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Theorem subgngp 18167
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 14640 . . 3  |-  ( A  e.  (SubGrp `  G
)  ->  H  e.  Grp )
32adantl 452 . 2  |-  ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  ->  H  e.  Grp )
4 ngpms 18138 . . . 4  |-  ( G  e. NrmGrp  ->  G  e.  MetSp )
5 ressms 18088 . . . 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 2380 . 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 14641 . . . . . . . 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 2373 . . . . . 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 2373 . . . . . 6  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  y  e.  A )
15 eqid 2296 . . . . . . 7  |-  ( -g `  G )  =  (
-g `  G )
16 eqid 2296 . . . . . . 7  |-  ( -g `  H )  =  (
-g `  H )
1715, 1, 16subgsub 14649 . . . . . 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 5545 . . . 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 2296 . . . . . . . 8  |-  ( dist `  G )  =  (
dist `  G )
211, 20ressds 13334 . . . . . . 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 5891 . . . . 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 2296 . . . . . . . . 9  |-  ( Base `  G )  =  (
Base `  G )
2625subgss 14638 . . . . . . . 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 3194 . . . . . 6  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  x  e.  ( Base `  G
) )
2927, 14sseldd 3194 . . . . . 6  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  y  e.  ( Base `  G
) )
30 eqid 2296 . . . . . . 7  |-  ( norm `  G )  =  (
norm `  G )
3130, 25, 15, 20ngpds 18141 . . . . . 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 2330 . . . 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 2296 . . . . . . . . 9  |-  ( Base `  H )  =  (
Base `  H )
3534, 16grpsubcl 14562 . . . . . . . 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 2373 . . . . 5  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  (
x ( -g `  H
) y )  e.  A )
39 eqid 2296 . . . . . 6  |-  ( norm `  H )  =  (
norm `  H )
401, 30, 39subgnm2 18166 . . . . 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 2338 . . 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 2648 . 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 2296 . . 3  |-  ( dist `  H )  =  (
dist `  H )
4539, 16, 44, 34isngp3 18136 . 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 1632    e. wcel 1696   A.wral 2556    C_ wss 3165   ` cfv 5271  (class class class)co 5874   Basecbs 13164   ↾s cress 13165   distcds 13233   Grpcgrp 14378   -gcsg 14381  SubGrpcsubg 14631   MetSpcmt 17899   normcnm 18115  NrmGrpcngp 18116
This theorem is referenced by:  subrgnrg  18200  lssnlm  18227
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-tset 13243  df-ds 13246  df-rest 13343  df-topn 13344  df-topgen 13360  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-xms 17901  df-ms 17902  df-nm 18121  df-ngp 18122
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