Theorem binomcxplemdvbinom 36604

Description: Lemma for binomcxp 36608. 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 36606 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 2585	. . . . . 6 |- F/_ b `' abs
3		nfcv 2585	. . . . . . 7 |- F/_ b 0
4		nfcv 2585	. . . . . . 7 |- F/_ b [,)
5		binomcxplem.r	. . . . . . . 8 |- R = sup ( { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } , RR* , < )
6		nfcv 2585	. . . . . . . . . . . 12 |- F/_ b +
7		binomcxplem.s	. . . . . . . . . . . . . 14 |- S = ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) )
8		nfmpt1 4511	. . . . . . . . . . . . . 14 |- F/_ b ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) )
9	7, 8	nfcxfr 2583	. . . . . . . . . . . . 13 |- F/_ b S
10		nfcv 2585	. . . . . . . . . . . . 13 |- F/_ b r
11	9, 10	nffv 5886	. . . . . . . . . . . 12 |- F/_ b ( S ` r )
12	3, 6, 11	nfseq 12224	. . . . . . . . . . 11 |- F/_ b seq 0 ( + , ( S ` r ) )
13	12	nfel1 2601	. . . . . . . . . 10 |- F/ b seq 0 ( + , ( S ` r ) ) e. dom ~~>
14		nfcv 2585	. . . . . . . . . 10 |- F/_ b RR
15	13, 14	nfrab 3011	. . . . . . . . 9 |- F/_ b { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> }
16		nfcv 2585	. . . . . . . . 9 |- F/_ b RR*
17		nfcv 2585	. . . . . . . . 9 |- F/_ b <
18	15, 16, 17	nfsup 7969	. . . . . . . 8 |- F/_ b sup ( { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } , RR* , < )
19	5, 18	nfcxfr 2583	. . . . . . 7 |- F/_ b R
20	3, 4, 19	nfov 6329	. . . . . 6 |- F/_ b ( 0 [,) R )
21	2, 20	nfima 5193	. . . . 5 |- F/_ b ( `' abs " ( 0 [,) R ) )
22	1, 21	nfcxfr 2583	. . . 4 |- F/_ b D
23		nfcv 2585	. . . 4 |- F/_ y D
24		nfcv 2585	. . . 4 |- F/_ y ( ( 1 + b ) ^c -u C )
25		nfcv 2585	. . . 4 |- F/_ b ( ( 1 + y ) ^c -u C )
26		oveq2 6311	. . . . 5 |- ( b = y -> ( 1 + b ) = ( 1 + y ) )
27	26	oveq1d 6318	. . . 4 |- ( b = y -> ( ( 1 + b ) ^c -u C ) = ( ( 1 + y ) ^c -u C ) )
28	22, 23, 24, 25, 27	cbvmptf 4512	. . 3 |- ( b e. D |-> ( ( 1 + b ) ^c -u C ) ) = ( y e. D |-> ( ( 1 + y ) ^c -u C ) )
29	28	oveq2i 6314	. 2 |- ( CC _D ( b e. D |-> ( ( 1 + b ) ^c -u C ) ) ) = ( CC _D ( y e. D |-> ( ( 1 + y ) ^c -u C ) ) )
30		cnelprrecn 9634	. . . . 5 |- CC e. { RR , CC }
31	30	a1i 11	. . . 4 |- ( ( ph /\ -. C e. NN0 ) -> CC e. { RR , CC } )
32		1cnd 9661	. . . . . 6 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> 1 e. CC )
33		cnvimass 5205	. . . . . . . . . 10 |- ( `' abs " ( 0 [,) R ) ) C_ dom abs
34	1, 33	eqsstri 3495	. . . . . . . . 9 |- D C_ dom abs
35		absf 13394	. . . . . . . . . 10 |- abs : CC --> RR
36	35	fdmi 5749	. . . . . . . . 9 |- dom abs = CC
37	34, 36	sseqtri 3497	. . . . . . . 8 |- D C_ CC
38	37	a1i 11	. . . . . . 7 |- ( ( ph /\ -. C e. NN0 ) -> D C_ CC )
39	38	sselda 3465	. . . . . 6 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> y e. CC )
40	32, 39	addcld 9664	. . . . 5 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> ( 1 + y ) e. CC )
41		simpr 463	. . . . . . 7 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ ( 1 + y ) e. RR ) -> ( 1 + y ) e. RR )
42		1cnd 9661	. . . . . . . . . 10 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ ( 1 + y ) e. RR ) -> 1 e. CC )
43	39	adantr 467	. . . . . . . . . 10 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ ( 1 + y ) e. RR ) -> y e. CC )
44	42, 43	pncan2d 9990	. . . . . . . . 9 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ ( 1 + y ) e. RR ) -> ( ( 1 + y ) - 1 ) = y )
