Theorem tayl0 24116

Description: The Taylor series is always defined at the basepoint, with value equal to the value 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
tayl0	|- ( ph -> ( B e. dom T /\ ( T ` B ) = ( F ` B ) ) )

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

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

Proof of Theorem tayl0

Step	Hyp	Ref	Expression
1		taylfval.a	. . . . 5 |- ( ph -> A C_ S )
2		taylfval.s	. . . . . 6 |- ( ph -> S e. { RR , CC } )
3		recnprss 23668	. . . . . 6 |- ( S e. { RR , CC } -> S C_ CC )
4	2, 3	syl 17	. . . . 5 |- ( ph -> S C_ CC )
5	1, 4	sstrd 3613	. . . 4 |- ( ph -> A C_ CC )
6		fveq2 6191	. . . . . . . 8 |- ( k = 0 -> ( ( S Dn F ) ` k ) = ( ( S Dn F ) ` 0 ) )
7	6	dmeqd 5326	. . . . . . 7 |- ( k = 0 -> dom ( ( S Dn F ) ` k ) = dom ( ( S Dn F ) ` 0 ) )
8	7	eleq2d 2687	. . . . . 6 |- ( k = 0 -> ( B e. dom ( ( S Dn F ) ` k ) <-> B e. dom ( ( S Dn F ) ` 0 ) ) )
9		taylfval.b	. . . . . . 7 |- ( ( ph /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> B e. dom ( ( S Dn F ) ` k ) )
10	9	ralrimiva 2966	. . . . . 6 |- ( ph -> A. k e. ( ( 0 [,] N ) i^i ZZ ) B e. dom ( ( S Dn F ) ` k ) )
11		taylfval.n	. . . . . . . 8 |- ( ph -> ( N e. NN0 \/ N = +oo ) )
12		elxnn0 11365	. . . . . . . . 9 |- ( N e. NN0* <-> ( N e. NN0 \/ N = +oo ) )
13		0xr 10086	. . . . . . . . . . 11 |- 0 e. RR*
14	13	a1i 11	. . . . . . . . . 10 |- ( N e. NN0* -> 0 e. RR* )
15		xnn0xr 11368	. . . . . . . . . 10 |- ( N e. NN0* -> N e. RR* )
16		xnn0ge0 11967	. . . . . . . . . 10 |- ( N e. NN0* -> 0 <_ N )
17		lbicc2 12288	. . . . . . . . . 10 |- ( ( 0 e. RR* /\ N e. RR* /\ 0 <_ N ) -> 0 e. ( 0 [,] N ) )
18	14, 15, 16, 17	syl3anc 1326	. . . . . . . . 9 |- ( N e. NN0* -> 0 e. ( 0 [,] N ) )
19	12, 18	sylbir 225	. . . . . . . 8 |- ( ( N e. NN0 \/ N = +oo ) -> 0 e. ( 0 [,] N ) )
20	11, 19	syl 17	. . . . . . 7 |- ( ph -> 0 e. ( 0 [,] N ) )
21		0zd 11389	. . . . . . 7 |- ( ph -> 0 e. ZZ )
22	20, 21	elind 3798	. . . . . 6 |- ( ph -> 0 e. ( ( 0 [,] N ) i^i ZZ ) )
23	8, 10, 22	rspcdva 3316	. . . . 5 |- ( ph -> B e. dom ( ( S Dn F ) ` 0 ) )
24		cnex 10017	. . . . . . . . . 10 |- CC e. _V
25	24	a1i 11	. . . . . . . . 9 |- ( ph -> CC e. _V )
26		taylfval.f	. . . . . . . . 9 |- ( ph -> F : A --> CC )
27		elpm2r 7875	. . . . . . . . 9 |- ( ( ( CC e. _V /\ S e. { RR , CC } ) /\ ( F : A --> CC /\ A C_ S ) ) -> F e. ( CC ^pm S ) )
28	25, 2, 26, 1, 27	syl22anc 1327	. . . . . . . 8 |- ( ph -> F e. ( CC ^pm S ) )
29		dvn0 23687	. . . . . . . 8 |- ( ( S C_ CC /\ F e. ( CC ^pm S ) ) -> ( ( S Dn F ) ` 0 ) = F )
30	4, 28, 29	syl2anc 693	. . . . . . 7 |- ( ph -> ( ( S Dn F ) ` 0 ) = F )
31	30	dmeqd 5326	. . . . . 6 |- ( ph -> dom ( ( S Dn F ) ` 0 ) = dom F )
32		fdm 6051	. . . . . . 7 |- ( F : A --> CC -> dom F = A )
33	26, 32	syl 17	. . . . . 6 |- ( ph -> dom F = A )
34	31, 33	eqtrd 2656	. . . . 5 |- ( ph -> dom ( ( S Dn F ) ` 0 ) = A )
35	23, 34	eleqtrd 2703	. . . 4 |- ( ph -> B e. A )
36	5, 35	sseldd 3604	. . 3 |- ( ph -> B e. CC )
37		cnfldbas 19750	. . . . . . 7 |- CC = ( Base ` ℂfld )
38		cnfld0 19770	. . . . . . 7 |- 0 = ( 0g ` ℂfld )
39		cnring 19768	. . . . . . . 8 |- ℂfld e. Ring
