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Theorem subgngp 18151
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 14624 . . 3  |-  ( A  e.  (SubGrp `  G
)  ->  H  e.  Grp )
32adantl 452 . 2  |-  ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  ->  H  e.  Grp )
4 ngpms 18122 . . . 4  |-  ( G  e. NrmGrp  ->  G  e.  MetSp )
5 ressms 18072 . . . 4  |-  ( ( G  e.  MetSp  /\  A  e.  (SubGrp `  G )
)  ->  ( Gs  A
)  e.  MetSp )
64, 5sylan 457 . . 3  |-  ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  ->  ( Gs  A
)  e.  MetSp )
71, 6syl5eqel 2367 . 2  |-  ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  ->  H  e.  MetSp
)
8 simplr 731 . . . . . 6  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  A  e.  (SubGrp `  G )
)
9 simprl 732 . . . . . . 7  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  x  e.  ( Base `  H
) )
101subgbas 14625 . . . . . . . 8  |-  ( A  e.  (SubGrp `  G
)  ->  A  =  ( Base `  H )
)
1110ad2antlr 707 . . . . . . 7  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  A  =  ( Base `  H
) )
129, 11eleqtrrd 2360 . . . . . 6  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  x  e.  A )
13 simprr 733 . . . . . . 7  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  y  e.  ( Base `  H
) )
1413, 11eleqtrrd 2360 . . . . . 6  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  y  e.  A )
15 eqid 2283 . . . . . . 7  |-  ( -g `  G )  =  (
-g `  G )
16 eqid 2283 . . . . . . 7  |-  ( -g `  H )  =  (
-g `  H )
1715, 1, 16subgsub 14633 . . . . . 6  |-  ( ( A  e.  (SubGrp `  G )  /\  x  e.  A  /\  y  e.  A )  ->  (
x ( -g `  G
) y )  =  ( x ( -g `  H ) y ) )
188, 12, 14, 17syl3anc 1182 . . . . 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 5529 . . . 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 2283 . . . . . . . 8  |-  ( dist `  G )  =  (
dist `  G )
211, 20ressds 13318 . . . . . . 7  |-  ( A  e.  (SubGrp `  G
)  ->  ( dist `  G )  =  (
dist `  H )
)
2221ad2antlr 707 . . . . . 6  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  ( dist `  G )  =  ( dist `  H
) )
2322oveqd 5875 . . . . 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 730 . . . . . 6  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  G  e. NrmGrp )
25 eqid 2283 . . . . . . . . 9  |-  ( Base `  G )  =  (
Base `  G )
2625subgss 14622 . . . . . . . 8  |-  ( A  e.  (SubGrp `  G
)  ->  A  C_  ( Base `  G ) )
2726ad2antlr 707 . . . . . . 7  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  A  C_  ( Base `  G
) )
2827, 12sseldd 3181 . . . . . 6  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  x  e.  ( Base `  G
) )
2927, 14sseldd 3181 . . . . . 6  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  y  e.  ( Base `  G
) )
30 eqid 2283 . . . . . . 7  |-  ( norm `  G )  =  (
norm `  G )
3130, 25, 15, 20ngpds 18125 . . . . . 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 1182 . . . . 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 2317 . . . 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 2283 . . . . . . . . 9  |-  ( Base `  H )  =  (
Base `  H )
3534, 16grpsubcl 14546 . . . . . . . 8  |-  ( ( H  e.  Grp  /\  x  e.  ( Base `  H )  /\  y  e.  ( Base `  H
) )  ->  (
x ( -g `  H
) y )  e.  ( Base `  H
) )
36353expb 1152 . . . . . . 7  |-  ( ( H  e.  Grp  /\  ( x  e.  ( Base `  H )  /\  y  e.  ( Base `  H ) ) )  ->  ( x (
-g `  H )
y )  e.  (
Base `  H )
)
373, 36sylan 457 . . . . . 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 2360 . . . . 5  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  (
x ( -g `  H
) y )  e.  A )
39 eqid 2283 . . . . . 6  |-  ( norm `  H )  =  (
norm `  H )
401, 30, 39subgnm2 18150 . . . . 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 642 . . . 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 2325 . . 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 2635 . 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 2283 . . 3  |-  ( dist `  H )  =  (
dist `  H )
4539, 16, 44, 34isngp3 18120 . 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 1136 1  |-  ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  ->  H  e. NrmGrp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   ` cfv 5255  (class class class)co 5858   Basecbs 13148   ↾s cress 13149   distcds 13217   Grpcgrp 14362   -gcsg 14365  SubGrpcsubg 14615   MetSpcmt 17883   normcnm 18099  NrmGrpcngp 18100
This theorem is referenced by:  subrgnrg  18184  lssnlm  18211
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-tset 13227  df-ds 13230  df-rest 13327  df-topn 13328  df-topgen 13344  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-xms 17885  df-ms 17886  df-nm 18105  df-ngp 18106
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