Theorem dvnfre 23715

Description: The N-th derivative of a real function is real. (Contributed by Mario Carneiro, 1-Jan-2017.)

Assertion

Ref	Expression
dvnfre	|- ( ( F : A --> RR /\ A C_ RR /\ N e. NN0 ) -> ( ( RR Dn F ) ` N ) : dom ( ( RR Dn F ) ` N ) --> RR )

Proof of Theorem dvnfre

Dummy variables x n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . . 6 |- ( x = 0 -> ( ( RR Dn F ) ` x ) = ( ( RR Dn F ) ` 0 ) )
2	1	dmeqd 5326	. . . . . 6 |- ( x = 0 -> dom ( ( RR Dn F ) ` x ) = dom ( ( RR Dn F ) ` 0 ) )
3	1, 2	feq12d 6033	. . . . 5 |- ( x = 0 -> ( ( ( RR Dn F ) ` x ) : dom ( ( RR Dn F ) ` x ) --> RR <-> ( ( RR Dn F ) ` 0 ) : dom ( ( RR Dn F ) ` 0 ) --> RR ) )
4	3	imbi2d 330	. . . 4 |- ( x = 0 -> ( ( ( F : A --> RR /\ A C_ RR ) -> ( ( RR Dn F ) ` x ) : dom ( ( RR Dn F ) ` x ) --> RR ) <-> ( ( F : A --> RR /\ A C_ RR ) -> ( ( RR Dn F ) ` 0 ) : dom ( ( RR Dn F ) ` 0 ) --> RR ) ) )
5		fveq2 6191	. . . . . 6 |- ( x = n -> ( ( RR Dn F ) ` x ) = ( ( RR Dn F ) ` n ) )
6	5	dmeqd 5326	. . . . . 6 |- ( x = n -> dom ( ( RR Dn F ) ` x ) = dom ( ( RR Dn F ) ` n ) )
7	5, 6	feq12d 6033	. . . . 5 |- ( x = n -> ( ( ( RR Dn F ) ` x ) : dom ( ( RR Dn F ) ` x ) --> RR <-> ( ( RR Dn F ) ` n ) : dom ( ( RR Dn F ) ` n ) --> RR ) )
8	7	imbi2d 330	. . . 4 |- ( x = n -> ( ( ( F : A --> RR /\ A C_ RR ) -> ( ( RR Dn F ) ` x ) : dom ( ( RR Dn F ) ` x ) --> RR ) <-> ( ( F : A --> RR /\ A C_ RR ) -> ( ( RR Dn F ) ` n ) : dom ( ( RR Dn F ) ` n ) --> RR ) ) )
9		fveq2 6191	. . . . . 6 |- ( x = ( n + 1 ) -> ( ( RR Dn F ) ` x ) = ( ( RR Dn F ) ` ( n + 1 ) ) )
10	9	dmeqd 5326	. . . . . 6 |- ( x = ( n + 1 ) -> dom ( ( RR Dn F ) ` x ) = dom ( ( RR Dn F ) ` ( n + 1 ) ) )
11	9, 10	feq12d 6033	. . . . 5 |- ( x = ( n + 1 ) -> ( ( ( RR Dn F ) ` x ) : dom ( ( RR Dn F ) ` x ) --> RR <-> ( ( RR Dn F ) ` ( n + 1 ) ) : dom ( ( RR Dn F ) ` ( n + 1 ) ) --> RR ) )
12	11	imbi2d 330	. . . 4 |- ( x = ( n + 1 ) -> ( ( ( F : A --> RR /\ A C_ RR ) -> ( ( RR Dn F ) ` x ) : dom ( ( RR Dn F ) ` x ) --> RR ) <-> ( ( F : A --> RR /\ A C_ RR ) -> ( ( RR Dn F ) ` ( n + 1 ) ) : dom ( ( RR Dn F ) ` ( n + 1 ) ) --> RR ) ) )
13		fveq2 6191	. . . . . 6 |- ( x = N -> ( ( RR Dn F ) ` x ) = ( ( RR Dn F ) ` N ) )
14	13	dmeqd 5326	. . . . . 6 |- ( x = N -> dom ( ( RR Dn F ) ` x ) = dom ( ( RR Dn F ) ` N ) )
15	13, 14	feq12d 6033	. . . . 5 |- ( x = N -> ( ( ( RR Dn F ) ` x ) : dom ( ( RR Dn F ) ` x ) --> RR <-> ( ( RR Dn F ) ` N ) : dom ( ( RR Dn F ) ` N ) --> RR ) )
