Theorem binomcxplemdvsum 36774

Description: Lemma for binomcxp 36776. The derivative of the generalized sum in binomcxplemnn0 36768. 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 2612	. . . . . . . 8 |- F/_ b `' abs
5		nfcv 2612	. . . . . . . . 9 |- F/_ b 0
6		nfcv 2612	. . . . . . . . 9 |- F/_ b [,)
7		binomcxplem.r	. . . . . . . . . 10 |- R = sup ( { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } , RR* , < )
8		nfcv 2612	. . . . . . . . . . . . . 14 |- F/_ b +
9		nfmpt1 4485	. . . . . . . . . . . . . . . 16 |- F/_ b ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) )
10	1, 9	nfcxfr 2610	. . . . . . . . . . . . . . 15 |- F/_ b S
11		nfcv 2612	. . . . . . . . . . . . . . 15 |- F/_ b r
12	10, 11	nffv 5886	. . . . . . . . . . . . . 14 |- F/_ b ( S ` r )
13	5, 8, 12	nfseq 12261	. . . . . . . . . . . . 13 |- F/_ b seq 0 ( + , ( S ` r ) )
14	13	nfel1 2626	. . . . . . . . . . . 12 |- F/ b seq 0 ( + , ( S ` r ) ) e. dom ~~>
15		nfcv 2612	. . . . . . . . . . . 12 |- F/_ b RR
16	14, 15	nfrab 2958	. . . . . . . . . . 11 |- F/_ b { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> }
17		nfcv 2612	. . . . . . . . . . 11 |- F/_ b RR*
18		nfcv 2612	. . . . . . . . . . 11 |- F/_ b <
19	16, 17, 18	nfsup 7983	. . . . . . . . . 10 |- F/_ b sup ( { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } , RR* , < )
20	7, 19	nfcxfr 2610	. . . . . . . . 9 |- F/_ b R
21	5, 6, 20	nfov 6334	. . . . . . . 8 |- F/_ b ( 0 [,) R )
22	4, 21	nfima 5182	. . . . . . 7 |- F/_ b ( `' abs " ( 0 [,) R ) )
23	3, 22	nfcxfr 2610	. . . . . 6 |- F/_ b D
24		nfcv 2612	. . . . . 6 |- F/_ y D
25		nfcv 2612	. . . . . 6 |- F/_ y sum_ k e. NN0 ( ( S ` b ) ` k )
26		nfcv 2612	. . . . . . 7 |- F/_ b NN0
27		nfcv 2612	. . . . . . . . 9 |- F/_ b y
28	10, 27	nffv 5886	. . . . . . . 8 |- F/_ b ( S ` y )
29		nfcv 2612	. . . . . . . 8 |- F/_ b m
30	28, 29	nffv 5886	. . . . . . 7 |- F/_ b ( ( S ` y ) ` m )
31	26, 30	nfsum 13834	. . . . . 6 |- F/_ b sum_ m e. NN0 ( ( S ` y ) ` m )
32		simpl 464	. . . . . . . . . 10 |- ( ( b = y /\ k e. NN0 ) -> b = y )
33	32	fveq2d 5883	. . . . . . . . 9 |- ( ( b = y /\ k e. NN0 ) -> ( S ` b ) = ( S ` y ) )
34	33	fveq1d 5881	. . . . . . . 8 |- ( ( b = y /\ k e. NN0 ) -> ( ( S ` b ) ` k ) = ( ( S ` y ) ` k ) )
35	34	sumeq2dv 13846	. . . . . . 7 |- ( b = y -> sum_ k e. NN0 ( ( S ` b ) ` k ) = sum_ k e. NN0 ( ( S ` y ) ` k ) )
36		nfcv 2612	. . . . . . . 8 |- F/_ m ( ( S ` y ) ` k )
37		nfcv 2612	. . . . . . . . . . . 12 |- F/_ k CC
38		nfmpt1 4485	. . . . . . . . . . . 12 |- F/_ k ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) )
39	37, 38	nfmpt 4484	. . . . . . . . . . 11 |- F/_ k ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) )
40	1, 39	nfcxfr 2610	. . . . . . . . . 10 |- F/_ k S
41		nfcv 2612	. . . . . . . . . 10 |- F/_ k y
42	40, 41	nffv 5886	. . . . . . . . 9 |- F/_ k ( S ` y )
43		nfcv 2612	. . . . . . . . 9 |- F/_ k m
44	42, 43	nffv 5886	. . . . . . . 8 |- F/_ k ( ( S ` y ) ` m )
45		fveq2 5879	. . . . . . . 8 |- ( k = m -> ( ( S ` y ) ` k ) = ( ( S ` y ) ` m ) )
46	36, 44, 45	cbvsumi 13840	. . . . . . 7 |- sum_ k e. NN0 ( ( S ` y ) ` k ) = sum_ m e. NN0 ( ( S ` y ) ` m )
47	35, 46	syl6eq 2521	. . . . . 6 |- ( b = y -> sum_ k e. NN0 ( ( S ` b ) ` k ) = sum_ m e. NN0 ( ( S ` y ) ` m ) )
