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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > erng0g | Structured version Visualization version GIF version |
Description: The division ring zero of an endomorphism ring. (Contributed by NM, 5-Nov-2013.) (Revised by Mario Carneiro, 23-Jun-2014.) |
Ref | Expression |
---|---|
erng0g.b | β’ π΅ = (BaseβπΎ) |
erng0g.h | β’ π» = (LHypβπΎ) |
erng0g.t | β’ π = ((LTrnβπΎ)βπ) |
erng0g.d | β’ π· = ((EDRingβπΎ)βπ) |
erng0g.o | β’ π = (π β π β¦ ( I βΎ π΅)) |
erng0g.z | β’ 0 = (0gβπ·) |
Ref | Expression |
---|---|
erng0g | β’ ((πΎ β HL β§ π β π») β 0 = π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | erng0g.h | . . . . 5 β’ π» = (LHypβπΎ) | |
2 | erng0g.t | . . . . 5 β’ π = ((LTrnβπΎ)βπ) | |
3 | eqid 2733 | . . . . 5 β’ ((TEndoβπΎ)βπ) = ((TEndoβπΎ)βπ) | |
4 | erng0g.d | . . . . 5 β’ π· = ((EDRingβπΎ)βπ) | |
5 | eqid 2733 | . . . . 5 β’ (+gβπ·) = (+gβπ·) | |
6 | 1, 2, 3, 4, 5 | erngfplus 39315 | . . . 4 β’ ((πΎ β HL β§ π β π») β (+gβπ·) = (π β ((TEndoβπΎ)βπ), π‘ β ((TEndoβπΎ)βπ) β¦ (π β π β¦ ((π βπ) β (π‘βπ))))) |
7 | 6 | oveqd 7378 | . . 3 β’ ((πΎ β HL β§ π β π») β (π(+gβπ·)π) = (π(π β ((TEndoβπΎ)βπ), π‘ β ((TEndoβπΎ)βπ) β¦ (π β π β¦ ((π βπ) β (π‘βπ))))π)) |
8 | erng0g.b | . . . . 5 β’ π΅ = (BaseβπΎ) | |
9 | erng0g.o | . . . . 5 β’ π = (π β π β¦ ( I βΎ π΅)) | |
10 | 8, 1, 2, 3, 9 | tendo0cl 39303 | . . . 4 β’ ((πΎ β HL β§ π β π») β π β ((TEndoβπΎ)βπ)) |
11 | eqid 2733 | . . . . 5 β’ (π β ((TEndoβπΎ)βπ), π‘ β ((TEndoβπΎ)βπ) β¦ (π β π β¦ ((π βπ) β (π‘βπ)))) = (π β ((TEndoβπΎ)βπ), π‘ β ((TEndoβπΎ)βπ) β¦ (π β π β¦ ((π βπ) β (π‘βπ)))) | |
12 | 8, 1, 2, 3, 9, 11 | tendo0pl 39304 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ π β ((TEndoβπΎ)βπ)) β (π(π β ((TEndoβπΎ)βπ), π‘ β ((TEndoβπΎ)βπ) β¦ (π β π β¦ ((π βπ) β (π‘βπ))))π) = π) |
13 | 10, 12 | mpdan 686 | . . 3 β’ ((πΎ β HL β§ π β π») β (π(π β ((TEndoβπΎ)βπ), π‘ β ((TEndoβπΎ)βπ) β¦ (π β π β¦ ((π βπ) β (π‘βπ))))π) = π) |
14 | 7, 13 | eqtrd 2773 | . 2 β’ ((πΎ β HL β§ π β π») β (π(+gβπ·)π) = π) |
15 | 1, 4 | eringring 39505 | . . . 4 β’ ((πΎ β HL β§ π β π») β π· β Ring) |
16 | ringgrp 19977 | . . . 4 β’ (π· β Ring β π· β Grp) | |
17 | 15, 16 | syl 17 | . . 3 β’ ((πΎ β HL β§ π β π») β π· β Grp) |
18 | eqid 2733 | . . . . 5 β’ (Baseβπ·) = (Baseβπ·) | |
19 | 1, 2, 3, 4, 18 | erngbase 39314 | . . . 4 β’ ((πΎ β HL β§ π β π») β (Baseβπ·) = ((TEndoβπΎ)βπ)) |
20 | 10, 19 | eleqtrrd 2837 | . . 3 β’ ((πΎ β HL β§ π β π») β π β (Baseβπ·)) |
21 | erng0g.z | . . . 4 β’ 0 = (0gβπ·) | |
22 | 18, 5, 21 | grpid 18794 | . . 3 β’ ((π· β Grp β§ π β (Baseβπ·)) β ((π(+gβπ·)π) = π β 0 = π)) |
23 | 17, 20, 22 | syl2anc 585 | . 2 β’ ((πΎ β HL β§ π β π») β ((π(+gβπ·)π) = π β 0 = π)) |
24 | 14, 23 | mpbid 231 | 1 β’ ((πΎ β HL β§ π β π») β 0 = π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 β¦ cmpt 5192 I cid 5534 βΎ cres 5639 β ccom 5641 βcfv 6500 (class class class)co 7361 β cmpo 7363 Basecbs 17091 +gcplusg 17141 0gc0g 17329 Grpcgrp 18756 Ringcrg 19972 HLchlt 37862 LHypclh 38497 LTrncltrn 38614 TEndoctendo 39265 EDRingcedring 39266 |
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 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-riotaBAD 37465 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4870 df-iun 4960 df-iin 4961 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-undef 8208 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-2 12224 df-3 12225 df-n0 12422 df-z 12508 df-uz 12772 df-fz 13434 df-struct 17027 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-plusg 17154 df-mulr 17155 df-0g 17331 df-proset 18192 df-poset 18210 df-plt 18227 df-lub 18243 df-glb 18244 df-join 18245 df-meet 18246 df-p0 18322 df-p1 18323 df-lat 18329 df-clat 18396 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-grp 18759 df-mgp 19905 df-ring 19974 df-oposet 37688 df-ol 37690 df-oml 37691 df-covers 37778 df-ats 37779 df-atl 37810 df-cvlat 37834 df-hlat 37863 df-llines 38011 df-lplanes 38012 df-lvols 38013 df-lines 38014 df-psubsp 38016 df-pmap 38017 df-padd 38309 df-lhyp 38501 df-laut 38502 df-ldil 38617 df-ltrn 38618 df-trl 38672 df-tendo 39268 df-edring 39270 |
This theorem is referenced by: erng1r 39508 dvalveclem 39538 tendoinvcl 39617 tendolinv 39618 tendorinv 39619 cdlemn4 39711 |
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