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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > istendod | Structured version Visualization version GIF version |
Description: Deduce the predicate "is 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) |
istendod.1 | β’ (π β (πΎ β π β§ π β π»)) |
istendod.2 | β’ (π β π:πβΆπ) |
istendod.3 | β’ ((π β§ π β π β§ π β π) β (πβ(π β π)) = ((πβπ) β (πβπ))) |
istendod.4 | β’ ((π β§ π β π) β (π β(πβπ)) β€ (π βπ)) |
Ref | Expression |
---|---|
istendod | β’ (π β π β πΈ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | istendod.2 | . 2 β’ (π β π:πβΆπ) | |
2 | istendod.3 | . . . 4 β’ ((π β§ π β π β§ π β π) β (πβ(π β π)) = ((πβπ) β (πβπ))) | |
3 | 2 | 3expb 1117 | . . 3 β’ ((π β§ (π β π β§ π β π)) β (πβ(π β π)) = ((πβπ) β (πβπ))) |
4 | 3 | ralrimivva 3192 | . 2 β’ (π β βπ β π βπ β π (πβ(π β π)) = ((πβπ) β (πβπ))) |
5 | istendod.4 | . . 3 β’ ((π β§ π β π) β (π β(πβπ)) β€ (π βπ)) | |
6 | 5 | ralrimiva 3138 | . 2 β’ (π β βπ β π (π β(πβπ)) β€ (π βπ)) |
7 | istendod.1 | . . 3 β’ (π β (πΎ β π β§ π β π»)) | |
8 | tendoset.l | . . . 4 β’ β€ = (leβπΎ) | |
9 | tendoset.h | . . . 4 β’ π» = (LHypβπΎ) | |
10 | tendoset.t | . . . 4 β’ π = ((LTrnβπΎ)βπ) | |
11 | tendoset.r | . . . 4 β’ π = ((trLβπΎ)βπ) | |
12 | tendoset.e | . . . 4 β’ πΈ = ((TEndoβπΎ)βπ) | |
13 | 8, 9, 10, 11, 12 | istendo 40087 | . . 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)))) |
14 | 7, 13 | syl 17 | . 2 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)))) |
15 | 1, 4, 6, 14 | mpbir3and 1339 | 1 β’ (π β π β πΈ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3053 class class class wbr 5138 β ccom 5670 βΆwf 6529 βcfv 6533 lecple 17200 LHypclh 39311 LTrncltrn 39428 trLctrl 39485 TEndoctendo 40079 |
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 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7404 df-oprab 7405 df-mpo 7406 df-map 8817 df-tendo 40082 |
This theorem is referenced by: tendoidcl 40096 tendococl 40099 tendoplcl 40108 tendo0cl 40117 tendoicl 40123 cdlemk56 40298 |
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