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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ltrnlaut | Structured version Visualization version GIF version |
Description: A lattice translation is a lattice automorphism. (Contributed by NM, 20-May-2012.) |
Ref | Expression |
---|---|
ltrnlaut.h | β’ π» = (LHypβπΎ) |
ltrnlaut.i | β’ πΌ = (LAutβπΎ) |
ltrnlaut.t | β’ π = ((LTrnβπΎ)βπ) |
Ref | Expression |
---|---|
ltrnlaut | β’ (((πΎ β π β§ π β π») β§ πΉ β π) β πΉ β πΌ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltrnlaut.h | . . 3 β’ π» = (LHypβπΎ) | |
2 | eqid 2728 | . . 3 β’ ((LDilβπΎ)βπ) = ((LDilβπΎ)βπ) | |
3 | ltrnlaut.t | . . 3 β’ π = ((LTrnβπΎ)βπ) | |
4 | 1, 2, 3 | ltrnldil 39635 | . 2 β’ (((πΎ β π β§ π β π») β§ πΉ β π) β πΉ β ((LDilβπΎ)βπ)) |
5 | ltrnlaut.i | . . 3 β’ πΌ = (LAutβπΎ) | |
6 | 1, 5, 2 | ldillaut 39624 | . 2 β’ (((πΎ β π β§ π β π») β§ πΉ β ((LDilβπΎ)βπ)) β πΉ β πΌ) |
7 | 4, 6 | syldan 589 | 1 β’ (((πΎ β π β§ π β π») β§ πΉ β π) β πΉ β πΌ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 βcfv 6553 LHypclh 39497 LAutclaut 39498 LDilcldil 39613 LTrncltrn 39614 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-ldil 39617 df-ltrn 39618 |
This theorem is referenced by: ltrn1o 39637 ltrncl 39638 ltrn11 39639 ltrnle 39642 ltrncnvleN 39643 ltrnm 39644 ltrnj 39645 ltrncvr 39646 ltrnid 39648 ltrneq2 39661 |
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