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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 2821 | . . 3 ⊢ ((LDil‘𝐾)‘𝑊) = ((LDil‘𝐾)‘𝑊) | |
3 | ltrnlaut.t | . . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
4 | 1, 2, 3 | ltrnldil 37140 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ ((LDil‘𝐾)‘𝑊)) |
5 | ltrnlaut.i | . . 3 ⊢ 𝐼 = (LAut‘𝐾) | |
6 | 1, 5, 2 | ldillaut 37129 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ ((LDil‘𝐾)‘𝑊)) → 𝐹 ∈ 𝐼) |
7 | 4, 6 | syldan 591 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ‘cfv 6349 LHypclh 37002 LAutclaut 37003 LDilcldil 37118 LTrncltrn 37119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7148 df-ldil 37122 df-ltrn 37123 |
This theorem is referenced by: ltrn1o 37142 ltrncl 37143 ltrn11 37144 ltrnle 37147 ltrncnvleN 37148 ltrnm 37149 ltrnj 37150 ltrncvr 37151 ltrnid 37153 ltrneq2 37166 |
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