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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 786 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐾 ∈ 𝑉) | |
2 | ltrn1o.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | eqid 2610 | . . 3 ⊢ (LAut‘𝐾) = (LAut‘𝐾) | |
4 | ltrn1o.t | . . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
5 | 2, 3, 4 | ltrnlaut 34427 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ (LAut‘𝐾)) |
6 | ltrn1o.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
7 | 6, 3 | laut1o 34389 | . 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ (LAut‘𝐾)) → 𝐹:𝐵–1-1-onto→𝐵) |
8 | 1, 5, 7 | syl2anc 691 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:𝐵–1-1-onto→𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 –1-1-onto→wf1o 5803 ‘cfv 5804 Basecbs 15695 LHypclh 34288 LAutclaut 34289 LTrncltrn 34405 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-map 7746 df-laut 34293 df-ldil 34408 df-ltrn 34409 |
This theorem is referenced by: ltrncnvnid 34431 ltrncoidN 34432 ltrnid 34439 ltrncnvatb 34442 ltrncnvel 34446 ltrncoval 34449 ltrncnv 34450 ltrneq2 34452 trlcnv 34470 ltrniotacnvval 34888 cdlemg17h 34974 trlcoabs2N 35028 trlcoat 35029 trlcone 35034 cdlemg47a 35040 cdlemg46 35041 cdlemg47 35042 trljco 35046 tgrpgrplem 35055 tendo0pl 35097 tendoipl 35103 cdlemi2 35125 cdlemk2 35138 cdlemk4 35140 cdlemk8 35144 cdlemkid2 35230 cdlemk45 35253 cdlemk53b 35262 cdlemk53 35263 cdlemk55a 35265 tendocnv 35328 dvhgrp 35414 dvhopN 35423 cdlemn3 35504 cdlemn8 35511 cdlemn9 35512 dihordlem7b 35522 dihopelvalcpre 35555 dihmeetlem1N 35597 dihglblem5apreN 35598 |
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