40		ringmnd 18556	. . . . . . . 8 |- (ℂfld e. Ring -> ℂfld e. Mnd )
41	39, 40	mp1i 13	. . . . . . 7 |- ( ph -> ℂfld e. Mnd )
42		ovex 6678	. . . . . . . . 9 |- ( 0 [,] N ) e. _V
43	42	inex1 4799	. . . . . . . 8 |- ( ( 0 [,] N ) i^i ZZ ) e. _V
44	43	a1i 11	. . . . . . 7 |- ( ph -> ( ( 0 [,] N ) i^i ZZ ) e. _V )
45	2	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> S e. { RR , CC } )
46	28	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> F e. ( CC ^pm S ) )
47		simpr 477	. . . . . . . . . . . . . 14 |- ( ( ph /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> k e. ( ( 0 [,] N ) i^i ZZ ) )
48	47	elin2d 3803	. . . . . . . . . . . . 13 |- ( ( ph /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> k e. ZZ )
49	47	elin1d 3802	. . . . . . . . . . . . . . 15 |- ( ( ph /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> k e. ( 0 [,] N ) )
50		nn0re 11301	. . . . . . . . . . . . . . . . . . . 20 |- ( N e. NN0 -> N e. RR )
51	50	rexrd 10089	. . . . . . . . . . . . . . . . . . 19 |- ( N e. NN0 -> N e. RR* )
52		id 22	. . . . . . . . . . . . . . . . . . . 20 |- ( N = +oo -> N = +oo )
53		pnfxr 10092	. . . . . . . . . . . . . . . . . . . 20 |- +oo e. RR*
54	52, 53	syl6eqel 2709	. . . . . . . . . . . . . . . . . . 19 |- ( N = +oo -> N e. RR* )
55	51, 54	jaoi 394	. . . . . . . . . . . . . . . . . 18 |- ( ( N e. NN0 \/ N = +oo ) -> N e. RR* )
56	11, 55	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ph -> N e. RR* )
57	56	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> N e. RR* )
58		elicc1 12219	. . . . . . . . . . . . . . . 16 |- ( ( 0 e. RR* /\ N e. RR* ) -> ( k e. ( 0 [,] N ) <-> ( k e. RR* /\ 0 <_ k /\ k <_ N ) ) )
59	13, 57, 58	sylancr 695	. . . . . . . . . . . . . . 15 |- ( ( ph /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> ( k e. ( 0 [,] N ) <-> ( k e. RR* /\ 0 <_ k /\ k <_ N ) ) )
60	49, 59	mpbid 222	. . . . . . . . . . . . . 14 |- ( ( ph /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> ( k e. RR* /\ 0 <_ k /\ k <_ N ) )
61	60	simp2d 1074	. . . . . . . . . . . . 13 |- ( ( ph /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> 0 <_ k )
62		elnn0z 11390	. . . . . . . . . . . . 13 |- ( k e. NN0 <-> ( k e. ZZ /\ 0 <_ k ) )
63	48, 61, 62	sylanbrc 698	. . . . . . . . . . . 12 |- ( ( ph /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> k e. NN0 )
64		dvnf 23690	. . . . . . . . . . . 12 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) /\ k e. NN0 ) -> ( ( S Dn F ) ` k ) : dom ( ( S Dn F ) ` k ) --> CC )
65	45, 46, 63, 64	syl3anc 1326	. . . . . . . . . . 11 |- ( ( ph /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> ( ( S Dn F ) ` k ) : dom ( ( S Dn F ) ` k ) --> CC )
66	65, 9	ffvelrnd 6360	. . . . . . . . . 10 |- ( ( ph /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> ( ( ( S Dn F ) ` k ) ` B ) e. CC )
67	63	faccld 13071	. . . . . . . . . . 11 |- ( ( ph /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> ( ! ` k ) e. NN )
68	67	nncnd 11036	. . . . . . . . . 10 |- ( ( ph /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> ( ! ` k ) e. CC )
