Theorem binomcxplemdvsum 36088

Description: Lemma for binomcxp 36090. The derivative of the generalized sum in binomcxplemnn0 36082. 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 2564	. . . . . . . 8 |- F/_ b `' abs
5		nfcv 2564	. . . . . . . . 9 |- F/_ b 0
6		nfcv 2564	. . . . . . . . 9 |- F/_ b [,)
7		binomcxplem.r	. . . . . . . . . 10 |- R = sup ( { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } , RR* , < )
8		nfcv 2564	. . . . . . . . . . . . . 14 |- F/_ b +
9		nfmpt1 4483	. . . . . . . . . . . . . . . 16 |- F/_ b ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) )
10	1, 9	nfcxfr 2562	. . . . . . . . . . . . . . 15 |- F/_ b S
11		nfcv 2564	. . . . . . . . . . . . . . 15 |- F/_ b r
12	10, 11	nffv 5855	. . . . . . . . . . . . . 14 |- F/_ b ( S ` r )
13	5, 8, 12	nfseq 12159	. . . . . . . . . . . . 13 |- F/_ b seq 0 ( + , ( S ` r ) )
14	13	nfel1 2580	. . . . . . . . . . . 12 |- F/ b seq 0 ( + , ( S ` r ) ) e. dom ~~>
15		nfcv 2564	. . . . . . . . . . . 12 |- F/_ b RR
16	14, 15	nfrab 2988	. . . . . . . . . . 11 |- F/_ b { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> }
17		nfcv 2564	. . . . . . . . . . 11 |- F/_ b RR*
18		nfcv 2564	. . . . . . . . . . 11 |- F/_ b <
19	16, 17, 18	nfsup 7943	. . . . . . . . . 10 |- F/_ b sup ( { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } , RR* , < )
20	7, 19	nfcxfr 2562	. . . . . . . . 9 |- F/_ b R
21	5, 6, 20	nfov 6303	. . . . . . . 8 |- F/_ b ( 0 [,) R )
22	4, 21	nfima 5164	. . . . . . 7 |- F/_ b ( `' abs " ( 0 [,) R ) )
23	3, 22	nfcxfr 2562	. . . . . 6 |- F/_ b D
24		nfcv 2564	. . . . . 6 |- F/_ y D
25		nfcv 2564	. . . . . 6 |- F/_ y sum_ k e. NN0 ( ( S ` b ) ` k )
26		nfcv 2564	. . . . . . 7 |- F/_ b NN0
27		nfcv 2564	. . . . . . . . 9 |- F/_ b y
28	10, 27	nffv 5855	. . . . . . . 8 |- F/_ b ( S ` y )
29		nfcv 2564	. . . . . . . 8 |- F/_ b m
30	28, 29	nffv 5855	. . . . . . 7 |- F/_ b ( ( S ` y ) ` m )
31	26, 30	nfsum 13660	. . . . . 6 |- F/_ b sum_ m e. NN0 ( ( S ` y ) ` m )
32		simpl 455	. . . . . . . . . 10 |- ( ( b = y /\ k e. NN0 ) -> b = y )
33	32	fveq2d 5852	. . . . . . . . 9 |- ( ( b = y /\ k e. NN0 ) -> ( S ` b ) = ( S ` y ) )
34	33	fveq1d 5850	. . . . . . . 8 |- ( ( b = y /\ k e. NN0 ) -> ( ( S ` b ) ` k ) = ( ( S ` y ) ` k ) )
35	34	sumeq2dv 13672	. . . . . . 7 |- ( b = y -> sum_ k e. NN0 ( ( S ` b ) ` k ) = sum_ k e. NN0 ( ( S ` y ) ` k ) )
36		nfcv 2564	. . . . . . . 8 |- F/_ m ( ( S ` y ) ` k )
37		nfcv 2564	. . . . . . . . . . . 12 |- F/_ k CC
38		nfmpt1 4483	. . . . . . . . . . . 12 |- F/_ k ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) )
39	37, 38	nfmpt 4482	. . . . . . . . . . 11 |- F/_ k ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) )
40	1, 39	nfcxfr 2562	. . . . . . . . . 10 |- F/_ k S
41		nfcv 2564	. . . . . . . . . 10 |- F/_ k y
42	40, 41	nffv 5855	. . . . . . . . 9 |- F/_ k ( S ` y )
43		nfcv 2564	. . . . . . . . 9 |- F/_ k m
44	42, 43	nffv 5855	. . . . . . . 8 |- F/_ k ( ( S ` y ) ` m )
45		fveq2 5848	. . . . . . . 8 |- ( k = m -> ( ( S ` y ) ` k ) = ( ( S ` y ) ` m ) )
