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Theorem subgngp 18678
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 14949 . . 3  |-  ( A  e.  (SubGrp `  G
)  ->  H  e.  Grp )
32adantl 454 . 2  |-  ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  ->  H  e.  Grp )
4 ngpms 18649 . . . 4  |-  ( G  e. NrmGrp  ->  G  e.  MetSp )
5 ressms 18558 . . . 4  |-  ( ( G  e.  MetSp  /\  A  e.  (SubGrp `  G )
)  ->  ( Gs  A
)  e.  MetSp )
64, 5sylan 459 . . 3  |-  ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  ->  ( Gs  A
)  e.  MetSp )
71, 6syl5eqel 2522 . 2  |-  ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  ->  H  e.  MetSp
)
8 simplr 733 . . . . . 6  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  A  e.  (SubGrp `  G )
)
9 simprl 734 . . . . . . 7  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  x  e.  ( Base `  H
) )
101subgbas 14950 . . . . . . . 8  |-  ( A  e.  (SubGrp `  G
)  ->  A  =  ( Base `  H )
)
1110ad2antlr 709 . . . . . . 7  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  A  =  ( Base `  H
) )
129, 11eleqtrrd 2515 . . . . . 6  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  x  e.  A )
13 simprr 735 . . . . . . 7  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  y  e.  ( Base `  H
) )
1413, 11eleqtrrd 2515 . . . . . 6  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  y  e.  A )
15 eqid 2438 . . . . . . 7  |-  ( -g `  G )  =  (
-g `  G )
16 eqid 2438 . . . . . . 7  |-  ( -g `  H )  =  (
-g `  H )
1715, 1, 16subgsub 14958 . . . . . 6  |-  ( ( A  e.  (SubGrp `  G )  /\  x  e.  A  /\  y  e.  A )  ->  (
x ( -g `  G
) y )  =  ( x ( -g `  H ) y ) )
188, 12, 14, 17syl3anc 1185 . . . . 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 5734 . . . 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 2438 . . . . . . . 8  |-  ( dist `  G )  =  (
dist `  G )
211, 20ressds 13643 . . . . . . 7  |-  ( A  e.  (SubGrp `  G
)  ->  ( dist `  G )  =  (
dist `  H )
)
2221ad2antlr 709 . . . . . 6  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  ( dist `  G )  =  ( dist `  H
) )
2322oveqd 6100 . . . . 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 732 . . . . . 6  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  G  e. NrmGrp )
25 eqid 2438 . . . . . . . . 9  |-  ( Base `  G )  =  (
Base `  G )
2625subgss 14947 . . . . . . . 8  |-  ( A  e.  (SubGrp `  G
)  ->  A  C_  ( Base `  G ) )
2726ad2antlr 709 . . . . . . 7  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  A  C_  ( Base `  G
) )
2827, 12sseldd 3351 . . . . . 6  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  x  e.  ( Base `  G
) )
2927, 14sseldd 3351 . . . . . 6  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  y  e.  ( Base `  G
) )
30 eqid 2438 . . . . . . 7  |-  ( norm `  G )  =  (
norm `  G )
3130, 25, 15, 20ngpds 18652 . . . . . 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 1185 . . . . 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 2472 . . . 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 2438 . . . . . . . . 9  |-  ( Base `  H )  =  (
Base `  H )
3534, 16grpsubcl 14871 . . . . . . . 8  |-  ( ( H  e.  Grp  /\  x  e.  ( Base `  H )  /\  y  e.  ( Base `  H
) )  ->  (
x ( -g `  H
) y )  e.  ( Base `  H
) )
36353expb 1155 . . . . . . 7  |-  ( ( H  e.  Grp  /\  ( x  e.  ( Base `  H )  /\  y  e.  ( Base `  H ) ) )  ->  ( x (
-g `  H )
y )  e.  (
Base `  H )
)
373, 36sylan 459 . . . . . 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 2515 . . . . 5  |-  ( ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  /\  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) )  ->  (
x ( -g `  H
) y )  e.  A )
39 eqid 2438 . . . . . 6  |-  ( norm `  H )  =  (
norm `  H )
401, 30, 39subgnm2 18677 . . . . 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 644 . . . 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 2480 . . 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 2800 . 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 2438 . . 3  |-  ( dist `  H )  =  (
dist `  H )
4539, 16, 44, 34isngp3 18647 . 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 1139 1  |-  ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G )
)  ->  H  e. NrmGrp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707    C_ wss 3322   ` cfv 5456  (class class class)co 6083   Basecbs 13471   ↾s cress 13472   distcds 13540   Grpcgrp 14687   -gcsg 14690  SubGrpcsubg 14940   MetSpcmt 18350   normcnm 18626  NrmGrpcngp 18627
This theorem is referenced by:  subrgnrg  18711  lssnlm  18738
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-q 10577  df-rp 10615  df-xneg 10712  df-xadd 10713  df-xmul 10714  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-tset 13550  df-ds 13553  df-rest 13652  df-topn 13653  df-topgen 13669  df-0g 13729  df-mnd 14692  df-grp 14814  df-minusg 14815  df-sbg 14816  df-subg 14943  df-psmet 16696  df-xmet 16697  df-met 16698  df-bl 16699  df-mopn 16700  df-top 16965  df-bases 16967  df-topon 16968  df-topsp 16969  df-xms 18352  df-ms 18353  df-nm 18632  df-ngp 18633
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