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Theorem subgnm 18165
Description: The norm in a subgroup. (Contributed by Mario Carneiro, 4-Oct-2015.)
Hypotheses
Ref Expression
subgngp.h  |-  H  =  ( Gs  A )
subgnm.n  |-  N  =  ( norm `  G
)
subgnm.m  |-  M  =  ( norm `  H
)
Assertion
Ref Expression
subgnm  |-  ( A  e.  (SubGrp `  G
)  ->  M  =  ( N  |`  A ) )

Proof of Theorem subgnm
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqid 2296 . . . . 5  |-  ( Base `  G )  =  (
Base `  G )
21subgss 14638 . . . 4  |-  ( A  e.  (SubGrp `  G
)  ->  A  C_  ( Base `  G ) )
3 resmpt 5016 . . . 4  |-  ( A 
C_  ( Base `  G
)  ->  ( (
x  e.  ( Base `  G )  |->  ( x ( dist `  G
) ( 0g `  G ) ) )  |`  A )  =  ( x  e.  A  |->  ( x ( dist `  G
) ( 0g `  G ) ) ) )
42, 3syl 15 . . 3  |-  ( A  e.  (SubGrp `  G
)  ->  ( (
x  e.  ( Base `  G )  |->  ( x ( dist `  G
) ( 0g `  G ) ) )  |`  A )  =  ( x  e.  A  |->  ( x ( dist `  G
) ( 0g `  G ) ) ) )
5 subgngp.h . . . . 5  |-  H  =  ( Gs  A )
65subgbas 14641 . . . 4  |-  ( A  e.  (SubGrp `  G
)  ->  A  =  ( Base `  H )
)
7 eqid 2296 . . . . . 6  |-  ( dist `  G )  =  (
dist `  G )
85, 7ressds 13334 . . . . 5  |-  ( A  e.  (SubGrp `  G
)  ->  ( dist `  G )  =  (
dist `  H )
)
9 eqidd 2297 . . . . 5  |-  ( A  e.  (SubGrp `  G
)  ->  x  =  x )
10 eqid 2296 . . . . . 6  |-  ( 0g
`  G )  =  ( 0g `  G
)
115, 10subg0 14643 . . . . 5  |-  ( A  e.  (SubGrp `  G
)  ->  ( 0g `  G )  =  ( 0g `  H ) )
128, 9, 11oveq123d 5895 . . . 4  |-  ( A  e.  (SubGrp `  G
)  ->  ( x
( dist `  G )
( 0g `  G
) )  =  ( x ( dist `  H
) ( 0g `  H ) ) )
136, 12mpteq12dv 4114 . . 3  |-  ( A  e.  (SubGrp `  G
)  ->  ( x  e.  A  |->  ( x ( dist `  G
) ( 0g `  G ) ) )  =  ( x  e.  ( Base `  H
)  |->  ( x (
dist `  H )
( 0g `  H
) ) ) )
144, 13eqtr2d 2329 . 2  |-  ( A  e.  (SubGrp `  G
)  ->  ( x  e.  ( Base `  H
)  |->  ( x (
dist `  H )
( 0g `  H
) ) )  =  ( ( x  e.  ( Base `  G
)  |->  ( x (
dist `  G )
( 0g `  G
) ) )  |`  A ) )
15 subgnm.m . . 3  |-  M  =  ( norm `  H
)
16 eqid 2296 . . 3  |-  ( Base `  H )  =  (
Base `  H )
17 eqid 2296 . . 3  |-  ( 0g
`  H )  =  ( 0g `  H
)
18 eqid 2296 . . 3  |-  ( dist `  H )  =  (
dist `  H )
1915, 16, 17, 18nmfval 18127 . 2  |-  M  =  ( x  e.  (
Base `  H )  |->  ( x ( dist `  H ) ( 0g
`  H ) ) )
20 subgnm.n . . . 4  |-  N  =  ( norm `  G
)
2120, 1, 10, 7nmfval 18127 . . 3  |-  N  =  ( x  e.  (
Base `  G )  |->  ( x ( dist `  G ) ( 0g
`  G ) ) )
2221reseq1i 4967 . 2  |-  ( N  |`  A )  =  ( ( x  e.  (
Base `  G )  |->  ( x ( dist `  G ) ( 0g
`  G ) ) )  |`  A )
2314, 19, 223eqtr4g 2353 1  |-  ( A  e.  (SubGrp `  G
)  ->  M  =  ( N  |`  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696    C_ wss 3165    e. cmpt 4093    |` cres 4707   ` cfv 5271  (class class class)co 5874   Basecbs 13164   ↾s cress 13165   distcds 13233   0gc0g 13416  SubGrpcsubg 14631   normcnm 18115
This theorem is referenced by:  subgnm2  18166  subrgnrg  18200
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-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
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-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  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-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-ds 13246  df-0g 13420  df-mnd 14383  df-grp 14505  df-subg 14634  df-nm 18121
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