69	67	nnne0d 11065	. . . . . . . . . 10 |- ( ( ph /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> ( ! ` k ) =/= 0 )
70	66, 68, 69	divcld 10801	. . . . . . . . 9 |- ( ( ph /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) e. CC )
71		0cnd 10033	. . . . . . . . . 10 |- ( ( ph /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> 0 e. CC )
72	71, 63	expcld 13008	. . . . . . . . 9 |- ( ( ph /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> ( 0 ^ k ) e. CC )
73	70, 72	mulcld 10060	. . . . . . . 8 |- ( ( ph /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( 0 ^ k ) ) e. CC )
74		eqid 2622	. . . . . . . 8 |- ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( 0 ^ k ) ) ) = ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( 0 ^ k ) ) )
75	73, 74	fmptd 6385	. . . . . . 7 |- ( ph -> ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( 0 ^ k ) ) ) : ( ( 0 [,] N ) i^i ZZ ) --> CC )
76		eldifi 3732	. . . . . . . . . . . . 13 |- ( k e. ( ( ( 0 [,] N ) i^i ZZ ) \ { 0 } ) -> k e. ( ( 0 [,] N ) i^i ZZ ) )
77	76, 63	sylan2 491	. . . . . . . . . . . 12 |- ( ( ph /\ k e. ( ( ( 0 [,] N ) i^i ZZ ) \ { 0 } ) ) -> k e. NN0 )
78		eldifsni 4320	. . . . . . . . . . . . 13 |- ( k e. ( ( ( 0 [,] N ) i^i ZZ ) \ { 0 } ) -> k =/= 0 )
79	78	adantl 482	. . . . . . . . . . . 12 |- ( ( ph /\ k e. ( ( ( 0 [,] N ) i^i ZZ ) \ { 0 } ) ) -> k =/= 0 )
80		elnnne0 11306	. . . . . . . . . . . 12 |- ( k e. NN <-> ( k e. NN0 /\ k =/= 0 ) )
81	77, 79, 80	sylanbrc 698	. . . . . . . . . . 11 |- ( ( ph /\ k e. ( ( ( 0 [,] N ) i^i ZZ ) \ { 0 } ) ) -> k e. NN )
82	81	0expd 13024	. . . . . . . . . 10 |- ( ( ph /\ k e. ( ( ( 0 [,] N ) i^i ZZ ) \ { 0 } ) ) -> ( 0 ^ k ) = 0 )
83	82	oveq2d 6666	. . . . . . . . 9 |- ( ( ph /\ k e. ( ( ( 0 [,] N ) i^i ZZ ) \ { 0 } ) ) -> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( 0 ^ k ) ) = ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. 0 ) )
84	70	mul01d 10235	. . . . . . . . . 10 |- ( ( ph /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. 0 ) = 0 )
85	76, 84	sylan2 491	. . . . . . . . 9 |- ( ( ph /\ k e. ( ( ( 0 [,] N ) i^i ZZ ) \ { 0 } ) ) -> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. 0 ) = 0 )
86	83, 85	eqtrd 2656	. . . . . . . 8 |- ( ( ph /\ k e. ( ( ( 0 [,] N ) i^i ZZ ) \ { 0 } ) ) -> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( 0 ^ k ) ) = 0 )
87		zex 11386	. . . . . . . . . 10 |- ZZ e. _V
88	87	inex2 4800	. . . . . . . . 9 |- ( ( 0 [,] N ) i^i ZZ ) e. _V
89	88	a1i 11	. . . . . . . 8 |- ( ph -> ( ( 0 [,] N ) i^i ZZ ) e. _V )
90	86, 89	suppss2 7329	. . . . . . 7 |- ( ph -> ( ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( 0 ^ k ) ) ) supp 0 ) C_ { 0 } )
91	37, 38, 41, 44, 22, 75, 90	gsumpt 18361	. . . . . 6 |- ( ph -> (ℂfld gsumg ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( 0 ^ k ) ) ) ) = ( ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( 0 ^ k ) ) ) ` 0 ) )
92	6	fveq1d 6193	. . . . . . . . . 10 |- ( k = 0 -> ( ( ( S Dn F ) ` k ) ` B ) = ( ( ( S Dn F ) ` 0 ) ` B ) )
93		fveq2 6191	. . . . . . . . . . 11 |- ( k = 0 -> ( ! ` k ) = ( ! ` 0 ) )
94		fac0 13063	. . . . . . . . . . 11 |- ( ! ` 0 ) = 1
