Theorem taylfval 24113

Description: Define the Taylor polynomial of a function. The constant Tayl is a function of five arguments: S is the base set with respect to evaluate the derivatives (generally RR or CC), F is the function we are approximating, at point B, to order N. The result is a polynomial function of x.

This "extended" version of taylpfval 24119 additionally handles the case N = +oo, in which case this is not a polynomial but an infinite series, the Taylor series of the function. (Contributed by Mario Carneiro, 30-Dec-2016.)

Hypotheses

Ref	Expression
taylfval.s	|- ( ph -> S e. { RR , CC } )
taylfval.f	|- ( ph -> F : A --> CC )
taylfval.a	|- ( ph -> A C_ S )
taylfval.n	|- ( ph -> ( N e. NN0 \/ N = +oo ) )
taylfval.b	|- ( ( ph /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> B e. dom ( ( S Dn F ) ` k ) )
taylfval.t	|- T = ( N ( S Tayl F ) B )

Assertion

Ref	Expression
taylfval	|- ( ph -> T = U_ x e. CC ( { x } X. (ℂfld tsums ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( x - B ) ^ k ) ) ) ) ) )

Distinct variable groups:   x, k, B   k, F, x   ph, k, x   k, N, x   S, k, x   x, T

Allowed substitution hints:   A( x, k)   T( k)

