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Mirrors > Home > MPE Home > Th. List > subgngp | Structured version Visualization version Unicode version |
Description: A normed group restricted to a subgroup is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.) |
Ref | Expression |
---|---|
subgngp.h | ↾s |
Ref | Expression |
---|---|
subgngp | NrmGrp SubGrp NrmGrp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subgngp.h | . . . 4 ↾s | |
2 | 1 | subggrp 17597 | . . 3 SubGrp |
3 | 2 | adantl 482 | . 2 NrmGrp SubGrp |
4 | ngpms 22404 | . . . 4 NrmGrp | |
5 | ressms 22331 | . . . 4 SubGrp ↾s | |
6 | 4, 5 | sylan 488 | . . 3 NrmGrp SubGrp ↾s |
7 | 1, 6 | syl5eqel 2705 | . 2 NrmGrp SubGrp |
8 | simplr 792 | . . . . . 6 NrmGrp SubGrp SubGrp | |
9 | simprl 794 | . . . . . . 7 NrmGrp SubGrp | |
10 | 1 | subgbas 17598 | . . . . . . . 8 SubGrp |
11 | 10 | ad2antlr 763 | . . . . . . 7 NrmGrp SubGrp |
12 | 9, 11 | eleqtrrd 2704 | . . . . . 6 NrmGrp SubGrp |
13 | simprr 796 | . . . . . . 7 NrmGrp SubGrp | |
14 | 13, 11 | eleqtrrd 2704 | . . . . . 6 NrmGrp SubGrp |
15 | eqid 2622 | . . . . . . 7 | |
16 | eqid 2622 | . . . . . . 7 | |
17 | 15, 1, 16 | subgsub 17606 | . . . . . 6 SubGrp |
18 | 8, 12, 14, 17 | syl3anc 1326 | . . . . 5 NrmGrp SubGrp |
19 | 18 | fveq2d 6195 | . . . 4 NrmGrp SubGrp |
20 | eqid 2622 | . . . . . . . 8 | |
21 | 1, 20 | ressds 16073 | . . . . . . 7 SubGrp |
22 | 21 | ad2antlr 763 | . . . . . 6 NrmGrp SubGrp |
23 | 22 | oveqd 6667 | . . . . 5 NrmGrp SubGrp |
24 | simpll 790 | . . . . . 6 NrmGrp SubGrp NrmGrp | |
25 | eqid 2622 | . . . . . . . . 9 | |
26 | 25 | subgss 17595 | . . . . . . . 8 SubGrp |
27 | 26 | ad2antlr 763 | . . . . . . 7 NrmGrp SubGrp |
28 | 27, 12 | sseldd 3604 | . . . . . 6 NrmGrp SubGrp |
29 | 27, 14 | sseldd 3604 | . . . . . 6 NrmGrp SubGrp |
30 | eqid 2622 | . . . . . . 7 | |
31 | 30, 25, 15, 20 | ngpds 22408 | . . . . . 6 NrmGrp |
32 | 24, 28, 29, 31 | syl3anc 1326 | . . . . 5 NrmGrp SubGrp |
33 | 23, 32 | eqtr3d 2658 | . . . 4 NrmGrp SubGrp |
34 | eqid 2622 | . . . . . . . . 9 | |
35 | 34, 16 | grpsubcl 17495 | . . . . . . . 8 |
36 | 35 | 3expb 1266 | . . . . . . 7 |
37 | 3, 36 | sylan 488 | . . . . . 6 NrmGrp SubGrp |
38 | 37, 11 | eleqtrrd 2704 | . . . . 5 NrmGrp SubGrp |
39 | eqid 2622 | . . . . . 6 | |
40 | 1, 30, 39 | subgnm2 22438 | . . . . 5 SubGrp |
41 | 8, 38, 40 | syl2anc 693 | . . . 4 NrmGrp SubGrp |
42 | 19, 33, 41 | 3eqtr4d 2666 | . . 3 NrmGrp SubGrp |
43 | 42 | ralrimivva 2971 | . 2 NrmGrp SubGrp |
44 | eqid 2622 | . . 3 | |
45 | 39, 16, 44, 34 | isngp3 22402 | . 2 NrmGrp |
46 | 3, 7, 43, 45 | syl3anbrc 1246 | 1 NrmGrp SubGrp NrmGrp |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 wral 2912 wss 3574 cfv 5888 (class class class)co 6650 cbs 15857 ↾s cress 15858 cds 15950 cgrp 17422 csg 17424 SubGrpcsubg 17588 cmt 22123 cnm 22381 NrmGrpcngp 22382 |
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 ax-pre-sup 10014 |
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-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-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 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-div 10685 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-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-tset 15960 df-ds 15964 df-rest 16083 df-topn 16084 df-0g 16102 df-topgen 16104 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-sbg 17427 df-subg 17591 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-xms 22125 df-ms 22126 df-nm 22387 df-ngp 22388 |
This theorem is referenced by: subrgnrg 22477 lssnlm 22505 |
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