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Mirrors > Home > MPE Home > Th. List > rngqiprngimf | Structured version Visualization version GIF version |
Description: πΉ is a function from (the base set of) a non-unital ring to the product of the (base set πΆ of the) quotient with a two-sided ideal and the (base set πΌ of the) two-sided ideal (because of 2idlbas 21156, (Baseβπ½) = πΌ!) (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 |
---|---|
rngqiprngimf | β’ (π β πΉ:π΅βΆ(πΆ Γ πΌ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rngqiprngim.g | . . . . . . 7 β’ βΌ = (π ~QG πΌ) | |
2 | 1 | ovexi 7447 | . . . . . 6 β’ βΌ β V |
3 | 2 | ecelqsi 8785 | . . . . 5 β’ (π₯ β π΅ β [π₯] βΌ β (π΅ / βΌ )) |
4 | 3 | adantl 480 | . . . 4 β’ ((π β§ π₯ β π΅) β [π₯] βΌ β (π΅ / βΌ )) |
5 | rngqiprngim.q | . . . . . . 7 β’ π = (π /s βΌ ) | |
6 | 5 | a1i 11 | . . . . . 6 β’ ((π β§ π₯ β π΅) β π = (π /s βΌ )) |
7 | rng2idlring.b | . . . . . . 7 β’ π΅ = (Baseβπ ) | |
8 | 7 | a1i 11 | . . . . . 6 β’ ((π β§ π₯ β π΅) β π΅ = (Baseβπ )) |
9 | 2 | a1i 11 | . . . . . 6 β’ ((π β§ π₯ β π΅) β βΌ β V) |
10 | rng2idlring.r | . . . . . . 7 β’ (π β π β Rng) | |
11 | 10 | adantr 479 | . . . . . 6 β’ ((π β§ π₯ β π΅) β π β Rng) |
12 | 6, 8, 9, 11 | qusbas 17521 | . . . . 5 β’ ((π β§ π₯ β π΅) β (π΅ / βΌ ) = (Baseβπ)) |
13 | rngqiprngim.c | . . . . 5 β’ πΆ = (Baseβπ) | |
14 | 12, 13 | eqtr4di 2783 | . . . 4 β’ ((π β§ π₯ β π΅) β (π΅ / βΌ ) = πΆ) |
15 | 4, 14 | eleqtrd 2827 | . . 3 β’ ((π β§ π₯ β π΅) β [π₯] βΌ β πΆ) |
16 | rng2idlring.i | . . . . . . 7 β’ (π β πΌ β (2Idealβπ )) | |
17 | rng2idlring.j | . . . . . . 7 β’ π½ = (π βΎs πΌ) | |
18 | eqid 2725 | . . . . . . 7 β’ (Baseβπ½) = (Baseβπ½) | |
19 | 16, 17, 18 | 2idlbas 21156 | . . . . . 6 β’ (π β (Baseβπ½) = πΌ) |
20 | 16, 17, 18 | 2idlelbas 21157 | . . . . . . 7 β’ (π β ((Baseβπ½) β (LIdealβπ ) β§ (Baseβπ½) β (LIdealβ(opprβπ )))) |
21 | 20 | simprd 494 | . . . . . 6 β’ (π β (Baseβπ½) β (LIdealβ(opprβπ ))) |
22 | 19, 21 | eqeltrrd 2826 | . . . . 5 β’ (π β πΌ β (LIdealβ(opprβπ ))) |
23 | rng2idlring.u | . . . . . . . 8 β’ (π β π½ β Ring) | |
24 | ringrng 20220 | . . . . . . . 8 β’ (π½ β Ring β π½ β Rng) | |
25 | 23, 24 | syl 17 | . . . . . . 7 β’ (π β π½ β Rng) |
26 | 17, 25 | eqeltrrid 2830 | . . . . . 6 β’ (π β (π βΎs πΌ) β Rng) |
27 | 10, 16, 26 | rng2idl0 21160 | . . . . 5 β’ (π β (0gβπ ) β πΌ) |
28 | 10, 22, 27 | 3jca 1125 | . . . 4 β’ (π β (π β Rng β§ πΌ β (LIdealβ(opprβπ )) β§ (0gβπ ) β πΌ)) |
29 | rng2idlring.1 | . . . . . . . 8 β’ 1 = (1rβπ½) | |
30 | 18, 29 | ringidcl 20201 | . . . . . . 7 β’ (π½ β Ring β 1 β (Baseβπ½)) |
31 | 23, 30 | syl 17 | . . . . . 6 β’ (π β 1 β (Baseβπ½)) |
32 | 31, 19 | eleqtrd 2827 | . . . . 5 β’ (π β 1 β πΌ) |
33 | 32 | anim1ci 614 | . . . 4 β’ ((π β§ π₯ β π΅) β (π₯ β π΅ β§ 1 β πΌ)) |
34 | eqid 2725 | . . . . 5 β’ (0gβπ ) = (0gβπ ) | |
35 | rng2idlring.t | . . . . 5 β’ Β· = (.rβπ ) | |
36 | eqid 2725 | . . . . 5 β’ (LIdealβ(opprβπ )) = (LIdealβ(opprβπ )) | |
37 | 34, 7, 35, 36 | rngridlmcl 21112 | . . . 4 β’ (((π β Rng β§ πΌ β (LIdealβ(opprβπ )) β§ (0gβπ ) β πΌ) β§ (π₯ β π΅ β§ 1 β πΌ)) β ( 1 Β· π₯) β πΌ) |
38 | 28, 33, 37 | syl2an2r 683 | . . 3 β’ ((π β§ π₯ β π΅) β ( 1 Β· π₯) β πΌ) |
39 | 15, 38 | opelxpd 5712 | . 2 β’ ((π β§ π₯ β π΅) β β¨[π₯] βΌ , ( 1 Β· π₯)β© β (πΆ Γ πΌ)) |
40 | rngqiprngim.f | . 2 β’ πΉ = (π₯ β π΅ β¦ β¨[π₯] βΌ , ( 1 Β· π₯)β©) | |
41 | 39, 40 | fmptd 7117 | 1 β’ (π β πΉ:π΅βΆ(πΆ Γ πΌ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 Vcvv 3463 β¨cop 4631 β¦ cmpt 5227 Γ cxp 5671 βΆwf 6539 βcfv 6543 (class class class)co 7413 [cec 8716 / cqs 8717 Basecbs 17174 βΎs cress 17203 .rcmulr 17228 0gc0g 17415 /s cqus 17481 Γs cxps 17482 ~QG cqg 19076 Rngcrng 20091 1rcur 20120 Ringcrg 20172 opprcoppr 20271 LIdealclidl 21101 2Idealc2idl 21142 |
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 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
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 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7987 df-2nd 7988 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-ec 8720 df-qs 8724 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9460 df-inf 9461 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-z 12584 df-dec 12703 df-uz 12848 df-fz 13512 df-struct 17110 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-mulr 17241 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-0g 17417 df-imas 17484 df-qus 17485 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-grp 18892 df-minusg 18893 df-subg 19077 df-cmn 19736 df-abl 19737 df-mgp 20074 df-rng 20092 df-ur 20121 df-ring 20174 df-oppr 20272 df-subrng 20482 df-lss 20815 df-sra 21057 df-rgmod 21058 df-lidl 21103 df-2idl 21143 |
This theorem is referenced by: rngqiprngghm 21188 rngqiprngimfo 21190 |
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