Proof of Theorem taylfval

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

Step	Hyp	Ref	Expression
1		taylfval.t	. 2 |- T = ( N ( S Tayl F ) B )
2		df-tayl 24109	. . . . 5 |- Tayl = ( s e. { RR , CC } , f e. ( CC ^pm s ) |-> ( n e. ( NN0 u. { +oo } ) , a e. |^|_ k e. ( ( 0 [,] n ) i^i ZZ ) dom ( ( s Dn f ) ` k ) |-> U_ x e. CC ( { x } X. (ℂfld tsums ( k e. ( ( 0 [,] n ) i^i ZZ ) |-> ( ( ( ( ( s Dn f ) ` k ) ` a ) / ( ! ` k ) ) x. ( ( x - a ) ^ k ) ) ) ) ) ) )
3	2	a1i 11	. . . 4 |- ( ph -> Tayl = ( s e. { RR , CC } , f e. ( CC ^pm s ) |-> ( n e. ( NN0 u. { +oo } ) , a e. |^|_ k e. ( ( 0 [,] n ) i^i ZZ ) dom ( ( s Dn f ) ` k ) |-> U_ x e. CC ( { x } X. (ℂfld tsums ( k e. ( ( 0 [,] n ) i^i ZZ ) |-> ( ( ( ( ( s Dn f ) ` k ) ` a ) / ( ! ` k ) ) x. ( ( x - a ) ^ k ) ) ) ) ) ) ) )
4		eqidd 2623	. . . . 5 |- ( ( ph /\ ( s = S /\ f = F ) ) -> ( NN0 u. { +oo } ) = ( NN0 u. { +oo } ) )
5		oveq12 6659	. . . . . . . . 9 |- ( ( s = S /\ f = F ) -> ( s Dn f ) = ( S Dn F ) )
6	5	ad2antlr 763	. . . . . . . 8 |- ( ( ( ph /\ ( s = S /\ f = F ) ) /\ k e. ( ( 0 [,] n ) i^i ZZ ) ) -> ( s Dn f ) = ( S Dn F ) )
7	6	fveq1d 6193	. . . . . . 7 |- ( ( ( ph /\ ( s = S /\ f = F ) ) /\ k e. ( ( 0 [,] n ) i^i ZZ ) ) -> ( ( s Dn f ) ` k ) = ( ( S Dn F ) ` k ) )
8	7	dmeqd 5326	. . . . . 6 |- ( ( ( ph /\ ( s = S /\ f = F ) ) /\ k e. ( ( 0 [,] n ) i^i ZZ ) ) -> dom ( ( s Dn f ) ` k ) = dom ( ( S Dn F ) ` k ) )
9	8	iineq2dv 4543	. . . . 5 |- ( ( ph /\ ( s = S /\ f = F ) ) -> |^|_ k e. ( ( 0 [,] n ) i^i ZZ ) dom ( ( s Dn f ) ` k ) = |^|_ k e. ( ( 0 [,] n ) i^i ZZ ) dom ( ( S Dn F ) ` k ) )
10	7	fveq1d 6193	. . . . . . . . . . 11 |- ( ( ( ph /\ ( s = S /\ f = F ) ) /\ k e. ( ( 0 [,] n ) i^i ZZ ) ) -> ( ( ( s Dn f ) ` k ) ` a ) = ( ( ( S Dn F ) ` k ) ` a ) )
11	10	oveq1d 6665	. . . . . . . . . 10 |- ( ( ( ph /\ ( s = S /\ f = F ) ) /\ k e. ( ( 0 [,] n ) i^i ZZ ) ) -> ( ( ( ( s Dn f ) ` k ) ` a ) / ( ! ` k ) ) = ( ( ( ( S Dn F ) ` k ) ` a ) / ( ! ` k ) ) )
12	11	oveq1d 6665	. . . . . . . . 9 |- ( ( ( ph /\ ( s = S /\ f = F ) ) /\ k e. ( ( 0 [,] n ) i^i ZZ ) ) -> ( ( ( ( ( s Dn f ) ` k ) ` a ) / ( ! ` k ) ) x. ( ( x - a ) ^ k ) ) = ( ( ( ( ( S Dn F ) ` k ) ` a ) / ( ! ` k ) ) x. ( ( x - a ) ^ k ) ) )
13	12	mpteq2dva 4744	. . . . . . . 8 |- ( ( ph /\ ( s = S /\ f = F ) ) -> ( k e. ( ( 0 [,] n ) i^i ZZ ) |-> ( ( ( ( ( s Dn f ) ` k ) ` a ) / ( ! ` k ) ) x. ( ( x - a ) ^ k ) ) ) = ( k e. ( ( 0 [,] n ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` a ) / ( ! ` k ) ) x. ( ( x - a ) ^ k ) ) ) )
14	13	oveq2d 6666	. . . . . . 7 |- ( ( ph /\ ( s = S /\ f = F ) ) -> (ℂfld tsums ( k e. ( ( 0 [,] n ) i^i ZZ ) |-> ( ( ( ( ( s Dn f ) ` k ) ` a ) / ( ! ` k ) ) x. ( ( x - a ) ^ k ) ) ) ) = (ℂfld tsums ( k e. ( ( 0 [,] n ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` a ) / ( ! ` k ) ) x. ( ( x - a ) ^ k ) ) ) ) )
