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Theorem subgnm 18667
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 2436 . . . . 5  |-  ( Base `  G )  =  (
Base `  G )
21subgss 14938 . . . 4  |-  ( A  e.  (SubGrp `  G
)  ->  A  C_  ( Base `  G ) )
3 resmpt 5184 . . . 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 16 . . 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 14941 . . . 4  |-  ( A  e.  (SubGrp `  G
)  ->  A  =  ( Base `  H )
)
7 eqid 2436 . . . . . 6  |-  ( dist `  G )  =  (
dist `  G )
85, 7ressds 13634 . . . . 5  |-  ( A  e.  (SubGrp `  G
)  ->  ( dist `  G )  =  (
dist `  H )
)
9 eqidd 2437 . . . . 5  |-  ( A  e.  (SubGrp `  G
)  ->  x  =  x )
10 eqid 2436 . . . . . 6  |-  ( 0g
`  G )  =  ( 0g `  G
)
115, 10subg0 14943 . . . . 5  |-  ( A  e.  (SubGrp `  G
)  ->  ( 0g `  G )  =  ( 0g `  H ) )
128, 9, 11oveq123d 6095 . . . 4  |-  ( A  e.  (SubGrp `  G
)  ->  ( x
( dist `  G )
( 0g `  G
) )  =  ( x ( dist `  H
) ( 0g `  H ) ) )
136, 12mpteq12dv 4280 . . 3  |-  ( A  e.  (SubGrp `  G
)  ->  ( x  e.  A  |->  ( x ( dist `  G
) ( 0g `  G ) ) )  =  ( x  e.  ( Base `  H
)  |->  ( x (
dist `  H )
( 0g `  H
) ) ) )
144, 13eqtr2d 2469 . 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 2436 . . 3  |-  ( Base `  H )  =  (
Base `  H )
17 eqid 2436 . . 3  |-  ( 0g
`  H )  =  ( 0g `  H
)
18 eqid 2436 . . 3  |-  ( dist `  H )  =  (
dist `  H )
1915, 16, 17, 18nmfval 18629 . 2  |-  M  =  ( x  e.  (
Base `  H )  |->  ( x ( dist `  H ) ( 0g
`  H ) ) )
20 subgnm.n . . . 4  |-  N  =  ( norm `  G
)
2120, 1, 10, 7nmfval 18629 . . 3  |-  N  =  ( x  e.  (
Base `  G )  |->  ( x ( dist `  G ) ( 0g
`  G ) ) )
2221reseq1i 5135 . 2  |-  ( N  |`  A )  =  ( ( x  e.  (
Base `  G )  |->  ( x ( dist `  G ) ( 0g
`  G ) ) )  |`  A )
2314, 19, 223eqtr4g 2493 1  |-  ( A  e.  (SubGrp `  G
)  ->  M  =  ( N  |`  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725    C_ wss 3313    e. cmpt 4259    |` cres 4873   ` cfv 5447  (class class class)co 6074   Basecbs 13462   ↾s cress 13463   distcds 13531   0gc0g 13716  SubGrpcsubg 14931   normcnm 18617
This theorem is referenced by:  subgnm2  18668  subrgnrg  18702
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694  ax-cnex 9039  ax-resscn 9040  ax-1cn 9041  ax-icn 9042  ax-addcl 9043  ax-addrcl 9044  ax-mulcl 9045  ax-mulrcl 9046  ax-mulcom 9047  ax-addass 9048  ax-mulass 9049  ax-distr 9050  ax-i2m1 9051  ax-1ne0 9052  ax-1rid 9053  ax-rnegex 9054  ax-rrecex 9055  ax-cnre 9056  ax-pre-lttri 9057  ax-pre-lttrn 9058  ax-pre-ltadd 9059  ax-pre-mulgt0 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2703  df-rex 2704  df-reu 2705  df-rmo 2706  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-pss 3329  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-tp 3815  df-op 3816  df-uni 4009  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-tr 4296  df-eprel 4487  df-id 4491  df-po 4496  df-so 4497  df-fr 4534  df-we 4536  df-ord 4577  df-on 4578  df-lim 4579  df-suc 4580  df-om 4839  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-riota 6542  df-recs 6626  df-rdg 6661  df-er 6898  df-en 7103  df-dom 7104  df-sdom 7105  df-pnf 9115  df-mnf 9116  df-xr 9117  df-ltxr 9118  df-le 9119  df-sub 9286  df-neg 9287  df-nn 9994  df-2 10051  df-3 10052  df-4 10053  df-5 10054  df-6 10055  df-7 10056  df-8 10057  df-9 10058  df-10 10059  df-n0 10215  df-z 10276  df-dec 10376  df-ndx 13465  df-slot 13466  df-base 13467  df-sets 13468  df-ress 13469  df-plusg 13535  df-ds 13544  df-0g 13720  df-mnd 14683  df-grp 14805  df-subg 14934  df-nm 18623
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