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Mirrors > Home > MPE Home > Th. List > quscrng | Structured version Visualization version Unicode version |
Description: The quotient of a commutative ring by an ideal is a commutative ring. (Contributed by Mario Carneiro, 15-Jun-2015.) |
Ref | Expression |
---|---|
quscrng.u | s ~QG |
quscrng.i | LIdeal |
Ref | Expression |
---|---|
quscrng |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | crngring 18558 | . . . 4 | |
2 | 1 | adantr 481 | . . 3 |
3 | simpr 477 | . . . 4 | |
4 | quscrng.i | . . . . . 6 LIdeal | |
5 | 4 | crng2idl 19239 | . . . . 5 2Ideal |
6 | 5 | adantr 481 | . . . 4 2Ideal |
7 | 3, 6 | eleqtrd 2703 | . . 3 2Ideal |
8 | quscrng.u | . . . 4 s ~QG | |
9 | eqid 2622 | . . . 4 2Ideal 2Ideal | |
10 | 8, 9 | qusring 19236 | . . 3 2Ideal |
11 | 2, 7, 10 | syl2anc 693 | . 2 |
12 | 8 | a1i 11 | . . . . . . 7 s ~QG |
13 | eqidd 2623 | . . . . . . 7 | |
14 | ovexd 6680 | . . . . . . 7 ~QG | |
15 | 12, 13, 14, 2 | qusbas 16205 | . . . . . 6 ~QG |
16 | 15 | eleq2d 2687 | . . . . 5 ~QG |
17 | 15 | eleq2d 2687 | . . . . 5 ~QG |
18 | 16, 17 | anbi12d 747 | . . . 4 ~QG ~QG |
19 | eqid 2622 | . . . . . 6 ~QG ~QG | |
20 | oveq2 6658 | . . . . . . 7 ~QG ~QG | |
21 | oveq1 6657 | . . . . . . 7 ~QG ~QG | |
22 | 20, 21 | eqeq12d 2637 | . . . . . 6 ~QG ~QG ~QG |
23 | oveq1 6657 | . . . . . . . . 9 ~QG ~QG ~QG ~QG | |
24 | oveq2 6658 | . . . . . . . . 9 ~QG ~QG ~QG ~QG | |
25 | 23, 24 | eqeq12d 2637 | . . . . . . . 8 ~QG ~QG ~QG ~QG ~QG ~QG ~QG |
26 | eqid 2622 | . . . . . . . . . . . . 13 | |
27 | eqid 2622 | . . . . . . . . . . . . 13 | |
28 | 26, 27 | crngcom 18562 | . . . . . . . . . . . 12 |
29 | 28 | 3adant1r 1319 | . . . . . . . . . . 11 |
30 | 29 | 3expa 1265 | . . . . . . . . . 10 |
31 | 30 | eceq1d 7783 | . . . . . . . . 9 ~QG ~QG |
32 | 4 | lidlsubg 19215 | . . . . . . . . . . . . 13 SubGrp |
33 | 1, 32 | sylan 488 | . . . . . . . . . . . 12 SubGrp |
34 | eqid 2622 | . . . . . . . . . . . . 13 ~QG ~QG | |
35 | 26, 34 | eqger 17644 | . . . . . . . . . . . 12 SubGrp ~QG |
36 | 33, 35 | syl 17 | . . . . . . . . . . 11 ~QG |
37 | 26, 34, 9, 27 | 2idlcpbl 19234 | . . . . . . . . . . . 12 2Ideal ~QG ~QG ~QG |
38 | 2, 7, 37 | syl2anc 693 | . . . . . . . . . . 11 ~QG ~QG ~QG |
39 | 26, 27 | ringcl 18561 | . . . . . . . . . . . . 13 |
40 | 39 | 3expb 1266 | . . . . . . . . . . . 12 |
41 | 2, 40 | sylan 488 | . . . . . . . . . . 11 |
42 | eqid 2622 | . . . . . . . . . . 11 | |
43 | 12, 13, 36, 2, 38, 41, 27, 42 | qusmulval 16215 | . . . . . . . . . 10 ~QG ~QG ~QG |
44 | 43 | 3expa 1265 | . . . . . . . . 9 ~QG ~QG ~QG |
45 | 12, 13, 36, 2, 38, 41, 27, 42 | qusmulval 16215 | . . . . . . . . . . 11 ~QG ~QG ~QG |
46 | 45 | 3expa 1265 | . . . . . . . . . 10 ~QG ~QG ~QG |
47 | 46 | an32s 846 | . . . . . . . . 9 ~QG ~QG ~QG |
48 | 31, 44, 47 | 3eqtr4rd 2667 | . . . . . . . 8 ~QG ~QG ~QG ~QG |
49 | 19, 25, 48 | ectocld 7814 | . . . . . . 7 ~QG ~QG ~QG |
50 | 49 | an32s 846 | . . . . . 6 ~QG ~QG ~QG |
51 | 19, 22, 50 | ectocld 7814 | . . . . 5 ~QG ~QG |
52 | 51 | expl 648 | . . . 4 ~QG ~QG |
53 | 18, 52 | sylbird 250 | . . 3 |
54 | 53 | ralrimivv 2970 | . 2 |
55 | eqid 2622 | . . 3 | |
56 | 55, 42 | iscrng2 18563 | . 2 |
57 | 11, 54, 56 | sylanbrc 698 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 wral 2912 cvv 3200 class class class wbr 4653 cfv 5888 (class class class)co 6650 wer 7739 cec 7740 cqs 7741 cbs 15857 cmulr 15942 s cqus 16165 SubGrpcsubg 17588 ~QG cqg 17590 crg 18547 ccrg 18548 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-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-grp 17425 df-minusg 17426 df-sbg 17427 df-subg 17591 df-nsg 17592 df-eqg 17593 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-oppr 18623 df-subrg 18778 df-lmod 18865 df-lss 18933 df-lsp 18972 df-sra 19172 df-rgmod 19173 df-lidl 19174 df-rsp 19175 df-2idl 19232 |
This theorem is referenced by: zncrng2 19882 |
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