Theorem binomcxplemdvbinom 36106

Description: Lemma for binomcxp 36110. By the power and chain rules, calculate the derivative of ( ( 1 + b ) ^c -u C ), with respect to b in the disk of convergence D. We later multiply the derivative in the later binomcxplemdvsum 36108 by this derivative to show that ( ( 1 + b ) ^c C ) (with a non-negated C) and the later sum, since both at b = 0 equal one, are the same. (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 ) )

Assertion

Ref	Expression
binomcxplemdvbinom	|- ( ( ph /\ -. C e. NN0 ) -> ( CC _D ( b e. D |-> ( ( 1 + b ) ^c -u C ) ) ) = ( b e. D |-> ( -u C x. ( ( 1 + b ) ^c ( -u C - 1 ) ) ) ) )

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

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

Proof of Theorem binomcxplemdvbinom

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		binomcxplem.d	. . . . 5 |- D = ( `' abs " ( 0 [,) R ) )
2		nfcv 2564	. . . . . 6 |- F/_ b `' abs
3		nfcv 2564	. . . . . . 7 |- F/_ b 0
4		nfcv 2564	. . . . . . 7 |- F/_ b [,)
5		binomcxplem.r	. . . . . . . 8 |- R = sup ( { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } , RR* , < )
6		nfcv 2564	. . . . . . . . . . . 12 |- F/_ b +
7		binomcxplem.s	. . . . . . . . . . . . . 14 |- S = ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) )
8		nfmpt1 4484	. . . . . . . . . . . . . 14 |- F/_ b ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) )
9	7, 8	nfcxfr 2562	. . . . . . . . . . . . 13 |- F/_ b S
10		nfcv 2564	. . . . . . . . . . . . 13 |- F/_ b r
11	9, 10	nffv 5856	. . . . . . . . . . . 12 |- F/_ b ( S ` r )
12	3, 6, 11	nfseq 12161	. . . . . . . . . . 11 |- F/_ b seq 0 ( + , ( S ` r ) )
13	12	nfel1 2580	. . . . . . . . . 10 |- F/ b seq 0 ( + , ( S ` r ) ) e. dom ~~>
14		nfcv 2564	. . . . . . . . . 10 |- F/_ b RR
15	13, 14	nfrab 2989	. . . . . . . . 9 |- F/_ b { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> }
16		nfcv 2564	. . . . . . . . 9 |- F/_ b RR*
17		nfcv 2564	. . . . . . . . 9 |- F/_ b <
18	15, 16, 17	nfsup 7944	. . . . . . . 8 |- F/_ b sup ( { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } , RR* , < )
19	5, 18	nfcxfr 2562	. . . . . . 7 |- F/_ b R
20	3, 4, 19	nfov 6304	. . . . . 6 |- F/_ b ( 0 [,) R )
21	2, 20	nfima 5165	. . . . 5 |- F/_ b ( `' abs " ( 0 [,) R ) )
22	1, 21	nfcxfr 2562	. . . 4 |- F/_ b D
23		nfcv 2564	. . . 4 |- F/_ y D
24		nfcv 2564	. . . 4 |- F/_ y ( ( 1 + b ) ^c -u C )
25		nfcv 2564	. . . 4 |- F/_ b ( ( 1 + y ) ^c -u C )
26		oveq2 6286	. . . . 5 |- ( b = y -> ( 1 + b ) = ( 1 + y ) )
27	26	oveq1d 6293	. . . 4 |- ( b = y -> ( ( 1 + b ) ^c -u C ) = ( ( 1 + y ) ^c -u C ) )
28	22, 23, 24, 25, 27	cbvmptf 4485	. . 3 |- ( b e. D |-> ( ( 1 + b ) ^c -u C ) ) = ( y e. D |-> ( ( 1 + y ) ^c -u C ) )
29	28	oveq2i 6289	. 2 |- ( CC _D ( b e. D |-> ( ( 1 + b ) ^c -u C ) ) ) = ( CC _D ( y e. D |-> ( ( 1 + y ) ^c -u C ) ) )
