Theorem binomcxplemdvsum 38554

Description: Lemma for binomcxp 38556. The derivative of the generalized sum in binomcxplemnn0 38548. Part of remark "This convergence allows us to apply term-by-term differentiation..." in the Wikibooks proof. (Contributed by Steve Rodriguez, 22-Apr-2020.)

Hypotheses

Ref	Expression
binomcxp.a	|- ( ph -> A e. RR+ )
binomcxp.b	|- ( ph -> B e. RR )
binomcxp.lt	|- ( ph -> ( abs ` B ) < ( abs ` A ) )
binomcxp.c	|- ( ph -> C e. CC )
binomcxplem.f	|- F = ( j e. NN0 |-> ( CC𝑐 j ) )
binomcxplem.s	|- S = ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) )
binomcxplem.r	|- R = sup ( { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } , RR* , < )
binomcxplem.e	|- E = ( b e. CC |-> ( k e. NN |-> ( ( k x. ( F ` k ) ) x. ( b ^ ( k - 1 ) ) ) ) )
binomcxplem.d	|- D = ( `' abs " ( 0 [,) R ) )
binomcxplem.p	|- P = ( b e. D |-> sum_ k e. NN0 ( ( S ` b ) ` k ) )

Assertion

Ref	Expression
binomcxplemdvsum	|- ( ph -> ( CC _D P ) = ( b e. D |-> sum_ k e. NN ( ( E ` b ) ` k ) ) )

Distinct variable groups:   k, b, F   ph, b, k   r, b, k, F   j, k, ph   C, j

Allowed substitution hints:   ph( r)   A( j, k, r, b)   B( j, k, r, b)   C( k, r, b)   D( j, k, r, b)   P( j, k, r, b)   R( j, k, r, b)   S( j, k, r, b)   E( j, k, r, b)   F( j)

Proof of Theorem binomcxplemdvsum

Dummy variables m n x y z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		binomcxplem.s	. . . 4 |- S = ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) )
2		binomcxplem.p	. . . . 5 |- P = ( b e. D |-> sum_ k e. NN0 ( ( S ` b ) ` k ) )
3		binomcxplem.d	. . . . . . 7 |- D = ( `' abs " ( 0 [,) R ) )
4		nfcv 2764	. . . . . . . 8 |- F/_ b `' abs
5		nfcv 2764	. . . . . . . . 9 |- F/_ b 0
6		nfcv 2764	. . . . . . . . 9 |- F/_ b [,)
7		binomcxplem.r	. . . . . . . . . 10 |- R = sup ( { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } , RR* , < )
8		nfcv 2764	. . . . . . . . . . . . . 14 |- F/_ b +
9		nfmpt1 4747	. . . . . . . . . . . . . . . 16 |- F/_ b ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) )
10	1, 9	nfcxfr 2762	. . . . . . . . . . . . . . 15 |- F/_ b S
11		nfcv 2764	. . . . . . . . . . . . . . 15 |- F/_ b r
12	10, 11	nffv 6198	. . . . . . . . . . . . . 14 |- F/_ b ( S ` r )
13	5, 8, 12	nfseq 12811	. . . . . . . . . . . . 13 |- F/_ b seq 0 ( + , ( S ` r ) )
14	13	nfel1 2779	. . . . . . . . . . . 12 |- F/ b seq 0 ( + , ( S ` r ) ) e. dom ~~>
15		nfcv 2764	. . . . . . . . . . . 12 |- F/_ b RR
16	14, 15	nfrab 3123	. . . . . . . . . . 11 |- F/_ b { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> }
17		nfcv 2764	. . . . . . . . . . 11 |- F/_ b RR*
18		nfcv 2764	. . . . . . . . . . 11 |- F/_ b <
19	16, 17, 18	nfsup 8357	. . . . . . . . . 10 |- F/_ b sup ( { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } , RR* , < )
20	7, 19	nfcxfr 2762	. . . . . . . . 9 |- F/_ b R
21	5, 6, 20	nfov 6676	. . . . . . . 8 |- F/_ b ( 0 [,) R )
