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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 2724 | . . . . 5 β’ ((TEndoβπΎ)βπ) = ((TEndoβπΎ)βπ) | |
4 | erng0g.d | . . . . 5 β’ π· = ((EDRingβπΎ)βπ) | |
5 | eqid 2724 | . . . . 5 β’ (+gβπ·) = (+gβπ·) | |
6 | 1, 2, 3, 4, 5 | erngfplus 40177 | . . . 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 7419 | . . 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 40165 | . . . 4 β’ ((πΎ β HL β§ π β π») β π β ((TEndoβπΎ)βπ)) |
11 | eqid 2724 | . . . . 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 40166 | . . . 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 684 | . . 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 2764 | . 2 β’ ((πΎ β HL β§ π β π») β (π(+gβπ·)π) = π) |
15 | 1, 4 | eringring 40367 | . . . 4 β’ ((πΎ β HL β§ π β π») β π· β Ring) |
16 | ringgrp 20139 | . . . 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 2724 | . . . . 5 β’ (Baseβπ·) = (Baseβπ·) | |
19 | 1, 2, 3, 4, 18 | erngbase 40176 | . . . 4 β’ ((πΎ β HL β§ π β π») β (Baseβπ·) = ((TEndoβπΎ)βπ)) |
20 | 10, 19 | eleqtrrd 2828 | . . 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 18901 | . . 3 β’ ((π· β Grp β§ π β (Baseβπ·)) β ((π(+gβπ·)π) = π β 0 = π)) |
23 | 17, 20, 22 | syl2anc 583 | . 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 395 = wceq 1533 β wcel 2098 β¦ cmpt 5222 I cid 5564 βΎ cres 5669 β ccom 5671 βcfv 6534 (class class class)co 7402 β cmpo 7404 Basecbs 17149 +gcplusg 17202 0gc0g 17390 Grpcgrp 18859 Ringcrg 20134 HLchlt 38724 LHypclh 39359 LTrncltrn 39476 TEndoctendo 40127 EDRingcedring 40128 |
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 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-riotaBAD 38327 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 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-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-iun 4990 df-iin 4991 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-undef 8254 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-n0 12472 df-z 12558 df-uz 12822 df-fz 13486 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-plusg 17215 df-mulr 17216 df-0g 17392 df-proset 18256 df-poset 18274 df-plt 18291 df-lub 18307 df-glb 18308 df-join 18309 df-meet 18310 df-p0 18386 df-p1 18387 df-lat 18393 df-clat 18460 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-grp 18862 df-mgp 20036 df-ring 20136 df-oposet 38550 df-ol 38552 df-oml 38553 df-covers 38640 df-ats 38641 df-atl 38672 df-cvlat 38696 df-hlat 38725 df-llines 38873 df-lplanes 38874 df-lvols 38875 df-lines 38876 df-psubsp 38878 df-pmap 38879 df-padd 39171 df-lhyp 39363 df-laut 39364 df-ldil 39479 df-ltrn 39480 df-trl 39534 df-tendo 40130 df-edring 40132 |
This theorem is referenced by: erng1r 40370 dvalveclem 40400 tendoinvcl 40479 tendolinv 40480 tendorinv 40481 cdlemn4 40573 |
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