30		cnelprrecn 9615	. . . . 5 |- CC e. { RR , CC }
31	30	a1i 11	. . . 4 |- ( ( ph /\ -. C e. NN0 ) -> CC e. { RR , CC } )
32		1cnd 9642	. . . . . 6 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> 1 e. CC )
33		cnvimass 5177	. . . . . . . . . 10 |- ( `' abs " ( 0 [,) R ) ) C_ dom abs
34	1, 33	eqsstri 3472	. . . . . . . . 9 |- D C_ dom abs
35		absf 13319	. . . . . . . . . 10 |- abs : CC --> RR
36	35	fdmi 5719	. . . . . . . . 9 |- dom abs = CC
37	34, 36	sseqtri 3474	. . . . . . . 8 |- D C_ CC
38	37	a1i 11	. . . . . . 7 |- ( ( ph /\ -. C e. NN0 ) -> D C_ CC )
39	38	sselda 3442	. . . . . 6 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> y e. CC )
40	32, 39	addcld 9645	. . . . 5 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> ( 1 + y ) e. CC )
41		simpr 459	. . . . . . 7 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ ( 1 + y ) e. RR ) -> ( 1 + y ) e. RR )
42		1cnd 9642	. . . . . . . . . 10 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ ( 1 + y ) e. RR ) -> 1 e. CC )
43	39	adantr 463	. . . . . . . . . 10 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ ( 1 + y ) e. RR ) -> y e. CC )
44	42, 43	pncan2d 9969	. . . . . . . . 9 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ ( 1 + y ) e. RR ) -> ( ( 1 + y ) - 1 ) = y )
45		1red 9641	. . . . . . . . . 10 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ ( 1 + y ) e. RR ) -> 1 e. RR )
46	41, 45	resubcld 10028	. . . . . . . . 9 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ ( 1 + y ) e. RR ) -> ( ( 1 + y ) - 1 ) e. RR )
47	44, 46	eqeltrrd 2491	. . . . . . . 8 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ ( 1 + y ) e. RR ) -> y e. RR )
48		1pneg1e0 10685	. . . . . . . . 9 |- ( 1 + -u 1 ) = 0
49		1red 9641	. . . . . . . . . . 11 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ y e. RR ) -> 1 e. RR )
50	49	renegcld 10027	. . . . . . . . . 10 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ y e. RR ) -> -u 1 e. RR )
51		simpr 459	. . . . . . . . . 10 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ y e. RR ) -> y e. RR )
52		ffn 5714	. . . . . . . . . . . . . . . . . . . 20 |- ( abs : CC --> RR -> abs Fn CC )
53		elpreima 5985	. . . . . . . . . . . . . . . . . . . 20 |- ( abs Fn CC -> ( y e. ( `' abs " ( 0 [,) R ) ) <-> ( y e. CC /\ ( abs ` y ) e. ( 0 [,) R ) ) ) )
54	35, 52, 53	mp2b 10	. . . . . . . . . . . . . . . . . . 19 |- ( y e. ( `' abs " ( 0 [,) R ) ) <-> ( y e. CC /\ ( abs ` y ) e. ( 0 [,) R ) ) )
55	54	simprbi 462	. . . . . . . . . . . . . . . . . 18 |- ( y e. ( `' abs " ( 0 [,) R ) ) -> ( abs ` y ) e. ( 0 [,) R ) )
56	55, 1	eleq2s 2510	. . . . . . . . . . . . . . . . 17 |- ( y e. D -> ( abs ` y ) e. ( 0 [,) R ) )
57		0re 9626	. . . . . . . . . . . . . . . . . 18 |- 0 e. RR
58		ssrab2 3524	. . . . . . . . . . . . . . . . . . . . 21 |- { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } C_ RR
