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Mirrors > Home > MPE Home > Th. List > qusgrp | Structured version Visualization version Unicode version |
Description: If is a normal subgroup of , then is a group, called the quotient of by . (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) |
Ref | Expression |
---|---|
qusgrp.h | s ~QG |
Ref | Expression |
---|---|
qusgrp | NrmSGrp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qusgrp.h | . . . 4 s ~QG | |
2 | 1 | a1i 11 | . . 3 NrmSGrp s ~QG |
3 | eqidd 2623 | . . 3 NrmSGrp | |
4 | eqidd 2623 | . . 3 NrmSGrp | |
5 | nsgsubg 17626 | . . . 4 NrmSGrp SubGrp | |
6 | eqid 2622 | . . . . 5 | |
7 | eqid 2622 | . . . . 5 ~QG ~QG | |
8 | 6, 7 | eqger 17644 | . . . 4 SubGrp ~QG |
9 | 5, 8 | syl 17 | . . 3 NrmSGrp ~QG |
10 | subgrcl 17599 | . . . 4 SubGrp | |
11 | 5, 10 | syl 17 | . . 3 NrmSGrp |
12 | eqid 2622 | . . . 4 | |
13 | 6, 7, 12 | eqgcpbl 17648 | . . 3 NrmSGrp ~QG ~QG ~QG |
14 | 6, 12 | grpcl 17430 | . . . 4 |
15 | 11, 14 | syl3an1 1359 | . . 3 NrmSGrp |
16 | 9 | adantr 481 | . . . . 5 NrmSGrp ~QG |
17 | 11 | adantr 481 | . . . . . 6 NrmSGrp |
18 | simpr1 1067 | . . . . . . 7 NrmSGrp | |
19 | simpr2 1068 | . . . . . . 7 NrmSGrp | |
20 | 17, 18, 19, 14 | syl3anc 1326 | . . . . . 6 NrmSGrp |
21 | simpr3 1069 | . . . . . 6 NrmSGrp | |
22 | 6, 12 | grpcl 17430 | . . . . . 6 |
23 | 17, 20, 21, 22 | syl3anc 1326 | . . . . 5 NrmSGrp |
24 | 16, 23 | erref 7762 | . . . 4 NrmSGrp ~QG |
25 | 6, 12 | grpass 17431 | . . . . 5 |
26 | 11, 25 | sylan 488 | . . . 4 NrmSGrp |
27 | 24, 26 | breqtrd 4679 | . . 3 NrmSGrp ~QG |
28 | eqid 2622 | . . . . 5 | |
29 | 6, 28 | grpidcl 17450 | . . . 4 |
30 | 11, 29 | syl 17 | . . 3 NrmSGrp |
31 | 6, 12, 28 | grplid 17452 | . . . . 5 |
32 | 11, 31 | sylan 488 | . . . 4 NrmSGrp |
33 | 9 | adantr 481 | . . . . 5 NrmSGrp ~QG |
34 | simpr 477 | . . . . 5 NrmSGrp | |
35 | 33, 34 | erref 7762 | . . . 4 NrmSGrp ~QG |
36 | 32, 35 | eqbrtrd 4675 | . . 3 NrmSGrp ~QG |
37 | eqid 2622 | . . . . 5 | |
38 | 6, 37 | grpinvcl 17467 | . . . 4 |
39 | 11, 38 | sylan 488 | . . 3 NrmSGrp |
40 | 6, 12, 28, 37 | grplinv 17468 | . . . . 5 |
41 | 11, 40 | sylan 488 | . . . 4 NrmSGrp |
42 | 30 | adantr 481 | . . . . 5 NrmSGrp |
43 | 33, 42 | erref 7762 | . . . 4 NrmSGrp ~QG |
44 | 41, 43 | eqbrtrd 4675 | . . 3 NrmSGrp ~QG |
45 | 2, 3, 4, 9, 11, 13, 15, 27, 30, 36, 39, 44 | qusgrp2 17533 | . 2 NrmSGrp ~QG |
46 | 45 | simpld 475 | 1 NrmSGrp |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 w3a 1037 wceq 1483 wcel 1990 cfv 5888 (class class class)co 6650 wer 7739 cec 7740 cbs 15857 cplusg 15941 c0g 16100 s cqus 16165 cgrp 17422 cminusg 17423 SubGrpcsubg 17588 NrmSGrpcnsg 17589 ~QG cqg 17590 |
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-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-subg 17591 df-nsg 17592 df-eqg 17593 |
This theorem is referenced by: qus0 17652 qusinv 17653 qusghm 17697 qusabl 18268 qustgplem 21924 rzgrp 24300 |
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