Theorem dvnadd 23692

Description: The N-th derivative of the M-th derivative of F is the same as the M + N-th derivative of F. (Contributed by Mario Carneiro, 11-Feb-2015.)

Assertion

Ref	Expression
dvnadd	|- ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) /\ ( M e. NN0 /\ N e. NN0 ) ) -> ( ( S Dn ( ( S Dn F ) ` M ) ) ` N ) = ( ( S Dn F ) ` ( M + N ) ) )

Proof of Theorem dvnadd

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

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . . 6 |- ( n = 0 -> ( ( S Dn ( ( S Dn F ) ` M ) ) ` n ) = ( ( S Dn ( ( S Dn F ) ` M ) ) ` 0 ) )
2		oveq2 6658	. . . . . . 7 |- ( n = 0 -> ( M + n ) = ( M + 0 ) )
3	2	fveq2d 6195	. . . . . 6 |- ( n = 0 -> ( ( S Dn F ) ` ( M + n ) ) = ( ( S Dn F ) ` ( M + 0 ) ) )
4	1, 3	eqeq12d 2637	. . . . 5 |- ( n = 0 -> ( ( ( S Dn ( ( S Dn F ) ` M ) ) ` n ) = ( ( S Dn F ) ` ( M + n ) ) <-> ( ( S Dn ( ( S Dn F ) ` M ) ) ` 0 ) = ( ( S Dn F ) ` ( M + 0 ) ) ) )
5	4	imbi2d 330	. . . 4 |- ( n = 0 -> ( ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) /\ M e. NN0 ) -> ( ( S Dn ( ( S Dn F ) ` M ) ) ` n ) = ( ( S Dn F ) ` ( M + n ) ) ) <-> ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) /\ M e. NN0 ) -> ( ( S Dn ( ( S Dn F ) ` M ) ) ` 0 ) = ( ( S Dn F ) ` ( M + 0 ) ) ) ) )
6		fveq2 6191	. . . . . 6 |- ( n = k -> ( ( S Dn ( ( S Dn F ) ` M ) ) ` n ) = ( ( S Dn ( ( S Dn F ) ` M ) ) ` k ) )
7		oveq2 6658	. . . . . . 7 |- ( n = k -> ( M + n ) = ( M + k ) )
8	7	fveq2d 6195	. . . . . 6 |- ( n = k -> ( ( S Dn F ) ` ( M + n ) ) = ( ( S Dn F ) ` ( M + k ) ) )
9	6, 8	eqeq12d 2637	. . . . 5 |- ( n = k -> ( ( ( S Dn ( ( S Dn F ) ` M ) ) ` n ) = ( ( S Dn F ) ` ( M + n ) ) <-> ( ( S Dn ( ( S Dn F ) ` M ) ) ` k ) = ( ( S Dn F ) ` ( M + k ) ) ) )
10	9	imbi2d 330	. . . 4 |- ( n = k -> ( ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) /\ M e. NN0 ) -> ( ( S Dn ( ( S Dn F ) ` M ) ) ` n ) = ( ( S Dn F ) ` ( M + n ) ) ) <-> ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) /\ M e. NN0 ) -> ( ( S Dn ( ( S Dn F ) ` M ) ) ` k ) = ( ( S Dn F ) ` ( M + k ) ) ) ) )
11		fveq2 6191	. . . . . 6 |- ( n = ( k + 1 ) -> ( ( S Dn ( ( S Dn F ) ` M ) ) ` n ) = ( ( S Dn ( ( S Dn F ) ` M ) ) ` ( k + 1 ) ) )
12		oveq2 6658	. . . . . . 7 |- ( n = ( k + 1 ) -> ( M + n ) = ( M + ( k + 1 ) ) )
13	12	fveq2d 6195	. . . . . 6 |- ( n = ( k + 1 ) -> ( ( S Dn F ) ` ( M + n ) ) = ( ( S Dn F ) ` ( M + ( k + 1 ) ) ) )
14	11, 13	eqeq12d 2637	. . . . 5 |- ( n = ( k + 1 ) -> ( ( ( S Dn ( ( S Dn F ) ` M ) ) ` n ) = ( ( S Dn F ) ` ( M + n ) ) <-> ( ( S Dn ( ( S Dn F ) ` M ) ) ` ( k + 1 ) ) = ( ( S Dn F ) ` ( M + ( k + 1 ) ) ) ) )
15	14	imbi2d 330	. . . 4 |- ( n = ( k + 1 ) -> ( ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) /\ M e. NN0 ) -> ( ( S Dn ( ( S Dn F ) ` M ) ) ` n ) = ( ( S Dn F ) ` ( M + n ) ) ) <-> ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) /\ M e. NN0 ) -> ( ( S Dn ( ( S Dn F ) ` M ) ) ` ( k + 1 ) ) = ( ( S Dn F ) ` ( M + ( k + 1 ) ) ) ) ) )
