Theorem dvnfval 23685

Description: Value of the iterated derivative. (Contributed by Mario Carneiro, 11-Feb-2015.)

Hypothesis

Ref	Expression
dvnfval.1	|- G = ( x e. _V |-> ( S _D x ) )

Assertion

Ref	Expression
dvnfval	|- ( ( S C_ CC /\ F e. ( CC ^pm S ) ) -> ( S Dn F ) = seq 0 ( ( G o. 1st ) , ( NN0 X. { F } ) ) )

Distinct variable groups:   x, F   x, S

Allowed substitution hint:   G( x)

Proof of Theorem dvnfval

Dummy variables f s are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-dvn 23632	. . 3 |- Dn = ( s e. ~P CC , f e. ( CC ^pm s ) |-> seq 0 ( ( ( x e. _V |-> ( s _D x ) ) o. 1st ) , ( NN0 X. { f } ) ) )
2	1	a1i 11	. 2 |- ( ( S C_ CC /\ F e. ( CC ^pm S ) ) -> Dn = ( s e. ~P CC , f e. ( CC ^pm s ) |-> seq 0 ( ( ( x e. _V |-> ( s _D x ) ) o. 1st ) , ( NN0 X. { f } ) ) ) )
3		simprl 794	. . . . . . . 8 |- ( ( ( S C_ CC /\ F e. ( CC ^pm S ) ) /\ ( s = S /\ f = F ) ) -> s = S )
4	3	oveq1d 6665	. . . . . . 7 |- ( ( ( S C_ CC /\ F e. ( CC ^pm S ) ) /\ ( s = S /\ f = F ) ) -> ( s _D x ) = ( S _D x ) )
5	4	mpteq2dv 4745	. . . . . 6 |- ( ( ( S C_ CC /\ F e. ( CC ^pm S ) ) /\ ( s = S /\ f = F ) ) -> ( x e. _V |-> ( s _D x ) ) = ( x e. _V |-> ( S _D x ) ) )
6		dvnfval.1	. . . . . 6 |- G = ( x e. _V |-> ( S _D x ) )
7	5, 6	syl6eqr 2674	. . . . 5 |- ( ( ( S C_ CC /\ F e. ( CC ^pm S ) ) /\ ( s = S /\ f = F ) ) -> ( x e. _V |-> ( s _D x ) ) = G )
8	7	coeq1d 5283	. . . 4 |- ( ( ( S C_ CC /\ F e. ( CC ^pm S ) ) /\ ( s = S /\ f = F ) ) -> ( ( x e. _V |-> ( s _D x ) ) o. 1st ) = ( G o. 1st ) )
9	8	seqeq2d 12808	. . 3 |- ( ( ( S C_ CC /\ F e. ( CC ^pm S ) ) /\ ( s = S /\ f = F ) ) -> seq 0 ( ( ( x e. _V |-> ( s _D x ) ) o. 1st ) , ( NN0 X. { f } ) ) = seq 0 ( ( G o. 1st ) , ( NN0 X. { f } ) ) )
10		simprr 796	. . . . . 6 |- ( ( ( S C_ CC /\ F e. ( CC ^pm S ) ) /\ ( s = S /\ f = F ) ) -> f = F )
11	10	sneqd 4189	. . . . 5 |- ( ( ( S C_ CC /\ F e. ( CC ^pm S ) ) /\ ( s = S /\ f = F ) ) -> { f } = { F } )
12	11	xpeq2d 5139	. . . 4 |- ( ( ( S C_ CC /\ F e. ( CC ^pm S ) ) /\ ( s = S /\ f = F ) ) -> ( NN0 X. { f } ) = ( NN0 X. { F } ) )
13	12	seqeq3d 12809	. . 3 |- ( ( ( S C_ CC /\ F e. ( CC ^pm S ) ) /\ ( s = S /\ f = F ) ) -> seq 0 ( ( G o. 1st ) , ( NN0 X. { f } ) ) = seq 0 ( ( G o. 1st ) , ( NN0 X. { F } ) ) )
14	9, 13	eqtrd 2656	. 2 |- ( ( ( S C_ CC /\ F e. ( CC ^pm S ) ) /\ ( s = S /\ f = F ) ) -> seq 0 ( ( ( x e. _V |-> ( s _D x ) ) o. 1st ) , ( NN0 X. { f } ) ) = seq 0 ( ( G o. 1st ) , ( NN0 X. { F } ) ) )
15		simpr 477	. . 3 |- ( ( ( S C_ CC /\ F e. ( CC ^pm S ) ) /\ s = S ) -> s = S )
16	15	oveq2d 6666	. 2 |- ( ( ( S C_ CC /\ F e. ( CC ^pm S ) ) /\ s = S ) -> ( CC ^pm s ) = ( CC ^pm S ) )
17		simpl 473	. . 3 |- ( ( S C_ CC /\ F e. ( CC ^pm S ) ) -> S C_ CC )
18		cnex 10017	. . . 4 |- CC e. _V
19	18	elpw2 4828	. . 3 |- ( S e. ~P CC <-> S C_ CC )
20	17, 19	sylibr 224	. 2 |- ( ( S C_ CC /\ F e. ( CC ^pm S ) ) -> S e. ~P CC )
21		simpr 477	. 2 |- ( ( S C_ CC /\ F e. ( CC ^pm S ) ) -> F e. ( CC ^pm S ) )
22		seqex 12803	. . 3 |- seq 0 ( ( G o. 1st ) , ( NN0 X. { F } ) ) e. _V
23	22	a1i 11	. 2 |- ( ( S C_ CC /\ F e. ( CC ^pm S ) ) -> seq 0 ( ( G o. 1st ) , ( NN0 X. { F } ) ) e. _V )
24	2, 14, 16, 20, 21, 23	ovmpt2dx 6787	1 |- ( ( S C_ CC /\ F e. ( CC ^pm S ) ) -> ( S Dn F ) = seq 0 ( ( G o. 1st ) , ( NN0 X. { F } ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   {csn 4177   |-> cmpt 4729   X. cxp 5112   o. ccom 5118  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   ^pm cpm 7858   CCcc 9934   0cc0 9936   NN0cn0 11292   seqcseq 12801   _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

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-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seq 12802  df-dvn 23632

This theorem is referenced by:  dvnff  23686  dvn0  23687  dvnp1  23688