Theorem cpnfval 23695

Description: Condition for n-times continuous differentiability. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Assertion

Ref	Expression
cpnfval	|- ( S C_ CC -> ( C^n ` S ) = ( n e. NN0 |-> { f e. ( CC ^pm S ) | ( ( S Dn f ) ` n ) e. ( dom f -cn-> CC ) } ) )

Distinct variable group:   f, n, S

Proof of Theorem cpnfval

Dummy variable s is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnex 10017	. . 3 |- CC e. _V
2	1	elpw2 4828	. 2 |- ( S e. ~P CC <-> S C_ CC )
3		oveq2 6658	. . . . 5 |- ( s = S -> ( CC ^pm s ) = ( CC ^pm S ) )
4		oveq1 6657	. . . . . . 7 |- ( s = S -> ( s Dn f ) = ( S Dn f ) )
5	4	fveq1d 6193	. . . . . 6 |- ( s = S -> ( ( s Dn f ) ` n ) = ( ( S Dn f ) ` n ) )
6	5	eleq1d 2686	. . . . 5 |- ( s = S -> ( ( ( s Dn f ) ` n ) e. ( dom f -cn-> CC ) <-> ( ( S Dn f ) ` n ) e. ( dom f -cn-> CC ) ) )
7	3, 6	rabeqbidv 3195	. . . 4 |- ( s = S -> { f e. ( CC ^pm s ) | ( ( s Dn f ) ` n ) e. ( dom f -cn-> CC ) } = { f e. ( CC ^pm S ) | ( ( S Dn f ) ` n ) e. ( dom f -cn-> CC ) } )
8	7	mpteq2dv 4745	. . 3 |- ( s = S -> ( n e. NN0 |-> { f e. ( CC ^pm s ) | ( ( s Dn f ) ` n ) e. ( dom f -cn-> CC ) } ) = ( n e. NN0 |-> { f e. ( CC ^pm S ) | ( ( S Dn f ) ` n ) e. ( dom f -cn-> CC ) } ) )
9		df-cpn 23633	. . 3 |- C^n = ( s e. ~P CC |-> ( n e. NN0 |-> { f e. ( CC ^pm s ) | ( ( s Dn f ) ` n ) e. ( dom f -cn-> CC ) } ) )
10		nn0ex 11298	. . . 4 |- NN0 e. _V
11	10	mptex 6486	. . 3 |- ( n e. NN0 |-> { f e. ( CC ^pm S ) | ( ( S Dn f ) ` n ) e. ( dom f -cn-> CC ) } ) e. _V
12	8, 9, 11	fvmpt 6282	. 2 |- ( S e. ~P CC -> ( C^n ` S ) = ( n e. NN0 |-> { f e. ( CC ^pm S ) | ( ( S Dn f ) ` n ) e. ( dom f -cn-> CC ) } ) )
13	2, 12	sylbir 225	1 |- ( S C_ CC -> ( C^n ` S ) = ( n e. NN0 |-> { f e. ( CC ^pm S ) | ( ( S Dn f ) ` n ) e. ( dom f -cn-> CC ) } ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {crab 2916   C_ wss 3574   ~Pcpw 4158   |-> cmpt 4729   dom cdm 5114   ` cfv 5888  (class class class)co 6650   ^pm cpm 7858   CCcc 9934   NN0cn0 11292   -cn->ccncf 22679   Dncdvn 23628   C^nccpn 23629

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-i2m1 10004  ax-1ne0 10005  ax-rrecex 10008  ax-cnre 10009

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-ral 2917  df-rex 2918  df-reu 2919  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-ov 6653  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-nn 11021  df-n0 11293  df-cpn 23633

This theorem is referenced by:  fncpn  23696  elcpn  23697