Theorem binomcxplemdvsum 36704

Description: Lemma for binomcxp 36706. The derivative of the generalized sum in binomcxplemnn0 36698. 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 2592	. . . . . . . 8 |- F/_ b `' abs
5		nfcv 2592	. . . . . . . . 9 |- F/_ b 0
6		nfcv 2592	. . . . . . . . 9 |- F/_ b [,)
7		binomcxplem.r	. . . . . . . . . 10 |- R = sup ( { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } , RR* , < )
8		nfcv 2592	. . . . . . . . . . . . . 14 |- F/_ b +
9		nfmpt1 4492	. . . . . . . . . . . . . . . 16 |- F/_ b ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) )
10	1, 9	nfcxfr 2590	. . . . . . . . . . . . . . 15 |- F/_ b S
11		nfcv 2592	. . . . . . . . . . . . . . 15 |- F/_ b r
12	10, 11	nffv 5872	. . . . . . . . . . . . . 14 |- F/_ b ( S ` r )
13	5, 8, 12	nfseq 12223	. . . . . . . . . . . . 13 |- F/_ b seq 0 ( + , ( S ` r ) )
14	13	nfel1 2606	. . . . . . . . . . . 12 |- F/ b seq 0 ( + , ( S ` r ) ) e. dom ~~>
15		nfcv 2592	. . . . . . . . . . . 12 |- F/_ b RR
16	14, 15	nfrab 2972	. . . . . . . . . . 11 |- F/_ b { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> }
17		nfcv 2592	. . . . . . . . . . 11 |- F/_ b RR*
18		nfcv 2592	. . . . . . . . . . 11 |- F/_ b <
19	16, 17, 18	nfsup 7965	. . . . . . . . . 10 |- F/_ b sup ( { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } , RR* , < )
20	7, 19	nfcxfr 2590	. . . . . . . . 9 |- F/_ b R
21	5, 6, 20	nfov 6316	. . . . . . . 8 |- F/_ b ( 0 [,) R )
22	4, 21	nfima 5176	. . . . . . 7 |- F/_ b ( `' abs " ( 0 [,) R ) )
23	3, 22	nfcxfr 2590	. . . . . 6 |- F/_ b D
24		nfcv 2592	. . . . . 6 |- F/_ y D
25		nfcv 2592	. . . . . 6 |- F/_ y sum_ k e. NN0 ( ( S ` b ) ` k )
26		nfcv 2592	. . . . . . 7 |- F/_ b NN0
27		nfcv 2592	. . . . . . . . 9 |- F/_ b y
28	10, 27	nffv 5872	. . . . . . . 8 |- F/_ b ( S ` y )
29		nfcv 2592	. . . . . . . 8 |- F/_ b m
30	28, 29	nffv 5872	. . . . . . 7 |- F/_ b ( ( S ` y ) ` m )
31	26, 30	nfsum 13757	. . . . . 6 |- F/_ b sum_ m e. NN0 ( ( S ` y ) ` m )
32		simpl 459	. . . . . . . . . 10 |- ( ( b = y /\ k e. NN0 ) -> b = y )
33	32	fveq2d 5869	. . . . . . . . 9 |- ( ( b = y /\ k e. NN0 ) -> ( S ` b ) = ( S ` y ) )
34	33	fveq1d 5867	. . . . . . . 8 |- ( ( b = y /\ k e. NN0 ) -> ( ( S ` b ) ` k ) = ( ( S ` y ) ` k ) )
35	34	sumeq2dv 13769	. . . . . . 7 |- ( b = y -> sum_ k e. NN0 ( ( S ` b ) ` k ) = sum_ k e. NN0 ( ( S ` y ) ` k ) )
36		nfcv 2592	. . . . . . . 8 |- F/_ m ( ( S ` y ) ` k )
37		nfcv 2592	. . . . . . . . . . . 12 |- F/_ k CC
38		nfmpt1 4492	. . . . . . . . . . . 12 |- F/_ k ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) )
39	37, 38	nfmpt 4491	. . . . . . . . . . 11 |- F/_ k ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) )
40	1, 39	nfcxfr 2590	. . . . . . . . . 10 |- F/_ k S