16		fveq2 6191	. . . . . 6 |- ( n = N -> ( ( S Dn ( ( S Dn F ) ` M ) ) ` n ) = ( ( S Dn ( ( S Dn F ) ` M ) ) ` N ) )
17		oveq2 6658	. . . . . . 7 |- ( n = N -> ( M + n ) = ( M + N ) )
18	17	fveq2d 6195	. . . . . 6 |- ( n = N -> ( ( S Dn F ) ` ( M + n ) ) = ( ( S Dn F ) ` ( M + N ) ) )
19	16, 18	eqeq12d 2637	. . . . 5 |- ( n = N -> ( ( ( S Dn ( ( S Dn F ) ` M ) ) ` n ) = ( ( S Dn F ) ` ( M + n ) ) <-> ( ( S Dn ( ( S Dn F ) ` M ) ) ` N ) = ( ( S Dn F ) ` ( M + N ) ) ) )
20	19	imbi2d 330	. . . 4 |- ( n = N -> ( ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) /\ M e. NN0 ) -> ( ( S Dn ( ( S Dn F ) ` M ) ) ` n ) = ( ( S Dn F ) ` ( M + n ) ) ) <-> ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) /\ M e. NN0 ) -> ( ( S Dn ( ( S Dn F ) ` M ) ) ` N ) = ( ( S Dn F ) ` ( M + N ) ) ) ) )
21		recnprss 23668	. . . . . . 7 |- ( S e. { RR , CC } -> S C_ CC )
22	21	ad2antrr 762	. . . . . 6 |- ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) /\ M e. NN0 ) -> S C_ CC )
23		ssid 3624	. . . . . . . . . 10 |- CC C_ CC
24	23	a1i 11	. . . . . . . . 9 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) -> CC C_ CC )
25		cnex 10017	. . . . . . . . . . 11 |- CC e. _V
26		elpm2g 7874	. . . . . . . . . . 11 |- ( ( CC e. _V /\ S e. { RR , CC } ) -> ( F e. ( CC ^pm S ) <-> ( F : dom F --> CC /\ dom F C_ S ) ) )
27	25, 26	mpan 706	. . . . . . . . . 10 |- ( S e. { RR , CC } -> ( F e. ( CC ^pm S ) <-> ( F : dom F --> CC /\ dom F C_ S ) ) )
28	27	simplbda 654	. . . . . . . . 9 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) -> dom F C_ S )
29	25	a1i 11	. . . . . . . . 9 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) -> CC e. _V )
30		simpl 473	. . . . . . . . 9 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) -> S e. { RR , CC } )
31		pmss12g 7884	. . . . . . . . 9 |- ( ( ( CC C_ CC /\ dom F C_ S ) /\ ( CC e. _V /\ S e. { RR , CC } ) ) -> ( CC ^pm dom F ) C_ ( CC ^pm S ) )
32	24, 28, 29, 30, 31	syl22anc 1327	. . . . . . . 8 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) -> ( CC ^pm dom F ) C_ ( CC ^pm S ) )
33	32	adantr 481	. . . . . . 7 |- ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) /\ M e. NN0 ) -> ( CC ^pm dom F ) C_ ( CC ^pm S ) )
34		dvnff 23686	. . . . . . . 8 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) -> ( S Dn F ) : NN0 --> ( CC ^pm dom F ) )
35	34	ffvelrnda 6359	. . . . . . 7 |- ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) /\ M e. NN0 ) -> ( ( S Dn F ) ` M ) e. ( CC ^pm dom F ) )
36	33, 35	sseldd 3604	. . . . . 6 |- ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) /\ M e. NN0 ) -> ( ( S Dn F ) ` M ) e. ( CC ^pm S ) )
37		dvn0 23687	. . . . . 6 |- ( ( S C_ CC /\ ( ( S Dn F ) ` M ) e. ( CC ^pm S ) ) -> ( ( S Dn ( ( S Dn F ) ` M ) ) ` 0 ) = ( ( S Dn F ) ` M ) )
