Theorem dvn2bss 23693

Description: An N-times differentiable point is an M-times differentiable point, if M <_ N. (Contributed by Mario Carneiro, 30-Dec-2016.)

Assertion

Ref	Expression
dvn2bss	|- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) /\ M e. ( 0 ... N ) ) -> dom ( ( S Dn F ) ` N ) C_ dom ( ( S Dn F ) ` M ) )

Proof of Theorem dvn2bss

Step	Hyp	Ref	Expression
1		simp1 1061	. . . . 5 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) /\ M e. ( 0 ... N ) ) -> S e. { RR , CC } )
2		simp2 1062	. . . . 5 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) /\ M e. ( 0 ... N ) ) -> F e. ( CC ^pm S ) )
3		elfznn0 12433	. . . . . 6 |- ( M e. ( 0 ... N ) -> M e. NN0 )
4	3	3ad2ant3 1084	. . . . 5 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) /\ M e. ( 0 ... N ) ) -> M e. NN0 )
5		elfzuz3 12339	. . . . . . 7 |- ( M e. ( 0 ... N ) -> N e. ( ZZ>= ` M ) )
6	5	3ad2ant3 1084	. . . . . 6 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) /\ M e. ( 0 ... N ) ) -> N e. ( ZZ>= ` M ) )
7		uznn0sub 11719	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> ( N - M ) e. NN0 )
8	6, 7	syl 17	. . . . 5 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) /\ M e. ( 0 ... N ) ) -> ( N - M ) e. NN0 )
9		dvnadd 23692	. . . . 5 |- ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) /\ ( M e. NN0 /\ ( N - M ) e. NN0 ) ) -> ( ( S Dn ( ( S Dn F ) ` M ) ) ` ( N - M ) ) = ( ( S Dn F ) ` ( M + ( N - M ) ) ) )
10	1, 2, 4, 8, 9	syl22anc 1327	. . . 4 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) /\ M e. ( 0 ... N ) ) -> ( ( S Dn ( ( S Dn F ) ` M ) ) ` ( N - M ) ) = ( ( S Dn F ) ` ( M + ( N - M ) ) ) )
11	4	nn0cnd 11353	. . . . . 6 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) /\ M e. ( 0 ... N ) ) -> M e. CC )
12		elfzuz2 12346	. . . . . . . . 9 |- ( M e. ( 0 ... N ) -> N e. ( ZZ>= ` 0 ) )
13	12	3ad2ant3 1084	. . . . . . . 8 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) /\ M e. ( 0 ... N ) ) -> N e. ( ZZ>= ` 0 ) )
14		nn0uz 11722	. . . . . . . 8 |- NN0 = ( ZZ>= ` 0 )
15	13, 14	syl6eleqr 2712	. . . . . . 7 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) /\ M e. ( 0 ... N ) ) -> N e. NN0 )
16	15	nn0cnd 11353	. . . . . 6 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) /\ M e. ( 0 ... N ) ) -> N e. CC )
17	11, 16	pncan3d 10395	. . . . 5 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) /\ M e. ( 0 ... N ) ) -> ( M + ( N - M ) ) = N )
18	17	fveq2d 6195	. . . 4 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) /\ M e. ( 0 ... N ) ) -> ( ( S Dn F ) ` ( M + ( N - M ) ) ) = ( ( S Dn F ) ` N ) )
19	10, 18	eqtrd 2656	. . 3 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) /\ M e. ( 0 ... N ) ) -> ( ( S Dn ( ( S Dn F ) ` M ) ) ` ( N - M ) ) = ( ( S Dn F ) ` N ) )
20	19	dmeqd 5326	. 2 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) /\ M e. ( 0 ... N ) ) -> dom ( ( S Dn ( ( S Dn F ) ` M ) ) ` ( N - M ) ) = dom ( ( S Dn F ) ` N ) )
21		cnex 10017	. . . . 5 |- CC e. _V
22	21	a1i 11	. . . 4 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) /\ M e. ( 0 ... N ) ) -> CC e. _V )
23		dvnf 23690	. . . . 5 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) /\ M e. NN0 ) -> ( ( S Dn F ) ` M ) : dom ( ( S Dn F ) ` M ) --> CC )
24	3, 23	syl3an3 1361	. . . 4 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) /\ M e. ( 0 ... N ) ) -> ( ( S Dn F ) ` M ) : dom ( ( S Dn F ) ` M ) --> CC )
25		dvnbss 23691	. . . . . 6 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) /\ M e. NN0 ) -> dom ( ( S Dn F ) ` M ) C_ dom F )
26	3, 25	syl3an3 1361	. . . . 5 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) /\ M e. ( 0 ... N ) ) -> dom ( ( S Dn F ) ` M ) C_ dom F )
27		elpmi 7876	. . . . . . 7 |- ( F e. ( CC ^pm S ) -> ( F : dom F --> CC /\ dom F C_ S ) )
28	27	3ad2ant2 1083	. . . . . 6 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) /\ M e. ( 0 ... N ) ) -> ( F : dom F --> CC /\ dom F C_ S ) )
29	28	simprd 479	. . . . 5 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) /\ M e. ( 0 ... N ) ) -> dom F C_ S )
30	26, 29	sstrd 3613	. . . 4 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) /\ M e. ( 0 ... N ) ) -> dom ( ( S Dn F ) ` M ) C_ S )
31		elpm2r 7875	. . . 4 |- ( ( ( CC e. _V /\ S e. { RR , CC } ) /\ ( ( ( S Dn F ) ` M ) : dom ( ( S Dn F ) ` M ) --> CC /\ dom ( ( S Dn F ) ` M ) C_ S ) ) -> ( ( S Dn F ) ` M ) e. ( CC ^pm S ) )
32	22, 1, 24, 30, 31	syl22anc 1327	. . 3 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) /\ M e. ( 0 ... N ) ) -> ( ( S Dn F ) ` M ) e. ( CC ^pm S ) )
33		dvnbss 23691	. . 3 |- ( ( S e. { RR , CC } /\ ( ( S Dn F ) ` M ) e. ( CC ^pm S ) /\ ( N - M ) e. NN0 ) -> dom ( ( S Dn ( ( S Dn F ) ` M ) ) ` ( N - M ) ) C_ dom ( ( S Dn F ) ` M ) )
34	1, 32, 8, 33	syl3anc 1326	. 2 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) /\ M e. ( 0 ... N ) ) -> dom ( ( S Dn ( ( S Dn F ) ` M ) ) ` ( N - M ) ) C_ dom ( ( S Dn F ) ` M ) )
35	20, 34	eqsstr3d 3640	1 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) /\ M e. ( 0 ... N ) ) -> dom ( ( S Dn F ) ` N ) C_ dom ( ( S Dn F ) ` M ) )

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   + caddc 9939   - cmin 10266   NN0cn0 11292   ZZ>=cuz 11687   ...cfz 12326   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-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-cnp 21032  df-haus 21119  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-limc 23630  df-dv 23631  df-dvn 23632

This theorem is referenced by:  taylplem1  24117  taylply2  24122  taylply  24123  taylthlem1  24127  taylthlem2  24128