22	4, 21	nfima 5474	. . . . . . 7 |- F/_ b ( `' abs " ( 0 [,) R ) )
23	3, 22	nfcxfr 2762	. . . . . 6 |- F/_ b D
24		nfcv 2764	. . . . . 6 |- F/_ y D
25		nfcv 2764	. . . . . 6 |- F/_ y sum_ k e. NN0 ( ( S ` b ) ` k )
26		nfcv 2764	. . . . . . 7 |- F/_ b NN0
27		nfcv 2764	. . . . . . . . 9 |- F/_ b y
28	10, 27	nffv 6198	. . . . . . . 8 |- F/_ b ( S ` y )
29		nfcv 2764	. . . . . . . 8 |- F/_ b m
30	28, 29	nffv 6198	. . . . . . 7 |- F/_ b ( ( S ` y ) ` m )
31	26, 30	nfsum 14421	. . . . . 6 |- F/_ b sum_ m e. NN0 ( ( S ` y ) ` m )
32		simpl 473	. . . . . . . . . 10 |- ( ( b = y /\ k e. NN0 ) -> b = y )
33	32	fveq2d 6195	. . . . . . . . 9 |- ( ( b = y /\ k e. NN0 ) -> ( S ` b ) = ( S ` y ) )
34	33	fveq1d 6193	. . . . . . . 8 |- ( ( b = y /\ k e. NN0 ) -> ( ( S ` b ) ` k ) = ( ( S ` y ) ` k ) )
35	34	sumeq2dv 14433	. . . . . . 7 |- ( b = y -> sum_ k e. NN0 ( ( S ` b ) ` k ) = sum_ k e. NN0 ( ( S ` y ) ` k ) )
36		nfcv 2764	. . . . . . . 8 |- F/_ m ( ( S ` y ) ` k )
37		nfcv 2764	. . . . . . . . . . . 12 |- F/_ k CC
38		nfmpt1 4747	. . . . . . . . . . . 12 |- F/_ k ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) )
39	37, 38	nfmpt 4746	. . . . . . . . . . 11 |- F/_ k ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) )
40	1, 39	nfcxfr 2762	. . . . . . . . . 10 |- F/_ k S
41		nfcv 2764	. . . . . . . . . 10 |- F/_ k y
42	40, 41	nffv 6198	. . . . . . . . 9 |- F/_ k ( S ` y )
43		nfcv 2764	. . . . . . . . 9 |- F/_ k m
44	42, 43	nffv 6198	. . . . . . . 8 |- F/_ k ( ( S ` y ) ` m )
45		fveq2 6191	. . . . . . . 8 |- ( k = m -> ( ( S ` y ) ` k ) = ( ( S ` y ) ` m ) )
46	36, 44, 45	cbvsumi 14427	. . . . . . 7 |- sum_ k e. NN0 ( ( S ` y ) ` k ) = sum_ m e. NN0 ( ( S ` y ) ` m )
47	35, 46	syl6eq 2672	. . . . . 6 |- ( b = y -> sum_ k e. NN0 ( ( S ` b ) ` k ) = sum_ m e. NN0 ( ( S ` y ) ` m ) )
48	23, 24, 25, 31, 47	cbvmptf 4748	. . . . 5 |- ( b e. D |-> sum_ k e. NN0 ( ( S ` b ) ` k ) ) = ( y e. D |-> sum_ m e. NN0 ( ( S ` y ) ` m ) )
49	2, 48	eqtri 2644	. . . 4 |- P = ( y e. D |-> sum_ m e. NN0 ( ( S ` y ) ` m ) )
50		ovexd 6680	. . . . 5 |- ( ( ph /\ j e. NN0 ) -> ( CC𝑐 j ) e. _V )
51		binomcxplem.f	. . . . . 6 |- F = ( j e. NN0 |-> ( CC𝑐 j ) )
52	51	a1i 11	. . . . 5 |- ( ph -> F = ( j e. NN0 |-> ( CC𝑐 j ) ) )
53	51	a1i 11	. . . . . . 7 |- ( ( ph /\ k e. NN0 ) -> F = ( j e. NN0 |-> ( CC𝑐 j ) ) )
54		simpr 477	. . . . . . . 8 |- ( ( ( ph /\ k e. NN0 ) /\ j = k ) -> j = k )
55	54	oveq2d 6666	. . . . . . 7 |- ( ( ( ph /\ k e. NN0 ) /\ j = k ) -> ( CC𝑐 j ) = ( CC𝑐 k ) )
56		simpr 477	. . . . . . 7 |- ( ( ph /\ k e. NN0 ) -> k e. NN0 )
57		binomcxp.c	. . . . . . . . 9 |- ( ph -> C e. CC )
58	57	adantr 481	. . . . . . . 8 |- ( ( ph /\ k e. NN0 ) -> C e. CC )
59	58, 56	bcccl 38538	. . . . . . 7 |- ( ( ph /\ k e. NN0 ) -> ( CC𝑐 k ) e. CC )
60	53, 55, 56, 59	fvmptd 6288	. . . . . 6 |- ( ( ph /\ k e. NN0 ) -> ( F ` k ) = ( CC𝑐 k ) )
61	60, 59	eqeltrd 2701	. . . . 5 |- ( ( ph /\ k e. NN0 ) -> ( F ` k ) e. CC )
