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Theorem subgngp 21643
Description: A normed group restricted to a subgroup is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
Hypothesis
Ref Expression
subgngp.h  |-  H  =  ( Gs  A )
Assertion
Ref Expression
subgngp  |-  ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  ->  H  e. NrmGrp )

Proof of Theorem subgngp
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 subgngp.h . . . 4  |-  H  =  ( Gs  A )
21subggrp 16820 . . 3  |-  ( A  e.  (SubGrp `  G
)  ->  H  e.  Grp )
32adantl 468 . 2  |-  ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  ->  H  e.  Grp )
4 ngpms 21614 . . . 4  |-  ( G  e. NrmGrp  ->  G  e.  MetSp )
5 ressms 21541 . . . 4  |-  ( ( G  e.  MetSp  /\  A  e.  (SubGrp `  G )
)  ->  ( Gs  A
)  e.  MetSp )
64, 5sylan 474 . . 3  |-  ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  ->  ( Gs  A
)  e.  MetSp )
71, 6syl5eqel 2533 . 2  |-  ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  ->  H  e.  MetSp
)
8 simplr 762 . . . . . 6  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  A  e.  (SubGrp `  G )
)
9 simprl 764 . . . . . . 7  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  x  e.  ( Base `  H
) )
101subgbas 16821 . . . . . . . 8  |-  ( A  e.  (SubGrp `  G
)  ->  A  =  ( Base `  H )
)
1110ad2antlr 733 . . . . . . 7  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  A  =  ( Base `  H
) )
129, 11eleqtrrd 2532 . . . . . 6  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  x  e.  A )
13 simprr 766 . . . . . . 7  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  y  e.  ( Base `  H
) )
1413, 11eleqtrrd 2532 . . . . . 6  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  y  e.  A )
15 eqid 2451 . . . . . . 7  |-  ( -g `  G )  =  (
-g `  G )
16 eqid 2451 . . . . . . 7  |-  ( -g `  H )  =  (
-g `  H )
1715, 1, 16subgsub 16829 . . . . . 6  |-  ( ( A  e.  (SubGrp `  G )  /\  x  e.  A  /\  y  e.  A )  ->  (
x ( -g `  G
) y )  =  ( x ( -g `  H ) y ) )
188, 12, 14, 17syl3anc 1268 . . . . 5  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  (
x ( -g `  G
) y )  =  ( x ( -g `  H ) y ) )
1918fveq2d 5869 . . . 4  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  (
( norm `  G ) `  ( x ( -g `  G ) y ) )  =  ( (
norm `  G ) `  ( x ( -g `  H ) y ) ) )
20 eqid 2451 . . . . . . . 8  |-  ( dist `  G )  =  (
dist `  G )
211, 20ressds 15311 . . . . . . 7  |-  ( A  e.  (SubGrp `  G
)  ->  ( dist `  G )  =  (
dist `  H )
)
2221ad2antlr 733 . . . . . 6  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  ( dist `  G )  =  ( dist `  H
) )
2322oveqd 6307 . . . . 5  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  (
x ( dist `  G
) y )  =  ( x ( dist `  H ) y ) )
24 simpll 760 . . . . . 6  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  G  e. NrmGrp )
25 eqid 2451 . . . . . . . . 9  |-  ( Base `  G )  =  (
Base `  G )
2625subgss 16818 . . . . . . . 8  |-  ( A  e.  (SubGrp `  G
)  ->  A  C_  ( Base `  G ) )
2726ad2antlr 733 . . . . . . 7  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  A  C_  ( Base `  G
) )
2827, 12sseldd 3433 . . . . . 6  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  x  e.  ( Base `  G
) )
2927, 14sseldd 3433 . . . . . 6  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  y  e.  ( Base `  G
