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Mirrors > Home > MPE Home > Th. List > Mathboxes > ltrn1o | Structured version Visualization version GIF version |
Description: A lattice translation is a one-to-one onto function. (Contributed by NM, 20-May-2012.) |
Ref | Expression |
---|---|
ltrn1o.b | ⊢ 𝐵 = (Base‘𝐾) |
ltrn1o.h | ⊢ 𝐻 = (LHyp‘𝐾) |
ltrn1o.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
ltrn1o | ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:𝐵–1-1-onto→𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 765 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐾 ∈ 𝑉) | |
2 | ltrn1o.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | eqid 2821 | . . 3 ⊢ (LAut‘𝐾) = (LAut‘𝐾) | |
4 | ltrn1o.t | . . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
5 | 2, 3, 4 | ltrnlaut 37253 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ (LAut‘𝐾)) |
6 | ltrn1o.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
7 | 6, 3 | laut1o 37215 | . 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ (LAut‘𝐾)) → 𝐹:𝐵–1-1-onto→𝐵) |
8 | 1, 5, 7 | syl2anc 586 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:𝐵–1-1-onto→𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 –1-1-onto→wf1o 6348 ‘cfv 6349 Basecbs 16477 LHypclh 37114 LAutclaut 37115 LTrncltrn 37231 |
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 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 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 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 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 7153 df-oprab 7154 df-mpo 7155 df-map 8402 df-laut 37119 df-ldil 37234 df-ltrn 37235 |
This theorem is referenced by: ltrncnvnid 37257 ltrncoidN 37258 ltrnid 37265 ltrncnvatb 37268 ltrncnvel 37272 ltrncoval 37275 ltrncnv 37276 ltrneq2 37278 trlcnv 37295 ltrniotacnvval 37712 cdlemg17h 37798 trlcoabs2N 37852 trlcoat 37853 trlcone 37858 cdlemg47a 37864 cdlemg46 37865 cdlemg47 37866 trljco 37870 tgrpgrplem 37879 tendo0pl 37921 tendoipl 37927 cdlemi2 37949 cdlemk2 37962 cdlemk4 37964 cdlemk8 37968 cdlemkid2 38054 cdlemk45 38077 cdlemk53b 38086 cdlemk53 38087 cdlemk55a 38089 tendocnv 38151 dvhgrp 38237 dvhopN 38246 cdlemn3 38327 cdlemn8 38334 cdlemn9 38335 dihordlem7b 38345 dihopelvalcpre 38378 dihmeetlem1N 38420 dihglblem5apreN 38421 |
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