41		nfcv 2592	. . . . . . . . . 10 |- F/_ k y
42	40, 41	nffv 5872	. . . . . . . . 9 |- F/_ k ( S ` y )
43		nfcv 2592	. . . . . . . . 9 |- F/_ k m
44	42, 43	nffv 5872	. . . . . . . 8 |- F/_ k ( ( S ` y ) ` m )
45		fveq2 5865	. . . . . . . 8 |- ( k = m -> ( ( S ` y ) ` k ) = ( ( S ` y ) ` m ) )
46	36, 44, 45	cbvsumi 13763	. . . . . . 7 |- sum_ k e. NN0 ( ( S ` y ) ` k ) = sum_ m e. NN0 ( ( S ` y ) ` m )
47	35, 46	syl6eq 2501	. . . . . 6 |- ( b = y -> sum_ k e. NN0 ( ( S ` b ) ` k ) = sum_ m e. NN0 ( ( S ` y ) ` m ) )
48	23, 24, 25, 31, 47	cbvmptf 4493	. . . . 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 2473	. . . 4 |- P = ( y e. D |-> sum_ m e. NN0 ( ( S ` y ) ` m ) )
50		ovex 6318	. . . . . 6 |- ( CC𝑐 j ) e. _V
51	50	a1i 11	. . . . 5 |- ( ( ph /\ j e. NN0 ) -> ( CC𝑐 j ) e. _V )
52		binomcxplem.f	. . . . . 6 |- F = ( j e. NN0 |-> ( CC𝑐 j ) )
53	52	a1i 11	. . . . 5 |- ( ph -> F = ( j e. NN0 |-> ( CC𝑐 j ) ) )
54	52	a1i 11	. . . . . . 7 |- ( ( ph /\ k e. NN0 ) -> F = ( j e. NN0 |-> ( CC𝑐 j ) ) )
55		simpr 463	. . . . . . . 8 |- ( ( ( ph /\ k e. NN0 ) /\ j = k ) -> j = k )
56	55	oveq2d 6306	. . . . . . 7 |- ( ( ( ph /\ k e. NN0 ) /\ j = k ) -> ( CC𝑐 j ) = ( CC𝑐 k ) )
57		simpr 463	. . . . . . 7 |- ( ( ph /\ k e. NN0 ) -> k e. NN0 )
58		binomcxp.c	. . . . . . . . 9 |- ( ph -> C e. CC )
59	58	adantr 467	. . . . . . . 8 |- ( ( ph /\ k e. NN0 ) -> C e. CC )
60	59, 57	bcccl 36688	. . . . . . 7 |- ( ( ph /\ k e. NN0 ) -> ( CC𝑐 k ) e. CC )
61	54, 56, 57, 60	fvmptd 5954	. . . . . 6 |- ( ( ph /\ k e. NN0 ) -> ( F ` k ) = ( CC𝑐 k ) )
62	61, 60	eqeltrd 2529	. . . . 5 |- ( ( ph /\ k e. NN0 ) -> ( F ` k ) e. CC )
63	51, 53, 62	fmpt2d 6053	. . . 4 |- ( ph -> F : NN0 --> CC )
64		nfcv 2592	. . . . . . 7 |- F/_ r RR
65		nfcv 2592	. . . . . . 7 |- F/_ z RR
66		nfv 1761	. . . . . . 7 |- F/ z seq 0 ( + , ( S ` r ) ) e. dom ~~>
67		nfcv 2592	. . . . . . . . 9 |- F/_ r 0
68		nfcv 2592	. . . . . . . . 9 |- F/_ r +
69		nfcv 2592	. . . . . . . . . . 11 |- F/_ r ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) )
70	1, 69	nfcxfr 2590	. . . . . . . . . 10 |- F/_ r S
71		nfcv 2592	. . . . . . . . . 10 |- F/_ r z
72	70, 71	nffv 5872	. . . . . . . . 9 |- F/_ r ( S ` z )
73	67, 68, 72	nfseq 12223	. . . . . . . 8 |- F/_ r seq 0 ( + , ( S ` z ) )
74	73	nfel1 2606	. . . . . . 7 |- F/ r seq 0 ( + , ( S ` z ) ) e. dom ~~>
75		fveq2 5865	. . . . . . . . 9 |- ( r = z -> ( S ` r ) = ( S ` z ) )
76	75	seqeq3d 12221	. . . . . . . 8 |- ( r = z -> seq 0 ( + , ( S ` r ) ) = seq 0 ( + , ( S ` z ) ) )
77	76	eleq1d 2513	. . . . . . 7 |- ( r = z -> ( seq 0 ( + , ( S ` r ) ) e. dom ~~> <-> seq 0 ( + , ( S ` z ) ) e. dom ~~> ) )
78	64, 65, 66, 74, 77	cbvrab 3043	. . . . . 6 |- { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } = { z e. RR | seq 0 ( + , ( S ` z ) ) e. dom ~~> }
