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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 2777 | . . . 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 36315 | . . 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 36310 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ((𝑅‘𝐹) ∈ (Atoms‘𝐾) ↔ (𝑅‘𝐹) ≠ 0 )) |
9 | 6, 8 | bitrd 271 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝐹 ≠ ( I ↾ 𝐵) ↔ (𝑅‘𝐹) ≠ 0 )) |
10 | 9 | necon4bid 3013 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝐹 = ( I ↾ 𝐵) ↔ (𝑅‘𝐹) = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2106 ≠ wne 2968 I cid 5260 ↾ cres 5357 ‘cfv 6135 Basecbs 16255 0.cp0 17423 Atomscatm 35401 HLchlt 35488 LHypclh 36122 LTrncltrn 36239 trLctrl 36296 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-map 8142 df-proset 17314 df-poset 17332 df-plt 17344 df-lub 17360 df-glb 17361 df-join 17362 df-meet 17363 df-p0 17425 df-p1 17426 df-lat 17432 df-clat 17494 df-oposet 35314 df-ol 35316 df-oml 35317 df-covers 35404 df-ats 35405 df-atl 35436 df-cvlat 35460 df-hlat 35489 df-lhyp 36126 df-laut 36127 df-ldil 36242 df-ltrn 36243 df-trl 36297 |
This theorem is referenced by: trlnid 36317 trlcoat 36861 trlcone 36866 trljco 36878 tendoid 36911 tendoex 37113 dia0 37190 |
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