45		1red 9660	. . . . . . . . . 10 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ ( 1 + y ) e. RR ) -> 1 e. RR )
46	41, 45	resubcld 10049	. . . . . . . . 9 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ ( 1 + y ) e. RR ) -> ( ( 1 + y ) - 1 ) e. RR )
47	44, 46	eqeltrrd 2512	. . . . . . . 8 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ ( 1 + y ) e. RR ) -> y e. RR )
48		1pneg1e0 10720	. . . . . . . . 9 |- ( 1 + -u 1 ) = 0
49		1red 9660	. . . . . . . . . . 11 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ y e. RR ) -> 1 e. RR )
50	49	renegcld 10048	. . . . . . . . . 10 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ y e. RR ) -> -u 1 e. RR )
51		simpr 463	. . . . . . . . . 10 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ y e. RR ) -> y e. RR )
52		ffn 5744	. . . . . . . . . . . . . . . . . . . 20 |- ( abs : CC --> RR -> abs Fn CC )
53		elpreima 6015	. . . . . . . . . . . . . . . . . . . 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 466	. . . . . . . . . . . . . . . . . 18 |- ( y e. ( `' abs " ( 0 [,) R ) ) -> ( abs ` y ) e. ( 0 [,) R ) )
56	55, 1	eleq2s 2531	. . . . . . . . . . . . . . . . 17 |- ( y e. D -> ( abs ` y ) e. ( 0 [,) R ) )
57		0re 9645	. . . . . . . . . . . . . . . . . 18 |- 0 e. RR
58		ssrab2 3547	. . . . . . . . . . . . . . . . . . . . 21 |- { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } C_ RR
59		ressxr 9686	. . . . . . . . . . . . . . . . . . . . 21 |- RR C_ RR*
60	58, 59	sstri 3474	. . . . . . . . . . . . . . . . . . . 20 |- { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } C_ RR*
61		supxrcl 11602	. . . . . . . . . . . . . . . . . . . 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 2507	. . . . . . . . . . . . . . . . . 18 |- R e. RR*
64		elico2 11700	. . . . . . . . . . . . . . . . . 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 677	. . . . . . . . . . . . . . . . 17 |- ( ( abs ` y ) e. ( 0 [,) R ) <-> ( ( abs ` y ) e. RR /\ 0 <_ ( abs ` y ) /\ ( abs ` y ) < R ) )
66	56, 65	sylib 200	. . . . . . . . . . . . . . . 16 |- ( y e. D -> ( ( abs ` y ) e. RR /\ 0 <_ ( abs ` y ) /\ ( abs ` y ) < R ) )
67	66	simp3d 1020	. . . . . . . . . . . . . . 15 |- ( y e. D -> ( abs ` y ) < R )
68	67	adantl 468	. . . . . . . . . . . . . 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 36603	. . . . . . . . . . . . . . 15 |- ( ( ph /\ -. C e. NN0 ) -> R = 1 )
75	74	adantr 467	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> R = 1 )
76	68, 75	breqtrd 4446	. . . . . . . . . . . . 13 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> ( abs ` y ) < 1 )
77	76	adantr 467	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ y e. RR ) -> ( abs ` y ) < 1 )
78	51, 49	absltd 13485	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ y e. RR ) -> ( ( abs ` y ) < 1 <-> ( -u 1 < y /\ y < 1 ) ) )
79	77, 78	mpbid 214	. . . . . . . . . . 11 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ y e. RR ) -> ( -u 1 < y /\ y < 1 ) )
80	79	simpld 461	. . . . . . . . . 10 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ y e. RR ) -> -u 1 < y )
81	50, 51, 49, 80	ltadd2dd 9796	. . . . . . . . 9 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ y e. RR ) -> ( 1 + -u 1 ) < ( 1 + y ) )
82	48, 81	syl5eqbrr 4456	. . . . . . . 8 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ y e. RR ) -> 0 < ( 1 + y ) )
