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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 21139, (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 7448 | . . . . . 6 β’ βΌ β V |
3 | 2 | ecelqsi 8781 | . . . . 5 β’ (π₯ β π΅ β [π₯] βΌ β (π΅ / βΌ )) |
4 | 3 | adantl 481 | . . . 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 480 | . . . . . 6 β’ ((π β§ π₯ β π΅) β π β Rng) |
12 | 6, 8, 9, 11 | qusbas 17512 | . . . . 5 β’ ((π β§ π₯ β π΅) β (π΅ / βΌ ) = (Baseβπ)) |
13 | rngqiprngim.c | . . . . 5 β’ πΆ = (Baseβπ) | |
14 | 12, 13 | eqtr4di 2785 | . . . 4 β’ ((π β§ π₯ β π΅) β (π΅ / βΌ ) = πΆ) |
15 | 4, 14 | eleqtrd 2830 | . . 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 2727 | . . . . . . 7 β’ (Baseβπ½) = (Baseβπ½) | |
19 | 16, 17, 18 | 2idlbas 21139 | . . . . . 6 β’ (π β (Baseβπ½) = πΌ) |
20 | 16, 17, 18 | 2idlelbas 21140 | . . . . . . 7 β’ (π β ((Baseβπ½) β (LIdealβπ ) β§ (Baseβπ½) β (LIdealβ(opprβπ )))) |
21 | 20 | simprd 495 | . . . . . 6 β’ (π β (Baseβπ½) β (LIdealβ(opprβπ ))) |
22 | 19, 21 | eqeltrrd 2829 | . . . . 5 β’ (π β πΌ β (LIdealβ(opprβπ ))) |
23 | rng2idlring.u | . . . . . . . 8 β’ (π β π½ β Ring) | |
24 | ringrng 20203 | . . . . . . . 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 2833 | . . . . . 6 β’ (π β (π βΎs πΌ) β Rng) |
27 | 10, 16, 26 | rng2idl0 21143 | . . . . 5 β’ (π β (0gβπ ) β πΌ) |
28 | 10, 22, 27 | 3jca 1126 | . . . 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 20184 | . . . . . . 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 2830 | . . . . 5 β’ (π β 1 β πΌ) |
33 | 32 | anim1ci 615 | . . . 4 β’ ((π β§ π₯ β π΅) β (π₯ β π΅ β§ 1 β πΌ)) |
34 | eqid 2727 | . . . . 5 β’ (0gβπ ) = (0gβπ ) | |
35 | rng2idlring.t | . . . . 5 β’ Β· = (.rβπ ) | |
36 | eqid 2727 | . . . . 5 β’ (LIdealβ(opprβπ )) = (LIdealβ(opprβπ )) | |
37 | 34, 7, 35, 36 | rngridlmcl 21095 | . . . 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 684 | . . 3 β’ ((π β§ π₯ β π΅) β ( 1 Β· π₯) β πΌ) |
39 | 15, 38 | opelxpd 5711 | . 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 7118 | 1 β’ (π β πΉ:π΅βΆ(πΆ Γ πΌ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β§ w3a 1085 = wceq 1534 β wcel 2099 Vcvv 3469 β¨cop 4630 β¦ cmpt 5225 Γ cxp 5670 βΆwf 6538 βcfv 6542 (class class class)co 7414 [cec 8714 / cqs 8715 Basecbs 17165 βΎs cress 17194 .rcmulr 17219 0gc0g 17406 /s cqus 17472 Γs cxps 17473 ~QG cqg 19061 Rngcrng 20076 1rcur 20105 Ringcrg 20157 opprcoppr 20254 LIdealclidl 21084 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-er 8716 df-ec 8718 df-qs 8722 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-0g 17408 df-imas 17475 df-qus 17476 df-mgm 18585 df-sgrp 18664 df-mnd 18680 df-grp 18878 df-minusg 18879 df-subg 19062 df-cmn 19721 df-abl 19722 df-mgp 20059 df-rng 20077 df-ur 20106 df-ring 20159 df-oppr 20255 df-subrng 20465 df-lss 20798 df-sra 21040 df-rgmod 21041 df-lidl 21086 df-2idl 21126 |
This theorem is referenced by: rngqiprngghm 21171 rngqiprngimfo 21173 |
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