38	22, 36, 37	syl2anc 693	. . . . 5 |- ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) /\ M e. NN0 ) -> ( ( S Dn ( ( S Dn F ) ` M ) ) ` 0 ) = ( ( S Dn F ) ` M ) )
39		nn0cn 11302	. . . . . . . 8 |- ( M e. NN0 -> M e. CC )
40	39	adantl 482	. . . . . . 7 |- ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) /\ M e. NN0 ) -> M e. CC )
41	40	addid1d 10236	. . . . . 6 |- ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) /\ M e. NN0 ) -> ( M + 0 ) = M )
42	41	fveq2d 6195	. . . . 5 |- ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) /\ M e. NN0 ) -> ( ( S Dn F ) ` ( M + 0 ) ) = ( ( S Dn F ) ` M ) )
43	38, 42	eqtr4d 2659	. . . 4 |- ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) /\ M e. NN0 ) -> ( ( S Dn ( ( S Dn F ) ` M ) ) ` 0 ) = ( ( S Dn F ) ` ( M + 0 ) ) )
44		oveq2 6658	. . . . . . 7 |- ( ( ( S Dn ( ( S Dn F ) ` M ) ) ` k ) = ( ( S Dn F ) ` ( M + k ) ) -> ( S _D ( ( S Dn ( ( S Dn F ) ` M ) ) ` k ) ) = ( S _D ( ( S Dn F ) ` ( M + k ) ) ) )
45	22	adantr 481	. . . . . . . . 9 |- ( ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) /\ M e. NN0 ) /\ k e. NN0 ) -> S C_ CC )
46	36	adantr 481	. . . . . . . . 9 |- ( ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) /\ M e. NN0 ) /\ k e. NN0 ) -> ( ( S Dn F ) ` M ) e. ( CC ^pm S ) )
47		simpr 477	. . . . . . . . 9 |- ( ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) /\ M e. NN0 ) /\ k e. NN0 ) -> k e. NN0 )
48		dvnp1 23688	. . . . . . . . 9 |- ( ( S C_ CC /\ ( ( S Dn F ) ` M ) e. ( CC ^pm S ) /\ k e. NN0 ) -> ( ( S Dn ( ( S Dn F ) ` M ) ) ` ( k + 1 ) ) = ( S _D ( ( S Dn ( ( S Dn F ) ` M ) ) ` k ) ) )
49	45, 46, 47, 48	syl3anc 1326	. . . . . . . 8 |- ( ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) /\ M e. NN0 ) /\ k e. NN0 ) -> ( ( S Dn ( ( S Dn F ) ` M ) ) ` ( k + 1 ) ) = ( S _D ( ( S Dn ( ( S Dn F ) ` M ) ) ` k ) ) )
50	40	adantr 481	. . . . . . . . . . 11 |- ( ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) /\ M e. NN0 ) /\ k e. NN0 ) -> M e. CC )
51		nn0cn 11302	. . . . . . . . . . . 12 |- ( k e. NN0 -> k e. CC )
52	51	adantl 482	. . . . . . . . . . 11 |- ( ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) /\ M e. NN0 ) /\ k e. NN0 ) -> k e. CC )
53		1cnd 10056	. . . . . . . . . . 11 |- ( ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) /\ M e. NN0 ) /\ k e. NN0 ) -> 1 e. CC )
54	50, 52, 53	addassd 10062	. . . . . . . . . 10 |- ( ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) /\ M e. NN0 ) /\ k e. NN0 ) -> ( ( M + k ) + 1 ) = ( M + ( k + 1 ) ) )
55	54	fveq2d 6195	. . . . . . . . 9 |- ( ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) /\ M e. NN0 ) /\ k e. NN0 ) -> ( ( S Dn F ) ` ( ( M + k ) + 1 ) ) = ( ( S Dn F ) ` ( M + ( k + 1 ) ) ) )
56		simpllr 799	. . . . . . . . . 10 |- ( ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) /\ M e. NN0 ) /\ k e. NN0 ) -> F e. ( CC ^pm S ) )