16	15	imbi2d 330	. . . 4 |- ( x = N -> ( ( ( F : A --> RR /\ A C_ RR ) -> ( ( RR Dn F ) ` x ) : dom ( ( RR Dn F ) ` x ) --> RR ) <-> ( ( F : A --> RR /\ A C_ RR ) -> ( ( RR Dn F ) ` N ) : dom ( ( RR Dn F ) ` N ) --> RR ) ) )
17		simpl 473	. . . . 5 |- ( ( F : A --> RR /\ A C_ RR ) -> F : A --> RR )
18		ax-resscn 9993	. . . . . . 7 |- RR C_ CC
19		fss 6056	. . . . . . . . 9 |- ( ( F : A --> RR /\ RR C_ CC ) -> F : A --> CC )
20	18, 19	mpan2 707	. . . . . . . 8 |- ( F : A --> RR -> F : A --> CC )
21		cnex 10017	. . . . . . . . 9 |- CC e. _V
22		reex 10027	. . . . . . . . 9 |- RR e. _V
23		elpm2r 7875	. . . . . . . . 9 |- ( ( ( CC e. _V /\ RR e. _V ) /\ ( F : A --> CC /\ A C_ RR ) ) -> F e. ( CC ^pm RR ) )
24	21, 22, 23	mpanl12 718	. . . . . . . 8 |- ( ( F : A --> CC /\ A C_ RR ) -> F e. ( CC ^pm RR ) )
25	20, 24	sylan 488	. . . . . . 7 |- ( ( F : A --> RR /\ A C_ RR ) -> F e. ( CC ^pm RR ) )
26		dvn0 23687	. . . . . . 7 |- ( ( RR C_ CC /\ F e. ( CC ^pm RR ) ) -> ( ( RR Dn F ) ` 0 ) = F )
27	18, 25, 26	sylancr 695	. . . . . 6 |- ( ( F : A --> RR /\ A C_ RR ) -> ( ( RR Dn F ) ` 0 ) = F )
28	27	dmeqd 5326	. . . . . . 7 |- ( ( F : A --> RR /\ A C_ RR ) -> dom ( ( RR Dn F ) ` 0 ) = dom F )
29		fdm 6051	. . . . . . . 8 |- ( F : A --> RR -> dom F = A )
30	29	adantr 481	. . . . . . 7 |- ( ( F : A --> RR /\ A C_ RR ) -> dom F = A )
31	28, 30	eqtrd 2656	. . . . . 6 |- ( ( F : A --> RR /\ A C_ RR ) -> dom ( ( RR Dn F ) ` 0 ) = A )
32	27, 31	feq12d 6033	. . . . 5 |- ( ( F : A --> RR /\ A C_ RR ) -> ( ( ( RR Dn F ) ` 0 ) : dom ( ( RR Dn F ) ` 0 ) --> RR <-> F : A --> RR ) )
33	17, 32	mpbird 247	. . . 4 |- ( ( F : A --> RR /\ A C_ RR ) -> ( ( RR Dn F ) ` 0 ) : dom ( ( RR Dn F ) ` 0 ) --> RR )
34		simprr 796	. . . . . . . . 9 |- ( ( ( F : A --> RR /\ A C_ RR ) /\ ( n e. NN0 /\ ( ( RR Dn F ) ` n ) : dom ( ( RR Dn F ) ` n ) --> RR ) ) -> ( ( RR Dn F ) ` n ) : dom ( ( RR Dn F ) ` n ) --> RR )
35	22	prid1 4297	. . . . . . . . . . . . 13 |- RR e. { RR , CC }
36	35	a1i 11	. . . . . . . . . . . 12 |- ( ( ( F : A --> RR /\ A C_ RR ) /\ ( n e. NN0 /\ ( ( RR Dn F ) ` n ) : dom ( ( RR Dn F ) ` n ) --> RR ) ) -> RR e. { RR , CC } )
37	25	adantr 481	. . . . . . . . . . . 12 |- ( ( ( F : A --> RR /\ A C_ RR ) /\ ( n e. NN0 /\ ( ( RR Dn F ) ` n ) : dom ( ( RR Dn F ) ` n ) --> RR ) ) -> F e. ( CC ^pm RR ) )
38		simprl 794	. . . . . . . . . . . 12 |- ( ( ( F : A --> RR /\ A C_ RR ) /\ ( n e. NN0 /\ ( ( RR Dn F ) ` n ) : dom ( ( RR Dn F ) ` n ) --> RR ) ) -> n e. NN0 )