62	50, 52, 61	fmpt2d 6393	. . . 4 |- ( ph -> F : NN0 --> CC )
63		nfcv 2764	. . . . . . 7 |- F/_ r RR
64		nfcv 2764	. . . . . . 7 |- F/_ z RR
65		nfv 1843	. . . . . . 7 |- F/ z seq 0 ( + , ( S ` r ) ) e. dom ~~>
66		nfcv 2764	. . . . . . . . 9 |- F/_ r 0
67		nfcv 2764	. . . . . . . . 9 |- F/_ r +
68		nfcv 2764	. . . . . . . . . . 11 |- F/_ r ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) )
69	1, 68	nfcxfr 2762	. . . . . . . . . 10 |- F/_ r S
70		nfcv 2764	. . . . . . . . . 10 |- F/_ r z
71	69, 70	nffv 6198	. . . . . . . . 9 |- F/_ r ( S ` z )
72	66, 67, 71	nfseq 12811	. . . . . . . 8 |- F/_ r seq 0 ( + , ( S ` z ) )
73	72	nfel1 2779	. . . . . . 7 |- F/ r seq 0 ( + , ( S ` z ) ) e. dom ~~>
74		fveq2 6191	. . . . . . . . 9 |- ( r = z -> ( S ` r ) = ( S ` z ) )
75	74	seqeq3d 12809	. . . . . . . 8 |- ( r = z -> seq 0 ( + , ( S ` r ) ) = seq 0 ( + , ( S ` z ) ) )
76	75	eleq1d 2686	. . . . . . 7 |- ( r = z -> ( seq 0 ( + , ( S ` r ) ) e. dom ~~> <-> seq 0 ( + , ( S ` z ) ) e. dom ~~> ) )
77	63, 64, 65, 73, 76	cbvrab 3198	. . . . . 6 |- { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } = { z e. RR | seq 0 ( + , ( S ` z ) ) e. dom ~~> }
78	77	supeq1i 8353	. . . . 5 |- sup ( { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } , RR* , < ) = sup ( { z e. RR | seq 0 ( + , ( S ` z ) ) e. dom ~~> } , RR* , < )
79	7, 78	eqtri 2644	. . . 4 |- R = sup ( { z e. RR | seq 0 ( + , ( S ` z ) ) e. dom ~~> } , RR* , < )
80	1	fveq1i 6192	. . . . . . . . . . . 12 |- ( S ` z ) = ( ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) ) ` z )
81		seqeq3 12806	. . . . . . . . . . . 12 |- ( ( S ` z ) = ( ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) ) ` z ) -> seq 0 ( + , ( S ` z ) ) = seq 0 ( + , ( ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) ) ` z ) ) )
82	80, 81	ax-mp 5	. . . . . . . . . . 11 |- seq 0 ( + , ( S ` z ) ) = seq 0 ( + , ( ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) ) ` z ) )
83	82	eleq1i 2692	. . . . . . . . . 10 |- ( seq 0 ( + , ( S ` z ) ) e. dom ~~> <-> seq 0 ( + , ( ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) ) ` z ) ) e. dom ~~> )
84	83	a1i 11	. . . . . . . . 9 |- ( z e. RR -> ( seq 0 ( + , ( S ` z ) ) e. dom ~~> <-> seq 0 ( + , ( ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) ) ` z ) ) e. dom ~~> ) )
85	84	rabbiia 3185	. . . . . . . 8 |- { z e. RR | seq 0 ( + , ( S ` z ) ) e. dom ~~> } = { z e. RR | seq 0 ( + , ( ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) ) ` z ) ) e. dom ~~> }
86	85	supeq1i 8353	. . . . . . 7 |- sup ( { z e. RR | seq 0 ( + , ( S ` z ) ) e. dom ~~> } , RR* , < ) = sup ( { z e. RR | seq 0 ( + , ( ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) ) ` z ) ) e. dom ~~> } , RR* , < )
87	7, 78, 86	3eqtrri 2649	. . . . . 6 |- sup ( { z e. RR | seq 0 ( + , ( ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) ) ` z ) ) e. dom ~~> } , RR* , < ) = R
