Theorem tngnm 22455

Description: The topology generated by a normed structure. (Contributed by Mario Carneiro, 4-Oct-2015.)

Hypotheses

Ref	Expression
tngnm.t	|- T = ( G toNrmGrp N )
tngnm.x	|- X = ( Base ` G )
tngnm.a	|- A e. _V

Assertion

Ref	Expression
tngnm	|- ( ( G e. Grp /\ N : X --> A ) -> N = ( norm ` T ) )

Proof of Theorem tngnm

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 477	. . 3 |- ( ( G e. Grp /\ N : X --> A ) -> N : X --> A )
2	1	feqmptd 6249	. 2 |- ( ( G e. Grp /\ N : X --> A ) -> N = ( x e. X |-> ( N ` x ) ) )
3		tngnm.x	. . . . . . . 8 |- X = ( Base ` G )
4		eqid 2622	. . . . . . . 8 |- ( -g ` G ) = ( -g ` G )
5	3, 4	grpsubf 17494	. . . . . . 7 |- ( G e. Grp -> ( -g ` G ) : ( X X. X ) --> X )
6	5	ad2antrr 762	. . . . . 6 |- ( ( ( G e. Grp /\ N : X --> A ) /\ x e. X ) -> ( -g ` G ) : ( X X. X ) --> X )
7		simpr 477	. . . . . . 7 |- ( ( ( G e. Grp /\ N : X --> A ) /\ x e. X ) -> x e. X )
8		eqid 2622	. . . . . . . . 9 |- ( 0g ` G ) = ( 0g ` G )
9	3, 8	grpidcl 17450	. . . . . . . 8 |- ( G e. Grp -> ( 0g ` G ) e. X )
10	9	ad2antrr 762	. . . . . . 7 |- ( ( ( G e. Grp /\ N : X --> A ) /\ x e. X ) -> ( 0g ` G ) e. X )
11		opelxpi 5148	. . . . . . 7 |- ( ( x e. X /\ ( 0g ` G ) e. X ) -> <. x , ( 0g ` G ) >. e. ( X X. X ) )
12	7, 10, 11	syl2anc 693	. . . . . 6 |- ( ( ( G e. Grp /\ N : X --> A ) /\ x e. X ) -> <. x , ( 0g ` G ) >. e. ( X X. X ) )
13		fvco3 6275	. . . . . 6 |- ( ( ( -g ` G ) : ( X X. X ) --> X /\ <. x , ( 0g ` G ) >. e. ( X X. X ) ) -> ( ( N o. ( -g ` G ) ) ` <. x , ( 0g ` G ) >. ) = ( N ` ( ( -g ` G ) ` <. x , ( 0g ` G ) >. ) ) )
14	6, 12, 13	syl2anc 693	. . . . 5 |- ( ( ( G e. Grp /\ N : X --> A ) /\ x e. X ) -> ( ( N o. ( -g ` G ) ) ` <. x , ( 0g ` G ) >. ) = ( N ` ( ( -g ` G ) ` <. x , ( 0g ` G ) >. ) ) )
15		df-ov 6653	. . . . 5 |- ( x ( N o. ( -g ` G ) ) ( 0g ` G ) ) = ( ( N o. ( -g ` G ) ) ` <. x , ( 0g ` G ) >. )
16		df-ov 6653	. . . . . 6 |- ( x ( -g ` G ) ( 0g ` G ) ) = ( ( -g ` G ) ` <. x , ( 0g ` G ) >. )
17	16	fveq2i 6194	. . . . 5 |- ( N ` ( x ( -g ` G ) ( 0g ` G ) ) ) = ( N ` ( ( -g ` G ) ` <. x , ( 0g ` G ) >. ) )
18	14, 15, 17	3eqtr4g 2681	. . . 4 |- ( ( ( G e. Grp /\ N : X --> A ) /\ x e. X ) -> ( x ( N o. ( -g ` G ) ) ( 0g ` G ) ) = ( N ` ( x ( -g ` G ) ( 0g ` G ) ) ) )
19	3, 8, 4	grpsubid1 17500	. . . . . 6 |- ( ( G e. Grp /\ x e. X ) -> ( x ( -g ` G ) ( 0g ` G ) ) = x )
20	19	adantlr 751	. . . . 5 |- ( ( ( G e. Grp /\ N : X --> A ) /\ x e. X ) -> ( x ( -g ` G ) ( 0g ` G ) ) = x )
21	20	fveq2d 6195	. . . 4 |- ( ( ( G e. Grp /\ N : X --> A ) /\ x e. X ) -> ( N ` ( x ( -g ` G ) ( 0g ` G ) ) ) = ( N ` x ) )
22	18, 21	eqtr2d 2657	. . 3 |- ( ( ( G e. Grp /\ N : X --> A ) /\ x e. X ) -> ( N ` x ) = ( x ( N o. ( -g ` G ) ) ( 0g ` G ) ) )