46	36, 44, 45	cbvsumi 13666	. . . . . . 7 |- sum_ k e. NN0 ( ( S ` y ) ` k ) = sum_ m e. NN0 ( ( S ` y ) ` m )
47	35, 46	syl6eq 2459	. . . . . 6 |- ( b = y -> sum_ k e. NN0 ( ( S ` b ) ` k ) = sum_ m e. NN0 ( ( S ` y ) ` m ) )
48	23, 24, 25, 31, 47	cbvmptf 4484	. . . . 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 2431	. . . 4 |- P = ( y e. D |-> sum_ m e. NN0 ( ( S ` y ) ` m ) )
50		ovex 6305	. . . . . 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 459	. . . . . . . 8 |- ( ( ( ph /\ k e. NN0 ) /\ j = k ) -> j = k )
56	55	oveq2d 6293	. . . . . . 7 |- ( ( ( ph /\ k e. NN0 ) /\ j = k ) -> ( CC𝑐 j ) = ( CC𝑐 k ) )
57		simpr 459	. . . . . . 7 |- ( ( ph /\ k e. NN0 ) -> k e. NN0 )
58		binomcxp.c	. . . . . . . . 9 |- ( ph -> C e. CC )
59	58	adantr 463	. . . . . . . 8 |- ( ( ph /\ k e. NN0 ) -> C e. CC )
60	59, 57	bcccl 36072	. . . . . . 7 |- ( ( ph /\ k e. NN0 ) -> ( CC𝑐 k ) e. CC )
61	54, 56, 57, 60	fvmptd 5937	. . . . . 6 |- ( ( ph /\ k e. NN0 ) -> ( F ` k ) = ( CC𝑐 k ) )
62	61, 60	eqeltrd 2490	. . . . 5 |- ( ( ph /\ k e. NN0 ) -> ( F ` k ) e. CC )
63	51, 53, 62	fmpt2d 6039	. . . 4 |- ( ph -> F : NN0 --> CC )
64		nfcv 2564	. . . . . . 7 |- F/_ r RR
65		nfcv 2564	. . . . . . 7 |- F/_ z RR
66		nfv 1728	. . . . . . 7 |- F/ z seq 0 ( + , ( S ` r ) ) e. dom ~~>
67		nfcv 2564	. . . . . . . . 9 |- F/_ r 0
68		nfcv 2564	. . . . . . . . 9 |- F/_ r +
69		nfcv 2564	. . . . . . . . . . 11 |- F/_ r ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) )
70	1, 69	nfcxfr 2562	. . . . . . . . . 10 |- F/_ r S
71		nfcv 2564	. . . . . . . . . 10 |- F/_ r z
72	70, 71	nffv 5855	. . . . . . . . 9 |- F/_ r ( S ` z )
73	67, 68, 72	nfseq 12159	. . . . . . . 8 |- F/_ r seq 0 ( + , ( S ` z ) )
74	73	nfel1 2580	. . . . . . 7 |- F/ r seq 0 ( + , ( S ` z ) ) e. dom ~~>
75		fveq2 5848	. . . . . . . . 9 |- ( r = z -> ( S ` r ) = ( S ` z ) )
76	75	seqeq3d 12157	. . . . . . . 8 |- ( r = z -> seq 0 ( + , ( S ` r ) ) = seq 0 ( + , ( S ` z ) ) )
77	76	eleq1d 2471	. . . . . . 7 |- ( r = z -> ( seq 0 ( + , ( S ` r ) ) e. dom ~~> <-> seq 0 ( + , ( S ` z ) ) e. dom ~~> ) )
78	64, 65, 66, 74, 77	cbvrab 3056	. . . . . 6 |- { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } = { z e. RR | seq 0 ( + , ( S ` z ) ) e. dom ~~> }
79	78	supeq1i 7939	. . . . 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 2431	. . . 4 |- R = sup ( { z e. RR | seq 0 ( + , ( S ` z ) ) e. dom ~~> } , RR* , < )
81	1	fveq1i 5849	. . . . . . . . . . . 12 |- ( S ` z ) = ( ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) ) ` z )
82		seqeq3 12154	. . . . . . . . . . . 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 2479	. . . . . . . . . 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 3047	. . . . . . . 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 7939	. . . . . . 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 2436	. . . . . 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 2479	. . . . 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 6288	. . . . . 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 6287	. . . . 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 2402	. . . . 5 |- ( ( abs ` x ) + 1 ) = ( ( abs ` x ) + 1 )