59		ressxr 9667	. . . . . . . . . . . . . . . . . . . . 21 |- RR C_ RR*
60	58, 59	sstri 3451	. . . . . . . . . . . . . . . . . . . 20 |- { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } C_ RR*
61		supxrcl 11559	. . . . . . . . . . . . . . . . . . . 20 |- ( { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } C_ RR* -> sup ( { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } , RR* , < ) e. RR* )
62	60, 61	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 |- sup ( { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } , RR* , < ) e. RR*
63	5, 62	eqeltri 2486	. . . . . . . . . . . . . . . . . 18 |- R e. RR*
64		elico2 11642	. . . . . . . . . . . . . . . . . 18 |- ( ( 0 e. RR /\ R e. RR* ) -> ( ( abs ` y ) e. ( 0 [,) R ) <-> ( ( abs ` y ) e. RR /\ 0 <_ ( abs ` y ) /\ ( abs ` y ) < R ) ) )
65	57, 63, 64	mp2an 670	. . . . . . . . . . . . . . . . 17 |- ( ( abs ` y ) e. ( 0 [,) R ) <-> ( ( abs ` y ) e. RR /\ 0 <_ ( abs ` y ) /\ ( abs ` y ) < R ) )
66	56, 65	sylib 196	. . . . . . . . . . . . . . . 16 |- ( y e. D -> ( ( abs ` y ) e. RR /\ 0 <_ ( abs ` y ) /\ ( abs ` y ) < R ) )
67	66	simp3d 1011	. . . . . . . . . . . . . . 15 |- ( y e. D -> ( abs ` y ) < R )
68	67	adantl 464	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> ( abs ` y ) < R )
69		binomcxp.a	. . . . . . . . . . . . . . . 16 |- ( ph -> A e. RR+ )
70		binomcxp.b	. . . . . . . . . . . . . . . 16 |- ( ph -> B e. RR )
71		binomcxp.lt	. . . . . . . . . . . . . . . 16 |- ( ph -> ( abs ` B ) < ( abs ` A ) )
72		binomcxp.c	. . . . . . . . . . . . . . . 16 |- ( ph -> C e. CC )
73		binomcxplem.f	. . . . . . . . . . . . . . . 16 |- F = ( j e. NN0 |-> ( CC𝑐 j ) )
74	69, 70, 71, 72, 73, 7, 5	binomcxplemradcnv 36105	. . . . . . . . . . . . . . 15 |- ( ( ph /\ -. C e. NN0 ) -> R = 1 )
75	74	adantr 463	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> R = 1 )
76	68, 75	breqtrd 4419	. . . . . . . . . . . . 13 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> ( abs ` y ) < 1 )
77	76	adantr 463	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ y e. RR ) -> ( abs ` y ) < 1 )
78	51, 49	absltd 13410	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ y e. RR ) -> ( ( abs ` y ) < 1 <-> ( -u 1 < y /\ y < 1 ) ) )
79	77, 78	mpbid 210	. . . . . . . . . . 11 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ y e. RR ) -> ( -u 1 < y /\ y < 1 ) )
80	79	simpld 457	. . . . . . . . . 10 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ y e. RR ) -> -u 1 < y )
81	50, 51, 49, 80	ltadd2dd 9775	. . . . . . . . 9 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ y e. RR ) -> ( 1 + -u 1 ) < ( 1 + y ) )
82	48, 81	syl5eqbrr 4429	. . . . . . . 8 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ y e. RR ) -> 0 < ( 1 + y ) )
83	47, 82	syldan 468	. . . . . . 7 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ ( 1 + y ) e. RR ) -> 0 < ( 1 + y ) )
