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Mirrors > Home > MPE Home > Th. List > rngqiprngim | Structured version Visualization version GIF version |
Description: πΉ is an isomorphism of non-unital rings. (Contributed by AV, 21-Feb-2025.) |
Ref | Expression |
---|---|
rng2idlring.r | β’ (π β π β Rng) |
rng2idlring.i | β’ (π β πΌ β (2Idealβπ )) |
rng2idlring.j | β’ π½ = (π βΎs πΌ) |
rng2idlring.u | β’ (π β π½ β Ring) |
rng2idlring.b | β’ π΅ = (Baseβπ ) |
rng2idlring.t | β’ Β· = (.rβπ ) |
rng2idlring.1 | β’ 1 = (1rβπ½) |
rngqiprngim.g | β’ βΌ = (π ~QG πΌ) |
rngqiprngim.q | β’ π = (π /s βΌ ) |
rngqiprngim.c | β’ πΆ = (Baseβπ) |
rngqiprngim.p | β’ π = (π Γs π½) |
rngqiprngim.f | β’ πΉ = (π₯ β π΅ β¦ β¨[π₯] βΌ , ( 1 Β· π₯)β©) |
Ref | Expression |
---|---|
rngqiprngim | β’ (π β πΉ β (π RngIso π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rng2idlring.r | . . 3 β’ (π β π β Rng) | |
2 | rng2idlring.i | . . 3 β’ (π β πΌ β (2Idealβπ )) | |
3 | rng2idlring.j | . . 3 β’ π½ = (π βΎs πΌ) | |
4 | rng2idlring.u | . . 3 β’ (π β π½ β Ring) | |
5 | rng2idlring.b | . . 3 β’ π΅ = (Baseβπ ) | |
6 | rng2idlring.t | . . 3 β’ Β· = (.rβπ ) | |
7 | rng2idlring.1 | . . 3 β’ 1 = (1rβπ½) | |
8 | rngqiprngim.g | . . 3 β’ βΌ = (π ~QG πΌ) | |
9 | rngqiprngim.q | . . 3 β’ π = (π /s βΌ ) | |
10 | rngqiprngim.c | . . 3 β’ πΆ = (Baseβπ) | |
11 | rngqiprngim.p | . . 3 β’ π = (π Γs π½) | |
12 | rngqiprngim.f | . . 3 β’ πΉ = (π₯ β π΅ β¦ β¨[π₯] βΌ , ( 1 Β· π₯)β©) | |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | rngqiprngho 21195 | . 2 β’ (π β πΉ β (π RngHom π)) |
14 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | rngqiprngimf1 21192 | . . . 4 β’ (π β πΉ:π΅β1-1β(πΆ Γ πΌ)) |
15 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | rngqiprngimfo 21193 | . . . 4 β’ (π β πΉ:π΅βontoβ(πΆ Γ πΌ)) |
16 | df-f1o 6549 | . . . 4 β’ (πΉ:π΅β1-1-ontoβ(πΆ Γ πΌ) β (πΉ:π΅β1-1β(πΆ Γ πΌ) β§ πΉ:π΅βontoβ(πΆ Γ πΌ))) | |
17 | 14, 15, 16 | sylanbrc 581 | . . 3 β’ (π β πΉ:π΅β1-1-ontoβ(πΆ Γ πΌ)) |
18 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 | rngqipbas 21187 | . . . 4 β’ (π β (Baseβπ) = (πΆ Γ πΌ)) |
19 | 18 | f1oeq3d 6830 | . . 3 β’ (π β (πΉ:π΅β1-1-ontoβ(Baseβπ) β πΉ:π΅β1-1-ontoβ(πΆ Γ πΌ))) |
20 | 17, 19 | mpbird 256 | . 2 β’ (π β πΉ:π΅β1-1-ontoβ(Baseβπ)) |
21 | 11 | ovexi 7449 | . . 3 β’ π β V |
22 | eqid 2725 | . . . 4 β’ (Baseβπ) = (Baseβπ) | |
23 | 5, 22 | isrngim2 20394 | . . 3 β’ ((π β Rng β§ π β V) β (πΉ β (π RngIso π) β (πΉ β (π RngHom π) β§ πΉ:π΅β1-1-ontoβ(Baseβπ)))) |
24 | 1, 21, 23 | sylancl 584 | . 2 β’ (π β (πΉ β (π RngIso π) β (πΉ β (π RngHom π) β§ πΉ:π΅β1-1-ontoβ(Baseβπ)))) |
25 | 13, 20, 24 | mpbir2and 711 | 1 β’ (π β πΉ β (π RngIso π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 = wceq 1533 β wcel 2098 Vcvv 3463 β¨cop 4630 β¦ cmpt 5226 Γ cxp 5670 β1-1βwf1 6539 βontoβwfo 6540 β1-1-ontoβwf1o 6541 βcfv 6542 (class class class)co 7415 [cec 8719 Basecbs 17177 βΎs cress 17206 .rcmulr 17231 /s cqus 17484 Γs cxps 17485 ~QG cqg 19079 Rngcrng 20094 1rcur 20123 Ringcrg 20175 RngHom crnghm 20375 RngIso crngim 20376 2Idealc2idl 21145 |
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 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 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 5144 df-opab 5206 df-mpt 5227 df-tr 5261 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 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 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 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-tpos 8228 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-2o 8484 df-er 8721 df-ec 8723 df-qs 8727 df-map 8843 df-ixp 8913 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-sup 9463 df-inf 9464 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12501 df-z 12587 df-dec 12706 df-uz 12851 df-fz 13515 df-struct 17113 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-mulr 17244 df-sca 17246 df-vsca 17247 df-ip 17248 df-tset 17249 df-ple 17250 df-ds 17252 df-hom 17254 df-cco 17255 df-0g 17420 df-prds 17426 df-imas 17487 df-qus 17488 df-xps 17489 df-mgm 18597 df-mgmhm 18649 df-sgrp 18676 df-mnd 18692 df-grp 18895 df-minusg 18896 df-sbg 18897 df-subg 19080 df-nsg 19081 df-eqg 19082 df-ghm 19170 df-cmn 19739 df-abl 19740 df-mgp 20077 df-rng 20095 df-ur 20124 df-ring 20177 df-oppr 20275 df-dvdsr 20298 df-unit 20299 df-invr 20329 df-rnghm 20377 df-rngim 20378 df-subrng 20485 df-lss 20818 df-sra 21060 df-rgmod 21061 df-lidl 21106 df-2idl 21146 |
This theorem is referenced by: rngringbdlem2 21199 rngqiprngu 21210 |
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