15	14	xpeq2d 5139	. . . . . 6 |- ( ( ph /\ ( s = S /\ f = F ) ) -> ( { x } X. (ℂfld tsums ( k e. ( ( 0 [,] n ) i^i ZZ ) |-> ( ( ( ( ( s Dn f ) ` k ) ` a ) / ( ! ` k ) ) x. ( ( x - a ) ^ k ) ) ) ) ) = ( { x } X. (ℂfld tsums ( k e. ( ( 0 [,] n ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` a ) / ( ! ` k ) ) x. ( ( x - a ) ^ k ) ) ) ) ) )
16	15	iuneq2d 4547	. . . . 5 |- ( ( ph /\ ( s = S /\ f = F ) ) -> U_ x e. CC ( { x } X. (ℂfld tsums ( k e. ( ( 0 [,] n ) i^i ZZ ) |-> ( ( ( ( ( s Dn f ) ` k ) ` a ) / ( ! ` k ) ) x. ( ( x - a ) ^ k ) ) ) ) ) = U_ x e. CC ( { x } X. (ℂfld tsums ( k e. ( ( 0 [,] n ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` a ) / ( ! ` k ) ) x. ( ( x - a ) ^ k ) ) ) ) ) )
17	4, 9, 16	mpt2eq123dv 6717	. . . 4 |- ( ( ph /\ ( s = S /\ f = F ) ) -> ( n e. ( NN0 u. { +oo } ) , a e. |^|_ k e. ( ( 0 [,] n ) i^i ZZ ) dom ( ( s Dn f ) ` k ) |-> U_ x e. CC ( { x } X. (ℂfld tsums ( k e. ( ( 0 [,] n ) i^i ZZ ) |-> ( ( ( ( ( s Dn f ) ` k ) ` a ) / ( ! ` k ) ) x. ( ( x - a ) ^ k ) ) ) ) ) ) = ( n e. ( NN0 u. { +oo } ) , a e. |^|_ k e. ( ( 0 [,] n ) i^i ZZ ) dom ( ( S Dn F ) ` k ) |-> U_ x e. CC ( { x } X. (ℂfld tsums ( k e. ( ( 0 [,] n ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` a ) / ( ! ` k ) ) x. ( ( x - a ) ^ k ) ) ) ) ) ) )
18		simpr 477	. . . . 5 |- ( ( ph /\ s = S ) -> s = S )
19	18	oveq2d 6666	. . . 4 |- ( ( ph /\ s = S ) -> ( CC ^pm s ) = ( CC ^pm S ) )
20		taylfval.s	. . . 4 |- ( ph -> S e. { RR , CC } )
21		cnex 10017	. . . . . 6 |- CC e. _V
22	21	a1i 11	. . . . 5 |- ( ph -> CC e. _V )
23		taylfval.f	. . . . 5 |- ( ph -> F : A --> CC )
24		taylfval.a	. . . . 5 |- ( ph -> A C_ S )
25		elpm2r 7875	. . . . 5 |- ( ( ( CC e. _V /\ S e. { RR , CC } ) /\ ( F : A --> CC /\ A C_ S ) ) -> F e. ( CC ^pm S ) )
26	22, 20, 23, 24, 25	syl22anc 1327	. . . 4 |- ( ph -> F e. ( CC ^pm S ) )
27		nn0ex 11298	. . . . . . 7 |- NN0 e. _V
28		snex 4908	. . . . . . 7 |- { +oo } e. _V
29	27, 28	unex 6956	. . . . . 6 |- ( NN0 u. { +oo } ) e. _V
30		0xr 10086	. . . . . . . . . . 11 |- 0 e. RR*
31	30	a1i 11	. . . . . . . . . 10 |- ( n e. ( NN0 u. { +oo } ) -> 0 e. RR* )
32		nn0ssre 11296	. . . . . . . . . . . . 13 |- NN0 C_ RR
33		ressxr 10083	. . . . . . . . . . . . 13 |- RR C_ RR*
34	32, 33	sstri 3612	. . . . . . . . . . . 12 |- NN0 C_ RR*
35		pnfxr 10092	. . . . . . . . . . . . 13 |- +oo e. RR*
36		snssi 4339	. . . . . . . . . . . . 13 |- ( +oo e. RR* -> { +oo } C_ RR* )
37	35, 36	ax-mp 5	. . . . . . . . . . . 12 |- { +oo } C_ RR*
38	34, 37	unssi 3788	. . . . . . . . . . 11 |- ( NN0 u. { +oo } ) C_ RR*
39	38	sseli 3599	. . . . . . . . . 10 |- ( n e. ( NN0 u. { +oo } ) -> n e. RR* )
40		elun 3753	. . . . . . . . . . 11 |- ( n e. ( NN0 u. { +oo } ) <-> ( n e. NN0 \/ n e. { +oo } ) )
41		nn0ge0 11318	. . . . . . . . . . . 12 |- ( n e. NN0 -> 0 <_ n )
42		0lepnf 11966	. . . . . . . . . . . . 13 |- 0 <_ +oo
43		elsni 4194	. . . . . . . . . . . . 13 |- ( n e. { +oo } -> n = +oo )
44	42, 43	syl5breqr 4691	. . . . . . . . . . . 12 |- ( n e. { +oo } -> 0 <_ n )