88	87	eleq1i 2692	. . . . 5 |- ( sup ( { z e. RR | seq 0 ( + , ( ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) ) ` z ) ) e. dom ~~> } , RR* , < ) e. RR <-> R e. RR )
89	87	oveq2i 6661	. . . . . 6 |- ( ( abs ` x ) + sup ( { z e. RR | seq 0 ( + , ( ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) ) ` z ) ) e. dom ~~> } , RR* , < ) ) = ( ( abs ` x ) + R )
90	89	oveq1i 6660	. . . . 5 |- ( ( ( abs ` x ) + sup ( { z e. RR | seq 0 ( + , ( ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) ) ` z ) ) e. dom ~~> } , RR* , < ) ) / 2 ) = ( ( ( abs ` x ) + R ) / 2 )
91		eqid 2622	. . . . 5 |- ( ( abs ` x ) + 1 ) = ( ( abs ` x ) + 1 )
92	88, 90, 91	ifbieq12i 4112	. . . 4 |- if ( sup ( { z e. RR | seq 0 ( + , ( ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) ) ` z ) ) e. dom ~~> } , RR* , < ) e. RR , ( ( ( abs ` x ) + sup ( { z e. RR | seq 0 ( + , ( ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) ) ` z ) ) e. dom ~~> } , RR* , < ) ) / 2 ) , ( ( abs ` x ) + 1 ) ) = if ( R e. RR , ( ( ( abs ` x ) + R ) / 2 ) , ( ( abs ` x ) + 1 ) )
93		oveq1 6657	. . . . . . . . . . . . . . . . . 18 |- ( w = b -> ( w ^ k ) = ( b ^ k ) )
94	93	oveq2d 6666	. . . . . . . . . . . . . . . . 17 |- ( w = b -> ( ( F ` k ) x. ( w ^ k ) ) = ( ( F ` k ) x. ( b ^ k ) ) )
95	94	mpteq2dv 4745	. . . . . . . . . . . . . . . 16 |- ( w = b -> ( k e. NN0 |-> ( ( F ` k ) x. ( w ^ k ) ) ) = ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) )
96	95	cbvmptv 4750	. . . . . . . . . . . . . . 15 |- ( w e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( w ^ k ) ) ) ) = ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) )
97	96	fveq1i 6192	. . . . . . . . . . . . . 14 |- ( ( w e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( w ^ k ) ) ) ) ` z ) = ( ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) ) ` z )
98		seqeq3 12806	. . . . . . . . . . . . . 14 |- ( ( ( w e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( w ^ k ) ) ) ) ` z ) = ( ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) ) ` z ) -> seq 0 ( + , ( ( w e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( w ^ k ) ) ) ) ` z ) ) = seq 0 ( + , ( ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) ) ` z ) ) )
99	97, 98	ax-mp 5	. . . . . . . . . . . . 13 |- seq 0 ( + , ( ( w e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( w ^ k ) ) ) ) ` z ) ) = seq 0 ( + , ( ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) ) ` z ) )
100	99	eleq1i 2692	. . . . . . . . . . . 12 |- ( seq 0 ( + , ( ( w e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( w ^ k ) ) ) ) ` z ) ) e. dom ~~> <-> seq 0 ( + , ( ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) ) ` z ) ) e. dom ~~> )
101	100	a1i 11	. . . . . . . . . . 11 |- ( z e. RR -> ( seq 0 ( + , ( ( w e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( w ^ k ) ) ) ) ` z ) ) e. dom ~~> <-> seq 0 ( + , ( ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) ) ` z ) ) e. dom ~~> ) )
