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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 39226 | . . 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 6848 | . . . . . 6 β’ (π = πΉ β (π β(πβπ)) = (π β(πβπΉ))) | |
8 | fveq2 6843 | . . . . . 6 β’ (π = πΉ β (π βπ) = (π βπΉ)) | |
9 | 7, 8 | breq12d 5119 | . . . . 5 β’ (π = πΉ β ((π β(πβπ)) β€ (π βπ) β (π β(πβπΉ)) β€ (π βπΉ))) |
10 | 9 | rspccv 3579 | . . . 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 1136 | . . 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 | syl6bi 253 | . 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 1112 | 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 397 β§ w3a 1088 = wceq 1542 β wcel 2107 βwral 3065 class class class wbr 5106 β ccom 5638 βΆwf 6493 βcfv 6497 lecple 17141 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 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 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-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-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-ov 7361 df-oprab 7362 df-mpo 7363 df-map 8768 df-tendo 39221 |
This theorem is referenced by: tendococl 39238 tendoid 39239 tendopltp 39246 tendoicl 39262 cdlemi1 39284 tendotr 39296 cdleml1N 39442 dva1dim 39451 dialss 39512 diblss 39636 |
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