57		nn0addcl 11328	. . . . . . . . . . 11 |- ( ( M e. NN0 /\ k e. NN0 ) -> ( M + k ) e. NN0 )
58	57	adantll 750	. . . . . . . . . 10 |- ( ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) /\ M e. NN0 ) /\ k e. NN0 ) -> ( M + k ) e. NN0 )
59		dvnp1 23688	. . . . . . . . . 10 |- ( ( S C_ CC /\ F e. ( CC ^pm S ) /\ ( M + k ) e. NN0 ) -> ( ( S Dn F ) ` ( ( M + k ) + 1 ) ) = ( S _D ( ( S Dn F ) ` ( M + k ) ) ) )
60	45, 56, 58, 59	syl3anc 1326	. . . . . . . . 9 |- ( ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) /\ M e. NN0 ) /\ k e. NN0 ) -> ( ( S Dn F ) ` ( ( M + k ) + 1 ) ) = ( S _D ( ( S Dn F ) ` ( M + k ) ) ) )
61	55, 60	eqtr3d 2658	. . . . . . . 8 |- ( ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) /\ M e. NN0 ) /\ k e. NN0 ) -> ( ( S Dn F ) ` ( M + ( k + 1 ) ) ) = ( S _D ( ( S Dn F ) ` ( M + k ) ) ) )
62	49, 61	eqeq12d 2637	. . . . . . 7 |- ( ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) /\ M e. NN0 ) /\ k e. NN0 ) -> ( ( ( S Dn ( ( S Dn F ) ` M ) ) ` ( k + 1 ) ) = ( ( S Dn F ) ` ( M + ( k + 1 ) ) ) <-> ( S _D ( ( S Dn ( ( S Dn F ) ` M ) ) ` k ) ) = ( S _D ( ( S Dn F ) ` ( M + k ) ) ) ) )
63	44, 62	syl5ibr 236	. . . . . 6 |- ( ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) /\ M e. NN0 ) /\ k e. NN0 ) -> ( ( ( S Dn ( ( S Dn F ) ` M ) ) ` k ) = ( ( S Dn F ) ` ( M + k ) ) -> ( ( S Dn ( ( S Dn F ) ` M ) ) ` ( k + 1 ) ) = ( ( S Dn F ) ` ( M + ( k + 1 ) ) ) ) )
64	63	expcom 451	. . . . 5 |- ( k e. NN0 -> ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) /\ M e. NN0 ) -> ( ( ( S Dn ( ( S Dn F ) ` M ) ) ` k ) = ( ( S Dn F ) ` ( M + k ) ) -> ( ( S Dn ( ( S Dn F ) ` M ) ) ` ( k + 1 ) ) = ( ( S Dn F ) ` ( M + ( k + 1 ) ) ) ) ) )
65	64	a2d 29	. . . 4 |- ( k e. NN0 -> ( ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) /\ M e. NN0 ) -> ( ( S Dn ( ( S Dn F ) ` M ) ) ` k ) = ( ( S Dn F ) ` ( M + k ) ) ) -> ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) /\ M e. NN0 ) -> ( ( S Dn ( ( S Dn F ) ` M ) ) ` ( k + 1 ) ) = ( ( S Dn F ) ` ( M + ( k + 1 ) ) ) ) ) )
66	5, 10, 15, 20, 43, 65	nn0ind 11472	. . 3 |- ( N e. NN0 -> ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) /\ M e. NN0 ) -> ( ( S Dn ( ( S Dn F ) ` M ) ) ` N ) = ( ( S Dn F ) ` ( M + N ) ) ) )
67	66	com12 32	. 2 |- ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) /\ M e. NN0 ) -> ( N e. NN0 -> ( ( S Dn ( ( S Dn F ) ` M ) ) ` N ) = ( ( S Dn F ) ` ( M + N ) ) ) )
68	67	impr 649	1 |- ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) /\ ( M e. NN0 /\ N e. NN0 ) ) -> ( ( S Dn ( ( S Dn F ) ` M ) ) ` N ) = ( ( S Dn F ) ` ( M + N ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = 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-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:  dvn2bss  23693  dvtaylp  24124  dvntaylp  24125  dvntaylp0  24126