83	47, 82	syldan 473	. . . . . . 7 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ ( 1 + y ) e. RR ) -> 0 < ( 1 + y ) )
84	41, 83	elrpd 11340	. . . . . 6 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ ( 1 + y ) e. RR ) -> ( 1 + y ) e. RR+ )
85	84	ex 436	. . . . 5 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> ( ( 1 + y ) e. RR -> ( 1 + y ) e. RR+ ) )
86		eqid 2423	. . . . . 6 |- ( CC \ ( -oo (,] 0 ) ) = ( CC \ ( -oo (,] 0 ) )
87	86	ellogdm 23576	. . . . 5 |- ( ( 1 + y ) e. ( CC \ ( -oo (,] 0 ) ) <-> ( ( 1 + y ) e. CC /\ ( ( 1 + y ) e. RR -> ( 1 + y ) e. RR+ ) ) )
88	40, 85, 87	sylanbrc 669	. . . 4 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> ( 1 + y ) e. ( CC \ ( -oo (,] 0 ) ) )
89		eldifi 3588	. . . . . 6 |- ( x e. ( CC \ ( -oo (,] 0 ) ) -> x e. CC )
90	89	adantl 468	. . . . 5 |- ( ( ( ph /\ -. C e. NN0 ) /\ x e. ( CC \ ( -oo (,] 0 ) ) ) -> x e. CC )
91	72	adantr 467	. . . . . . 7 |- ( ( ph /\ -. C e. NN0 ) -> C e. CC )
92	91	negcld 9975	. . . . . 6 |- ( ( ph /\ -. C e. NN0 ) -> -u C e. CC )
93	92	adantr 467	. . . . 5 |- ( ( ( ph /\ -. C e. NN0 ) /\ x e. ( CC \ ( -oo (,] 0 ) ) ) -> -u C e. CC )
94	90, 93	cxpcld 23645	. . . 4 |- ( ( ( ph /\ -. C e. NN0 ) /\ x e. ( CC \ ( -oo (,] 0 ) ) ) -> ( x ^c -u C ) e. CC )
95		ovex 6331	. . . . 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 9661	. . . . . . 7 |- ( ( ( ph /\ -. C e. NN0 ) /\ x e. CC ) -> 1 e. CC )
98		simpr 463	. . . . . . 7 |- ( ( ( ph /\ -. C e. NN0 ) /\ x e. CC ) -> x e. CC )
99	97, 98	addcld 9664	. . . . . 6 |- ( ( ( ph /\ -. C e. NN0 ) /\ x e. CC ) -> ( 1 + x ) e. CC )
100		c0ex 9639	. . . . . . . . 9 |- 0 e. _V
101	100	a1i 11	. . . . . . . 8 |- ( ( ( ph /\ -. C e. NN0 ) /\ x e. CC ) -> 0 e. _V )
102		1cnd 9661	. . . . . . . . 9 |- ( ( ph /\ -. C e. NN0 ) -> 1 e. CC )
103	31, 102	dvmptc 22904	. . . . . . . 8 |- ( ( ph /\ -. C e. NN0 ) -> ( CC _D ( x e. CC |-> 1 ) ) = ( x e. CC |-> 0 ) )
104	31	dvmptid 22903	. . . . . . . 8 |- ( ( ph /\ -. C e. NN0 ) -> ( CC _D ( x e. CC |-> x ) ) = ( x e. CC |-> 1 ) )
105	31, 97, 101, 103, 98, 97, 104	dvmptadd 22906	. . . . . . 7 |- ( ( ph /\ -. C e. NN0 ) -> ( CC _D ( x e. CC |-> ( 1 + x ) ) ) = ( x e. CC |-> ( 0 + 1 ) ) )
106		0p1e1 10723	. . . . . . . 8 |- ( 0 + 1 ) = 1
107	106	mpteq2i 4505	. . . . . . 7 |- ( x e. CC |-> ( 0 + 1 ) ) = ( x e. CC |-> 1 )
108	105, 107	syl6eq 2480	. . . . . 6 |- ( ( ph /\ -. C e. NN0 ) -> ( CC _D ( x e. CC |-> ( 1 + x ) ) ) = ( x e. CC |-> 1 ) )
109		fvex 5889	. . . . . . . 8 |- ( TopOpen ` ℂfld ) e. _V
110		cnfldtps 21790	. . . . . . . . . 10 |- ℂfld e. TopSp
111		cnfldbas 18967	. . . . . . . . . . 11 |- CC = ( Base ` ℂfld )
112		eqid 2423	. . . . . . . . . . 11 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
113	111, 112	tpsuni 19945	. . . . . . . . . 10 |- (ℂfld e. TopSp -> CC = U. ( TopOpen ` ℂfld ) )
114	110, 113	ax-mp 5	. . . . . . . . 9 |- CC = U. ( TopOpen ` ℂfld )
115	114	restid 15325	. . . . . . . 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 2436	. . . . . 6 |- ( TopOpen ` ℂfld ) = ( ( TopOpen ` ℂfld ) ↾t CC )
118	112	cnfldtop 21796	. . . . . . . 8 |- ( TopOpen ` ℂfld ) e. Top
119		eqid 2423	. . . . . . . . . . . 12 |- ( abs o. - ) = ( abs o. - )
120	119	cnbl0 21786	. . . . . . . . . . 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 2452	. . . . . . . . 9 |- D = ( 0 ( ball ` ( abs o. - ) ) R )
