| Mathbox for Norm Megill |
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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 767 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐾 ∈ 𝑉) | |
| 2 | ltrn1o.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
| 3 | eqid 2737 | . . 3 ⊢ (LAut‘𝐾) = (LAut‘𝐾) | |
| 4 | ltrn1o.t | . . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
| 5 | 2, 3, 4 | ltrnlaut 40125 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ (LAut‘𝐾)) |
| 6 | ltrn1o.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
| 7 | 6, 3 | laut1o 40087 | . 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 395 = wceq 1540 ∈ wcel 2108 –1-1-onto→wf1o 6560 ‘cfv 6561 Basecbs 17247 LHypclh 39986 LAutclaut 39987 LTrncltrn 40103 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8868 df-laut 39991 df-ldil 40106 df-ltrn 40107 |
| This theorem is referenced by: ltrncnvnid 40129 ltrncoidN 40130 ltrnid 40137 ltrncnvatb 40140 ltrncnvel 40144 ltrncoval 40147 ltrncnv 40148 ltrneq2 40150 trlcnv 40167 ltrniotacnvval 40584 cdlemg17h 40670 trlcoabs2N 40724 trlcoat 40725 trlcone 40730 cdlemg47a 40736 cdlemg46 40737 cdlemg47 40738 trljco 40742 tgrpgrplem 40751 tendo0pl 40793 tendoipl 40799 cdlemi2 40821 cdlemk2 40834 cdlemk4 40836 cdlemk8 40840 cdlemkid2 40926 cdlemk45 40949 cdlemk53b 40958 cdlemk53 40959 cdlemk55a 40961 tendocnv 41023 dvhgrp 41109 dvhopN 41118 cdlemn3 41199 cdlemn8 41206 cdlemn9 41207 dihordlem7b 41217 dihopelvalcpre 41250 dihmeetlem1N 41292 dihglblem5apreN 41293 |
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