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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 | β’ π΅ = (BaseβπΎ) |
ltrn1o.h | β’ π» = (LHypβπΎ) |
ltrn1o.t | β’ π = ((LTrnβπΎ)βπ) |
Ref | Expression |
---|---|
ltrn1o | β’ (((πΎ β π β§ π β π») β§ πΉ β π) β πΉ:π΅β1-1-ontoβπ΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 766 | . 2 β’ (((πΎ β π β§ π β π») β§ πΉ β π) β πΎ β π) | |
2 | ltrn1o.h | . . 3 β’ π» = (LHypβπΎ) | |
3 | eqid 2733 | . . 3 β’ (LAutβπΎ) = (LAutβπΎ) | |
4 | ltrn1o.t | . . 3 β’ π = ((LTrnβπΎ)βπ) | |
5 | 2, 3, 4 | ltrnlaut 38994 | . 2 β’ (((πΎ β π β§ π β π») β§ πΉ β π) β πΉ β (LAutβπΎ)) |
6 | ltrn1o.b | . . 3 β’ π΅ = (BaseβπΎ) | |
7 | 6, 3 | laut1o 38956 | . 2 β’ ((πΎ β π β§ πΉ β (LAutβπΎ)) β πΉ:π΅β1-1-ontoβπ΅) |
8 | 1, 5, 7 | syl2anc 585 | 1 β’ (((πΎ β π β§ π β π») β§ πΉ β π) β πΉ:π΅β1-1-ontoβπ΅) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 β1-1-ontoβwf1o 6543 βcfv 6544 Basecbs 17144 LHypclh 38855 LAutclaut 38856 LTrncltrn 38972 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-map 8822 df-laut 38860 df-ldil 38975 df-ltrn 38976 |
This theorem is referenced by: ltrncnvnid 38998 ltrncoidN 38999 ltrnid 39006 ltrncnvatb 39009 ltrncnvel 39013 ltrncoval 39016 ltrncnv 39017 ltrneq2 39019 trlcnv 39036 ltrniotacnvval 39453 cdlemg17h 39539 trlcoabs2N 39593 trlcoat 39594 trlcone 39599 cdlemg47a 39605 cdlemg46 39606 cdlemg47 39607 trljco 39611 tgrpgrplem 39620 tendo0pl 39662 tendoipl 39668 cdlemi2 39690 cdlemk2 39703 cdlemk4 39705 cdlemk8 39709 cdlemkid2 39795 cdlemk45 39818 cdlemk53b 39827 cdlemk53 39828 cdlemk55a 39830 tendocnv 39892 dvhgrp 39978 dvhopN 39987 cdlemn3 40068 cdlemn8 40075 cdlemn9 40076 dihordlem7b 40086 dihopelvalcpre 40119 dihmeetlem1N 40161 dihglblem5apreN 40162 |
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