102	101	rabbiia 3185	. . . . . . . . . 10 |- { z e. RR | seq 0 ( + , ( ( w e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( w ^ k ) ) ) ) ` z ) ) e. dom ~~> } = { z e. RR | seq 0 ( + , ( ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) ) ` z ) ) e. dom ~~> }
103	102	supeq1i 8353	. . . . . . . . 9 |- sup ( { z e. RR | seq 0 ( + , ( ( w e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( w ^ k ) ) ) ) ` z ) ) e. dom ~~> } , RR* , < ) = sup ( { z e. RR | seq 0 ( + , ( ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) ) ` z ) ) e. dom ~~> } , RR* , < )
104	103	eleq1i 2692	. . . . . . . 8 |- ( sup ( { z e. RR | seq 0 ( + , ( ( w e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( w ^ k ) ) ) ) ` z ) ) e. dom ~~> } , RR* , < ) e. RR <-> sup ( { z e. RR | seq 0 ( + , ( ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) ) ` z ) ) e. dom ~~> } , RR* , < ) e. RR )
105	103	oveq2i 6661	. . . . . . . . 9 |- ( ( abs ` x ) + sup ( { z e. RR | seq 0 ( + , ( ( w e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( w ^ k ) ) ) ) ` z ) ) e. dom ~~> } , RR* , < ) ) = ( ( abs ` x ) + sup ( { z e. RR | seq 0 ( + , ( ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) ) ` z ) ) e. dom ~~> } , RR* , < ) )
106	105	oveq1i 6660	. . . . . . . 8 |- ( ( ( abs ` x ) + sup ( { z e. RR | seq 0 ( + , ( ( w e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( w ^ k ) ) ) ) ` z ) ) e. dom ~~> } , RR* , < ) ) / 2 ) = ( ( ( abs ` x ) + sup ( { z e. RR | seq 0 ( + , ( ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) ) ` z ) ) e. dom ~~> } , RR* , < ) ) / 2 )
107	104, 106, 91	ifbieq12i 4112	. . . . . . 7 |- if ( sup ( { z e. RR | seq 0 ( + , ( ( w e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( w ^ k ) ) ) ) ` z ) ) e. dom ~~> } , RR* , < ) e. RR , ( ( ( abs ` x ) + sup ( { z e. RR | seq 0 ( + , ( ( w e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( w ^ k ) ) ) ) ` z ) ) e. dom ~~> } , RR* , < ) ) / 2 ) , ( ( abs ` x ) + 1 ) ) = if ( sup ( { z e. RR | seq 0 ( + , ( ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) ) ` z ) ) e. dom ~~> } , RR* , < ) e. RR , ( ( ( abs ` x ) + sup ( { z e. RR | seq 0 ( + , ( ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) ) ` z ) ) e. dom ~~> } , RR* , < ) ) / 2 ) , ( ( abs ` x ) + 1 ) )
108	107	oveq2i 6661	. . . . . 6 |- ( ( abs ` x ) + if ( sup ( { z e. RR | seq 0 ( + , ( ( w e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( w ^ k ) ) ) ) ` z ) ) e. dom ~~> } , RR* , < ) e. RR , ( ( ( abs ` x ) + sup ( { z e. RR | seq 0 ( + , ( ( w e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( w ^ k ) ) ) ) ` z ) ) e. dom ~~> } , RR* , < ) ) / 2 ) , ( ( abs ` x ) + 1 ) ) ) = ( ( abs ` x ) + if ( sup ( { z e. RR | seq 0 ( + , ( ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) ) ` z ) ) e. dom ~~> } , RR* , < ) e. RR , ( ( ( abs ` x ) + sup ( { z e. RR | seq 0 ( + , ( ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) ) ` z ) ) e. dom ~~> } , RR* , < ) ) / 2 ) , ( ( abs ` x ) + 1 ) ) )
