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Mirrors > Home > MPE Home > Th. List > Mathboxes > islpln3 | Structured version Visualization version GIF version |
Description: The predicate "is a lattice plane". (Contributed by NM, 17-Jun-2012.) |
Ref | Expression |
---|---|
islpln3.b | ⊢ 𝐵 = (Base‘𝐾) |
islpln3.l | ⊢ ≤ = (le‘𝐾) |
islpln3.j | ⊢ ∨ = (join‘𝐾) |
islpln3.a | ⊢ 𝐴 = (Atoms‘𝐾) |
islpln3.n | ⊢ 𝑁 = (LLines‘𝐾) |
islpln3.p | ⊢ 𝑃 = (LPlanes‘𝐾) |
Ref | Expression |
---|---|
islpln3 | ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∈ 𝑃 ↔ ∃𝑦 ∈ 𝑁 ∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑝)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | islpln3.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | eqid 2736 | . . 3 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾) | |
3 | islpln3.n | . . 3 ⊢ 𝑁 = (LLines‘𝐾) | |
4 | islpln3.p | . . 3 ⊢ 𝑃 = (LPlanes‘𝐾) | |
5 | 1, 2, 3, 4 | islpln4 37994 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∈ 𝑃 ↔ ∃𝑦 ∈ 𝑁 𝑦( ⋖ ‘𝐾)𝑋)) |
6 | simpll 765 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝑁) → 𝐾 ∈ HL) | |
7 | 1, 3 | llnbase 37972 | . . . . . 6 ⊢ (𝑦 ∈ 𝑁 → 𝑦 ∈ 𝐵) |
8 | 7 | adantl 482 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝑁) → 𝑦 ∈ 𝐵) |
9 | simplr 767 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝑁) → 𝑋 ∈ 𝐵) | |
10 | islpln3.l | . . . . . 6 ⊢ ≤ = (le‘𝐾) | |
11 | islpln3.j | . . . . . 6 ⊢ ∨ = (join‘𝐾) | |
12 | islpln3.a | . . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾) | |
13 | 1, 10, 11, 2, 12 | cvrval3 37876 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑦 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑦( ⋖ ‘𝐾)𝑋 ↔ ∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑦 ∧ (𝑦 ∨ 𝑝) = 𝑋))) |
14 | 6, 8, 9, 13 | syl3anc 1371 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝑁) → (𝑦( ⋖ ‘𝐾)𝑋 ↔ ∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑦 ∧ (𝑦 ∨ 𝑝) = 𝑋))) |
15 | eqcom 2743 | . . . . . . 7 ⊢ ((𝑦 ∨ 𝑝) = 𝑋 ↔ 𝑋 = (𝑦 ∨ 𝑝)) | |
16 | 15 | a1i 11 | . . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝑁) ∧ 𝑝 ∈ 𝐴) → ((𝑦 ∨ 𝑝) = 𝑋 ↔ 𝑋 = (𝑦 ∨ 𝑝))) |
17 | 16 | anbi2d 629 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝑁) ∧ 𝑝 ∈ 𝐴) → ((¬ 𝑝 ≤ 𝑦 ∧ (𝑦 ∨ 𝑝) = 𝑋) ↔ (¬ 𝑝 ≤ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑝)))) |
18 | 17 | rexbidva 3173 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝑁) → (∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑦 ∧ (𝑦 ∨ 𝑝) = 𝑋) ↔ ∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑝)))) |
19 | 14, 18 | bitrd 278 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝑁) → (𝑦( ⋖ ‘𝐾)𝑋 ↔ ∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑝)))) |
20 | 19 | rexbidva 3173 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (∃𝑦 ∈ 𝑁 𝑦( ⋖ ‘𝐾)𝑋 ↔ ∃𝑦 ∈ 𝑁 ∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑝)))) |
21 | 5, 20 | bitrd 278 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∈ 𝑃 ↔ ∃𝑦 ∈ 𝑁 ∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑝)))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∃wrex 3073 class class class wbr 5105 ‘cfv 6496 (class class class)co 7357 Basecbs 17083 lecple 17140 joincjn 18200 ⋖ ccvr 37724 Atomscatm 37725 HLchlt 37812 LLinesclln 37954 LPlanesclpl 37955 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-proset 18184 df-poset 18202 df-plt 18219 df-lub 18235 df-glb 18236 df-join 18237 df-meet 18238 df-p0 18314 df-lat 18321 df-clat 18388 df-oposet 37638 df-ol 37640 df-oml 37641 df-covers 37728 df-ats 37729 df-atl 37760 df-cvlat 37784 df-hlat 37813 df-llines 37961 df-lplanes 37962 |
This theorem is referenced by: islpln5 37998 lplnexllnN 38027 |
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