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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 2821 | . . . 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 37307 | . . 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 37302 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ((𝑅‘𝐹) ∈ (Atoms‘𝐾) ↔ (𝑅‘𝐹) ≠ 0 )) |
9 | 6, 8 | bitrd 281 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝐹 ≠ ( I ↾ 𝐵) ↔ (𝑅‘𝐹) ≠ 0 )) |
10 | 9 | necon4bid 3061 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝐹 = ( I ↾ 𝐵) ↔ (𝑅‘𝐹) = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 I cid 5454 ↾ cres 5552 ‘cfv 6350 Basecbs 16477 0.cp0 17641 Atomscatm 36393 HLchlt 36480 LHypclh 37114 LTrncltrn 37231 trLctrl 37288 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-map 8402 df-proset 17532 df-poset 17550 df-plt 17562 df-lub 17578 df-glb 17579 df-join 17580 df-meet 17581 df-p0 17643 df-p1 17644 df-lat 17650 df-clat 17712 df-oposet 36306 df-ol 36308 df-oml 36309 df-covers 36396 df-ats 36397 df-atl 36428 df-cvlat 36452 df-hlat 36481 df-lhyp 37118 df-laut 37119 df-ldil 37234 df-ltrn 37235 df-trl 37289 |
This theorem is referenced by: trlnid 37309 trlcoat 37853 trlcone 37858 trljco 37870 tendoid 37903 tendoex 38105 dia0 38182 |
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