95	93, 94	syl6eq 2672	. . . . . . . . . 10 |- ( k = 0 -> ( ! ` k ) = 1 )
96	92, 95	oveq12d 6668	. . . . . . . . 9 |- ( k = 0 -> ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) = ( ( ( ( S Dn F ) ` 0 ) ` B ) / 1 ) )
97		oveq2 6658	. . . . . . . . . 10 |- ( k = 0 -> ( 0 ^ k ) = ( 0 ^ 0 ) )
98		0exp0e1 12865	. . . . . . . . . 10 |- ( 0 ^ 0 ) = 1
99	97, 98	syl6eq 2672	. . . . . . . . 9 |- ( k = 0 -> ( 0 ^ k ) = 1 )
100	96, 99	oveq12d 6668	. . . . . . . 8 |- ( k = 0 -> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( 0 ^ k ) ) = ( ( ( ( ( S Dn F ) ` 0 ) ` B ) / 1 ) x. 1 ) )
101		ovex 6678	. . . . . . . 8 |- ( ( ( ( ( S Dn F ) ` 0 ) ` B ) / 1 ) x. 1 ) e. _V
102	100, 74, 101	fvmpt 6282	. . . . . . 7 |- ( 0 e. ( ( 0 [,] N ) i^i ZZ ) -> ( ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( 0 ^ k ) ) ) ` 0 ) = ( ( ( ( ( S Dn F ) ` 0 ) ` B ) / 1 ) x. 1 ) )
103	22, 102	syl 17	. . . . . 6 |- ( ph -> ( ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( 0 ^ k ) ) ) ` 0 ) = ( ( ( ( ( S Dn F ) ` 0 ) ` B ) / 1 ) x. 1 ) )
104	30	fveq1d 6193	. . . . . . . . . 10 |- ( ph -> ( ( ( S Dn F ) ` 0 ) ` B ) = ( F ` B ) )
105	104	oveq1d 6665	. . . . . . . . 9 |- ( ph -> ( ( ( ( S Dn F ) ` 0 ) ` B ) / 1 ) = ( ( F ` B ) / 1 ) )
106	26, 35	ffvelrnd 6360	. . . . . . . . . 10 |- ( ph -> ( F ` B ) e. CC )
107	106	div1d 10793	. . . . . . . . 9 |- ( ph -> ( ( F ` B ) / 1 ) = ( F ` B ) )
108	105, 107	eqtrd 2656	. . . . . . . 8 |- ( ph -> ( ( ( ( S Dn F ) ` 0 ) ` B ) / 1 ) = ( F ` B ) )
109	108	oveq1d 6665	. . . . . . 7 |- ( ph -> ( ( ( ( ( S Dn F ) ` 0 ) ` B ) / 1 ) x. 1 ) = ( ( F ` B ) x. 1 ) )
110	106	mulid1d 10057	. . . . . . 7 |- ( ph -> ( ( F ` B ) x. 1 ) = ( F ` B ) )
111	109, 110	eqtrd 2656	. . . . . 6 |- ( ph -> ( ( ( ( ( S Dn F ) ` 0 ) ` B ) / 1 ) x. 1 ) = ( F ` B ) )
112	91, 103, 111	3eqtrd 2660	. . . . 5 |- ( ph -> (ℂfld gsumg ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( 0 ^ k ) ) ) ) = ( F ` B ) )
113		ringcmn 18581	. . . . . . 7 |- (ℂfld e. Ring -> ℂfld e. CMnd )
114	39, 113	mp1i 13	. . . . . 6 |- ( ph -> ℂfld e. CMnd )
115		cnfldtps 22581	. . . . . . 7 |- ℂfld e. TopSp
116	115	a1i 11	. . . . . 6 |- ( ph -> ℂfld e. TopSp )
117		mptexg 6484	. . . . . . . 8 |- ( ( ( 0 [,] N ) i^i ZZ ) e. _V -> ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( 0 ^ k ) ) ) e. _V )
118	88, 117	mp1i 13	. . . . . . 7 |- ( ph -> ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( 0 ^ k ) ) ) e. _V )
119		funmpt 5926	. . . . . . . 8 |- Fun ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( 0 ^ k ) ) )
120	119	a1i 11	. . . . . . 7 |- ( ph -> Fun ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( 0 ^ k ) ) ) )
121		c0ex 10034	. . . . . . . 8 |- 0 e. _V
122	121	a1i 11	. . . . . . 7 |- ( ph -> 0 e. _V )
123		snfi 8038	. . . . . . . 8 |- { 0 } e. Fin
124	123	a1i 11	. . . . . . 7 |- ( ph -> { 0 } e. Fin )
125		suppssfifsupp 8290	. . . . . . 7 |- ( ( ( ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( 0 ^ k ) ) ) e. _V /\ Fun ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( 0 ^ k ) ) ) /\ 0 e. _V ) /\ ( { 0 } e. Fin /\ ( ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( 0 ^ k ) ) ) supp 0 ) C_ { 0 } ) ) -> ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( 0 ^ k ) ) ) finSupp 0 )
