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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > tendo0cl | Structured version Visualization version GIF version |
Description: The additive identity is a trace-preserving endormorphism. (Contributed by NM, 12-Jun-2013.) |
Ref | Expression |
---|---|
tendo0.b | β’ π΅ = (BaseβπΎ) |
tendo0.h | β’ π» = (LHypβπΎ) |
tendo0.t | β’ π = ((LTrnβπΎ)βπ) |
tendo0.e | β’ πΈ = ((TEndoβπΎ)βπ) |
tendo0.o | β’ π = (π β π β¦ ( I βΎ π΅)) |
Ref | Expression |
---|---|
tendo0cl | β’ ((πΎ β HL β§ π β π») β π β πΈ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2728 | . 2 β’ (leβπΎ) = (leβπΎ) | |
2 | tendo0.h | . 2 β’ π» = (LHypβπΎ) | |
3 | tendo0.t | . 2 β’ π = ((LTrnβπΎ)βπ) | |
4 | eqid 2728 | . 2 β’ ((trLβπΎ)βπ) = ((trLβπΎ)βπ) | |
5 | tendo0.e | . 2 β’ πΈ = ((TEndoβπΎ)βπ) | |
6 | id 22 | . 2 β’ ((πΎ β HL β§ π β π») β (πΎ β HL β§ π β π»)) | |
7 | tendo0.b | . . . . 5 β’ π΅ = (BaseβπΎ) | |
8 | 7, 2, 3 | idltrn 39663 | . . . 4 β’ ((πΎ β HL β§ π β π») β ( I βΎ π΅) β π) |
9 | 8 | adantr 479 | . . 3 β’ (((πΎ β HL β§ π β π») β§ π β π) β ( I βΎ π΅) β π) |
10 | tendo0.o | . . . 4 β’ π = (π β π β¦ ( I βΎ π΅)) | |
11 | 10 | tendo0cbv 40299 | . . 3 β’ π = (π β π β¦ ( I βΎ π΅)) |
12 | 9, 11 | fmptd 7129 | . 2 β’ ((πΎ β HL β§ π β π») β π:πβΆπ) |
13 | 7, 2, 3, 5, 10 | tendo0co2 40301 | . 2 β’ (((πΎ β HL β§ π β π») β§ π β π β§ β β π) β (πβ(π β β)) = ((πβπ) β (πββ))) |
14 | 7, 2, 3, 5, 10, 1, 4 | tendo0tp 40302 | . 2 β’ (((πΎ β HL β§ π β π») β§ π β π) β (((trLβπΎ)βπ)β(πβπ))(leβπΎ)(((trLβπΎ)βπ)βπ)) |
15 | 1, 2, 3, 4, 5, 6, 12, 13, 14 | istendod 40275 | 1 β’ ((πΎ β HL β§ π β π») β π β πΈ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 β¦ cmpt 5235 I cid 5579 βΎ cres 5684 βcfv 6553 Basecbs 17189 lecple 17249 HLchlt 38862 LHypclh 39497 LTrncltrn 39614 trLctrl 39671 TEndoctendo 40265 |
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-pow 5369 ax-pr 5433 ax-un 7748 ax-riotaBAD 38465 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 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-rmo 3374 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-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-iin 5003 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-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 8001 df-2nd 8002 df-undef 8287 df-map 8855 df-proset 18296 df-poset 18314 df-plt 18331 df-lub 18347 df-glb 18348 df-join 18349 df-meet 18350 df-p0 18426 df-p1 18427 df-lat 18433 df-clat 18500 df-oposet 38688 df-ol 38690 df-oml 38691 df-covers 38778 df-ats 38779 df-atl 38810 df-cvlat 38834 df-hlat 38863 df-llines 39011 df-lplanes 39012 df-lvols 39013 df-lines 39014 df-psubsp 39016 df-pmap 39017 df-padd 39309 df-lhyp 39501 df-laut 39502 df-ldil 39617 df-ltrn 39618 df-trl 39672 df-tendo 40268 |
This theorem is referenced by: tendo0pl 40304 tendo0plr 40305 tendoipl 40310 tendoid0 40338 tendo0mul 40339 tendo0mulr 40340 tendoex 40488 cdleml5N 40493 erngdvlem1 40501 erngdvlem4 40504 erng0g 40507 erngdvlem1-rN 40509 erngdvlem4-rN 40512 dvh0g 40624 dvhopN 40629 dib1dim 40678 dib1dim2 40681 dibss 40682 diblss 40683 diblsmopel 40684 dicn0 40705 cdlemn4 40711 cdlemn4a 40712 cdlemn6 40715 dihopelvalcpre 40761 dihmeetlem4preN 40819 dihatlat 40847 dihatexv 40851 |
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