109	108	oveq1i 6660	. . . . 5 |- ( ( ( abs ` x ) + if ( sup ( { z e. RR | seq 0 ( + , ( ( w e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( w ^ k ) ) ) ) ` z ) ) e. dom ~~> } , RR* , < ) e. RR , ( ( ( abs ` x ) + sup ( { z e. RR | seq 0 ( + , ( ( w e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( w ^ k ) ) ) ) ` z ) ) e. dom ~~> } , RR* , < ) ) / 2 ) , ( ( abs ` x ) + 1 ) ) ) / 2 ) = ( ( ( abs ` x ) + if ( sup ( { z e. RR | seq 0 ( + , ( ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) ) ` z ) ) e. dom ~~> } , RR* , < ) e. RR , ( ( ( abs ` x ) + sup ( { z e. RR | seq 0 ( + , ( ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) ) ` z ) ) e. dom ~~> } , RR* , < ) ) / 2 ) , ( ( abs ` x ) + 1 ) ) ) / 2 )
110	109	oveq2i 6661	. . . 4 |- ( 0 ( ball ` ( abs o. - ) ) ( ( ( abs ` x ) + if ( sup ( { z e. RR | seq 0 ( + , ( ( w e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( w ^ k ) ) ) ) ` z ) ) e. dom ~~> } , RR* , < ) e. RR , ( ( ( abs ` x ) + sup ( { z e. RR | seq 0 ( + , ( ( w e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( w ^ k ) ) ) ) ` z ) ) e. dom ~~> } , RR* , < ) ) / 2 ) , ( ( abs ` x ) + 1 ) ) ) / 2 ) ) = ( 0 ( ball ` ( abs o. - ) ) ( ( ( abs ` x ) + if ( sup ( { z e. RR | seq 0 ( + , ( ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) ) ` z ) ) e. dom ~~> } , RR* , < ) e. RR , ( ( ( abs ` x ) + sup ( { z e. RR | seq 0 ( + , ( ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) ) ` z ) ) e. dom ~~> } , RR* , < ) ) / 2 ) , ( ( abs ` x ) + 1 ) ) ) / 2 ) )
111	1, 49, 62, 79, 3, 92, 110	pserdv2 24184	. . 3 |- ( ph -> ( CC _D P ) = ( y e. D |-> sum_ n e. NN ( ( n x. ( F ` n ) ) x. ( y ^ ( n - 1 ) ) ) ) )
112		cnvimass 5485	. . . . . . . 8 |- ( `' abs " ( 0 [,) R ) ) C_ dom abs
113	3, 112	eqsstri 3635	. . . . . . 7 |- D C_ dom abs
114		absf 14077	. . . . . . . 8 |- abs : CC --> RR
115	114	fdmi 6052	. . . . . . 7 |- dom abs = CC
116	113, 115	sseqtri 3637	. . . . . 6 |- D C_ CC
117	116	sseli 3599	. . . . 5 |- ( y e. D -> y e. CC )
118		binomcxplem.e	. . . . . . . . . 10 |- E = ( b e. CC |-> ( k e. NN |-> ( ( k x. ( F ` k ) ) x. ( b ^ ( k - 1 ) ) ) ) )
119	118	a1i 11	. . . . . . . . 9 |- ( ( ph /\ y e. CC ) -> E = ( b e. CC |-> ( k e. NN |-> ( ( k x. ( F ` k ) ) x. ( b ^ ( k - 1 ) ) ) ) ) )
120		simplr 792	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ y e. CC ) /\ b = y ) /\ k e. NN ) -> b = y )
121	120	oveq1d 6665	. . . . . . . . . . 11 |- ( ( ( ( ph /\ y e. CC ) /\ b = y ) /\ k e. NN ) -> ( b ^ ( k - 1 ) ) = ( y ^ ( k - 1 ) ) )
122	121	oveq2d 6666	. . . . . . . . . 10 |- ( ( ( ( ph /\ y e. CC ) /\ b = y ) /\ k e. NN ) -> ( ( k x. ( F ` k ) ) x. ( b ^ ( k - 1 ) ) ) = ( ( k x. ( F ` k ) ) x. ( y ^ ( k - 1 ) ) ) )