39		dvnbss 23691	. . . . . . . . . . . 12 |- ( ( RR e. { RR , CC } /\ F e. ( CC ^pm RR ) /\ n e. NN0 ) -> dom ( ( RR Dn F ) ` n ) C_ dom F )
40	36, 37, 38, 39	syl3anc 1326	. . . . . . . . . . 11 |- ( ( ( F : A --> RR /\ A C_ RR ) /\ ( n e. NN0 /\ ( ( RR Dn F ) ` n ) : dom ( ( RR Dn F ) ` n ) --> RR ) ) -> dom ( ( RR Dn F ) ` n ) C_ dom F )
41	30	adantr 481	. . . . . . . . . . 11 |- ( ( ( F : A --> RR /\ A C_ RR ) /\ ( n e. NN0 /\ ( ( RR Dn F ) ` n ) : dom ( ( RR Dn F ) ` n ) --> RR ) ) -> dom F = A )
42	40, 41	sseqtrd 3641	. . . . . . . . . 10 |- ( ( ( F : A --> RR /\ A C_ RR ) /\ ( n e. NN0 /\ ( ( RR Dn F ) ` n ) : dom ( ( RR Dn F ) ` n ) --> RR ) ) -> dom ( ( RR Dn F ) ` n ) C_ A )
43		simplr 792	. . . . . . . . . 10 |- ( ( ( F : A --> RR /\ A C_ RR ) /\ ( n e. NN0 /\ ( ( RR Dn F ) ` n ) : dom ( ( RR Dn F ) ` n ) --> RR ) ) -> A C_ RR )
44	42, 43	sstrd 3613	. . . . . . . . 9 |- ( ( ( F : A --> RR /\ A C_ RR ) /\ ( n e. NN0 /\ ( ( RR Dn F ) ` n ) : dom ( ( RR Dn F ) ` n ) --> RR ) ) -> dom ( ( RR Dn F ) ` n ) C_ RR )
45		dvfre 23714	. . . . . . . . 9 |- ( ( ( ( RR Dn F ) ` n ) : dom ( ( RR Dn F ) ` n ) --> RR /\ dom ( ( RR Dn F ) ` n ) C_ RR ) -> ( RR _D ( ( RR Dn F ) ` n ) ) : dom ( RR _D ( ( RR Dn F ) ` n ) ) --> RR )
46	34, 44, 45	syl2anc 693	. . . . . . . 8 |- ( ( ( F : A --> RR /\ A C_ RR ) /\ ( n e. NN0 /\ ( ( RR Dn F ) ` n ) : dom ( ( RR Dn F ) ` n ) --> RR ) ) -> ( RR _D ( ( RR Dn F ) ` n ) ) : dom ( RR _D ( ( RR Dn F ) ` n ) ) --> RR )
47	18	a1i 11	. . . . . . . . . 10 |- ( ( ( F : A --> RR /\ A C_ RR ) /\ ( n e. NN0 /\ ( ( RR Dn F ) ` n ) : dom ( ( RR Dn F ) ` n ) --> RR ) ) -> RR C_ CC )
48		dvnp1 23688	. . . . . . . . . 10 |- ( ( RR C_ CC /\ F e. ( CC ^pm RR ) /\ n e. NN0 ) -> ( ( RR Dn F ) ` ( n + 1 ) ) = ( RR _D ( ( RR Dn F ) ` n ) ) )
49	47, 37, 38, 48	syl3anc 1326	. . . . . . . . 9 |- ( ( ( F : A --> RR /\ A C_ RR ) /\ ( n e. NN0 /\ ( ( RR Dn F ) ` n ) : dom ( ( RR Dn F ) ` n ) --> RR ) ) -> ( ( RR Dn F ) ` ( n + 1 ) ) = ( RR _D ( ( RR Dn F ) ` n ) ) )
50	49	dmeqd 5326	. . . . . . . . 9 |- ( ( ( F : A --> RR /\ A C_ RR ) /\ ( n e. NN0 /\ ( ( RR Dn F ) ` n ) : dom ( ( RR Dn F ) ` n ) --> RR ) ) -> dom ( ( RR Dn F ) ` ( n + 1 ) ) = dom ( RR _D ( ( RR Dn F ) ` n ) ) )
51	49, 50	feq12d 6033	. . . . . . . 8 |- ( ( ( F : A --> RR /\ A C_ RR ) /\ ( n e. NN0 /\ ( ( RR Dn F ) ` n ) : dom ( ( RR Dn F ) ` n ) --> RR ) ) -> ( ( ( RR Dn F ) ` ( n + 1 ) ) : dom ( ( RR Dn F ) ` ( n + 1 ) ) --> RR <-> ( RR _D ( ( RR Dn F ) ` n ) ) : dom ( RR _D ( ( RR Dn F ) ` n ) ) --> RR ) )