48	23, 24, 25, 31, 47	cbvmptf 4486	. . . . 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 2493	. . . 4 |- P = ( y e. D |-> sum_ m e. NN0 ( ( S ` y ) ` m ) )
50		ovex 6336	. . . . . 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 468	. . . . . . . 8 |- ( ( ( ph /\ k e. NN0 ) /\ j = k ) -> j = k )
56	55	oveq2d 6324	. . . . . . 7 |- ( ( ( ph /\ k e. NN0 ) /\ j = k ) -> ( CC𝑐 j ) = ( CC𝑐 k ) )
57		simpr 468	. . . . . . 7 |- ( ( ph /\ k e. NN0 ) -> k e. NN0 )
58		binomcxp.c	. . . . . . . . 9 |- ( ph -> C e. CC )
59	58	adantr 472	. . . . . . . 8 |- ( ( ph /\ k e. NN0 ) -> C e. CC )
60	59, 57	bcccl 36758	. . . . . . 7 |- ( ( ph /\ k e. NN0 ) -> ( CC𝑐 k ) e. CC )
61	54, 56, 57, 60	fvmptd 5969	. . . . . 6 |- ( ( ph /\ k e. NN0 ) -> ( F ` k ) = ( CC𝑐 k ) )
62	61, 60	eqeltrd 2549	. . . . 5 |- ( ( ph /\ k e. NN0 ) -> ( F ` k ) e. CC )
63	51, 53, 62	fmpt2d 6069	. . . 4 |- ( ph -> F : NN0 --> CC )
64		nfcv 2612	. . . . . . 7 |- F/_ r RR
65		nfcv 2612	. . . . . . 7 |- F/_ z RR
66		nfv 1769	. . . . . . 7 |- F/ z seq 0 ( + , ( S ` r ) ) e. dom ~~>
67		nfcv 2612	. . . . . . . . 9 |- F/_ r 0
68		nfcv 2612	. . . . . . . . 9 |- F/_ r +
69		nfcv 2612	. . . . . . . . . . 11 |- F/_ r ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) )
70	1, 69	nfcxfr 2610	. . . . . . . . . 10 |- F/_ r S
71		nfcv 2612	. . . . . . . . . 10 |- F/_ r z
72	70, 71	nffv 5886	. . . . . . . . 9 |- F/_ r ( S ` z )
73	67, 68, 72	nfseq 12261	. . . . . . . 8 |- F/_ r seq 0 ( + , ( S ` z ) )
74	73	nfel1 2626	. . . . . . 7 |- F/ r seq 0 ( + , ( S ` z ) ) e. dom ~~>
75		fveq2 5879	. . . . . . . . 9 |- ( r = z -> ( S ` r ) = ( S ` z ) )
76	75	seqeq3d 12259	. . . . . . . 8 |- ( r = z -> seq 0 ( + , ( S ` r ) ) = seq 0 ( + , ( S ` z ) ) )
77	76	eleq1d 2533	. . . . . . 7 |- ( r = z -> ( seq 0 ( + , ( S ` r ) ) e. dom ~~> <-> seq 0 ( + , ( S ` z ) ) e. dom ~~> ) )
78	64, 65, 66, 74, 77	cbvrab 3029	. . . . . 6 |- { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } = { z e. RR | seq 0 ( + , ( S ` z ) ) e. dom ~~> }
79	78	supeq1i 7979	. . . . 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 2493	. . . 4 |- R = sup ( { z e. RR | seq 0 ( + , ( S ` z ) ) e. dom ~~> } , RR* , < )
81	1	fveq1i 5880	. . . . . . . . . . . 12 |- ( S ` z ) = ( ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) ) ` z )
82		seqeq3 12256	. . . . . . . . . . . 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 2540	. . . . . . . . . 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 3019	. . . . . . . 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 7979	. . . . . . 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 2498	. . . . . 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 2540	. . . . 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 6319	. . . . . 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 6318	. . . . 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 2471	. . . . 5 |- ( ( abs ` x ) + 1 ) = ( ( abs ` x ) + 1 )
93	89, 91, 92	ifbieq12i 3898	. . . 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 6315	. . . . . . . . . . . . . . . . . 18 |- ( w = b -> ( w ^ k ) = ( b ^ k ) )
95	94	oveq2d 6324	. . . . . . . . . . . . . . . . 17 |- ( w = b -> ( ( F ` k ) x. ( w ^ k ) ) = ( ( F ` k ) x. ( b ^ k ) ) )
96	95	mpteq2dv 4483	. . . . . . . . . . . . . . . 16 |- ( w = b -> ( k e. NN0 |-> ( ( F ` k ) x. ( w ^ k ) ) ) = ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) )
