln3 | ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∈ 𝑁 ↔ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝 ∨ 𝑞)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | islln3.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | eqid 2735 | . . 3 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾) | |
3 | islln3.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | islln3.n | . . 3 ⊢ 𝑁 = (LLines‘𝐾) | |
5 | 1, 2, 3, 4 | islln4 39490 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∈ 𝑁 ↔ ∃𝑝 ∈ 𝐴 𝑝( ⋖ ‘𝐾)𝑋)) |
6 | simpll 767 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → 𝐾 ∈ HL) | |
7 | 1, 3 | atbase 39271 | . . . . . 6 ⊢ (𝑝 ∈ 𝐴 → 𝑝 ∈ 𝐵) |
8 | 7 | adantl 481 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ 𝐵) |
9 | simplr 769 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → 𝑋 ∈ 𝐵) | |
10 | eqid 2735 | . . . . . 6 ⊢ (le‘𝐾) = (le‘𝐾) | |
11 | islln3.j | . . . . . 6 ⊢ ∨ = (join‘𝐾) | |
12 | 1, 10, 11, 2, 3 | cvrval3 39396 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑝 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑝( ⋖ ‘𝐾)𝑋 ↔ ∃𝑞 ∈ 𝐴 (¬ 𝑞(le‘𝐾)𝑝 ∧ (𝑝 ∨ 𝑞) = 𝑋))) |
13 | 6, 8, 9, 12 | syl3anc 1370 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → (𝑝( ⋖ ‘𝐾)𝑋 ↔ ∃𝑞 ∈ 𝐴 (¬ 𝑞(le‘𝐾)𝑝 ∧ (𝑝 ∨ 𝑞) = 𝑋))) |
14 | hlatl 39342 | . . . . . . . . 9 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat) | |
15 | 14 | ad3antrrr 730 | . . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) ∧ 𝑞 ∈ 𝐴) → 𝐾 ∈ AtLat) |
16 | simpr 484 | . . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) ∧ 𝑞 ∈ 𝐴) → 𝑞 ∈ 𝐴) | |
17 | simplr 769 | . . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) ∧ 𝑞 ∈ 𝐴) → 𝑝 ∈ 𝐴) | |
18 | 10, 3 | atncmp 39294 | . . . . . . . 8 ⊢ ((𝐾 ∈ AtLat ∧ 𝑞 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) → (¬ 𝑞(le‘𝐾)𝑝 ↔ 𝑞 ≠ 𝑝)) |
19 | 15, 16, 17, 18 | syl3anc 1370 | . . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) ∧ 𝑞 ∈ 𝐴) → (¬ 𝑞(le‘𝐾)𝑝 ↔ 𝑞 ≠ 𝑝)) |
20 | necom 2992 | . . . . . . 7 ⊢ (𝑞 ≠ 𝑝 ↔ 𝑝 ≠ 𝑞) | |
21 | 19, 20 | bitrdi 287 | . . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) ∧ 𝑞 ∈ 𝐴) → (¬ 𝑞(le‘𝐾)𝑝 ↔ 𝑝 ≠ 𝑞)) |
22 | eqcom 2742 | . . . . . . 7 ⊢ ((𝑝 ∨ 𝑞) = 𝑋 ↔ 𝑋 = (𝑝 ∨ 𝑞)) | |
23 | 22 | a1i 11 | . . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) ∧ 𝑞 ∈ 𝐴) → ((𝑝 ∨ 𝑞) = 𝑋 ↔ 𝑋 = (𝑝 ∨ 𝑞))) |
24 | 21, 23 | anbi12d 632 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) ∧ 𝑞 ∈ 𝐴) → ((¬ 𝑞(le‘𝐾)𝑝 ∧ (𝑝 ∨ 𝑞) = 𝑋) ↔ (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝 ∨ 𝑞)))) |
25 | 24 | rexbidva 3175 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → (∃𝑞 ∈ 𝐴 (¬ 𝑞(le‘𝐾)𝑝 ∧ (𝑝 ∨ 𝑞) = 𝑋) ↔ ∃𝑞 ∈ 𝐴 (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝 ∨ 𝑞)))) |
26 | 13, 25 | bitrd 279 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → (𝑝( ⋖ ‘𝐾)𝑋 ↔ ∃𝑞 ∈ 𝐴 (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝 ∨ 𝑞)))) |
27 | 26 | rexbidva 3175 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (∃𝑝 ∈ 𝐴 𝑝( ⋖ ‘𝐾)𝑋 ↔ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝 ∨ 𝑞)))) |
28 | 5, 27 | bitrd 279 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∈ 𝑁 ↔ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝 ∨ 𝑞)))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∃wrex 3068 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 lecple 17305 joincjn 18369 ⋖ ccvr 39244 Atomscatm 39245 AtLatcal 39246 HLchlt 39332 LLinesclln 39474 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-proset 18352 df-poset 18371 df-plt 18388 df-lub 18404 df-glb 18405 df-join 18406 df-meet 18407 df-p0 18483 df-lat 18490 df-clat 18557 df-oposet 39158 df-ol 39160 df-oml 39161 df-covers 39248 df-ats 39249 df-atl 39280 df-cvlat 39304 df-hlat 39333 df-llines 39481 |
This theorem is referenced by: islln2 39494 llni2 39495 atcvrlln2 39502 atcvrlln 39503 llnexchb2 39852 |
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