123	122	mpteq2dva 4744	. . . . . . . . 9 |- ( ( ( ph /\ y e. CC ) /\ b = y ) -> ( k e. NN |-> ( ( k x. ( F ` k ) ) x. ( b ^ ( k - 1 ) ) ) ) = ( k e. NN |-> ( ( k x. ( F ` k ) ) x. ( y ^ ( k - 1 ) ) ) ) )
124		simpr 477	. . . . . . . . 9 |- ( ( ph /\ y e. CC ) -> y e. CC )
125		nnex 11026	. . . . . . . . . . 11 |- NN e. _V
126	125	mptex 6486	. . . . . . . . . 10 |- ( k e. NN |-> ( ( k x. ( F ` k ) ) x. ( y ^ ( k - 1 ) ) ) ) e. _V
127	126	a1i 11	. . . . . . . . 9 |- ( ( ph /\ y e. CC ) -> ( k e. NN |-> ( ( k x. ( F ` k ) ) x. ( y ^ ( k - 1 ) ) ) ) e. _V )
128	119, 123, 124, 127	fvmptd 6288	. . . . . . . 8 |- ( ( ph /\ y e. CC ) -> ( E ` y ) = ( k e. NN |-> ( ( k x. ( F ` k ) ) x. ( y ^ ( k - 1 ) ) ) ) )
129	128	adantr 481	. . . . . . 7 |- ( ( ( ph /\ y e. CC ) /\ n e. NN ) -> ( E ` y ) = ( k e. NN |-> ( ( k x. ( F ` k ) ) x. ( y ^ ( k - 1 ) ) ) ) )
130		simpr 477	. . . . . . . . 9 |- ( ( ( ( ph /\ y e. CC ) /\ n e. NN ) /\ k = n ) -> k = n )
131	130	fveq2d 6195	. . . . . . . . 9 |- ( ( ( ( ph /\ y e. CC ) /\ n e. NN ) /\ k = n ) -> ( F ` k ) = ( F ` n ) )
132	130, 131	oveq12d 6668	. . . . . . . 8 |- ( ( ( ( ph /\ y e. CC ) /\ n e. NN ) /\ k = n ) -> ( k x. ( F ` k ) ) = ( n x. ( F ` n ) ) )
133	130	oveq1d 6665	. . . . . . . . 9 |- ( ( ( ( ph /\ y e. CC ) /\ n e. NN ) /\ k = n ) -> ( k - 1 ) = ( n - 1 ) )
134	133	oveq2d 6666	. . . . . . . 8 |- ( ( ( ( ph /\ y e. CC ) /\ n e. NN ) /\ k = n ) -> ( y ^ ( k - 1 ) ) = ( y ^ ( n - 1 ) ) )
135	132, 134	oveq12d 6668	. . . . . . 7 |- ( ( ( ( ph /\ y e. CC ) /\ n e. NN ) /\ k = n ) -> ( ( k x. ( F ` k ) ) x. ( y ^ ( k - 1 ) ) ) = ( ( n x. ( F ` n ) ) x. ( y ^ ( n - 1 ) ) ) )
136		simpr 477	. . . . . . 7 |- ( ( ( ph /\ y e. CC ) /\ n e. NN ) -> n e. NN )
137		ovexd 6680	. . . . . . 7 |- ( ( ( ph /\ y e. CC ) /\ n e. NN ) -> ( ( n x. ( F ` n ) ) x. ( y ^ ( n - 1 ) ) ) e. _V )
138	129, 135, 136, 137	fvmptd 6288	. . . . . 6 |- ( ( ( ph /\ y e. CC ) /\ n e. NN ) -> ( ( E ` y ) ` n ) = ( ( n x. ( F ` n ) ) x. ( y ^ ( n - 1 ) ) ) )
139	138	sumeq2dv 14433	. . . . 5 |- ( ( ph /\ y e. CC ) -> sum_ n e. NN ( ( E ` y ) ` n ) = sum_ n e. NN ( ( n x. ( F ` n ) ) x. ( y ^ ( n - 1 ) ) ) )
140	117, 139	sylan2 491	. . . 4 |- ( ( ph /\ y e. D ) -> sum_ n e. NN ( ( E ` y ) ` n ) = sum_ n e. NN ( ( n x. ( F ` n ) ) x. ( y ^ ( n - 1 ) ) ) )
141	140	mpteq2dva 4744	. . 3 |- ( ph -> ( y e. D |-> sum_ n e. NN ( ( E ` y ) ` n ) ) = ( y e. D |-> sum_ n e. NN ( ( n x. ( F ` n ) ) x. ( y ^ ( n - 1 ) ) ) ) )
142	111, 141	eqtr4d 2659	. 2 |- ( ph -> ( CC _D P ) = ( y e. D |-> sum_ n e. NN ( ( E ` y ) ` n ) ) )
143		nfcv 2764	. . . 4 |- F/_ b NN
144		nfmpt1 4747	. . . . . . 7 |- F/_ b ( b e. CC |-> ( k e. NN |-> ( ( k x. ( F ` k ) ) x. ( b ^ ( k - 1 ) ) ) ) )
145	118, 144	nfcxfr 2762	. . . . . 6 |- F/_ b E
146	145, 27	nffv 6198	. . . . 5 |- F/_ b ( E ` y )
147		nfcv 2764	. . . . 5 |- F/_ b n
148	146, 147	nffv 6198	. . . 4 |- F/_ b ( ( E ` y ) ` n )
