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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > tendo0tp | Structured version Visualization version GIF version |
Description: Trace-preserving property of endomorphism additive identity. (Contributed by NM, 11-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΅)) |
tendo0tp.l | β’ β€ = (leβπΎ) |
tendo0tp.r | β’ π = ((trLβπΎ)βπ) |
Ref | Expression |
---|---|
tendo0tp | β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β (π β(πβπΉ)) β€ (π βπΉ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tendo0.o | . . . . . 6 β’ π = (π β π β¦ ( I βΎ π΅)) | |
2 | tendo0.b | . . . . . 6 β’ π΅ = (BaseβπΎ) | |
3 | 1, 2 | tendo02 40171 | . . . . 5 β’ (πΉ β π β (πβπΉ) = ( I βΎ π΅)) |
4 | 3 | adantl 481 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β (πβπΉ) = ( I βΎ π΅)) |
5 | 4 | fveq2d 6889 | . . 3 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β (π β(πβπΉ)) = (π β( I βΎ π΅))) |
6 | eqid 2726 | . . . . 5 β’ (0.βπΎ) = (0.βπΎ) | |
7 | tendo0.h | . . . . 5 β’ π» = (LHypβπΎ) | |
8 | tendo0tp.r | . . . . 5 β’ π = ((trLβπΎ)βπ) | |
9 | 2, 6, 7, 8 | trlid0 39560 | . . . 4 β’ ((πΎ β HL β§ π β π») β (π β( I βΎ π΅)) = (0.βπΎ)) |
10 | 9 | adantr 480 | . . 3 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β (π β( I βΎ π΅)) = (0.βπΎ)) |
11 | 5, 10 | eqtrd 2766 | . 2 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β (π β(πβπΉ)) = (0.βπΎ)) |
12 | hlop 38745 | . . . 4 β’ (πΎ β HL β πΎ β OP) | |
13 | 12 | ad2antrr 723 | . . 3 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β πΎ β OP) |
14 | tendo0.t | . . . 4 β’ π = ((LTrnβπΎ)βπ) | |
15 | 2, 7, 14, 8 | trlcl 39548 | . . 3 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β (π βπΉ) β π΅) |
16 | tendo0tp.l | . . . 4 β’ β€ = (leβπΎ) | |
17 | 2, 16, 6 | op0le 38569 | . . 3 β’ ((πΎ β OP β§ (π βπΉ) β π΅) β (0.βπΎ) β€ (π βπΉ)) |
18 | 13, 15, 17 | syl2anc 583 | . 2 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β (0.βπΎ) β€ (π βπΉ)) |
19 | 11, 18 | eqbrtrd 5163 | 1 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β (π β(πβπΉ)) β€ (π βπΉ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 class class class wbr 5141 β¦ cmpt 5224 I cid 5566 βΎ cres 5671 βcfv 6537 Basecbs 17153 lecple 17213 0.cp0 18388 OPcops 38555 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 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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-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-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-lhyp 39372 df-laut 39373 df-ldil 39488 df-ltrn 39489 df-trl 39543 |
This theorem is referenced by: tendo0cl 40174 |
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