84	41, 83	elrpd 11301	. . . . . 6 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ ( 1 + y ) e. RR ) -> ( 1 + y ) e. RR+ )
85	84	ex 432	. . . . 5 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> ( ( 1 + y ) e. RR -> ( 1 + y ) e. RR+ ) )
86		eqid 2402	. . . . . 6 |- ( CC \ ( -oo (,] 0 ) ) = ( CC \ ( -oo (,] 0 ) )
87	86	ellogdm 23314	. . . . 5 |- ( ( 1 + y ) e. ( CC \ ( -oo (,] 0 ) ) <-> ( ( 1 + y ) e. CC /\ ( ( 1 + y ) e. RR -> ( 1 + y ) e. RR+ ) ) )
88	40, 85, 87	sylanbrc 662	. . . 4 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> ( 1 + y ) e. ( CC \ ( -oo (,] 0 ) ) )
89		eldifi 3565	. . . . . 6 |- ( x e. ( CC \ ( -oo (,] 0 ) ) -> x e. CC )
90	89	adantl 464	. . . . 5 |- ( ( ( ph /\ -. C e. NN0 ) /\ x e. ( CC \ ( -oo (,] 0 ) ) ) -> x e. CC )
91	72	adantr 463	. . . . . . 7 |- ( ( ph /\ -. C e. NN0 ) -> C e. CC )
92	91	negcld 9954	. . . . . 6 |- ( ( ph /\ -. C e. NN0 ) -> -u C e. CC )
93	92	adantr 463	. . . . 5 |- ( ( ( ph /\ -. C e. NN0 ) /\ x e. ( CC \ ( -oo (,] 0 ) ) ) -> -u C e. CC )
94	90, 93	cxpcld 23383	. . . 4 |- ( ( ( ph /\ -. C e. NN0 ) /\ x e. ( CC \ ( -oo (,] 0 ) ) ) -> ( x ^c -u C ) e. CC )
95		ovex 6306	. . . . 5 |- ( -u C x. ( x ^c ( -u C - 1 ) ) ) e. _V
96	95	a1i 11	. . . 4 |- ( ( ( ph /\ -. C e. NN0 ) /\ x e. ( CC \ ( -oo (,] 0 ) ) ) -> ( -u C x. ( x ^c ( -u C - 1 ) ) ) e. _V )
97		1cnd 9642	. . . . . . 7 |- ( ( ( ph /\ -. C e. NN0 ) /\ x e. CC ) -> 1 e. CC )
98		simpr 459	. . . . . . 7 |- ( ( ( ph /\ -. C e. NN0 ) /\ x e. CC ) -> x e. CC )
99	97, 98	addcld 9645	. . . . . 6 |- ( ( ( ph /\ -. C e. NN0 ) /\ x e. CC ) -> ( 1 + x ) e. CC )
100		c0ex 9620	. . . . . . . . 9 |- 0 e. _V
101	100	a1i 11	. . . . . . . 8 |- ( ( ( ph /\ -. C e. NN0 ) /\ x e. CC ) -> 0 e. _V )
102		1cnd 9642	. . . . . . . . 9 |- ( ( ph /\ -. C e. NN0 ) -> 1 e. CC )
103	31, 102	dvmptc 22653	. . . . . . . 8 |- ( ( ph /\ -. C e. NN0 ) -> ( CC _D ( x e. CC |-> 1 ) ) = ( x e. CC |-> 0 ) )
104	31	dvmptid 22652	. . . . . . . 8 |- ( ( ph /\ -. C e. NN0 ) -> ( CC _D ( x e. CC |-> x ) ) = ( x e. CC |-> 1 ) )
105	31, 97, 101, 103, 98, 97, 104	dvmptadd 22655	. . . . . . 7 |- ( ( ph /\ -. C e. NN0 ) -> ( CC _D ( x e. CC |-> ( 1 + x ) ) ) = ( x e. CC |-> ( 0 + 1 ) ) )
106		0p1e1 10688	. . . . . . . 8 |- ( 0 + 1 ) = 1
107	106	mpteq2i 4478	. . . . . . 7 |- ( x e. CC |-> ( 0 + 1 ) ) = ( x e. CC |-> 1 )
108	105, 107	syl6eq 2459	. . . . . 6 |- ( ( ph /\ -. C e. NN0 ) -> ( CC _D ( x e. CC |-> ( 1 + x ) ) ) = ( x e. CC |-> 1 ) )
109		fvex 5859	. . . . . . . 8 |- ( TopOpen ` ℂfld ) e. _V
110		cnfldtps 21577	. . . . . . . . . 10 |- ℂfld e. TopSp
111		cnfldbas 18744	. . . . . . . . . . 11 |- CC = ( Base ` ℂfld )
112		eqid 2402	. . . . . . . . . . 11 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
113	111, 112	tpsuni 19731	. . . . . . . . . 10 |- (ℂfld e. TopSp -> CC = U. ( TopOpen ` ℂfld ) )
114	110, 113	ax-mp 5	. . . . . . . . 9 |- CC = U. ( TopOpen ` ℂfld )
