Theorem subgnm 22437

Description: The norm in a subgroup. (Contributed by Mario Carneiro, 4-Oct-2015.)

Hypotheses

Ref	Expression
subgngp.h	|- H = ( G ↾s 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.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . 5 |- ( Base ` G ) = ( Base ` G )
2	1	subgss 17595	. . . 4 |- ( A e. (SubGrp ` G ) -> A C_ ( Base ` G ) )
3	2	resmptd 5452	. . 3 |- ( A e. (SubGrp ` G ) -> ( ( x e. ( Base ` G ) |-> ( x ( dist ` G ) ( 0g ` G ) ) ) |` A ) = ( x e. A |-> ( x ( dist ` G ) ( 0g ` G ) ) ) )
4		subgngp.h	. . . . 5 |- H = ( G ↾s A )
5	4	subgbas 17598	. . . 4 |- ( A e. (SubGrp ` G ) -> A = ( Base ` H ) )
6		eqid 2622	. . . . . 6 |- ( dist ` G ) = ( dist ` G )
7	4, 6	ressds 16073	. . . . 5 |- ( A e. (SubGrp ` G ) -> ( dist ` G ) = ( dist ` H ) )
8		eqidd 2623	. . . . 5 |- ( A e. (SubGrp ` G ) -> x = x )
9		eqid 2622	. . . . . 6 |- ( 0g ` G ) = ( 0g ` G )
10	4, 9	subg0 17600	. . . . 5 |- ( A e. (SubGrp ` G ) -> ( 0g ` G ) = ( 0g ` H ) )
11	7, 8, 10	oveq123d 6671	. . . 4 |- ( A e. (SubGrp ` G ) -> ( x ( dist ` G ) ( 0g ` G ) ) = ( x ( dist ` H ) ( 0g ` H ) ) )
12	5, 11	mpteq12dv 4733	. . 3 |- ( A e. (SubGrp ` G ) -> ( x e. A |-> ( x ( dist ` G ) ( 0g ` G ) ) ) = ( x e. ( Base ` H ) |-> ( x ( dist ` H ) ( 0g ` H ) ) ) )
13	3, 12	eqtr2d 2657	. 2 |- ( A e. (SubGrp ` G ) -> ( x e. ( Base ` H ) |-> ( x ( dist ` H ) ( 0g ` H ) ) ) = ( ( x e. ( Base ` G ) |-> ( x ( dist ` G ) ( 0g ` G ) ) ) |` A ) )
14		subgnm.m	. . 3 |- M = ( norm ` H )
15		eqid 2622	. . 3 |- ( Base ` H ) = ( Base ` H )
16		eqid 2622	. . 3 |- ( 0g ` H ) = ( 0g ` H )
17		eqid 2622	. . 3 |- ( dist ` H ) = ( dist ` H )
18	14, 15, 16, 17	nmfval 22393	. 2 |- M = ( x e. ( Base ` H ) |-> ( x ( dist ` H ) ( 0g ` H ) ) )
19		subgnm.n	. . . 4 |- N = ( norm ` G )
20	19, 1, 9, 6	nmfval 22393	. . 3 |- N = ( x e. ( Base ` G ) |-> ( x ( dist ` G ) ( 0g ` G ) ) )
21	20	reseq1i 5392	. 2 |- ( N |` A ) = ( ( x e. ( Base ` G ) |-> ( x ( dist ` G ) ( 0g ` G ) ) ) |` A )
22	13, 18, 21	3eqtr4g 2681	1 |- ( A e. (SubGrp ` G ) -> M = ( N |` A ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   |-> cmpt 4729   |` cres 5116   ` cfv 5888  (class class class)co 6650   Basecbs 15857   ↾_s cress 15858   distcds 15950   0gc0g 16100  SubGrpcsubg 17588   normcnm 22381

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-ds 15964  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-subg 17591  df-nm 22387

This theorem is referenced by:  subgnm2  22438  subrgnrg  22477  isncvsngp  22949