79	78	supeq1i 7961	. . . . 5 |- sup ( { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } , RR* , < ) = sup ( { z e. RR | seq 0 ( + , ( S ` z ) ) e. dom ~~> } , RR* , < )
80	7, 79	eqtri 2473	. . . 4 |- R = sup ( { z e. RR | seq 0 ( + , ( S ` z ) ) e. dom ~~> } , RR* , < )
81	1	fveq1i 5866	. . . . . . . . . . . 12 |- ( S ` z ) = ( ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) ) ` z )
82		seqeq3 12218	. . . . . . . . . . . 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 ) ) )
83	81, 82	ax-mp 5	. . . . . . . . . . 11 |- seq 0 ( + , ( S ` z ) ) = seq 0 ( + , ( ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) ) ` z ) )
84	83	eleq1i 2520	. . . . . . . . . 10 |- ( seq 0 ( + , ( S ` z ) ) e. dom ~~> <-> seq 0 ( + , ( ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) ) ` z ) ) e. dom ~~> )
85	84	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 ~~> ) )
86	85	rabbiia 3033	. . . . . . . 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 ~~> }
87	86	supeq1i 7961	. . . . . . 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* , < )
88	7, 79, 87	3eqtrri 2478	. . . . . 6 |- sup ( { z e. RR | seq 0 ( + , ( ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) ) ` z ) ) e. dom ~~> } , RR* , < ) = R
89	88	eleq1i 2520	. . . . 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 )
90	88	oveq2i 6301	. . . . . 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 )
91	90	oveq1i 6300	. . . . 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 )
92		eqid 2451	. . . . 5 |- ( ( abs ` x ) + 1 ) = ( ( abs ` x ) + 1 )
93	89, 91, 92	ifbieq12i 3907	. . . 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 ) )
94		oveq1 6297	. . . . . . . . . . . . . . . . . 18 |- ( w = b -> ( w ^ k ) = ( b ^ k ) )
95	94	oveq2d 6306	. . . . . . . . . . . . . . . . 17 |- ( w = b -> ( ( F ` k ) x. ( w ^ k ) ) = ( ( F ` k ) x. ( b ^ k ) ) )
96	95	mpteq2dv 4490	. . . . . . . . . . . . . . . 16 |- ( w = b -> ( k e. NN0 |-> ( ( F ` k ) x. ( w ^ k ) ) ) = ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) )
97	96	cbvmptv 4495	. . . . . . . . . . . . . . 15 |- ( w e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( w ^ k ) ) ) ) = ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) )
98	97	fveq1i 5866	. . . . . . . . . . . . . 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 )
99		seqeq3 12218	. . . . . . . . . . . . . 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 ) ) )
100	98, 99	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 ) )
101	100	eleq1i 2520	. . . . . . . . . . . 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 ~~> )
102	101	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 ~~> ) )