115	114	restid 15048	. . . . . . . 8 |- ( ( TopOpen ` ℂfld ) e. _V -> ( ( TopOpen ` ℂfld ) ↾t CC ) = ( TopOpen ` ℂfld ) )
116	109, 115	ax-mp 5	. . . . . . 7 |- ( ( TopOpen ` ℂfld ) ↾t CC ) = ( TopOpen ` ℂfld )
117	116	eqcomi 2415	. . . . . 6 |- ( TopOpen ` ℂfld ) = ( ( TopOpen ` ℂfld ) ↾t CC )
118	112	cnfldtop 21583	. . . . . . . 8 |- ( TopOpen ` ℂfld ) e. Top
119		eqid 2402	. . . . . . . . . . . 12 |- ( abs o. - ) = ( abs o. - )
120	119	cnbl0 21573	. . . . . . . . . . 11 |- ( R e. RR* -> ( `' abs " ( 0 [,) R ) ) = ( 0 ( ball ` ( abs o. - ) ) R ) )
121	63, 120	ax-mp 5	. . . . . . . . . 10 |- ( `' abs " ( 0 [,) R ) ) = ( 0 ( ball ` ( abs o. - ) ) R )
122	1, 121	eqtri 2431	. . . . . . . . 9 |- D = ( 0 ( ball ` ( abs o. - ) ) R )
123		cnxmet 21572	. . . . . . . . . 10 |- ( abs o. - ) e. ( *Met ` CC )
124		0cn 9618	. . . . . . . . . 10 |- 0 e. CC
125	112	cnfldtopn 21581	. . . . . . . . . . 11 |- ( TopOpen ` ℂfld ) = ( MetOpen ` ( abs o. - ) )
126	125	blopn 21295	. . . . . . . . . 10 |- ( ( ( abs o. - ) e. ( *Met ` CC ) /\ 0 e. CC /\ R e. RR* ) -> ( 0 ( ball ` ( abs o. - ) ) R ) e. ( TopOpen ` ℂfld ) )
127	123, 124, 63, 126	mp3an 1326	. . . . . . . . 9 |- ( 0 ( ball ` ( abs o. - ) ) R ) e. ( TopOpen ` ℂfld )
128	122, 127	eqeltri 2486	. . . . . . . 8 |- D e. ( TopOpen ` ℂfld )
129		isopn3i 19876	. . . . . . . 8 |- ( ( ( TopOpen ` ℂfld ) e. Top /\ D e. ( TopOpen ` ℂfld ) ) -> ( ( int ` ( TopOpen ` ℂfld ) ) ` D ) = D )
130	118, 128, 129	mp2an 670	. . . . . . 7 |- ( ( int ` ( TopOpen ` ℂfld ) ) ` D ) = D
131	130	a1i 11	. . . . . 6 |- ( ( ph /\ -. C e. NN0 ) -> ( ( int ` ( TopOpen ` ℂfld ) ) ` D ) = D )
132	31, 99, 97, 108, 38, 117, 112, 131	dvmptres2 22657	. . . . 5 |- ( ( ph /\ -. C e. NN0 ) -> ( CC _D ( x e. D |-> ( 1 + x ) ) ) = ( x e. D |-> 1 ) )
133		oveq2 6286	. . . . . . 7 |- ( x = y -> ( 1 + x ) = ( 1 + y ) )
134	133	cbvmptv 4487	. . . . . 6 |- ( x e. D |-> ( 1 + x ) ) = ( y e. D |-> ( 1 + y ) )
135	134	oveq2i 6289	. . . . 5 |- ( CC _D ( x e. D |-> ( 1 + x ) ) ) = ( CC _D ( y e. D |-> ( 1 + y ) ) )
136		eqidd 2403	. . . . . 6 |- ( x = y -> 1 = 1 )
137	136	cbvmptv 4487	. . . . 5 |- ( x e. D |-> 1 ) = ( y e. D |-> 1 )
138	132, 135, 137	3eqtr3g 2466	. . . 4 |- ( ( ph /\ -. C e. NN0 ) -> ( CC _D ( y e. D |-> ( 1 + y ) ) ) = ( y e. D |-> 1 ) )
139	86	dvcncxp1 23413	. . . . 5 |- ( -u C e. CC -> ( CC _D ( x e. ( CC \ ( -oo (,] 0 ) ) |-> ( x ^c -u C ) ) ) = ( x e. ( CC \ ( -oo (,] 0 ) ) |-> ( -u C x. ( x ^c ( -u C - 1 ) ) ) ) )
140	92, 139	syl 17	. . . 4 |- ( ( ph /\ -. C e. NN0 ) -> ( CC _D ( x e. ( CC \ ( -oo (,] 0 ) ) |-> ( x ^c -u C ) ) ) = ( x e. ( CC \ ( -oo (,] 0 ) ) |-> ( -u C x. ( x ^c ( -u C - 1 ) ) ) ) )
141		oveq1 6285	. . . 4 |- ( x = ( 1 + y ) -> ( x ^c -u C ) = ( ( 1 + y ) ^c -u C ) )
142		oveq1 6285	. . . . 5 |- ( x = ( 1 + y ) -> ( x ^c ( -u C - 1 ) ) = ( ( 1 + y ) ^c ( -u C - 1 ) ) )
