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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > tendof | Structured version Visualization version GIF version |
Description: Functionality of a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.) |
Ref | Expression |
---|---|
tendof.h | β’ π» = (LHypβπΎ) |
tendof.t | β’ π = ((LTrnβπΎ)βπ) |
tendof.e | β’ πΈ = ((TEndoβπΎ)βπ) |
Ref | Expression |
---|---|
tendof | β’ (((πΎ β π β§ π β π») β§ π β πΈ) β π:πβΆπ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2728 | . . . 4 β’ (leβπΎ) = (leβπΎ) | |
2 | tendof.h | . . . 4 β’ π» = (LHypβπΎ) | |
3 | tendof.t | . . . 4 β’ π = ((LTrnβπΎ)βπ) | |
4 | eqid 2728 | . . . 4 β’ ((trLβπΎ)βπ) = ((trLβπΎ)βπ) | |
5 | tendof.e | . . . 4 β’ πΈ = ((TEndoβπΎ)βπ) | |
6 | 1, 2, 3, 4, 5 | istendo 40233 | . . 3 β’ ((πΎ β π β§ π β π») β (π β πΈ β (π:πβΆπ β§ βπ β π βπ β π (πβ(π β π)) = ((πβπ) β (πβπ)) β§ βπ β π (((trLβπΎ)βπ)β(πβπ))(leβπΎ)(((trLβπΎ)βπ)βπ)))) |
7 | simp1 1134 | . . 3 β’ ((π:πβΆπ β§ βπ β π βπ β π (πβ(π β π)) = ((πβπ) β (πβπ)) β§ βπ β π (((trLβπΎ)βπ)β(πβπ))(leβπΎ)(((trLβπΎ)βπ)βπ)) β π:πβΆπ) | |
8 | 6, 7 | biimtrdi 252 | . 2 β’ ((πΎ β π β§ π β π») β (π β πΈ β π:πβΆπ)) |
9 | 8 | imp 406 | 1 β’ (((πΎ β π β§ π β π») β§ π β πΈ) β π:πβΆπ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β§ w3a 1085 = wceq 1534 β wcel 2099 βwral 3058 class class class wbr 5148 β ccom 5682 βΆwf 6544 βcfv 6548 lecple 17240 LHypclh 39457 LTrncltrn 39574 trLctrl 39631 TEndoctendo 40225 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-map 8847 df-tendo 40228 |
This theorem is referenced by: tendoeq1 40237 tendocoval 40239 tendocl 40240 tendo1mul 40243 tendo1mulr 40244 tendococl 40245 tendoconid 40302 tendospass 40492 dvhlveclem 40581 dicvscacl 40664 |
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