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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 763 | . 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 37141 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ (LAut‘𝐾)) |
6 | ltrn1o.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
7 | 6, 3 | laut1o 37103 | . 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ (LAut‘𝐾)) → 𝐹:𝐵–1-1-onto→𝐵) |
8 | 1, 5, 7 | syl2anc 584 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:𝐵–1-1-onto→𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 –1-1-onto→wf1o 6348 ‘cfv 6349 Basecbs 16473 LHypclh 37002 LAutclaut 37003 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-pow 5258 ax-pr 5321 ax-un 7450 |
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-pw 4539 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-oprab 7149 df-mpo 7150 df-map 8398 df-laut 37007 df-ldil 37122 df-ltrn 37123 |
This theorem is referenced by: ltrncnvnid 37145 ltrncoidN 37146 ltrnid 37153 ltrncnvatb 37156 ltrncnvel 37160 ltrncoval 37163 ltrncnv 37164 ltrneq2 37166 trlcnv 37183 ltrniotacnvval 37600 cdlemg17h 37686 trlcoabs2N 37740 trlcoat 37741 trlcone 37746 cdlemg47a 37752 cdlemg46 37753 cdlemg47 37754 trljco 37758 tgrpgrplem 37767 tendo0pl 37809 tendoipl 37815 cdlemi2 37837 cdlemk2 37850 cdlemk4 37852 cdlemk8 37856 cdlemkid2 37942 cdlemk45 37965 cdlemk53b 37974 cdlemk53 37975 cdlemk55a 37977 tendocnv 38039 dvhgrp 38125 dvhopN 38134 cdlemn3 38215 cdlemn8 38222 cdlemn9 38223 dihordlem7b 38233 dihopelvalcpre 38266 dihmeetlem1N 38308 dihglblem5apreN 38309 |
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