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Mirrors > Home > MPE Home > Th. List > rngringbdlem2 | Structured version Visualization version GIF version |
Description: A non-unital ring is unital if and only if there is a (two-sided) ideal of the ring which is unital, and the quotient of the ring and the ideal is unital. (Proposed by GL, 12-Feb-2025.) (Contributed by AV, 14-Feb-2025.) |
Ref | Expression |
---|---|
rngringbd.r | β’ (π β π β Rng) |
rngringbd.i | β’ (π β πΌ β (2Idealβπ )) |
rngringbd.j | β’ π½ = (π βΎs πΌ) |
rngringbd.u | β’ (π β π½ β Ring) |
rngringbd.q | β’ π = (π /s (π ~QG πΌ)) |
Ref | Expression |
---|---|
rngringbdlem2 | β’ ((π β§ π β Ring) β π β Ring) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2727 | . . 3 β’ (π Γs π½) = (π Γs π½) | |
2 | simpr 484 | . . 3 β’ ((π β§ π β Ring) β π β Ring) | |
3 | rngringbd.u | . . . 4 β’ (π β π½ β Ring) | |
4 | 3 | adantr 480 | . . 3 β’ ((π β§ π β Ring) β π½ β Ring) |
5 | 1, 2, 4 | xpsringd 20250 | . 2 β’ ((π β§ π β Ring) β (π Γs π½) β Ring) |
6 | rngringbd.r | . . 3 β’ (π β π β Rng) | |
7 | 6 | adantr 480 | . 2 β’ ((π β§ π β Ring) β π β Rng) |
8 | rngringbd.i | . . . . 5 β’ (π β πΌ β (2Idealβπ )) | |
9 | 8 | adantr 480 | . . . 4 β’ ((π β§ π β Ring) β πΌ β (2Idealβπ )) |
10 | rngringbd.j | . . . 4 β’ π½ = (π βΎs πΌ) | |
11 | eqid 2727 | . . . 4 β’ (Baseβπ ) = (Baseβπ ) | |
12 | eqid 2727 | . . . 4 β’ (.rβπ ) = (.rβπ ) | |
13 | eqid 2727 | . . . 4 β’ (1rβπ½) = (1rβπ½) | |
14 | eqid 2727 | . . . 4 β’ (π ~QG πΌ) = (π ~QG πΌ) | |
15 | rngringbd.q | . . . 4 β’ π = (π /s (π ~QG πΌ)) | |
16 | eqid 2727 | . . . 4 β’ (Baseβπ) = (Baseβπ) | |
17 | eqid 2727 | . . . 4 β’ (π₯ β (Baseβπ ) β¦ β¨[π₯](π ~QG πΌ), ((1rβπ½)(.rβπ )π₯)β©) = (π₯ β (Baseβπ ) β¦ β¨[π₯](π ~QG πΌ), ((1rβπ½)(.rβπ )π₯)β©) | |
18 | 7, 9, 10, 4, 11, 12, 13, 14, 15, 16, 1, 17 | rngqiprngim 21176 | . . 3 β’ ((π β§ π β Ring) β (π₯ β (Baseβπ ) β¦ β¨[π₯](π ~QG πΌ), ((1rβπ½)(.rβπ )π₯)β©) β (π RngIso (π Γs π½))) |
19 | rngimcnv 20377 | . . 3 β’ ((π₯ β (Baseβπ ) β¦ β¨[π₯](π ~QG πΌ), ((1rβπ½)(.rβπ )π₯)β©) β (π RngIso (π Γs π½)) β β‘(π₯ β (Baseβπ ) β¦ β¨[π₯](π ~QG πΌ), ((1rβπ½)(.rβπ )π₯)β©) β ((π Γs π½) RngIso π )) | |
20 | 18, 19 | syl 17 | . 2 β’ ((π β§ π β Ring) β β‘(π₯ β (Baseβπ ) β¦ β¨[π₯](π ~QG πΌ), ((1rβπ½)(.rβπ )π₯)β©) β ((π Γs π½) RngIso π )) |
21 | rngisomring 20388 | . 2 β’ (((π Γs π½) β Ring β§ π β Rng β§ β‘(π₯ β (Baseβπ ) β¦ β¨[π₯](π ~QG πΌ), ((1rβπ½)(.rβπ )π₯)β©) β ((π Γs π½) RngIso π )) β π β Ring) | |
22 | 5, 7, 20, 21 | syl3anc 1369 | 1 β’ ((π β§ π β Ring) β π β Ring) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 β¨cop 4630 β¦ cmpt 5225 β‘ccnv 5671 βcfv 6542 (class class class)co 7414 [cec 8714 Basecbs 17165 βΎs cress 17194 .rcmulr 17219 /s cqus 17472 Γs cxps 17473 ~QG cqg 19061 Rngcrng 20076 1rcur 20105 Ringcrg 20157 RngIso crngim 20356 2Idealc2idl 21125 |
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 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 |
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 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-1st 7985 df-2nd 7986 df-tpos 8223 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-2o 8479 df-er 8716 df-ec 8718 df-qs 8722 df-map 8836 df-ixp 8906 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-sup 9451 df-inf 9452 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-2 12291 df-3 12292 df-4 12293 df-5 12294 df-6 12295 df-7 12296 df-8 12297 df-9 12298 df-n0 12489 df-z 12575 df-dec 12694 df-uz 12839 df-fz 13503 df-struct 17101 df-sets 17118 df-slot 17136 df-ndx 17148 df-base 17166 df-ress 17195 df-plusg 17231 df-mulr 17232 df-sca 17234 df-vsca 17235 df-ip 17236 df-tset 17237 df-ple 17238 df-ds 17240 df-hom 17242 df-cco 17243 df-0g 17408 df-prds 17414 df-imas 17475 df-qus 17476 df-xps 17477 df-mgm 18585 df-mgmhm 18637 df-sgrp 18664 df-mnd 18680 df-grp 18878 df-minusg 18879 df-sbg 18880 df-subg 19062 df-nsg 19063 df-eqg 19064 df-ghm 19152 df-cmn 19721 df-abl 19722 df-mgp 20059 df-rng 20077 df-ur 20106 df-ring 20159 df-oppr 20255 df-dvdsr 20278 df-unit 20279 df-invr 20309 df-rnghm 20357 df-rngim 20358 df-subrng 20465 df-lss 20798 df-sra 21040 df-rgmod 21041 df-lidl 21086 df-2idl 21126 |
This theorem is referenced by: rngringbd 21180 ring2idlqusb 21182 ring2idlqus1 21191 |
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