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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 2728 | . . . . 5 β’ ((TEndoβπΎ)βπ) = ((TEndoβπΎ)βπ) | |
4 | erng0g.d | . . . . 5 β’ π· = ((EDRingβπΎ)βπ) | |
5 | eqid 2728 | . . . . 5 β’ (+gβπ·) = (+gβπ·) | |
6 | 1, 2, 3, 4, 5 | erngfplus 40275 | . . . 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 7437 | . . 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 40263 | . . . 4 β’ ((πΎ β HL β§ π β π») β π β ((TEndoβπΎ)βπ)) |
11 | eqid 2728 | . . . . 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 40264 | . . . 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 2768 | . 2 β’ ((πΎ β HL β§ π β π») β (π(+gβπ·)π) = π) |
15 | 1, 4 | eringring 40465 | . . . 4 β’ ((πΎ β HL β§ π β π») β π· β Ring) |
16 | ringgrp 20178 | . . . 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 2728 | . . . . 5 β’ (Baseβπ·) = (Baseβπ·) | |
19 | 1, 2, 3, 4, 18 | erngbase 40274 | . . . 4 β’ ((πΎ β HL β§ π β π») β (Baseβπ·) = ((TEndoβπΎ)βπ)) |
20 | 10, 19 | eleqtrrd 2832 | . . 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 18932 | . . 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 1534 β wcel 2099 β¦ cmpt 5231 I cid 5575 βΎ cres 5680 β ccom 5682 βcfv 6548 (class class class)co 7420 β cmpo 7422 Basecbs 17180 +gcplusg 17233 0gc0g 17421 Grpcgrp 18890 Ringcrg 20173 HLchlt 38822 LHypclh 39457 LTrncltrn 39574 TEndoctendo 40225 EDRingcedring 40226 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-riotaBAD 38425 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-undef 8279 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-n0 12504 df-z 12590 df-uz 12854 df-fz 13518 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-plusg 17246 df-mulr 17247 df-0g 17423 df-proset 18287 df-poset 18305 df-plt 18322 df-lub 18338 df-glb 18339 df-join 18340 df-meet 18341 df-p0 18417 df-p1 18418 df-lat 18424 df-clat 18491 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-grp 18893 df-mgp 20075 df-ring 20175 df-oposet 38648 df-ol 38650 df-oml 38651 df-covers 38738 df-ats 38739 df-atl 38770 df-cvlat 38794 df-hlat 38823 df-llines 38971 df-lplanes 38972 df-lvols 38973 df-lines 38974 df-psubsp 38976 df-pmap 38977 df-padd 39269 df-lhyp 39461 df-laut 39462 df-ldil 39577 df-ltrn 39578 df-trl 39632 df-tendo 40228 df-edring 40230 |
This theorem is referenced by: erng1r 40468 dvalveclem 40498 tendoinvcl 40577 tendolinv 40578 tendorinv 40579 cdlemn4 40671 |
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