23	22	mpteq2dva 4744	. 2 |- ( ( G e. Grp /\ N : X --> A ) -> ( x e. X |-> ( N ` x ) ) = ( x e. X |-> ( x ( N o. ( -g ` G ) ) ( 0g ` G ) ) ) )
24		fvex 6201	. . . . . . . 8 |- ( Base ` G ) e. _V
25	3, 24	eqeltri 2697	. . . . . . 7 |- X e. _V
26		tngnm.a	. . . . . . 7 |- A e. _V
27		fex2 7121	. . . . . . 7 |- ( ( N : X --> A /\ X e. _V /\ A e. _V ) -> N e. _V )
28	25, 26, 27	mp3an23 1416	. . . . . 6 |- ( N : X --> A -> N e. _V )
29	28	adantl 482	. . . . 5 |- ( ( G e. Grp /\ N : X --> A ) -> N e. _V )
30		tngnm.t	. . . . . 6 |- T = ( G toNrmGrp N )
31	30, 3	tngbas 22445	. . . . 5 |- ( N e. _V -> X = ( Base ` T ) )
32	29, 31	syl 17	. . . 4 |- ( ( G e. Grp /\ N : X --> A ) -> X = ( Base ` T ) )
33	30, 4	tngds 22452	. . . . . 6 |- ( N e. _V -> ( N o. ( -g ` G ) ) = ( dist ` T ) )
34	29, 33	syl 17	. . . . 5 |- ( ( G e. Grp /\ N : X --> A ) -> ( N o. ( -g ` G ) ) = ( dist ` T ) )
35		eqidd 2623	. . . . 5 |- ( ( G e. Grp /\ N : X --> A ) -> x = x )
36	30, 8	tng0 22447	. . . . . 6 |- ( N e. _V -> ( 0g ` G ) = ( 0g ` T ) )
37	29, 36	syl 17	. . . . 5 |- ( ( G e. Grp /\ N : X --> A ) -> ( 0g ` G ) = ( 0g ` T ) )
38	34, 35, 37	oveq123d 6671	. . . 4 |- ( ( G e. Grp /\ N : X --> A ) -> ( x ( N o. ( -g ` G ) ) ( 0g ` G ) ) = ( x ( dist ` T ) ( 0g ` T ) ) )
39	32, 38	mpteq12dv 4733	. . 3 |- ( ( G e. Grp /\ N : X --> A ) -> ( x e. X |-> ( x ( N o. ( -g ` G ) ) ( 0g ` G ) ) ) = ( x e. ( Base ` T ) |-> ( x ( dist ` T ) ( 0g ` T ) ) ) )
40		eqid 2622	. . . 4 |- ( norm ` T ) = ( norm ` T )
41		eqid 2622	. . . 4 |- ( Base ` T ) = ( Base ` T )
42		eqid 2622	. . . 4 |- ( 0g ` T ) = ( 0g ` T )
43		eqid 2622	. . . 4 |- ( dist ` T ) = ( dist ` T )
44	40, 41, 42, 43	nmfval 22393	. . 3 |- ( norm ` T ) = ( x e. ( Base ` T ) |-> ( x ( dist ` T ) ( 0g ` T ) ) )
45	39, 44	syl6eqr 2674	. 2 |- ( ( G e. Grp /\ N : X --> A ) -> ( x e. X |-> ( x ( N o. ( -g ` G ) ) ( 0g ` G ) ) ) = ( norm ` T ) )
46	2, 23, 45	3eqtrd 2660	1 |- ( ( G e. Grp /\ N : X --> A ) -> N = ( norm ` T ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183   |-> cmpt 4729   X. cxp 5112   o. ccom 5118   -->wf 5884   ` cfv 5888  (class class class)co 6650   Basecbs 15857   distcds 15950   0gc0g 16100   Grpcgrp 17422   -gcsg 17424   normcnm 22381   toNrmGrp ctng 22383

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-rep 4771  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-1st 7168  df-2nd 7169  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-plusg 15954  df-tset 15960  df-ds 15964  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-sbg 17427  df-nm 22387  df-tng 22389

This theorem is referenced by:  tngngp2  22456  tngngp  22458  tngngp3  22460  nrmtngnrm  22462  tngnrg  22478  tchnmfval  23027  tchcph  23036