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Theorem subgngp 21439
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 16526 . . 3  |-  ( A  e.  (SubGrp `  G
)  ->  H  e.  Grp )
32adantl 464 . 2  |-  ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  ->  H  e.  Grp )
4 ngpms 21410 . . . 4  |-  ( G  e. NrmGrp  ->  G  e.  MetSp )
5 ressms 21319 . . . 4  |-  ( ( G  e.  MetSp  /\  A  e.  (SubGrp `  G )
)  ->  ( Gs  A
)  e.  MetSp )
64, 5sylan 469 . . 3  |-  ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  ->  ( Gs  A
)  e.  MetSp )
71, 6syl5eqel 2494 . 2  |-  ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  ->  H  e.  MetSp
)
8 simplr 754 . . . . . 6  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  A  e.  (SubGrp `  G )
)
9 simprl 756 . . . . . . 7  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  x  e.  ( Base `  H
) )
101subgbas 16527 . . . . . . . 8  |-  ( A  e.  (SubGrp `  G
)  ->  A  =  ( Base `  H )
)
1110ad2antlr 725 . . . . . . 7  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  A  =  ( Base `  H
) )
129, 11eleqtrrd 2493 . . . . . 6  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  x  e.  A )
13 simprr 758 . . . . . . 7  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  y  e.  ( Base `  H
) )
1413, 11eleqtrrd 2493 . . . . . 6  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  y  e.  A )
15 eqid 2402 . . . . . . 7  |-  ( -g `  G )  =  (
-g `  G )
16 eqid 2402 . . . . . . 7  |-  ( -g `  H )  =  (
-g `  H )
1715, 1, 16subgsub 16535 . . . . . 6  |-  ( ( A  e.  (SubGrp `  G )  /\  x  e.  A  /\  y  e.  A )  ->  (
x ( -g `  G
) y )  =  ( x ( -g `  H ) y ) )
188, 12, 14, 17syl3anc 1230 . . . . 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 5852 . . . 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 2402 . . . . . . . 8  |-  ( dist `  G )  =  (
dist `  G )
211, 20ressds 15025 . . . . . . 7  |-  ( A  e.  (SubGrp `  G
)  ->  ( dist `  G )  =  (
dist `  H )
)
2221ad2antlr 725 . . . . . 6  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  ( dist `  G )  =  ( dist `  H
) )
2322oveqd 6294 . . . . 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 752 . . . . . 6  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  G  e. NrmGrp )
25 eqid 2402 . . . . . . . . 9  |-  ( Base `  G )  =  (
Base `  G )
2625subgss 16524 . . . . . . . 8  |-  ( A  e.  (SubGrp `  G
)  ->  A  C_  ( Base `  G ) )
2726ad2antlr 725 . . . . . . 7  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  A  C_  ( Base `  G
) )
2827, 12sseldd 3442 . . . . . 6  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  x  e.  ( Base `  G
) )
2927, 14sseldd 3442 . . . . . 6  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  y  e.  ( Base `  G
) )
30 eqid 2402 . . . . . . 7  |-  ( norm `  G )  =  (
norm `  G )
3130, 25, 15, 20ngpds 21413 . . . . . 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 1230 . . . . 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 2445 . . . 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 2402 . . . . . . . . 9  |-  ( Base `  H )  =  (
Base `  H )
3534, 16grpsubcl 16440 . . . . . . . 8  |-  ( ( H  e.  Grp  /\  x  e.  ( Base `  H )  /\  y  e.  ( Base `  H
) )  ->  (
x ( -g `  H
) y )  e.  ( Base `  H
) )
36353expb 1198 . . . . . . 7  |-  ( ( H  e.  Grp  /\  ( x  e.  ( Base `  H )  /\  y  e.  ( Base `  H ) ) )  ->  ( x (
-g `  H )
y )  e.  (
Base `  H )
)
373, 36sylan 469 . . . . . 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 2493 . . . . 5  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  (
x ( -g `  H
) y )  e.  A )
39 eqid 2402 . . . . . 6  |-  ( norm `  H )  =  (
norm `  H )
401, 30, 39subgnm2 21438 . . . . 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 659 . . . 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 2453 . . 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 2824 . 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 2402 . . 3  |-  ( dist `  H )  =  (
dist `  H )
4539, 16, 44, 34isngp3 21408 . 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 1181 1  |-  ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  ->  H  e. NrmGrp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2753    C_ wss 3413   ` cfv 5568  (class class class)co 6277   Basecbs 14839   ↾s cress 14840   distcds 14916   Grpcgrp 16375   -gcsg 16377  SubGrpcsubg 16517   MetSpcmt 21111   normcnm 21387  NrmGrpcngp 21388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-pre-sup 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-er 7347  df-map 7458  df-en 7554  df-dom 7555  df-sdom 7556  df-sup 7934  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-div 10247  df-nn 10576  df-2 10634  df-3 10635  df-4 10636  df-5 10637  df-6 10638  df-7 10639  df-8 10640  df-9 10641  df-10 10642  df-n0 10836  df-z 10905  df-dec 11019  df-uz 11127  df-q 11227  df-rp 11265  df-xneg 11370  df-xadd 11371  df-xmul 11372  df-ndx 14842  df-slot 14843  df-base 14844  df-sets 14845  df-ress 14846  df-plusg 14920  df-tset 14926  df-ds 14929  df-rest 15035  df-topn 15036  df-0g 15054  df-topgen 15056  df-mgm 16194  df-sgrp 16233  df-mnd 16243  df-grp 16379  df-minusg 16380  df-sbg 16381  df-subg 16520  df-psmet 18729  df-xmet 18730  df-met 18731  df-bl 18732  df-mopn 18733  df-top 19689  df-bases 19691  df-topon 19692  df-topsp 19693  df-xms 21113  df-ms 21114  df-nm 21393  df-ngp 21394
This theorem is referenced by:  subrgnrg  21472  lssnlm  21499
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