45	41, 44	jaoi 394	. . . . . . . . . . 11 |- ( ( n e. NN0 \/ n e. { +oo } ) -> 0 <_ n )
46	40, 45	sylbi 207	. . . . . . . . . 10 |- ( n e. ( NN0 u. { +oo } ) -> 0 <_ n )
47		lbicc2 12288	. . . . . . . . . 10 |- ( ( 0 e. RR* /\ n e. RR* /\ 0 <_ n ) -> 0 e. ( 0 [,] n ) )
48	31, 39, 46, 47	syl3anc 1326	. . . . . . . . 9 |- ( n e. ( NN0 u. { +oo } ) -> 0 e. ( 0 [,] n ) )
49		0z 11388	. . . . . . . . 9 |- 0 e. ZZ
50		inelcm 4032	. . . . . . . . 9 |- ( ( 0 e. ( 0 [,] n ) /\ 0 e. ZZ ) -> ( ( 0 [,] n ) i^i ZZ ) =/= (/) )
51	48, 49, 50	sylancl 694	. . . . . . . 8 |- ( n e. ( NN0 u. { +oo } ) -> ( ( 0 [,] n ) i^i ZZ ) =/= (/) )
52		fvex 6201	. . . . . . . . . 10 |- ( ( S Dn F ) ` k ) e. _V
53	52	dmex 7099	. . . . . . . . 9 |- dom ( ( S Dn F ) ` k ) e. _V
54	53	rgenw 2924	. . . . . . . 8 |- A. k e. ( ( 0 [,] n ) i^i ZZ ) dom ( ( S Dn F ) ` k ) e. _V
55		iinexg 4824	. . . . . . . 8 |- ( ( ( ( 0 [,] n ) i^i ZZ ) =/= (/) /\ A. k e. ( ( 0 [,] n ) i^i ZZ ) dom ( ( S Dn F ) ` k ) e. _V ) -> |^|_ k e. ( ( 0 [,] n ) i^i ZZ ) dom ( ( S Dn F ) ` k ) e. _V )
56	51, 54, 55	sylancl 694	. . . . . . 7 |- ( n e. ( NN0 u. { +oo } ) -> |^|_ k e. ( ( 0 [,] n ) i^i ZZ ) dom ( ( S Dn F ) ` k ) e. _V )
57	56	rgen 2922	. . . . . 6 |- A. n e. ( NN0 u. { +oo } ) |^|_ k e. ( ( 0 [,] n ) i^i ZZ ) dom ( ( S Dn F ) ` k ) e. _V
58		eqid 2622	. . . . . . 7 |- ( n e. ( NN0 u. { +oo } ) , a e. |^|_ k e. ( ( 0 [,] n ) i^i ZZ ) dom ( ( S Dn F ) ` k ) |-> U_ x e. CC ( { x } X. (ℂfld tsums ( k e. ( ( 0 [,] n ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` a ) / ( ! ` k ) ) x. ( ( x - a ) ^ k ) ) ) ) ) ) = ( n e. ( NN0 u. { +oo } ) , a e. |^|_ k e. ( ( 0 [,] n ) i^i ZZ ) dom ( ( S Dn F ) ` k ) |-> U_ x e. CC ( { x } X. (ℂfld tsums ( k e. ( ( 0 [,] n ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` a ) / ( ! ` k ) ) x. ( ( x - a ) ^ k ) ) ) ) ) )
59	58	mpt2exxg 7244	. . . . . 6 |- ( ( ( NN0 u. { +oo } ) e. _V /\ A. n e. ( NN0 u. { +oo } ) |^|_ k e. ( ( 0 [,] n ) i^i ZZ ) dom ( ( S Dn F ) ` k ) e. _V ) -> ( n e. ( NN0 u. { +oo } ) , a e. |^|_ k e. ( ( 0 [,] n ) i^i ZZ ) dom ( ( S Dn F ) ` k ) |-> U_ x e. CC ( { x } X. (ℂfld tsums ( k e. ( ( 0 [,] n ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` a ) / ( ! ` k ) ) x. ( ( x - a ) ^ k ) ) ) ) ) ) e. _V )
60	29, 57, 59	mp2an 708	. . . . 5 |- ( n e. ( NN0 u. { +oo } ) , a e. |^|_ k e. ( ( 0 [,] n ) i^i ZZ ) dom ( ( S Dn F ) ` k ) |-> U_ x e. CC ( { x } X. (ℂfld tsums ( k e. ( ( 0 [,] n ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` a ) / ( ! ` k ) ) x. ( ( x - a ) ^ k ) ) ) ) ) ) e. _V
61	60	a1i 11	. . . 4 |- ( ph -> ( n e. ( NN0 u. { +oo } ) , a e. |^|_ k e. ( ( 0 [,] n ) i^i ZZ ) dom ( ( S Dn F ) ` k ) |-> U_ x e. CC ( { x } X. (ℂfld tsums ( k e. ( ( 0 [,] n ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` a ) / ( ! ` k ) ) x. ( ( x - a ) ^ k ) ) ) ) ) ) e. _V )
