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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ltrnlaut | Structured version Visualization version GIF version |
Description: A lattice translation is a lattice automorphism. (Contributed by NM, 20-May-2012.) |
Ref | Expression |
---|---|
ltrnlaut.h | ⊢ 𝐻 = (LHyp‘𝐾) |
ltrnlaut.i | ⊢ 𝐼 = (LAut‘𝐾) |
ltrnlaut.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
ltrnlaut | ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltrnlaut.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | eqid 2760 | . . 3 ⊢ ((LDil‘𝐾)‘𝑊) = ((LDil‘𝐾)‘𝑊) | |
3 | ltrnlaut.t | . . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
4 | 1, 2, 3 | ltrnldil 35911 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ ((LDil‘𝐾)‘𝑊)) |
5 | ltrnlaut.i | . . 3 ⊢ 𝐼 = (LAut‘𝐾) | |
6 | 1, 5, 2 | ldillaut 35900 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ ((LDil‘𝐾)‘𝑊)) → 𝐹 ∈ 𝐼) |
7 | 4, 6 | syldan 488 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 ‘cfv 6049 LHypclh 35773 LAutclaut 35774 LDilcldil 35889 LTrncltrn 35890 |
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-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-pr 5055 |
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-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-ov 6816 df-ldil 35893 df-ltrn 35894 |
This theorem is referenced by: ltrn1o 35913 ltrncl 35914 ltrn11 35915 ltrnle 35918 ltrncnvleN 35919 ltrnm 35920 ltrnj 35921 ltrncvr 35922 ltrnid 35924 ltrneq2 35937 |
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