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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 2737 | . 2 β’ (leβπΎ) = (leβπΎ) | |
2 | tendo0.h | . 2 β’ π» = (LHypβπΎ) | |
3 | tendo0.t | . 2 β’ π = ((LTrnβπΎ)βπ) | |
4 | eqid 2737 | . 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 38616 | . . . 4 β’ ((πΎ β HL β§ π β π») β ( I βΎ π΅) β π) |
9 | 8 | adantr 482 | . . 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 39252 | . . 3 β’ π = (π β π β¦ ( I βΎ π΅)) |
12 | 9, 11 | fmptd 7063 | . 2 β’ ((πΎ β HL β§ π β π») β π:πβΆπ) |
13 | 7, 2, 3, 5, 10 | tendo0co2 39254 | . 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 39255 | . 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 39228 | 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 397 = wceq 1542 β wcel 2107 β¦ cmpt 5189 I cid 5531 βΎ cres 5636 βcfv 6497 Basecbs 17084 lecple 17141 HLchlt 37815 LHypclh 38450 LTrncltrn 38567 trLctrl 38624 TEndoctendo 39218 |
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 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-riotaBAD 37418 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-iin 4958 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-undef 8205 df-map 8768 df-proset 18185 df-poset 18203 df-plt 18220 df-lub 18236 df-glb 18237 df-join 18238 df-meet 18239 df-p0 18315 df-p1 18316 df-lat 18322 df-clat 18389 df-oposet 37641 df-ol 37643 df-oml 37644 df-covers 37731 df-ats 37732 df-atl 37763 df-cvlat 37787 df-hlat 37816 df-llines 37964 df-lplanes 37965 df-lvols 37966 df-lines 37967 df-psubsp 37969 df-pmap 37970 df-padd 38262 df-lhyp 38454 df-laut 38455 df-ldil 38570 df-ltrn 38571 df-trl 38625 df-tendo 39221 |
This theorem is referenced by: tendo0pl 39257 tendo0plr 39258 tendoipl 39263 tendoid0 39291 tendo0mul 39292 tendo0mulr 39293 tendoex 39441 cdleml5N 39446 erngdvlem1 39454 erngdvlem4 39457 erng0g 39460 erngdvlem1-rN 39462 erngdvlem4-rN 39465 dvh0g 39577 dvhopN 39582 dib1dim 39631 dib1dim2 39634 dibss 39635 diblss 39636 diblsmopel 39637 dicn0 39658 cdlemn4 39664 cdlemn4a 39665 cdlemn6 39668 dihopelvalcpre 39714 dihmeetlem4preN 39772 dihatlat 39800 dihatexv 39804 |
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