62	3, 17, 19, 20, 26, 61	ovmpt2dx 6787	. . 3 |- ( ph -> ( S Tayl F ) = ( n e. ( NN0 u. { +oo } ) , a e. |^|_ k e. ( ( 0 [,] n ) i^i ZZ ) dom ( ( S Dn F ) ` k ) |-> U_ x e. CC ( { x } X. (ℂfld tsums ( k e. ( ( 0 [,] n ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` a ) / ( ! ` k ) ) x. ( ( x - a ) ^ k ) ) ) ) ) ) )
63		simprl 794	. . . . . . . . 9 |- ( ( ph /\ ( n = N /\ a = B ) ) -> n = N )
64	63	oveq2d 6666	. . . . . . . 8 |- ( ( ph /\ ( n = N /\ a = B ) ) -> ( 0 [,] n ) = ( 0 [,] N ) )
65	64	ineq1d 3813	. . . . . . 7 |- ( ( ph /\ ( n = N /\ a = B ) ) -> ( ( 0 [,] n ) i^i ZZ ) = ( ( 0 [,] N ) i^i ZZ ) )
66		simprr 796	. . . . . . . . . 10 |- ( ( ph /\ ( n = N /\ a = B ) ) -> a = B )
67	66	fveq2d 6195	. . . . . . . . 9 |- ( ( ph /\ ( n = N /\ a = B ) ) -> ( ( ( S Dn F ) ` k ) ` a ) = ( ( ( S Dn F ) ` k ) ` B ) )
68	67	oveq1d 6665	. . . . . . . 8 |- ( ( ph /\ ( n = N /\ a = B ) ) -> ( ( ( ( S Dn F ) ` k ) ` a ) / ( ! ` k ) ) = ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) )
69	66	oveq2d 6666	. . . . . . . . 9 |- ( ( ph /\ ( n = N /\ a = B ) ) -> ( x - a ) = ( x - B ) )
70	69	oveq1d 6665	. . . . . . . 8 |- ( ( ph /\ ( n = N /\ a = B ) ) -> ( ( x - a ) ^ k ) = ( ( x - B ) ^ k ) )
71	68, 70	oveq12d 6668	. . . . . . 7 |- ( ( ph /\ ( n = N /\ a = B ) ) -> ( ( ( ( ( S Dn F ) ` k ) ` a ) / ( ! ` k ) ) x. ( ( x - a ) ^ k ) ) = ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( x - B ) ^ k ) ) )
72	65, 71	mpteq12dv 4733	. . . . . 6 |- ( ( ph /\ ( n = N /\ a = B ) ) -> ( k e. ( ( 0 [,] n ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` a ) / ( ! ` k ) ) x. ( ( x - a ) ^ k ) ) ) = ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( x - B ) ^ k ) ) ) )
73	72	oveq2d 6666	. . . . 5 |- ( ( ph /\ ( n = N /\ a = B ) ) -> (ℂfld tsums ( k e. ( ( 0 [,] n ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` a ) / ( ! ` k ) ) x. ( ( x - a ) ^ k ) ) ) ) = (ℂfld tsums ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( x - B ) ^ k ) ) ) ) )
74	73	xpeq2d 5139	. . . 4 |- ( ( ph /\ ( n = N /\ a = B ) ) -> ( { x } X. (ℂfld tsums ( k e. ( ( 0 [,] n ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` a ) / ( ! ` k ) ) x. ( ( x - a ) ^ k ) ) ) ) ) = ( { x } X. (ℂfld tsums ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( x - B ) ^ k ) ) ) ) ) )
75	74	iuneq2d 4547	. . 3 |- ( ( ph /\ ( n = N /\ a = B ) ) -> U_ x e. CC ( { x } X. (ℂfld tsums ( k e. ( ( 0 [,] n ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` a ) / ( ! ` k ) ) x. ( ( x - a ) ^ k ) ) ) ) ) = U_ x e. CC ( { x } X. (ℂfld tsums ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( x - B ) ^ k ) ) ) ) ) )
76		simpr 477	. . . . . 6 |- ( ( ph /\ n = N ) -> n = N )
77	76	oveq2d 6666	. . . . 5 |- ( ( ph /\ n = N ) -> ( 0 [,] n ) = ( 0 [,] N ) )
78	77	ineq1d 3813	. . . 4 |- ( ( ph /\ n = N ) -> ( ( 0 [,] n ) i^i ZZ ) = ( ( 0 [,] N ) i^i ZZ ) )
79		iineq1 4535	. . . 4 |- ( ( ( 0 [,] n ) i^i ZZ ) = ( ( 0 [,] N ) i^i ZZ ) -> |^|_ k e. ( ( 0 [,] n ) i^i ZZ ) dom ( ( S Dn F ) ` k ) = |^|_ k e. ( ( 0 [,] N ) i^i ZZ ) dom ( ( S Dn F ) ` k ) )
