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Mirrors > Home > MPE Home > Th. List > nsgconj | Structured version Visualization version Unicode version |
Description: The conjugation of an element of a normal subgroup is in the subgroup. (Contributed by Mario Carneiro, 4-Feb-2015.) |
Ref | Expression |
---|---|
isnsg3.1 | |
isnsg3.2 | |
isnsg3.3 |
Ref | Expression |
---|---|
nsgconj | NrmSGrp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nsgsubg 17626 | . . . . 5 NrmSGrp SubGrp | |
2 | 1 | 3ad2ant1 1082 | . . . 4 NrmSGrp SubGrp |
3 | subgrcl 17599 | . . . 4 SubGrp | |
4 | 2, 3 | syl 17 | . . 3 NrmSGrp |
5 | simp2 1062 | . . 3 NrmSGrp | |
6 | isnsg3.1 | . . . . . 6 | |
7 | 6 | subgss 17595 | . . . . 5 SubGrp |
8 | 2, 7 | syl 17 | . . . 4 NrmSGrp |
9 | simp3 1063 | . . . 4 NrmSGrp | |
10 | 8, 9 | sseldd 3604 | . . 3 NrmSGrp |
11 | isnsg3.2 | . . . 4 | |
12 | isnsg3.3 | . . . 4 | |
13 | 6, 11, 12 | grpaddsubass 17505 | . . 3 |
14 | 4, 5, 10, 5, 13 | syl13anc 1328 | . 2 NrmSGrp |
15 | 6, 11, 12 | grpnpcan 17507 | . . . . 5 |
16 | 4, 10, 5, 15 | syl3anc 1326 | . . . 4 NrmSGrp |
17 | 16, 9 | eqeltrd 2701 | . . 3 NrmSGrp |
18 | simp1 1061 | . . . 4 NrmSGrp NrmSGrp | |
19 | 6, 12 | grpsubcl 17495 | . . . . 5 |
20 | 4, 10, 5, 19 | syl3anc 1326 | . . . 4 NrmSGrp |
21 | 6, 11 | nsgbi 17625 | . . . 4 NrmSGrp |
22 | 18, 20, 5, 21 | syl3anc 1326 | . . 3 NrmSGrp |
23 | 17, 22 | mpbid 222 | . 2 NrmSGrp |
24 | 14, 23 | eqeltrd 2701 | 1 NrmSGrp |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 w3a 1037 wceq 1483 wcel 1990 wss 3574 cfv 5888 (class class class)co 6650 cbs 15857 cplusg 15941 cgrp 17422 csg 17424 SubGrpcsubg 17588 NrmSGrpcnsg 17589 |
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 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 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-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-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 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-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-1st 7168 df-2nd 7169 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-sbg 17427 df-subg 17591 df-nsg 17592 |
This theorem is referenced by: isnsg3 17628 ghmnsgima 17684 ghmnsgpreima 17685 clsnsg 21913 |
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