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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > tendof | Structured version Visualization version GIF version |
Description: Functionality of a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.) |
Ref | Expression |
---|---|
tendof.h | β’ π» = (LHypβπΎ) |
tendof.t | β’ π = ((LTrnβπΎ)βπ) |
tendof.e | β’ πΈ = ((TEndoβπΎ)βπ) |
Ref | Expression |
---|---|
tendof | β’ (((πΎ β π β§ π β π») β§ π β πΈ) β π:πβΆπ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2737 | . . . 4 β’ (leβπΎ) = (leβπΎ) | |
2 | tendof.h | . . . 4 β’ π» = (LHypβπΎ) | |
3 | tendof.t | . . . 4 β’ π = ((LTrnβπΎ)βπ) | |
4 | eqid 2737 | . . . 4 β’ ((trLβπΎ)βπ) = ((trLβπΎ)βπ) | |
5 | tendof.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 (((trLβπΎ)βπ)β(πβπ))(leβπΎ)(((trLβπΎ)βπ)βπ)))) |
7 | simp1 1137 | . . 3 β’ ((π:πβΆπ β§ βπ β π βπ β π (πβ(π β π)) = ((πβπ) β (πβπ)) β§ βπ β π (((trLβπΎ)βπ)β(πβπ))(leβπΎ)(((trLβπΎ)βπ)βπ)) β π:πβΆπ) | |
8 | 6, 7 | syl6bi 253 | . 2 β’ ((πΎ β π β§ π β π») β (π β πΈ β π:πβΆπ)) |
9 | 8 | imp 408 | 1 β’ (((πΎ β π β§ π β π») β§ π β πΈ) β π:πβΆπ) |
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: tendoeq1 39230 tendocoval 39232 tendocl 39233 tendo1mul 39236 tendo1mulr 39237 tendococl 39238 tendoconid 39295 tendospass 39485 dvhlveclem 39574 dicvscacl 39657 |
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