93	89, 91, 92	ifbieq12i 3910	. . . 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 6284	. . . . . . . . . . . . . . . . . 18 |- ( w = b -> ( w ^ k ) = ( b ^ k ) )
95	94	oveq2d 6293	. . . . . . . . . . . . . . . . 17 |- ( w = b -> ( ( F ` k ) x. ( w ^ k ) ) = ( ( F ` k ) x. ( b ^ k ) ) )
96	95	mpteq2dv 4481	. . . . . . . . . . . . . . . 16 |- ( w = b -> ( k e. NN0 |-> ( ( F ` k ) x. ( w ^ k ) ) ) = ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) )
97	96	cbvmptv 4486	. . . . . . . . . . . . . . 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 5849	. . . . . . . . . . . . . 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 12154	. . . . . . . . . . . . . 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 2479	. . . . . . . . . . . 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 3047	. . . . . . . . . 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 7939	. . . . . . . . 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 2479	. . . . . . . 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 6288	. . . . . . . . 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 6287	. . . . . . . 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 3910	. . . . . . 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 6288	. . . . . 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 6287	. . . . 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 6288	. . . 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 23115	. . 3 |- ( ph -> ( CC _D P ) = ( y e. D |-> sum_ n e. NN ( ( n x. ( F ` n ) ) x. ( y ^ ( n - 1 ) ) ) ) )
113		cnvimass 5176	. . . . . . . 8 |- ( `' abs " ( 0 [,) R ) ) C_ dom abs
114	3, 113	eqsstri 3471	. . . . . . 7 |- D C_ dom abs
115		absf 13317	. . . . . . . 8 |- abs : CC --> RR
116	115	fdmi 5718	. . . . . . 7 |- dom abs = CC
117	114, 116	sseqtri 3473	. . . . . 6 |- D C_ CC
118	117	sseli 3437	. . . . 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 754	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ y e. CC ) /\ b = y ) /\ k e. NN ) -> b = y )
122	121	oveq1d 6292	. . . . . . . . . . 11 |- ( ( ( ( ph /\ y e. CC ) /\ b = y ) /\ k e. NN ) -> ( b ^ ( k - 1 ) ) = ( y ^ ( k - 1 ) ) )
123	122	oveq2d 6293	. . . . . . . . . 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 4480	. . . . . . . . 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 459	. . . . . . . . 9 |- ( ( ph /\ y e. CC ) -> y e. CC )
126		nnex 10581	. . . . . . . . . . 11 |- NN e. _V
127	126	mptex 6123	. . . . . . . . . 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 5937	. . . . . . . 8 |- ( ( ph /\ y e. CC ) -> ( E ` y ) = ( k e. NN |-> ( ( k x. ( F ` k ) ) x. ( y ^ ( k - 1 ) ) ) ) )
130	129	adantr 463	. . . . . . 7 |- ( ( ( ph /\ y e. CC ) /\ n e. NN ) -> ( E ` y ) = ( k e. NN |-> ( ( k x. ( F ` k ) ) x. ( y ^ ( k - 1 ) ) ) ) )
131		simpr 459	. . . . . . . . 9 |- ( ( ( ( ph /\ y e. CC ) /\ n e. NN ) /\ k = n ) -> k = n )
