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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 2726 | . 2 β’ (leβπΎ) = (leβπΎ) | |
2 | tendo0.h | . 2 β’ π» = (LHypβπΎ) | |
3 | tendo0.t | . 2 β’ π = ((LTrnβπΎ)βπ) | |
4 | eqid 2726 | . 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 39534 | . . . 4 β’ ((πΎ β HL β§ π β π») β ( I βΎ π΅) β π) |
9 | 8 | adantr 480 | . . 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 40170 | . . 3 β’ π = (π β π β¦ ( I βΎ π΅)) |
12 | 9, 11 | fmptd 7109 | . 2 β’ ((πΎ β HL β§ π β π») β π:πβΆπ) |
13 | 7, 2, 3, 5, 10 | tendo0co2 40172 | . 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 40173 | . 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 40146 | 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 395 = wceq 1533 β wcel 2098 β¦ cmpt 5224 I cid 5566 βΎ cres 5671 βcfv 6537 Basecbs 17153 lecple 17213 HLchlt 38733 LHypclh 39368 LTrncltrn 39485 trLctrl 39542 TEndoctendo 40136 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-riotaBAD 38336 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7974 df-2nd 7975 df-undef 8259 df-map 8824 df-proset 18260 df-poset 18278 df-plt 18295 df-lub 18311 df-glb 18312 df-join 18313 df-meet 18314 df-p0 18390 df-p1 18391 df-lat 18397 df-clat 18464 df-oposet 38559 df-ol 38561 df-oml 38562 df-covers 38649 df-ats 38650 df-atl 38681 df-cvlat 38705 df-hlat 38734 df-llines 38882 df-lplanes 38883 df-lvols 38884 df-lines 38885 df-psubsp 38887 df-pmap 38888 df-padd 39180 df-lhyp 39372 df-laut 39373 df-ldil 39488 df-ltrn 39489 df-trl 39543 df-tendo 40139 |
This theorem is referenced by: tendo0pl 40175 tendo0plr 40176 tendoipl 40181 tendoid0 40209 tendo0mul 40210 tendo0mulr 40211 tendoex 40359 cdleml5N 40364 erngdvlem1 40372 erngdvlem4 40375 erng0g 40378 erngdvlem1-rN 40380 erngdvlem4-rN 40383 dvh0g 40495 dvhopN 40500 dib1dim 40549 dib1dim2 40552 dibss 40553 diblss 40554 diblsmopel 40555 dicn0 40576 cdlemn4 40582 cdlemn4a 40583 cdlemn6 40586 dihopelvalcpre 40632 dihmeetlem4preN 40690 dihatlat 40718 dihatexv 40722 |
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