103	102	rabbiia 3033	. . . . . . . . . 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 ~~> }
104	103	supeq1i 7961	. . . . . . . . 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* , < )
105	104	eleq1i 2520	. . . . . . . 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 )
106	104	oveq2i 6301	. . . . . . . . 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* , < ) )
107	106	oveq1i 6300	. . . . . . . 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 )
108	105, 107, 92	ifbieq12i 3907	. . . . . . 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 ) )
109	108	oveq2i 6301	. . . . . 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 ) ) )
110	109	oveq1i 6300	. . . . 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 )
111	110	oveq2i 6301	. . . 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 ) )
112	1, 49, 63, 80, 3, 93, 111	pserdv2 23385	. . 3 |- ( ph -> ( CC _D P ) = ( y e. D |-> sum_ n e. NN ( ( n x. ( F ` n ) ) x. ( y ^ ( n - 1 ) ) ) ) )
113		cnvimass 5188	. . . . . . . 8 |- ( `' abs " ( 0 [,) R ) ) C_ dom abs
114	3, 113	eqsstri 3462	. . . . . . 7 |- D C_ dom abs
115		absf 13400	. . . . . . . 8 |- abs : CC --> RR
116	115	fdmi 5734	. . . . . . 7 |- dom abs = CC
117	114, 116	sseqtri 3464	. . . . . 6 |- D C_ CC
118	117	sseli 3428	. . . . 5 |- ( y e. D -> y e. CC )
119		binomcxplem.e	. . . . . . . . . 10 |- E = ( b e. CC |-> ( k e. NN |-> ( ( k x. ( F ` k ) ) x. ( b ^ ( k - 1 ) ) ) ) )
120	119	a1i 11	. . . . . . . . 9 |- ( ( ph /\ y e. CC ) -> E = ( b e. CC |-> ( k e. NN |-> ( ( k x. ( F ` k ) ) x. ( b ^ ( k - 1 ) ) ) ) ) )
121		simplr 762	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ y e. CC ) /\ b = y ) /\ k e. NN ) -> b = y )
122	121	oveq1d 6305	. . . . . . . . . . 11 |- ( ( ( ( ph /\ y e. CC ) /\ b = y ) /\ k e. NN ) -> ( b ^ ( k - 1 ) ) = ( y ^ ( k - 1 ) ) )
123	122	oveq2d 6306	. . . . . . . . . 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 ) ) ) )
124	123	mpteq2dva 4489	. . . . . . . . 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 ) ) ) ) )
125		simpr 463	. . . . . . . . 9 |- ( ( ph /\ y e. CC ) -> y e. CC )
126		nnex 10615	. . . . . . . . . . 11 |- NN e. _V
127	126	mptex 6136	. . . . . . . . . 10 |- ( k e. NN |-> ( ( k x. ( F ` k ) ) x. ( y ^ ( k - 1 ) ) ) ) e. _V
128	127	a1i 11	. . . . . . . . 9 |- ( ( ph /\ y e. CC ) -> ( k e. NN |-> ( ( k x. ( F ` k ) ) x. ( y ^ ( k - 1 ) ) ) ) e. _V )
129	120, 124, 125, 128	fvmptd 5954	. . . . . . . 8 |- ( ( ph /\ y e. CC ) -> ( E ` y ) = ( k e. NN |-> ( ( k x. ( F ` k ) ) x. ( y ^ ( k - 1 ) ) ) ) )
130	129	adantr 467	. . . . . . 7 |- ( ( ( ph /\ y e. CC ) /\ n e. NN ) -> ( E ` y ) = ( k e. NN |-> ( ( k x. ( F ` k ) ) x. ( y ^ ( k - 1 ) ) ) ) )
131		simpr 463	. . . . . . . . 9 |- ( ( ( ( ph /\ y e. CC ) /\ n e. NN ) /\ k = n ) -> k = n )
