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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 40300 | . . . . 5 β’ (πΉ β π β (πβπΉ) = ( I βΎ π΅)) |
4 | 3 | adantl 480 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β (πβπΉ) = ( I βΎ π΅)) |
5 | 4 | fveq2d 6906 | . . 3 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β (π β(πβπΉ)) = (π β( I βΎ π΅))) |
6 | eqid 2728 | . . . . 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 39689 | . . . 4 β’ ((πΎ β HL β§ π β π») β (π β( I βΎ π΅)) = (0.βπΎ)) |
10 | 9 | adantr 479 | . . 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 2768 | . 2 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β (π β(πβπΉ)) = (0.βπΎ)) |
12 | hlop 38874 | . . . 4 β’ (πΎ β HL β πΎ β OP) | |
13 | 12 | ad2antrr 724 | . . 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 39677 | . . 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 38698 | . . 3 β’ ((πΎ β OP β§ (π βπΉ) β π΅) β (0.βπΎ) β€ (π βπΉ)) |
18 | 13, 15, 17 | syl2anc 582 | . 2 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β (0.βπΎ) β€ (π βπΉ)) |
19 | 11, 18 | eqbrtrd 5174 | 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 394 = wceq 1533 β wcel 2098 class class class wbr 5152 β¦ cmpt 5235 I cid 5579 βΎ cres 5684 βcfv 6553 Basecbs 17189 lecple 17249 0.cp0 18424 OPcops 38684 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 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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-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-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-lhyp 39501 df-laut 39502 df-ldil 39617 df-ltrn 39618 df-trl 39672 |
This theorem is referenced by: tendo0cl 40303 |
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