Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > qusrhm | Structured version Visualization version Unicode version |
Description: If is a two-sided ideal in , then the "natural map" from elements to their cosets is a ring homomorphism from to . (Contributed by Mario Carneiro, 15-Jun-2015.) |
Ref | Expression |
---|---|
qusring.u | s ~QG |
qusring.i | 2Ideal |
qusrhm.x | |
qusrhm.f | ~QG |
Ref | Expression |
---|---|
qusrhm | RingHom |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qusrhm.x | . 2 | |
2 | eqid 2622 | . 2 | |
3 | eqid 2622 | . 2 | |
4 | eqid 2622 | . 2 | |
5 | eqid 2622 | . 2 | |
6 | simpl 473 | . 2 | |
7 | qusring.u | . . 3 s ~QG | |
8 | qusring.i | . . 3 2Ideal | |
9 | 7, 8 | qusring 19236 | . 2 |
10 | eqid 2622 | . . . . . . . . 9 LIdeal LIdeal | |
11 | eqid 2622 | . . . . . . . . 9 oppr oppr | |
12 | eqid 2622 | . . . . . . . . 9 LIdealoppr LIdealoppr | |
13 | 10, 11, 12, 8 | 2idlval 19233 | . . . . . . . 8 LIdeal LIdealoppr |
14 | 13 | elin2 3801 | . . . . . . 7 LIdeal LIdealoppr |
15 | 14 | simplbi 476 | . . . . . 6 LIdeal |
16 | 10 | lidlsubg 19215 | . . . . . 6 LIdeal SubGrp |
17 | 15, 16 | sylan2 491 | . . . . 5 SubGrp |
18 | eqid 2622 | . . . . . 6 ~QG ~QG | |
19 | 1, 18 | eqger 17644 | . . . . 5 SubGrp ~QG |
20 | 17, 19 | syl 17 | . . . 4 ~QG |
21 | fvex 6201 | . . . . . 6 | |
22 | 1, 21 | eqeltri 2697 | . . . . 5 |
23 | 22 | a1i 11 | . . . 4 |
24 | qusrhm.f | . . . 4 ~QG | |
25 | 20, 23, 24 | divsfval 16207 | . . 3 ~QG |
26 | 7, 8, 2 | qus1 19235 | . . . 4 ~QG |
27 | 26 | simprd 479 | . . 3 ~QG |
28 | 25, 27 | eqtrd 2656 | . 2 |
29 | 7 | a1i 11 | . . . . 5 s ~QG |
30 | 1 | a1i 11 | . . . . 5 |
31 | 1, 18, 8, 4 | 2idlcpbl 19234 | . . . . 5 ~QG ~QG ~QG |
32 | 1, 4 | ringcl 18561 | . . . . . . . 8 |
33 | 32 | 3expb 1266 | . . . . . . 7 |
34 | 33 | adantlr 751 | . . . . . 6 |
35 | 34 | caovclg 6826 | . . . . 5 |
36 | 29, 30, 20, 6, 31, 35, 4, 5 | qusmulval 16215 | . . . 4 ~QG ~QG ~QG |
37 | 36 | 3expb 1266 | . . 3 ~QG ~QG ~QG |
38 | 20 | adantr 481 | . . . . 5 ~QG |
39 | 22 | a1i 11 | . . . . 5 |
40 | 38, 39, 24 | divsfval 16207 | . . . 4 ~QG |
41 | 38, 39, 24 | divsfval 16207 | . . . 4 ~QG |
42 | 40, 41 | oveq12d 6668 | . . 3 ~QG ~QG |
43 | 38, 39, 24 | divsfval 16207 | . . 3 ~QG |
44 | 37, 42, 43 | 3eqtr4rd 2667 | . 2 |
45 | ringabl 18580 | . . . . . 6 | |
46 | 45 | adantr 481 | . . . . 5 |
47 | ablnsg 18250 | . . . . 5 NrmSGrp SubGrp | |
48 | 46, 47 | syl 17 | . . . 4 NrmSGrp SubGrp |
49 | 17, 48 | eleqtrrd 2704 | . . 3 NrmSGrp |
50 | 1, 7, 24 | qusghm 17697 | . . 3 NrmSGrp |
51 | 49, 50 | syl 17 | . 2 |
52 | 1, 2, 3, 4, 5, 6, 9, 28, 44, 51 | isrhm2d 18728 | 1 RingHom |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 cvv 3200 cmpt 4729 cfv 5888 (class class class)co 6650 wer 7739 cec 7740 cbs 15857 cmulr 15942 s cqus 16165 SubGrpcsubg 17588 NrmSGrpcnsg 17589 ~QG cqg 17590 cghm 17657 cabl 18194 cur 18501 crg 18547 opprcoppr 18622 RingHom crh 18712 LIdealclidl 19170 2Idealc2idl 19231 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-ec 7744 df-qs 7748 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-0g 16102 df-imas 16168 df-qus 16169 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-grp 17425 df-minusg 17426 df-sbg 17427 df-subg 17591 df-nsg 17592 df-eqg 17593 df-ghm 17658 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-oppr 18623 df-rnghom 18715 df-subrg 18778 df-lmod 18865 df-lss 18933 df-sra 19172 df-rgmod 19173 df-lidl 19174 df-2idl 19232 |
This theorem is referenced by: znzrh2 19894 |
Copyright terms: Public domain | W3C validator |