80	78, 79	syl 17	. . 3 |- ( ( ph /\ n = N ) -> |^|_ k e. ( ( 0 [,] n ) i^i ZZ ) dom ( ( S Dn F ) ` k ) = |^|_ k e. ( ( 0 [,] N ) i^i ZZ ) dom ( ( S Dn F ) ` k ) )
81		taylfval.n	. . . . 5 |- ( ph -> ( N e. NN0 \/ N = +oo ) )
82		pnfex 10093	. . . . . . 7 |- +oo e. _V
83	82	elsn2 4211	. . . . . 6 |- ( N e. { +oo } <-> N = +oo )
84	83	orbi2i 541	. . . . 5 |- ( ( N e. NN0 \/ N e. { +oo } ) <-> ( N e. NN0 \/ N = +oo ) )
85	81, 84	sylibr 224	. . . 4 |- ( ph -> ( N e. NN0 \/ N e. { +oo } ) )
86		elun 3753	. . . 4 |- ( N e. ( NN0 u. { +oo } ) <-> ( N e. NN0 \/ N e. { +oo } ) )
87	85, 86	sylibr 224	. . 3 |- ( ph -> N e. ( NN0 u. { +oo } ) )
88		taylfval.b	. . . . 5 |- ( ( ph /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> B e. dom ( ( S Dn F ) ` k ) )
89	88	ralrimiva 2966	. . . 4 |- ( ph -> A. k e. ( ( 0 [,] N ) i^i ZZ ) B e. dom ( ( S Dn F ) ` k ) )
90		oveq2 6658	. . . . . . . . . 10 |- ( n = N -> ( 0 [,] n ) = ( 0 [,] N ) )
91	90	ineq1d 3813	. . . . . . . . 9 |- ( n = N -> ( ( 0 [,] n ) i^i ZZ ) = ( ( 0 [,] N ) i^i ZZ ) )
92	91	neeq1d 2853	. . . . . . . 8 |- ( n = N -> ( ( ( 0 [,] n ) i^i ZZ ) =/= (/) <-> ( ( 0 [,] N ) i^i ZZ ) =/= (/) ) )
93	92, 51	vtoclga 3272	. . . . . . 7 |- ( N e. ( NN0 u. { +oo } ) -> ( ( 0 [,] N ) i^i ZZ ) =/= (/) )
94	87, 93	syl 17	. . . . . 6 |- ( ph -> ( ( 0 [,] N ) i^i ZZ ) =/= (/) )
95		r19.2z 4060	. . . . . 6 |- ( ( ( ( 0 [,] N ) i^i ZZ ) =/= (/) /\ A. k e. ( ( 0 [,] N ) i^i ZZ ) B e. dom ( ( S Dn F ) ` k ) ) -> E. k e. ( ( 0 [,] N ) i^i ZZ ) B e. dom ( ( S Dn F ) ` k ) )
96	94, 89, 95	syl2anc 693	. . . . 5 |- ( ph -> E. k e. ( ( 0 [,] N ) i^i ZZ ) B e. dom ( ( S Dn F ) ` k ) )
97		elex 3212	. . . . . 6 |- ( B e. dom ( ( S Dn F ) ` k ) -> B e. _V )
98	97	rexlimivw 3029	. . . . 5 |- ( E. k e. ( ( 0 [,] N ) i^i ZZ ) B e. dom ( ( S Dn F ) ` k ) -> B e. _V )
99		eliin 4525	. . . . 5 |- ( B e. _V -> ( B e. |^|_ k e. ( ( 0 [,] N ) i^i ZZ ) dom ( ( S Dn F ) ` k ) <-> A. k e. ( ( 0 [,] N ) i^i ZZ ) B e. dom ( ( S Dn F ) ` k ) ) )
100	96, 98, 99	3syl 18	. . . 4 |- ( ph -> ( B e. |^|_ k e. ( ( 0 [,] N ) i^i ZZ ) dom ( ( S Dn F ) ` k ) <-> A. k e. ( ( 0 [,] N ) i^i ZZ ) B e. dom ( ( S Dn F ) ` k ) ) )
101	89, 100	mpbird 247	. . 3 |- ( ph -> B e. |^|_ k e. ( ( 0 [,] N ) i^i ZZ ) dom ( ( S Dn F ) ` k ) )
102		snssi 4339	. . . . . . . 8 |- ( x e. CC -> { x } C_ CC )
103	102	adantl 482	. . . . . . 7 |- ( ( ph /\ x e. CC ) -> { x } C_ CC )
104	20, 23, 24, 81, 88	taylfvallem 24112	. . . . . . 7 |- ( ( ph /\ x e. CC ) -> (ℂfld tsums ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( x - B ) ^ k ) ) ) ) C_ CC )
105		xpss12 5225	. . . . . . 7 |- ( ( { x } C_ CC /\ (ℂfld tsums ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( x - B ) ^ k ) ) ) ) C_ CC ) -> ( { x } X. (ℂfld tsums ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( x - B ) ^ k ) ) ) ) ) C_ ( CC X. CC ) )