132	131	fveq2d 5869	. . . . . . . . 9 |- ( ( ( ( ph /\ y e. CC ) /\ n e. NN ) /\ k = n ) -> ( F ` k ) = ( F ` n ) )
133	131, 132	oveq12d 6308	. . . . . . . 8 |- ( ( ( ( ph /\ y e. CC ) /\ n e. NN ) /\ k = n ) -> ( k x. ( F ` k ) ) = ( n x. ( F ` n ) ) )
134	131	oveq1d 6305	. . . . . . . . 9 |- ( ( ( ( ph /\ y e. CC ) /\ n e. NN ) /\ k = n ) -> ( k - 1 ) = ( n - 1 ) )
135	134	oveq2d 6306	. . . . . . . 8 |- ( ( ( ( ph /\ y e. CC ) /\ n e. NN ) /\ k = n ) -> ( y ^ ( k - 1 ) ) = ( y ^ ( n - 1 ) ) )
136	133, 135	oveq12d 6308	. . . . . . 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 ) ) ) )
137		simpr 463	. . . . . . 7 |- ( ( ( ph /\ y e. CC ) /\ n e. NN ) -> n e. NN )
138		ovex 6318	. . . . . . . 8 |- ( ( n x. ( F ` n ) ) x. ( y ^ ( n - 1 ) ) ) e. _V
139	138	a1i 11	. . . . . . 7 |- ( ( ( ph /\ y e. CC ) /\ n e. NN ) -> ( ( n x. ( F ` n ) ) x. ( y ^ ( n - 1 ) ) ) e. _V )
140	130, 136, 137, 139	fvmptd 5954	. . . . . 6 |- ( ( ( ph /\ y e. CC ) /\ n e. NN ) -> ( ( E ` y ) ` n ) = ( ( n x. ( F ` n ) ) x. ( y ^ ( n - 1 ) ) ) )
141	140	sumeq2dv 13769	. . . . 5 |- ( ( ph /\ y e. CC ) -> sum_ n e. NN ( ( E ` y ) ` n ) = sum_ n e. NN ( ( n x. ( F ` n ) ) x. ( y ^ ( n - 1 ) ) ) )
142	118, 141	sylan2 477	. . . 4 |- ( ( ph /\ y e. D ) -> sum_ n e. NN ( ( E ` y ) ` n ) = sum_ n e. NN ( ( n x. ( F ` n ) ) x. ( y ^ ( n - 1 ) ) ) )
143	142	mpteq2dva 4489	. . 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 ) ) ) ) )
144	112, 143	eqtr4d 2488	. 2 |- ( ph -> ( CC _D P ) = ( y e. D |-> sum_ n e. NN ( ( E ` y ) ` n ) ) )
145		nfcv 2592	. . . 4 |- F/_ b NN
146		nfmpt1 4492	. . . . . . 7 |- F/_ b ( b e. CC |-> ( k e. NN |-> ( ( k x. ( F ` k ) ) x. ( b ^ ( k - 1 ) ) ) ) )
147	119, 146	nfcxfr 2590	. . . . . 6 |- F/_ b E
148	147, 27	nffv 5872	. . . . 5 |- F/_ b ( E ` y )
149		nfcv 2592	. . . . 5 |- F/_ b n
150	148, 149	nffv 5872	. . . 4 |- F/_ b ( ( E ` y ) ` n )
151	145, 150	nfsum 13757	. . 3 |- F/_ b sum_ n e. NN ( ( E ` y ) ` n )
152		nfcv 2592	. . 3 |- F/_ y sum_ k e. NN ( ( E ` b ) ` k )
153		simpl 459	. . . . . . 7 |- ( ( y = b /\ n e. NN ) -> y = b )
154	153	fveq2d 5869	. . . . . 6 |- ( ( y = b /\ n e. NN ) -> ( E ` y ) = ( E ` b ) )
155	154	fveq1d 5867	. . . . 5 |- ( ( y = b /\ n e. NN ) -> ( ( E ` y ) ` n ) = ( ( E ` b ) ` n ) )
156	155	sumeq2dv 13769	. . . 4 |- ( y = b -> sum_ n e. NN ( ( E ` y ) ` n ) = sum_ n e. NN ( ( E ` b ) ` n ) )
157		nfmpt1 4492	. . . . . . . . 9 |- F/_ k ( k e. NN |-> ( ( k x. ( F ` k ) ) x. ( b ^ ( k - 1 ) ) ) )
158	37, 157	nfmpt 4491	. . . . . . . 8 |- F/_ k ( b e. CC |-> ( k e. NN |-> ( ( k x. ( F ` k ) ) x. ( b ^ ( k - 1 ) ) ) ) )
159	119, 158	nfcxfr 2590	. . . . . . 7 |- F/_ k E
160		nfcv 2592	. . . . . . 7 |- F/_ k b
161	159, 160	nffv 5872	. . . . . 6 |- F/_ k ( E ` b )
162		nfcv 2592	. . . . . 6 |- F/_ k n