143	142	oveq2d 6294	. . . 4 |- ( x = ( 1 + y ) -> ( -u C x. ( x ^c ( -u C - 1 ) ) ) = ( -u C x. ( ( 1 + y ) ^c ( -u C - 1 ) ) ) )
144	31, 31, 88, 32, 94, 96, 138, 140, 141, 143	dvmptco 22667	. . 3 |- ( ( ph /\ -. C e. NN0 ) -> ( CC _D ( y e. D |-> ( ( 1 + y ) ^c -u C ) ) ) = ( y e. D |-> ( ( -u C x. ( ( 1 + y ) ^c ( -u C - 1 ) ) ) x. 1 ) ) )
145	91	adantr 463	. . . . . . 7 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> C e. CC )
146	145	negcld 9954	. . . . . 6 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> -u C e. CC )
147	146, 32	subcld 9967	. . . . . . 7 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> ( -u C - 1 ) e. CC )
148	40, 147	cxpcld 23383	. . . . . 6 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> ( ( 1 + y ) ^c ( -u C - 1 ) ) e. CC )
149	146, 148	mulcld 9646	. . . . 5 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> ( -u C x. ( ( 1 + y ) ^c ( -u C - 1 ) ) ) e. CC )
150	149	mulid1d 9643	. . . 4 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> ( ( -u C x. ( ( 1 + y ) ^c ( -u C - 1 ) ) ) x. 1 ) = ( -u C x. ( ( 1 + y ) ^c ( -u C - 1 ) ) ) )
151	150	mpteq2dva 4481	. . 3 |- ( ( ph /\ -. C e. NN0 ) -> ( y e. D |-> ( ( -u C x. ( ( 1 + y ) ^c ( -u C - 1 ) ) ) x. 1 ) ) = ( y e. D |-> ( -u C x. ( ( 1 + y ) ^c ( -u C - 1 ) ) ) ) )
152		nfcv 2564	. . . . 5 |- F/_ b ( -u C x. ( ( 1 + y ) ^c ( -u C - 1 ) ) )
153		nfcv 2564	. . . . 5 |- F/_ y ( -u C x. ( ( 1 + b ) ^c ( -u C - 1 ) ) )
154		oveq2 6286	. . . . . . 7 |- ( y = b -> ( 1 + y ) = ( 1 + b ) )
155	154	oveq1d 6293	. . . . . 6 |- ( y = b -> ( ( 1 + y ) ^c ( -u C - 1 ) ) = ( ( 1 + b ) ^c ( -u C - 1 ) ) )
156	155	oveq2d 6294	. . . . 5 |- ( y = b -> ( -u C x. ( ( 1 + y ) ^c ( -u C - 1 ) ) ) = ( -u C x. ( ( 1 + b ) ^c ( -u C - 1 ) ) ) )
157	23, 22, 152, 153, 156	cbvmptf 4485	. . . 4 |- ( y e. D |-> ( -u C x. ( ( 1 + y ) ^c ( -u C - 1 ) ) ) ) = ( b e. D |-> ( -u C x. ( ( 1 + b ) ^c ( -u C - 1 ) ) ) )
158	157	a1i 11	. . 3 |- ( ( ph /\ -. C e. NN0 ) -> ( y e. D |-> ( -u C x. ( ( 1 + y ) ^c ( -u C - 1 ) ) ) ) = ( b e. D |-> ( -u C x. ( ( 1 + b ) ^c ( -u C - 1 ) ) ) ) )
159	144, 151, 158	3eqtrd 2447	. 2 |- ( ( ph /\ -. C e. NN0 ) -> ( CC _D ( y e. D |-> ( ( 1 + y ) ^c -u C ) ) ) = ( b e. D |-> ( -u C x. ( ( 1 + b ) ^c ( -u C - 1 ) ) ) ) )
160	29, 159	syl5eq 2455	1 |- ( ( ph /\ -. C e. NN0 ) -> ( CC _D ( b e. D |-> ( ( 1 + b ) ^c -u C ) ) ) = ( b e. D |-> ( -u C x. ( ( 1 + b ) ^c ( -u C - 1 ) ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 367   /\ w3a 974   = wceq 1405   e. wcel 1842   {crab 2758   _Vcvv 3059   \ cdif 3411   C_ wss 3414   {cpr 3974   U.cuni 4191   class class class wbr 4395   |-> cmpt 4453   `'ccnv 4822   dom cdm 4823   "cima 4826   o. ccom 4827   Fn wfn 5564   -->wf 5565   ` cfv 5569  (class class class)co 6278   supcsup 7934   CCcc 9520   RRcr 9521   0cc0 9522   1c1 9523   + caddc 9525   x. cmul 9527   -oocmnf 9656   RR*cxr 9657   < clt 9658   <_ cle 9659   - cmin 9841   -ucneg 9842   NNcn 10576   NN0cn0 10836   RR+crp 11265   (,]cioc 11583   [,)cico 11584   seqcseq 12151   ^cexp 12210   abscabs 13216   ~~> cli 13456   ↾_t crest 15035   TopOpenctopn 15036   *Metcxmt 18723   ballcbl 18725  ℂ_fldccnfld 18740   Topctop 19686   TopSpctps 19689   intcnt 19810   _D cdv 22559   ^c ccxp 23235  C_𝑐cbcc 36089