132	131	fveq2d 5852	. . . . . . . . 9 |- ( ( ( ( ph /\ y e. CC ) /\ n e. NN ) /\ k = n ) -> ( F ` k ) = ( F ` n ) )
133	131, 132	oveq12d 6295	. . . . . . . 8 |- ( ( ( ( ph /\ y e. CC ) /\ n e. NN ) /\ k = n ) -> ( k x. ( F ` k ) ) = ( n x. ( F ` n ) ) )
134	131	oveq1d 6292	. . . . . . . . 9 |- ( ( ( ( ph /\ y e. CC ) /\ n e. NN ) /\ k = n ) -> ( k - 1 ) = ( n - 1 ) )
135	134	oveq2d 6293	. . . . . . . 8 |- ( ( ( ( ph /\ y e. CC ) /\ n e. NN ) /\ k = n ) -> ( y ^ ( k - 1 ) ) = ( y ^ ( n - 1 ) ) )
136	133, 135	oveq12d 6295	. . . . . . 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 459	. . . . . . 7 |- ( ( ( ph /\ y e. CC ) /\ n e. NN ) -> n e. NN )
138		ovex 6305	. . . . . . . 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 5937	. . . . . 6 |- ( ( ( ph /\ y e. CC ) /\ n e. NN ) -> ( ( E ` y ) ` n ) = ( ( n x. ( F ` n ) ) x. ( y ^ ( n - 1 ) ) ) )
141	140	sumeq2dv 13672	. . . . 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 472	. . . 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 4480	. . 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 2446	. 2 |- ( ph -> ( CC _D P ) = ( y e. D |-> sum_ n e. NN ( ( E ` y ) ` n ) ) )
145		nfcv 2564	. . . 4 |- F/_ b NN
146		nfmpt1 4483	. . . . . . 7 |- F/_ b ( b e. CC |-> ( k e. NN |-> ( ( k x. ( F ` k ) ) x. ( b ^ ( k - 1 ) ) ) ) )
147	119, 146	nfcxfr 2562	. . . . . 6 |- F/_ b E
148	147, 27	nffv 5855	. . . . 5 |- F/_ b ( E ` y )
149		nfcv 2564	. . . . 5 |- F/_ b n
150	148, 149	nffv 5855	. . . 4 |- F/_ b ( ( E ` y ) ` n )
151	145, 150	nfsum 13660	. . 3 |- F/_ b sum_ n e. NN ( ( E ` y ) ` n )
152		nfcv 2564	. . 3 |- F/_ y sum_ k e. NN ( ( E ` b ) ` k )
153		simpl 455	. . . . . . 7 |- ( ( y = b /\ n e. NN ) -> y = b )
154	153	fveq2d 5852	. . . . . 6 |- ( ( y = b /\ n e. NN ) -> ( E ` y ) = ( E ` b ) )
155	154	fveq1d 5850	. . . . 5 |- ( ( y = b /\ n e. NN ) -> ( ( E ` y ) ` n ) = ( ( E ` b ) ` n ) )
156	155	sumeq2dv 13672	. . . 4 |- ( y = b -> sum_ n e. NN ( ( E ` y ) ` n ) = sum_ n e. NN ( ( E ` b ) ` n ) )
157		nfmpt1 4483	. . . . . . . . 9 |- F/_ k ( k e. NN |-> ( ( k x. ( F ` k ) ) x. ( b ^ ( k - 1 ) ) ) )
158	37, 157	nfmpt 4482	. . . . . . . 8 |- F/_ k ( b e. CC |-> ( k e. NN |-> ( ( k x. ( F ` k ) ) x. ( b ^ ( k - 1 ) ) ) ) )
159	119, 158	nfcxfr 2562	. . . . . . 7 |- F/_ k E
160		nfcv 2564	. . . . . . 7 |- F/_ k b
161	159, 160	nffv 5855	. . . . . 6 |- F/_ k ( E ` b )
162		nfcv 2564	. . . . . 6 |- F/_ k n
163	161, 162	nffv 5855	. . . . 5 |- F/_ k ( ( E ` b ) ` n )
164		nfcv 2564	. . . . 5 |- F/_ n ( ( E ` b ) ` k )
165		fveq2 5848	. . . . 5 |- ( n = k -> ( ( E ` b ) ` n ) = ( ( E ` b ) ` k ) )
166	163, 164, 165	cbvsumi 13666	. . . 4 |- sum_ n e. NN ( ( E ` b ) ` n ) = sum_ k e. NN ( ( E ` b ) ` k )
167	156, 166	syl6eq 2459	. . 3 |- ( y = b -> sum_ n e. NN ( ( E ` y ) ` n ) = sum_ k e. NN ( ( E ` b ) ` k ) )
