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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 2760 | . . . 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 35967 | . . 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 35962 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ((𝑅‘𝐹) ∈ (Atoms‘𝐾) ↔ (𝑅‘𝐹) ≠ 0 )) |
9 | 6, 8 | bitrd 268 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝐹 ≠ ( I ↾ 𝐵) ↔ (𝑅‘𝐹) ≠ 0 )) |
10 | 9 | necon4bid 2977 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝐹 = ( I ↾ 𝐵) ↔ (𝑅‘𝐹) = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1632 ∈ wcel 2139 ≠ wne 2932 I cid 5173 ↾ cres 5268 ‘cfv 6049 Basecbs 16059 0.cp0 17238 Atomscatm 35053 HLchlt 35140 LHypclh 35773 LTrncltrn 35890 trLctrl 35948 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-map 8025 df-preset 17129 df-poset 17147 df-plt 17159 df-lub 17175 df-glb 17176 df-join 17177 df-meet 17178 df-p0 17240 df-p1 17241 df-lat 17247 df-clat 17309 df-oposet 34966 df-ol 34968 df-oml 34969 df-covers 35056 df-ats 35057 df-atl 35088 df-cvlat 35112 df-hlat 35141 df-lhyp 35777 df-laut 35778 df-ldil 35893 df-ltrn 35894 df-trl 35949 |
This theorem is referenced by: trlnid 35969 trlcoat 36513 trlcone 36518 trljco 36530 tendoid 36563 tendoex 36765 dia0 36843 |
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