126	118, 120, 122, 124, 90, 125	syl32anc 1334	. . . . . 6 |- ( ph -> ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( 0 ^ k ) ) ) finSupp 0 )
127	37, 38, 114, 116, 44, 75, 126	tsmsid 21943	. . . . 5 |- ( ph -> (ℂfld gsumg ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( 0 ^ k ) ) ) ) e. (ℂfld tsums ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( 0 ^ k ) ) ) ) )
128	112, 127	eqeltrrd 2702	. . . 4 |- ( ph -> ( F ` B ) e. (ℂfld tsums ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( 0 ^ k ) ) ) ) )
129	36	subidd 10380	. . . . . . . 8 |- ( ph -> ( B - B ) = 0 )
130	129	oveq1d 6665	. . . . . . 7 |- ( ph -> ( ( B - B ) ^ k ) = ( 0 ^ k ) )
131	130	oveq2d 6666	. . . . . 6 |- ( ph -> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( B - B ) ^ k ) ) = ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( 0 ^ k ) ) )
132	131	mpteq2dv 4745	. . . . 5 |- ( ph -> ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( B - B ) ^ k ) ) ) = ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( 0 ^ k ) ) ) )
133	132	oveq2d 6666	. . . 4 |- ( ph -> (ℂfld tsums ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( B - B ) ^ k ) ) ) ) = (ℂfld tsums ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( 0 ^ k ) ) ) ) )
134	128, 133	eleqtrrd 2704	. . 3 |- ( ph -> ( F ` B ) e. (ℂfld tsums ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( B - B ) ^ k ) ) ) ) )
135		taylfval.t	. . . 4 |- T = ( N ( S Tayl F ) B )
136	2, 26, 1, 11, 9, 135	eltayl 24114	. . 3 |- ( ph -> ( B T ( F ` B ) <-> ( B e. CC /\ ( F ` B ) e. (ℂfld tsums ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( B - B ) ^ k ) ) ) ) ) ) )
137	36, 134, 136	mpbir2and 957	. 2 |- ( ph -> B T ( F ` B ) )
138	2, 26, 1, 11, 9, 135	taylf 24115	. . 3 |- ( ph -> T : dom T --> CC )
139		ffun 6048	. . 3 |- ( T : dom T --> CC -> Fun T )
140		funbrfv2b 6240	. . 3 |- ( Fun T -> ( B T ( F ` B ) <-> ( B e. dom T /\ ( T ` B ) = ( F ` B ) ) ) )
141	138, 139, 140	3syl 18	. 2 |- ( ph -> ( B T ( F ` B ) <-> ( B e. dom T /\ ( T ` B ) = ( F ` B ) ) ) )
142	137, 141	mpbid 222	1 |- ( ph -> ( B e. dom T /\ ( T ` B ) = ( F ` B ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   \ cdif 3571   i^i cin 3573   C_ wss 3574   {csn 4177   {cpr 4179   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   supp csupp 7295   ^pm cpm 7858   Fincfn 7955   finSupp cfsupp 8275   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   +oocpnf 10071   RR*cxr 10073   <_ cle 10075   - cmin 10266   / cdiv 10684   NNcn 11020   NN0cn0 11292  NN0*cxnn0 11363   ZZcz 11377   [,]cicc 12178   ^cexp 12860   !cfa 13060   gsum_g cgsu 16101   Mndcmnd 17294  CMndccmn 18193   Ringcrg 18547  ℂ_fldccnfld 19746   TopSpctps 20736   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-xnn0 11364  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-ress 15865  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-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-minusg 17426  df-mulg 17541  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:  dvntaylp0  24126