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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > tendotp | Structured version Visualization version GIF version |
Description: Trace-preserving property of a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.) |
Ref | Expression |
---|---|
tendoset.l | β’ β€ = (leβπΎ) |
tendoset.h | β’ π» = (LHypβπΎ) |
tendoset.t | β’ π = ((LTrnβπΎ)βπ) |
tendoset.r | β’ π = ((trLβπΎ)βπ) |
tendoset.e | β’ πΈ = ((TEndoβπΎ)βπ) |
Ref | Expression |
---|---|
tendotp | β’ (((πΎ β π β§ π β π») β§ π β πΈ β§ πΉ β π) β (π β(πβπΉ)) β€ (π βπΉ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tendoset.l | . . . 4 β’ β€ = (leβπΎ) | |
2 | tendoset.h | . . . 4 β’ π» = (LHypβπΎ) | |
3 | tendoset.t | . . . 4 β’ π = ((LTrnβπΎ)βπ) | |
4 | tendoset.r | . . . 4 β’ π = ((trLβπΎ)βπ) | |
5 | tendoset.e | . . . 4 β’ πΈ = ((TEndoβπΎ)βπ) | |
6 | 1, 2, 3, 4, 5 | istendo 40265 | . . 3 β’ ((πΎ β π β§ π β π») β (π β πΈ β (π:πβΆπ β§ βπ β π βπ β π (πβ(π β π)) = ((πβπ) β (πβπ)) β§ βπ β π (π β(πβπ)) β€ (π βπ)))) |
7 | 2fveq3 6907 | . . . . . 6 β’ (π = πΉ β (π β(πβπ)) = (π β(πβπΉ))) | |
8 | fveq2 6902 | . . . . . 6 β’ (π = πΉ β (π βπ) = (π βπΉ)) | |
9 | 7, 8 | breq12d 5165 | . . . . 5 β’ (π = πΉ β ((π β(πβπ)) β€ (π βπ) β (π β(πβπΉ)) β€ (π βπΉ))) |
10 | 9 | rspccv 3608 | . . . 4 β’ (βπ β π (π β(πβπ)) β€ (π βπ) β (πΉ β π β (π β(πβπΉ)) β€ (π βπΉ))) |
11 | 10 | 3ad2ant3 1132 | . . 3 β’ ((π:πβΆπ β§ βπ β π βπ β π (πβ(π β π)) = ((πβπ) β (πβπ)) β§ βπ β π (π β(πβπ)) β€ (π βπ)) β (πΉ β π β (π β(πβπΉ)) β€ (π βπΉ))) |
12 | 6, 11 | biimtrdi 252 | . 2 β’ ((πΎ β π β§ π β π») β (π β πΈ β (πΉ β π β (π β(πβπΉ)) β€ (π βπΉ)))) |
13 | 12 | 3imp 1108 | 1 β’ (((πΎ β π β§ π β π») β§ π β πΈ 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 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3058 class class class wbr 5152 β ccom 5686 βΆwf 6549 βcfv 6553 lecple 17247 LHypclh 39489 LTrncltrn 39606 trLctrl 39663 TEndoctendo 40257 |
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 7746 |
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-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-ov 7429 df-oprab 7430 df-mpo 7431 df-map 8853 df-tendo 40260 |
This theorem is referenced by: tendococl 40277 tendoid 40278 tendopltp 40285 tendoicl 40301 cdlemi1 40323 tendotr 40335 cdleml1N 40481 dva1dim 40490 dialss 40551 diblss 40675 |
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