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 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600  ax-addf 9601  ax-mulf 9602

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 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-iin 4274  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-of 6521  df-om 6684  df-1st 6784  df-2nd 6785  df-supp 6903  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-2o 7168  df-oadd 7171  df-er 7348  df-map 7459  df-pm 7460  df-ixp 7508  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-fsupp 7864  df-fi 7905  df-sup 7935  df-oi 7969  df-card 8352  df-cda 8580  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-5 10638  df-6 10639  df-7 10640  df-8 10641  df-9 10642  df-10 10643  df-n0 10837  df-z 10906  df-dec 11020  df-uz 11128  df-q 11228  df-rp 11266  df-xneg 11371  df-xadd 11372  df-xmul 11373  df-ioo 11586  df-ioc 11587  df-ico 11588  df-icc 11589  df-fz 11727  df-fzo 11855  df-fl 11966  df-mod 12035  df-seq 12152  df-exp 12211  df-fac 12398  df-bc 12425  df-hash 12453  df-shft 13049  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-limsup 13443  df-clim 13460  df-rlim 13461  df-sum 13658  df-prod 13865  df-fallfac 13952  df-ef 14012  df-sin 14014  df-cos 14015  df-tan 14016  df-pi 14017  df-struct 14843  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-ress 14848  df-plusg 14922  df-mulr 14923  df-starv 14924  df-sca 14925  df-vsca 14926  df-ip 14927  df-tset 14928  df-ple 14929  df-ds 14931  df-unif 14932  df-hom 14933  df-cco 14934  df-rest 15037  df-topn 15038  df-0g 15056  df-gsum 15057  df-topgen 15058  df-pt 15059  df-prds 15062  df-xrs 15116  df-qtop 15121  df-imas 15122  df-xps 15124  df-mre 15200  df-mrc 15201  df-acs 15203  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-submnd 16291  df-mulg 16384  df-cntz 16679  df-cmn 17124  df-psmet 18731  df-xmet 18732  df-met 18733  df-bl 18734  df-mopn 18735  df-fbas 18736  df-fg 18737  df-cnfld 18741  df-top 19691  df-bases 19693  df-topon 19694  df-topsp 19695  df-cld 19812  df-ntr 19813  df-cls 19814  df-nei 19892  df-lp 19930  df-perf 19931  df-cn 20021  df-cnp 20022  df-haus 20109  df-cmp 20180  df-tx 20355  df-hmeo 20548  df-fil 20639  df-fm 20731  df-flim 20732  df-flf 20733  df-xms 21115  df-ms 21116  df-tms 21117  df-cncf 21674  df-limc 22562  df-dv 22563  df-log 23236  df-cxp 23237  df-bcc 36090

This theorem is referenced by:  binomcxplemnotnn0  36109