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Mirrors > Home > MPE Home > Th. List > qusabl | Structured version Visualization version Unicode version |
Description: If is a subgroup of the abelian group , then is an abelian group. (Contributed by Mario Carneiro, 26-Apr-2016.) |
Ref | Expression |
---|---|
qusabl.h | s ~QG |
Ref | Expression |
---|---|
qusabl | SubGrp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ablnsg 18250 | . . . . 5 NrmSGrp SubGrp | |
2 | 1 | eleq2d 2687 | . . . 4 NrmSGrp SubGrp |
3 | 2 | biimpar 502 | . . 3 SubGrp NrmSGrp |
4 | qusabl.h | . . . 4 s ~QG | |
5 | 4 | qusgrp 17649 | . . 3 NrmSGrp |
6 | 3, 5 | syl 17 | . 2 SubGrp |
7 | vex 3203 | . . . . . . 7 | |
8 | 7 | elqs 7799 | . . . . . 6 ~QG ~QG |
9 | 4 | a1i 11 | . . . . . . . 8 SubGrp s ~QG |
10 | eqidd 2623 | . . . . . . . 8 SubGrp | |
11 | ovexd 6680 | . . . . . . . 8 SubGrp ~QG | |
12 | simpl 473 | . . . . . . . 8 SubGrp | |
13 | 9, 10, 11, 12 | qusbas 16205 | . . . . . . 7 SubGrp ~QG |
14 | 13 | eleq2d 2687 | . . . . . 6 SubGrp ~QG |
15 | 8, 14 | syl5bbr 274 | . . . . 5 SubGrp ~QG |
16 | vex 3203 | . . . . . . 7 | |
17 | 16 | elqs 7799 | . . . . . 6 ~QG ~QG |
18 | 13 | eleq2d 2687 | . . . . . 6 SubGrp ~QG |
19 | 17, 18 | syl5bbr 274 | . . . . 5 SubGrp ~QG |
20 | 15, 19 | anbi12d 747 | . . . 4 SubGrp ~QG ~QG |
21 | reeanv 3107 | . . . . 5 ~QG ~QG ~QG ~QG | |
22 | eqid 2622 | . . . . . . . . . . . 12 | |
23 | eqid 2622 | . . . . . . . . . . . 12 | |
24 | 22, 23 | ablcom 18210 | . . . . . . . . . . 11 |
25 | 24 | 3expb 1266 | . . . . . . . . . 10 |
26 | 25 | adantlr 751 | . . . . . . . . 9 SubGrp |
27 | 26 | eceq1d 7783 | . . . . . . . 8 SubGrp ~QG ~QG |
28 | 3 | adantr 481 | . . . . . . . . 9 SubGrp NrmSGrp |
29 | simprl 794 | . . . . . . . . 9 SubGrp | |
30 | simprr 796 | . . . . . . . . 9 SubGrp | |
31 | eqid 2622 | . . . . . . . . . 10 | |
32 | 4, 22, 23, 31 | qusadd 17651 | . . . . . . . . 9 NrmSGrp ~QG ~QG ~QG |
33 | 28, 29, 30, 32 | syl3anc 1326 | . . . . . . . 8 SubGrp ~QG ~QG ~QG |
34 | 4, 22, 23, 31 | qusadd 17651 | . . . . . . . . 9 NrmSGrp ~QG ~QG ~QG |
35 | 28, 30, 29, 34 | syl3anc 1326 | . . . . . . . 8 SubGrp ~QG ~QG ~QG |
36 | 27, 33, 35 | 3eqtr4d 2666 | . . . . . . 7 SubGrp ~QG ~QG ~QG ~QG |
37 | oveq12 6659 | . . . . . . . 8 ~QG ~QG ~QG ~QG | |
38 | oveq12 6659 | . . . . . . . . 9 ~QG ~QG ~QG ~QG | |
39 | 38 | ancoms 469 | . . . . . . . 8 ~QG ~QG ~QG ~QG |
40 | 37, 39 | eqeq12d 2637 | . . . . . . 7 ~QG ~QG ~QG ~QG ~QG ~QG |
41 | 36, 40 | syl5ibrcom 237 | . . . . . 6 SubGrp ~QG ~QG |
42 | 41 | rexlimdvva 3038 | . . . . 5 SubGrp ~QG ~QG |
43 | 21, 42 | syl5bir 233 | . . . 4 SubGrp ~QG ~QG |
44 | 20, 43 | sylbird 250 | . . 3 SubGrp |
45 | 44 | ralrimivv 2970 | . 2 SubGrp |
46 | eqid 2622 | . . 3 | |
47 | 46, 31 | isabl2 18201 | . 2 |
48 | 6, 45, 47 | sylanbrc 698 | 1 SubGrp |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 wral 2912 wrex 2913 cvv 3200 cfv 5888 (class class class)co 6650 cec 7740 cqs 7741 cbs 15857 cplusg 15941 s cqus 16165 cgrp 17422 SubGrpcsubg 17588 NrmSGrpcnsg 17589 ~QG cqg 17590 cabl 18194 |
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 df-cmn 18195 df-abl 18196 |
This theorem is referenced by: (None) |
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