106	103, 104, 105	syl2anc 693	. . . . . 6 |- ( ( ph /\ x e. CC ) -> ( { x } X. (ℂfld tsums ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( x - B ) ^ k ) ) ) ) ) C_ ( CC X. CC ) )
107	106	ralrimiva 2966	. . . . 5 |- ( ph -> A. x e. CC ( { x } X. (ℂfld tsums ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( x - B ) ^ k ) ) ) ) ) C_ ( CC X. CC ) )
108		iunss 4561	. . . . 5 |- ( U_ x e. CC ( { x } X. (ℂfld tsums ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( x - B ) ^ k ) ) ) ) ) C_ ( CC X. CC ) <-> A. x e. CC ( { x } X. (ℂfld tsums ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( x - B ) ^ k ) ) ) ) ) C_ ( CC X. CC ) )
109	107, 108	sylibr 224	. . . 4 |- ( ph -> U_ x e. CC ( { x } X. (ℂfld tsums ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( x - B ) ^ k ) ) ) ) ) C_ ( CC X. CC ) )
110	21, 21	xpex 6962	. . . . 5 |- ( CC X. CC ) e. _V
111	110	ssex 4802	. . . 4 |- ( U_ x e. CC ( { x } X. (ℂfld tsums ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( x - B ) ^ k ) ) ) ) ) C_ ( CC X. CC ) -> U_ x e. CC ( { x } X. (ℂfld tsums ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( x - B ) ^ k ) ) ) ) ) e. _V )
112	109, 111	syl 17	. . 3 |- ( ph -> U_ x e. CC ( { x } X. (ℂfld tsums ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( x - B ) ^ k ) ) ) ) ) e. _V )
113	62, 75, 80, 87, 101, 112	ovmpt2dx 6787	. 2 |- ( ph -> ( N ( S Tayl F ) B ) = U_ x e. CC ( { x } X. (ℂfld tsums ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( x - B ) ^ k ) ) ) ) ) )
114	1, 113	syl5eq 2668	1 |- ( ph -> T = U_ x e. CC ( { x } X. (ℂfld tsums ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( x - B ) ^ k ) ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   {cpr 4179   U_ciun 4520   |^|_ciin 4521   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   ^pm cpm 7858   CCcc 9934   RRcr 9935   0cc0 9936   x. cmul 9941   +oocpnf 10071   RR*cxr 10073   <_ cle 10075   - cmin 10266   / cdiv 10684   NN0cn0 11292   ZZcz 11377   [,]cicc 12178   ^cexp 12860   !cfa 13060  ℂ_fldccnfld 19746   tsums ctsu 21929   Dncdvn 23628   Tayl ctayl 24107

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  ax-addf 10015  ax-mulf 10016

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-se 5074  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  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-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  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-fzo 12466  df-seq 12802  df-exp 12861  df-fac 13061  df-hash 13118  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-sets 15864  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-0g 16102  df-gsum 16103  df-topgen 16104  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  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-tsms 21930  df-xms 22125  df-ms 22126  df-limc 23630  df-dv 23631  df-dvn 23632  df-tayl 24109

This theorem is referenced by:  eltayl  24114  taylf  24115  taylpfval  24119