97	96	cbvmptv 4488	. . . . . . . . . . . . . . 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 5880	. . . . . . . . . . . . . 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 12256	. . . . . . . . . . . . . 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 2540	. . . . . . . . . . . 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 3019	. . . . . . . . . 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 7979	. . . . . . . . 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 2540	. . . . . . . 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 6319	. . . . . . . . 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 6318	. . . . . . . 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 3898	. . . . . . 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 6319	. . . . . 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 6318	. . . . 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 6319	. . . 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 23464	. . 3 |- ( ph -> ( CC _D P ) = ( y e. D |-> sum_ n e. NN ( ( n x. ( F ` n ) ) x. ( y ^ ( n - 1 ) ) ) ) )
113		cnvimass 5194	. . . . . . . 8 |- ( `' abs " ( 0 [,) R ) ) C_ dom abs
114	3, 113	eqsstri 3448	. . . . . . 7 |- D C_ dom abs
115		absf 13477	. . . . . . . 8 |- abs : CC --> RR
116	115	fdmi 5746	. . . . . . 7 |- dom abs = CC
117	114, 116	sseqtri 3450	. . . . . 6 |- D C_ CC
118	117	sseli 3414	. . . . 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 770	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ y e. CC ) /\ b = y ) /\ k e. NN ) -> b = y )
122	121	oveq1d 6323	. . . . . . . . . . 11 |- ( ( ( ( ph /\ y e. CC ) /\ b = y ) /\ k e. NN ) -> ( b ^ ( k - 1 ) ) = ( y ^ ( k - 1 ) ) )
123	122	oveq2d 6324	. . . . . . . . . 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 4482	. . . . . . . . 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 468	. . . . . . . . 9 |- ( ( ph /\ y e. CC ) -> y e. CC )
126		nnex 10637	. . . . . . . . . . 11 |- NN e. _V
127	126	mptex 6152	. . . . . . . . . 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 5969	. . . . . . . 8 |- ( ( ph /\ y e. CC ) -> ( E ` y ) = ( k e. NN |-> ( ( k x. ( F ` k ) ) x. ( y ^ ( k - 1 ) ) ) ) )
130	129	adantr 472	. . . . . . 7 |- ( ( ( ph /\ y e. CC ) /\ n e. NN ) -> ( E ` y ) = ( k e. NN |-> ( ( k x. ( F ` k ) ) x. ( y ^ ( k - 1 ) ) ) ) )
131		simpr 468	. . . . . . . . 9 |- ( ( ( ( ph /\ y e. CC ) /\ n e. NN ) /\ k = n ) -> k = n )
132	131	fveq2d 5883	. . . . . . . . 9 |- ( ( ( ( ph /\ y e. CC ) /\ n e. NN ) /\ k = n ) -> ( F ` k ) = ( F ` n ) )
133	131, 132	oveq12d 6326	. . . . . . . 8 |- ( ( ( ( ph /\ y e. CC ) /\ n e. NN ) /\ k = n ) -> ( k x. ( F ` k ) ) = ( n x. ( F ` n ) ) )
134	131	oveq1d 6323	. . . . . . . . 9 |- ( ( ( ( ph /\ y e. CC ) /\ n e. NN ) /\ k = n ) -> ( k - 1 ) = ( n - 1 ) )
135	134	oveq2d 6324	. . . . . . . 8 |- ( ( ( ( ph /\ y e. CC ) /\ n e. NN ) /\ k = n ) -> ( y ^ ( k - 1 ) ) = ( y ^ ( n - 1 ) ) )
136	133, 135	oveq12d 6326	. . . . . . 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 468	. . . . . . 7 |- ( ( ( ph /\ y e. CC ) /\ n e. NN ) -> n e. NN )
138		ovex 6336	. . . . . . . 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 5969	. . . . . 6 |- ( ( ( ph /\ y e. CC ) /\ n e. NN ) -> ( ( E ` y ) ` n ) = ( ( n x. ( F ` n ) ) x. ( y ^ ( n - 1 ) ) ) )