168	24, 23, 151, 152, 167	cbvmptf 4484	. 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 2459	1 |- ( ph -> ( CC _D P ) = ( b e. D |-> sum_ k e. NN ( ( E ` b ) ` k ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   = wceq 1405   e. wcel 1842   {crab 2757   _Vcvv 3058   ifcif 3884   class class class wbr 4394   |-> cmpt 4452   `'ccnv 4821   dom cdm 4822   "cima 4825   o. ccom 4826   ` cfv 5568  (class class class)co 6277   supcsup 7933   CCcc 9519   RRcr 9520   0cc0 9521   1c1 9522   + caddc 9524   x. cmul 9526   RR*cxr 9656   < clt 9657   - cmin 9840   / cdiv 10246   NNcn 10575   2c2 10625   NN0cn0 10835   RR+crp 11264   [,)cico 11583   seqcseq 12149   ^cexp 12208   abscabs 13214   ~~> cli 13454   sum_csu 13655   ballcbl 18723   _D cdv 22557  C_𝑐cbcc 36069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-inf2 8090  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-pre-sup 9599  ax-addf 9600  ax-mulf 9601

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-iin 4273  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-isom 5577  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6520  df-om 6683  df-1st 6783  df-2nd 6784  df-supp 6902  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-2o 7167  df-oadd 7170  df-er 7347  df-map 7458  df-pm 7459  df-ixp 7507  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-fsupp 7863  df-fi 7904  df-sup 7934  df-oi 7968  df-card 8351  df-cda 8579  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-div 10247  df-nn 10576  df-2 10634  df-3 10635  df-4 10636  df-5 10637  df-6 10638  df-7 10639  df-8 10640  df-9 10641  df-10 10642  df-n0 10836  df-z 10905  df-dec 11019  df-uz 11127  df-q 11227  df-rp 11265  df-xneg 11370  df-xadd 11371  df-xmul 11372  df-ioo 11585  df-ico 11587  df-icc 11588  df-fz 11725  df-fzo 11853  df-fl 11964  df-seq 12150  df-exp 12209  df-fac 12396  df-hash 12451  df-shft 13047  df-cj 13079  df-re 13080  df-im 13081  df-sqrt 13215  df-abs 13216  df-limsup 13441  df-clim 13458  df-rlim 13459  df-sum 13656  df-prod 13863  df-fallfac 13950  df-struct 14841  df-ndx 14842  df-slot 14843  df-base 14844  df-sets 14845  df-ress 14846  df-plusg 14920  df-mulr 14921  df-starv 14922  df-sca 14923  df-vsca 14924  df-ip 14925  df-tset 14926  df-ple 14927  df-ds 14929  df-unif 14930  df-hom 14931  df-cco 14932  df-rest 15035  df-topn 15036  df-0g 15054  df-gsum 15055  df-topgen 15056  df-pt 15057  df-prds 15060  df-xrs 15114  df-qtop 15119  df-imas 15120  df-xps 15122  df-mre 15198  df-mrc 15199  df-acs 15201  df-mgm 16194  df-sgrp 16233  df-mnd 16243  df-submnd 16289  df-mulg 16382  df-cntz 16677  df-cmn 17122  df-psmet 18729  df-xmet 18730  df-met 18731  df-bl 18732  df-mopn 18733  df-fbas 18734  df-fg 18735  df-cnfld 18739  df-top 19689  df-bases 19691  df-topon 19692  df-topsp 19693  df-cld 19810  df-ntr 19811  df-cls 19812  df-nei 19890  df-lp 19928  df-perf 19929  df-cn 20019  df-cnp 20020  df-haus 20107  df-cmp 20178  df-tx 20353  df-hmeo 20546  df-fil 20637  df-fm 20729  df-flim 20730  df-flf 20731  df-xms 21113  df-ms 21114  df-tms 21115  df-cncf 21672  df-limc 22560  df-dv 22561  df-ulm 23062  df-bcc 36070

This theorem is referenced by:  binomcxplemnotnn0  36089