Theorem nmfval 22393

Description: The value of the norm function. (Contributed by Mario Carneiro, 2-Oct-2015.)

Hypotheses

Ref	Expression
nmfval.n	|- N = ( norm ` W )
nmfval.x	|- X = ( Base ` W )
nmfval.z	|- .0. = ( 0g ` W )
nmfval.d	|- D = ( dist ` W )

Assertion

Ref	Expression
nmfval	|- N = ( x e. X |-> ( x D .0. ) )

Distinct variable groups:   x, D   x, W   x, X   x, .0.

Allowed substitution hint:   N( x)

Proof of Theorem nmfval

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nmfval.n	. 2 |- N = ( norm ` W )
2		fveq2 6191	. . . . . 6 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
3		nmfval.x	. . . . . 6 |- X = ( Base ` W )
4	2, 3	syl6eqr 2674	. . . . 5 |- ( w = W -> ( Base ` w ) = X )
5		fveq2 6191	. . . . . . 7 |- ( w = W -> ( dist ` w ) = ( dist ` W ) )
6		nmfval.d	. . . . . . 7 |- D = ( dist ` W )
7	5, 6	syl6eqr 2674	. . . . . 6 |- ( w = W -> ( dist ` w ) = D )
8		eqidd 2623	. . . . . 6 |- ( w = W -> x = x )
9		fveq2 6191	. . . . . . 7 |- ( w = W -> ( 0g ` w ) = ( 0g ` W ) )
10		nmfval.z	. . . . . . 7 |- .0. = ( 0g ` W )
11	9, 10	syl6eqr 2674	. . . . . 6 |- ( w = W -> ( 0g ` w ) = .0. )
12	7, 8, 11	oveq123d 6671	. . . . 5 |- ( w = W -> ( x ( dist ` w ) ( 0g ` w ) ) = ( x D .0. ) )
13	4, 12	mpteq12dv 4733	. . . 4 |- ( w = W -> ( x e. ( Base ` w ) |-> ( x ( dist ` w ) ( 0g ` w ) ) ) = ( x e. X |-> ( x D .0. ) ) )
14		df-nm 22387	. . . 4 |- norm = ( w e. _V |-> ( x e. ( Base ` w ) |-> ( x ( dist ` w ) ( 0g ` w ) ) ) )
15		eqid 2622	. . . . . 6 |- ( x e. X |-> ( x D .0. ) ) = ( x e. X |-> ( x D .0. ) )
16		df-ov 6653	. . . . . . . 8 |- ( x D .0. ) = ( D ` <. x , .0. >. )
17		fvrn0 6216	. . . . . . . 8 |- ( D ` <. x , .0. >. ) e. ( ran D u. { (/) } )
18	16, 17	eqeltri 2697	. . . . . . 7 |- ( x D .0. ) e. ( ran D u. { (/) } )
19	18	a1i 11	. . . . . 6 |- ( x e. X -> ( x D .0. ) e. ( ran D u. { (/) } ) )
20	15, 19	fmpti 6383	. . . . 5 |- ( x e. X |-> ( x D .0. ) ) : X --> ( ran D u. { (/) } )
21		fvex 6201	. . . . . 6 |- ( Base ` W ) e. _V
22	3, 21	eqeltri 2697	. . . . 5 |- X e. _V
23		fvex 6201	. . . . . . . 8 |- ( dist ` W ) e. _V
24	6, 23	eqeltri 2697	. . . . . . 7 |- D e. _V
25	24	rnex 7100	. . . . . 6 |- ran D e. _V
26		p0ex 4853	. . . . . 6 |- { (/) } e. _V
27	25, 26	unex 6956	. . . . 5 |- ( ran D u. { (/) } ) e. _V
28		fex2 7121	. . . . 5 |- ( ( ( x e. X |-> ( x D .0. ) ) : X --> ( ran D u. { (/) } ) /\ X e. _V /\ ( ran D u. { (/) } ) e. _V ) -> ( x e. X |-> ( x D .0. ) ) e. _V )
29	20, 22, 27, 28	mp3an 1424	. . . 4 |- ( x e. X |-> ( x D .0. ) ) e. _V
30	13, 14, 29	fvmpt 6282	. . 3 |- ( W e. _V -> ( norm ` W ) = ( x e. X |-> ( x D .0. ) ) )
31		fvprc 6185	. . . . 5 |- ( -. W e. _V -> ( norm ` W ) = (/) )
32		mpt0 6021	. . . . 5 |- ( x e. (/) |-> ( x D .0. ) ) = (/)
33	31, 32	syl6eqr 2674	. . . 4 |- ( -. W e. _V -> ( norm ` W ) = ( x e. (/) |-> ( x D .0. ) ) )
34		fvprc 6185	. . . . . 6 |- ( -. W e. _V -> ( Base ` W ) = (/) )
35	3, 34	syl5eq 2668	. . . . 5 |- ( -. W e. _V -> X = (/) )
36	35	mpteq1d 4738	. . . 4 |- ( -. W e. _V -> ( x e. X |-> ( x D .0. ) ) = ( x e. (/) |-> ( x D .0. ) ) )
37	33, 36	eqtr4d 2659	. . 3 |- ( -. W e. _V -> ( norm ` W ) = ( x e. X |-> ( x D .0. ) ) )
38	30, 37	pm2.61i 176	. 2 |- ( norm ` W ) = ( x e. X |-> ( x D .0. ) )
39	1, 38	eqtri 2644	1 |- N = ( x e. X |-> ( x D .0. ) )

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   (/)c0 3915   {csn 4177   <.cop 4183   |-> cmpt 4729   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   Basecbs 15857   distcds 15950   0gc0g 16100   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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-nm 22387

This theorem is referenced by:  nmval  22394  nmfval2  22395  nmpropd  22398  subgnm  22437  tngnm  22455  cnfldnm  22582  nmcn  22647  ressnm  29651