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 1120 | . . 3 β’ ((π β§ (π β π β§ π β π)) β (πβ(π β π)) = ((πβπ) β (πβπ))) |
4 | 3 | ralrimivva 3194 | . 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 3140 | . 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 38813 | . . 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 1342 | 1 β’ (π β π β πΈ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 β§ w3a 1087 = wceq 1539 β wcel 2104 βwral 3062 class class class wbr 5081 β ccom 5600 βΆwf 6450 βcfv 6454 lecple 17010 LHypclh 38037 LTrncltrn 38154 trLctrl 38211 TEndoctendo 38805 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7616 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-id 5496 df-xp 5602 df-rel 5603 df-cnv 5604 df-co 5605 df-dm 5606 df-rn 5607 df-res 5608 df-ima 5609 df-iota 6406 df-fun 6456 df-fn 6457 df-f 6458 df-f1 6459 df-fo 6460 df-f1o 6461 df-fv 6462 df-ov 7306 df-oprab 7307 df-mpo 7308 df-map 8644 df-tendo 38808 |
This theorem is referenced by: tendoidcl 38822 tendococl 38825 tendoplcl 38834 tendo0cl 38843 tendoicl 38849 cdlemk56 39024 |
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