149	143, 148	nfsum 14421	. . 3 |- F/_ b sum_ n e. NN ( ( E ` y ) ` n )
150		nfcv 2764	. . 3 |- F/_ y sum_ k e. NN ( ( E ` b ) ` k )
151		simpl 473	. . . . . . 7 |- ( ( y = b /\ n e. NN ) -> y = b )
152	151	fveq2d 6195	. . . . . 6 |- ( ( y = b /\ n e. NN ) -> ( E ` y ) = ( E ` b ) )
153	152	fveq1d 6193	. . . . 5 |- ( ( y = b /\ n e. NN ) -> ( ( E ` y ) ` n ) = ( ( E ` b ) ` n ) )
154	153	sumeq2dv 14433	. . . 4 |- ( y = b -> sum_ n e. NN ( ( E ` y ) ` n ) = sum_ n e. NN ( ( E ` b ) ` n ) )
155		nfmpt1 4747	. . . . . . . . 9 |- F/_ k ( k e. NN |-> ( ( k x. ( F ` k ) ) x. ( b ^ ( k - 1 ) ) ) )
156	37, 155	nfmpt 4746	. . . . . . . 8 |- F/_ k ( b e. CC |-> ( k e. NN |-> ( ( k x. ( F ` k ) ) x. ( b ^ ( k - 1 ) ) ) ) )
157	118, 156	nfcxfr 2762	. . . . . . 7 |- F/_ k E
158		nfcv 2764	. . . . . . 7 |- F/_ k b
159	157, 158	nffv 6198	. . . . . 6 |- F/_ k ( E ` b )
160		nfcv 2764	. . . . . 6 |- F/_ k n
161	159, 160	nffv 6198	. . . . 5 |- F/_ k ( ( E ` b ) ` n )
162		nfcv 2764	. . . . 5 |- F/_ n ( ( E ` b ) ` k )
163		fveq2 6191	. . . . 5 |- ( n = k -> ( ( E ` b ) ` n ) = ( ( E ` b ) ` k ) )
164	161, 162, 163	cbvsumi 14427	. . . 4 |- sum_ n e. NN ( ( E ` b ) ` n ) = sum_ k e. NN ( ( E ` b ) ` k )
165	154, 164	syl6eq 2672	. . 3 |- ( y = b -> sum_ n e. NN ( ( E ` y ) ` n ) = sum_ k e. NN ( ( E ` b ) ` k ) )
166	24, 23, 149, 150, 165	cbvmptf 4748	. 2 |- ( y e. D |-> sum_ n e. NN ( ( E ` y ) ` n ) ) = ( b e. D |-> sum_ k e. NN ( ( E ` b ) ` k ) )
167	142, 166	syl6eq 2672	1 |- ( ph -> ( CC _D P ) = ( b e. D |-> sum_ k e. NN ( ( E ` b ) ` k ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   `'ccnv 5113   dom cdm 5114   "cima 5117   o. ccom 5118   ` cfv 5888  (class class class)co 6650   supcsup 8346   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   RR*cxr 10073   < clt 10074   - cmin 10266   / cdiv 10684   NNcn 11020   2c2 11070   NN0cn0 11292   RR+crp 11832   [,)cico 12177   seqcseq 12801   ^cexp 12860   abscabs 13974   ~~> cli 14215   sum_csu 14416   ballcbl 19733   _D cdv 23627  C_𝑐cbcc 38535

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-fal 1489  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-of 6897  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-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  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-cda 8990  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-ioo 12179  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-fac 13061  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-prod 14636  df-fallfac 14738  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-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  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-cn 21031  df-cnp 21032  df-haus 21119  df-cmp 21190  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631  df-ulm 24131  df-bcc 38536

This theorem is referenced by:  binomcxplemnotnn0  38555