141	140	sumeq2dv 13846	. . . . 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 482	. . . 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 4482	. . 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 2508	. 2 |- ( ph -> ( CC _D P ) = ( y e. D |-> sum_ n e. NN ( ( E ` y ) ` n ) ) )
145		nfcv 2612	. . . 4 |- F/_ b NN
146		nfmpt1 4485	. . . . . . 7 |- F/_ b ( b e. CC |-> ( k e. NN |-> ( ( k x. ( F ` k ) ) x. ( b ^ ( k - 1 ) ) ) ) )
147	119, 146	nfcxfr 2610	. . . . . 6 |- F/_ b E
148	147, 27	nffv 5886	. . . . 5 |- F/_ b ( E ` y )
149		nfcv 2612	. . . . 5 |- F/_ b n
150	148, 149	nffv 5886	. . . 4 |- F/_ b ( ( E ` y ) ` n )
151	145, 150	nfsum 13834	. . 3 |- F/_ b sum_ n e. NN ( ( E ` y ) ` n )
152		nfcv 2612	. . 3 |- F/_ y sum_ k e. NN ( ( E ` b ) ` k )
153		simpl 464	. . . . . . 7 |- ( ( y = b /\ n e. NN ) -> y = b )
154	153	fveq2d 5883	. . . . . 6 |- ( ( y = b /\ n e. NN ) -> ( E ` y ) = ( E ` b ) )
155	154	fveq1d 5881	. . . . 5 |- ( ( y = b /\ n e. NN ) -> ( ( E ` y ) ` n ) = ( ( E ` b ) ` n ) )
156	155	sumeq2dv 13846	. . . 4 |- ( y = b -> sum_ n e. NN ( ( E ` y ) ` n ) = sum_ n e. NN ( ( E ` b ) ` n ) )
157		nfmpt1 4485	. . . . . . . . 9 |- F/_ k ( k e. NN |-> ( ( k x. ( F ` k ) ) x. ( b ^ ( k - 1 ) ) ) )
158	37, 157	nfmpt 4484	. . . . . . . 8 |- F/_ k ( b e. CC |-> ( k e. NN |-> ( ( k x. ( F ` k ) ) x. ( b ^ ( k - 1 ) ) ) ) )
159	119, 158	nfcxfr 2610	. . . . . . 7 |- F/_ k E
160		nfcv 2612	. . . . . . 7 |- F/_ k b
161	159, 160	nffv 5886	. . . . . 6 |- F/_ k ( E ` b )
162		nfcv 2612	. . . . . 6 |- F/_ k n
163	161, 162	nffv 5886	. . . . 5 |- F/_ k ( ( E ` b ) ` n )
164		nfcv 2612	. . . . 5 |- F/_ n ( ( E ` b ) ` k )
165		fveq2 5879	. . . . 5 |- ( n = k -> ( ( E ` b ) ` n ) = ( ( E ` b ) ` k ) )
166	163, 164, 165	cbvsumi 13840	. . . 4 |- sum_ n e. NN ( ( E ` b ) ` n ) = sum_ k e. NN ( ( E ` b ) ` k )
167	156, 166	syl6eq 2521	. . 3 |- ( y = b -> sum_ n e. NN ( ( E ` y ) ` n ) = sum_ k e. NN ( ( E ` b ) ` k ) )
168	24, 23, 151, 152, 167	cbvmptf 4486	. 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 2521	1 |- ( ph -> ( CC _D P ) = ( b e. D |-> sum_ k e. NN ( ( E ` b ) ` k ) ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   = wceq 1452   e. wcel 1904   {crab 2760   _Vcvv 3031   ifcif 3872   class class class wbr 4395   |-> cmpt 4454   `'ccnv 4838   dom cdm 4839   "cima 4842   o. ccom 4843   ` cfv 5589  (class class class)co 6308   supcsup 7972   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558   + caddc 9560   x. cmul 9562   RR*cxr 9692   < clt 9693   - cmin 9880   / cdiv 10291   NNcn 10631   2c2 10681   NN0cn0 10893   RR+crp 11325   [,)cico 11662   seqcseq 12251   ^cexp 12310   abscabs 13374   ~~> cli 13625   sum_csu 13829   ballcbl 19034   _D cdv 22897  C_𝑐cbcc 36755

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-seq 12252  df-exp 12311  df-fac 12498  df-hash 12554  df-shft 13207  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-limsup 13603  df-clim 13629  df-rlim 13630  df-sum 13830  df-prod 14037  df-fallfac 14137  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-haus 20408  df-cmp 20479  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-limc 22900  df-dv 22901  df-ulm 23411  df-bcc 36756

This theorem is referenced by:  binomcxplemnotnn0  36775