![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > islvol3 | Structured version Visualization version GIF version |
Description: The predicate "is a 3-dim lattice volume". (Contributed by NM, 1-Jul-2012.) |
Ref | Expression |
---|---|
islvol3.b | ⊢ 𝐵 = (Base‘𝐾) |
islvol3.l | ⊢ ≤ = (le‘𝐾) |
islvol3.j | ⊢ ∨ = (join‘𝐾) |
islvol3.a | ⊢ 𝐴 = (Atoms‘𝐾) |
islvol3.p | ⊢ 𝑃 = (LPlanes‘𝐾) |
islvol3.v | ⊢ 𝑉 = (LVols‘𝐾) |
Ref | Expression |
---|---|
islvol3 | ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∈ 𝑉 ↔ ∃𝑦 ∈ 𝑃 ∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑝)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | islvol3.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | eqid 2726 | . . 3 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾) | |
3 | islvol3.p | . . 3 ⊢ 𝑃 = (LPlanes‘𝐾) | |
4 | islvol3.v | . . 3 ⊢ 𝑉 = (LVols‘𝐾) | |
5 | 1, 2, 3, 4 | islvol4 39285 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∈ 𝑉 ↔ ∃𝑦 ∈ 𝑃 𝑦( ⋖ ‘𝐾)𝑋)) |
6 | simpll 765 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝑃) → 𝐾 ∈ HL) | |
7 | 1, 3 | lplnbase 39245 | . . . . . 6 ⊢ (𝑦 ∈ 𝑃 → 𝑦 ∈ 𝐵) |
8 | 7 | adantl 480 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝑃) → 𝑦 ∈ 𝐵) |
9 | simplr 767 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝑃) → 𝑋 ∈ 𝐵) | |
10 | islvol3.l | . . . . . 6 ⊢ ≤ = (le‘𝐾) | |
11 | islvol3.j | . . . . . 6 ⊢ ∨ = (join‘𝐾) | |
12 | islvol3.a | . . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾) | |
13 | 1, 10, 11, 2, 12 | cvrval3 39124 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑦 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑦( ⋖ ‘𝐾)𝑋 ↔ ∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑦 ∧ (𝑦 ∨ 𝑝) = 𝑋))) |
14 | 6, 8, 9, 13 | syl3anc 1368 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝑃) → (𝑦( ⋖ ‘𝐾)𝑋 ↔ ∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑦 ∧ (𝑦 ∨ 𝑝) = 𝑋))) |
15 | eqcom 2733 | . . . . . . 7 ⊢ ((𝑦 ∨ 𝑝) = 𝑋 ↔ 𝑋 = (𝑦 ∨ 𝑝)) | |
16 | 15 | a1i 11 | . . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝑃) ∧ 𝑝 ∈ 𝐴) → ((𝑦 ∨ 𝑝) = 𝑋 ↔ 𝑋 = (𝑦 ∨ 𝑝))) |
17 | 16 | anbi2d 628 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝑃) ∧ 𝑝 ∈ 𝐴) → ((¬ 𝑝 ≤ 𝑦 ∧ (𝑦 ∨ 𝑝) = 𝑋) ↔ (¬ 𝑝 ≤ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑝)))) |
18 | 17 | rexbidva 3167 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝑃) → (∃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 ∈ 𝑃) → (𝑦( ⋖ ‘𝐾)𝑋 ↔ ∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑝)))) |
20 | 19 | rexbidva 3167 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (∃𝑦 ∈ 𝑃 𝑦( ⋖ ‘𝐾)𝑋 ↔ ∃𝑦 ∈ 𝑃 ∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑝)))) |
21 | 5, 20 | bitrd 278 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∈ 𝑉 ↔ ∃𝑦 ∈ 𝑃 ∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑝)))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ∃wrex 3060 class class class wbr 5145 ‘cfv 6545 (class class class)co 7415 Basecbs 17207 lecple 17267 joincjn 18330 ⋖ ccvr 38972 Atomscatm 38973 HLchlt 39060 LPlanesclpl 39203 LVolsclvol 39204 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7737 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3365 df-reu 3366 df-rab 3421 df-v 3466 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4325 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4908 df-iun 4997 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6497 df-fun 6547 df-fn 6548 df-f 6549 df-f1 6550 df-fo 6551 df-f1o 6552 df-fv 6553 df-riota 7371 df-ov 7418 df-oprab 7419 df-proset 18314 df-poset 18332 df-plt 18349 df-lub 18365 df-glb 18366 df-join 18367 df-meet 18368 df-p0 18444 df-lat 18451 df-clat 18518 df-oposet 38886 df-ol 38888 df-oml 38889 df-covers 38976 df-ats 38977 df-atl 39008 df-cvlat 39032 df-hlat 39061 df-lplanes 39210 df-lvols 39211 |
This theorem is referenced by: lvoli3 39288 islvol5 39290 |
Copyright terms: Public domain | W3C validator |