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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > trlid0b | Structured version Visualization version GIF version |
Description: A lattice translation is the identity iff its trace is zero. (Contributed by NM, 14-Jun-2013.) |
Ref | Expression |
---|---|
trlid0b.b | ⊢ 𝐵 = (Base‘𝐾) |
trlid0b.z | ⊢ 0 = (0.‘𝐾) |
trlid0b.h | ⊢ 𝐻 = (LHyp‘𝐾) |
trlid0b.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
trlid0b.r | ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
trlid0b | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝐹 = ( I ↾ 𝐵) ↔ (𝑅‘𝐹) = 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trlid0b.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
2 | eqid 2731 | . . . 4 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
3 | trlid0b.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | trlid0b.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
5 | trlid0b.r | . . . 4 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) | |
6 | 1, 2, 3, 4, 5 | trlnidatb 39512 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝐹 ≠ ( I ↾ 𝐵) ↔ (𝑅‘𝐹) ∈ (Atoms‘𝐾))) |
7 | trlid0b.z | . . . 4 ⊢ 0 = (0.‘𝐾) | |
8 | 7, 2, 3, 4, 5 | trlatn0 39507 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ((𝑅‘𝐹) ∈ (Atoms‘𝐾) ↔ (𝑅‘𝐹) ≠ 0 )) |
9 | 6, 8 | bitrd 279 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝐹 ≠ ( I ↾ 𝐵) ↔ (𝑅‘𝐹) ≠ 0 )) |
10 | 9 | necon4bid 2985 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝐹 = ( I ↾ 𝐵) ↔ (𝑅‘𝐹) = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 I cid 5573 ↾ cres 5678 ‘cfv 6543 Basecbs 17151 0.cp0 18386 Atomscatm 38597 HLchlt 38684 LHypclh 39319 LTrncltrn 39436 trLctrl 39493 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-map 8828 df-proset 18258 df-poset 18276 df-plt 18293 df-lub 18309 df-glb 18310 df-join 18311 df-meet 18312 df-p0 18388 df-p1 18389 df-lat 18395 df-clat 18462 df-oposet 38510 df-ol 38512 df-oml 38513 df-covers 38600 df-ats 38601 df-atl 38632 df-cvlat 38656 df-hlat 38685 df-lhyp 39323 df-laut 39324 df-ldil 39439 df-ltrn 39440 df-trl 39494 |
This theorem is referenced by: trlnid 39514 trlcoat 40058 trlcone 40063 trljco 40075 tendoid 40108 tendoex 40310 dia0 40387 |
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