) )
30 eqid 2451 . . . . . . 7  |-  ( norm `  G )  =  (
norm `  G )
3130, 25, 15, 20ngpds 21617 . . . . . 6  |-  ( ( G  e. NrmGrp  /\  x  e.  ( Base `  G
)  /\  y  e.  ( Base `  G )
)  ->  ( x
( dist `  G )
y )  =  ( ( norm `  G
) `  ( x
( -g `  G ) y ) ) )
3224, 28, 29, 31syl3anc 1268 . . . . 5  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  (
x ( dist `  G
) y )  =  ( ( norm `  G
) `  ( x
( -g `  G ) y ) ) )
3323, 32eqtr3d 2487 . . . 4  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  (
x ( dist `  H
) y )  =  ( ( norm `  G
) `  ( x
( -g `  G ) y ) ) )
34 eqid 2451 . . . . . . . . 9  |-  ( Base `  H )  =  (
Base `  H )
3534, 16grpsubcl 16734 . . . . . . . 8  |-  ( ( H  e.  Grp  /\  x  e.  ( Base `  H )  /\  y  e.  ( Base `  H
) )  ->  (
x ( -g `  H
) y )  e.  ( Base `  H
) )
36353expb 1209 . . . . . . 7  |-  ( ( H  e.  Grp  /\  ( x  e.  ( Base `  H )  /\  y  e.  ( Base `  H ) ) )  ->  ( x (
-g `  H )
y )  e.  (
Base `  H )
)
373, 36sylan 474 . . . . . 6  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  (
x ( -g `  H
) y )  e.  ( Base `  H
) )
3837, 11eleqtrrd 2532 . . . . 5  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  (
x ( -g `  H
) y )  e.  A )
39 eqid 2451 . . . . . 6  |-  ( norm `  H )  =  (
norm `  H )
401, 30, 39subgnm2 21642 . . . . 5  |-  ( ( A  e.  (SubGrp `  G )  /\  (
x ( -g `  H
) y )  e.  A )  ->  (
( norm `  H ) `  ( x ( -g `  H ) y ) )  =  ( (
norm `  G ) `  ( x ( -g `  H ) y ) ) )
418, 38, 40syl2anc 667 . . . 4  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  (
( norm `  H ) `  ( x ( -g `  H ) y ) )  =  ( (
norm `  G ) `  ( x ( -g `  H ) y ) ) )
4219, 33, 413eqtr4d 2495 . . 3  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  (
x ( dist `  H
) y )  =  ( ( norm `  H
) `  ( x
( -g `  H ) y ) ) )
4342ralrimivva 2809 . 2  |-  ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  ->  A. x  e.  ( Base `  H
) A. y  e.  ( Base `  H
) ( x (
dist `  H )
y )  =  ( ( norm `  H
) `  ( x
( -g `  H ) y ) ) )
44 eqid 2451 . . 3  |-  ( dist `  H )  =  (
dist `  H )
4539, 16, 44, 34isngp3 21612 . 2  |-  ( H  e. NrmGrp 
<->  ( H  e.  Grp  /\  H  e.  MetSp  /\  A. x  e.  ( Base `  H ) A. y  e.  ( Base `  H
) ( x (
dist `  H )
y )  =  ( ( norm `  H
) `  ( x
( -g `  H ) y ) ) ) )
463, 7, 43, 45syl3anbrc 1192 1  |-  ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  ->  H  e. NrmGrp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1444    e. wcel 1887   A.wral 2737    C_ wss 3404   ` cfv 5582  (class class class)co 6290   Basecbs 15121   ↾s cress 15122   distcds 15199   Grpcgrp 16669   -gcsg 16671  SubGrpcsubg 16811   MetSpcmt 21333   normcnm 21591  NrmGrpcngp 21592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-sup 7956  df-inf 7957  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-tset 15209  df-ds 15212  df-rest 15321  df-topn 15322  df-0g 15340  df-topgen 15342  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-grp 16673  df-minusg 16674  df-sbg 16675  df-subg 16814  df-psmet 18962  df-xmet 18963  df-met 18964  df-bl 18965  df-mopn 18966  df-top 19921  df-bases 19922  df-topon 19923  df-topsp 19924  df-xms 21335  df-ms 21336  df-nm 21597  df-ngp 21598
This theorem is referenced by:  subrgnrg  21676  lssnlm  21703
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