52	46, 51	mpbird 247	. . . . . . 7 |- ( ( ( F : A --> RR /\ A C_ RR ) /\ ( n e. NN0 /\ ( ( RR Dn F ) ` n ) : dom ( ( RR Dn F ) ` n ) --> RR ) ) -> ( ( RR Dn F ) ` ( n + 1 ) ) : dom ( ( RR Dn F ) ` ( n + 1 ) ) --> RR )
53	52	expr 643	. . . . . 6 |- ( ( ( F : A --> RR /\ A C_ RR ) /\ n e. NN0 ) -> ( ( ( RR Dn F ) ` n ) : dom ( ( RR Dn F ) ` n ) --> RR -> ( ( RR Dn F ) ` ( n + 1 ) ) : dom ( ( RR Dn F ) ` ( n + 1 ) ) --> RR ) )
54	53	expcom 451	. . . . 5 |- ( n e. NN0 -> ( ( F : A --> RR /\ A C_ RR ) -> ( ( ( RR Dn F ) ` n ) : dom ( ( RR Dn F ) ` n ) --> RR -> ( ( RR Dn F ) ` ( n + 1 ) ) : dom ( ( RR Dn F ) ` ( n + 1 ) ) --> RR ) ) )
55	54	a2d 29	. . . 4 |- ( n e. NN0 -> ( ( ( F : A --> RR /\ A C_ RR ) -> ( ( RR Dn F ) ` n ) : dom ( ( RR Dn F ) ` n ) --> RR ) -> ( ( F : A --> RR /\ A C_ RR ) -> ( ( RR Dn F ) ` ( n + 1 ) ) : dom ( ( RR Dn F ) ` ( n + 1 ) ) --> RR ) ) )
56	4, 8, 12, 16, 33, 55	nn0ind 11472	. . 3 |- ( N e. NN0 -> ( ( F : A --> RR /\ A C_ RR ) -> ( ( RR Dn F ) ` N ) : dom ( ( RR Dn F ) ` N ) --> RR ) )
57	56	com12 32	. 2 |- ( ( F : A --> RR /\ A C_ RR ) -> ( N e. NN0 -> ( ( RR Dn F ) ` N ) : dom ( ( RR Dn F ) ` N ) --> RR ) )
58	57	3impia 1261	1 |- ( ( F : A --> RR /\ A C_ RR /\ N e. NN0 ) -> ( ( RR Dn F ) ` N ) : dom ( ( RR Dn F ) ` N ) --> RR )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   {cpr 4179   dom cdm 5114   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^pm cpm 7858   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   NN0cn0 11292   _D cdv 23627   Dncdvn 23628

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-inf2 8538  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  ax-pre-sup 10014

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-int 4476  df-iun 4522  df-iin 4523  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-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  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-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-icc 12182  df-fz 12327  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-mulr 15955  df-starv 15956  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-rest 16083  df-topn 16084  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-cncf 22681  df-limc 23630  df-dv 23631  df-dvn 23632

This theorem is referenced by:  taylthlem2  24128