163	161, 162	nffv 5872	. . . . 5 |- F/_ k ( ( E ` b ) ` n )
164		nfcv 2592	. . . . 5 |- F/_ n ( ( E ` b ) ` k )
165		fveq2 5865	. . . . 5 |- ( n = k -> ( ( E ` b ) ` n ) = ( ( E ` b ) ` k ) )
166	163, 164, 165	cbvsumi 13763	. . . 4 |- sum_ n e. NN ( ( E ` b ) ` n ) = sum_ k e. NN ( ( E ` b ) ` k )
167	156, 166	syl6eq 2501	. . 3 |- ( y = b -> sum_ n e. NN ( ( E ` y ) ` n ) = sum_ k e. NN ( ( E ` b ) ` k ) )
168	24, 23, 151, 152, 167	cbvmptf 4493	. 2 |- ( y e. D |-> sum_ n e. NN ( ( E ` y ) ` n ) ) = ( b e. D |-> sum_ k e. NN ( ( E ` b ) ` k ) )
169	144, 168	syl6eq 2501	1 |- ( ph -> ( CC _D P ) = ( b e. D |-> sum_ k e. NN ( ( E ` b ) ` k ) ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   = wceq 1444   e. wcel 1887   {crab 2741   _Vcvv 3045   ifcif 3881   class class class wbr 4402   |-> cmpt 4461   `'ccnv 4833   dom cdm 4834   "cima 4837   o. ccom 4838   ` cfv 5582  (class class class)co 6290   supcsup 7954   CCcc 9537   RRcr 9538   0cc0 9539   1c1 9540   + caddc 9542   x. cmul 9544   RR*cxr 9674   < clt 9675   - cmin 9860   / cdiv 10269   NNcn 10609   2c2 10659   NN0cn0 10869   RR+crp 11302   [,)cico 11637   seqcseq 12213   ^cexp 12272   abscabs 13297   ~~> cli 13548   sum_csu 13752   ballcbl 18957   _D cdv 22818  C_𝑐cbcc 36685

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-ixp 7523  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-fi 7925  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-fl 12028  df-seq 12214  df-exp 12273  df-fac 12460  df-hash 12516  df-shft 13130  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-limsup 13526  df-clim 13552  df-rlim 13553  df-sum 13753  df-prod 13960  df-fallfac 14060  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-starv 15205  df-sca 15206  df-vsca 15207  df-ip 15208  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-hom 15214  df-cco 15215  df-rest 15321  df-topn 15322  df-0g 15340  df-gsum 15341  df-topgen 15342  df-pt 15343  df-prds 15346  df-xrs 15400  df-qtop 15406  df-imas 15407  df-xps 15410  df-mre 15492  df-mrc 15493  df-acs 15495  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-submnd 16583  df-mulg 16676  df-cntz 16971  df-cmn 17432  df-psmet 18962  df-xmet 18963  df-met 18964  df-bl 18965  df-mopn 18966  df-fbas 18967  df-fg 18968  df-cnfld 18971  df-top 19921  df-bases 19922  df-topon 19923  df-topsp 19924  df-cld 20034  df-ntr 20035  df-cls 20036  df-nei 20114  df-lp 20152  df-perf 20153  df-cn 20243  df-cnp 20244  df-haus 20331  df-cmp 20402  df-tx 20577  df-hmeo 20770  df-fil 20861  df-fm 20953  df-flim 20954  df-flf 20955  df-xms 21335  df-ms 21336  df-tms 21337  df-cncf 21910  df-limc 22821  df-dv 22822  df-ulm 23332  df-bcc 36686

This theorem is referenced by:  binomcxplemnotnn0  36705