123		cnxmet 21785	. . . . . . . . . 10 |- ( abs o. - ) e. ( *Met ` CC )
124		0cn 9637	. . . . . . . . . 10 |- 0 e. CC
125	112	cnfldtopn 21794	. . . . . . . . . . 11 |- ( TopOpen ` ℂfld ) = ( MetOpen ` ( abs o. - ) )
126	125	blopn 21507	. . . . . . . . . 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 1361	. . . . . . . . 9 |- ( 0 ( ball ` ( abs o. - ) ) R ) e. ( TopOpen ` ℂfld )
128	122, 127	eqeltri 2507	. . . . . . . 8 |- D e. ( TopOpen ` ℂfld )
129		isopn3i 20090	. . . . . . . 8 |- ( ( ( TopOpen ` ℂfld ) e. Top /\ D e. ( TopOpen ` ℂfld ) ) -> ( ( int ` ( TopOpen ` ℂfld ) ) ` D ) = D )
130	118, 128, 129	mp2an 677	. . . . . . 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 22908	. . . . 5 |- ( ( ph /\ -. C e. NN0 ) -> ( CC _D ( x e. D |-> ( 1 + x ) ) ) = ( x e. D |-> 1 ) )
133		oveq2 6311	. . . . . . 7 |- ( x = y -> ( 1 + x ) = ( 1 + y ) )
134	133	cbvmptv 4514	. . . . . 6 |- ( x e. D |-> ( 1 + x ) ) = ( y e. D |-> ( 1 + y ) )
135	134	oveq2i 6314	. . . . 5 |- ( CC _D ( x e. D |-> ( 1 + x ) ) ) = ( CC _D ( y e. D |-> ( 1 + y ) ) )
136		eqidd 2424	. . . . . 6 |- ( x = y -> 1 = 1 )
137	136	cbvmptv 4514	. . . . 5 |- ( x e. D |-> 1 ) = ( y e. D |-> 1 )
138	132, 135, 137	3eqtr3g 2487	. . . 4 |- ( ( ph /\ -. C e. NN0 ) -> ( CC _D ( y e. D |-> ( 1 + y ) ) ) = ( y e. D |-> 1 ) )
139	86	dvcncxp1 23675	. . . . 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 6310	. . . 4 |- ( x = ( 1 + y ) -> ( x ^c -u C ) = ( ( 1 + y ) ^c -u C ) )
142		oveq1 6310	. . . . 5 |- ( x = ( 1 + y ) -> ( x ^c ( -u C - 1 ) ) = ( ( 1 + y ) ^c ( -u C - 1 ) ) )
143	142	oveq2d 6319	. . . 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 22918	. . 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 467	. . . . . . 7 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> C e. CC )
146	145	negcld 9975	. . . . . 6 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> -u C e. CC )
147	146, 32	subcld 9988	. . . . . . 7 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> ( -u C - 1 ) e. CC )
148	40, 147	cxpcld 23645	. . . . . 6 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> ( ( 1 + y ) ^c ( -u C - 1 ) ) e. CC )
149	146, 148	mulcld 9665	. . . . 5 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> ( -u C x. ( ( 1 + y ) ^c ( -u C - 1 ) ) ) e. CC )
150	149	mulid1d 9662	. . . 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 4508	. . 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 2585	. . . . 5 |- F/_ b ( -u C x. ( ( 1 + y ) ^c ( -u C - 1 ) ) )
153		nfcv 2585	. . . . 5 |- F/_ y ( -u C x. ( ( 1 + b ) ^c ( -u C - 1 ) ) )
154		oveq2 6311	. . . . . . 7 |- ( y = b -> ( 1 + y ) = ( 1 + b ) )
155	154	oveq1d 6318	. . . . . 6 |- ( y = b -> ( ( 1 + y ) ^c ( -u C - 1 ) ) = ( ( 1 + b ) ^c ( -u C - 1 ) ) )
156	155	oveq2d 6319	. . . . 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 4512	. . . 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 2468	. 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 2476	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 188   /\ wa 371   /\ w3a 983   = wceq 1438   e. wcel 1869   {crab 2780   _Vcvv 3082   \ cdif 3434   C_ wss 3437   {cpr 3999   U.cuni 4217   class class class wbr 4421   |-> cmpt 4480   `'ccnv 4850   dom cdm 4851   "cima 4854   o. ccom 4855   Fn wfn 5594   -->wf 5595   ` cfv 5599  (class class class)co 6303   supcsup 7958   CCcc 9539   RRcr 9540   0cc0 9541   1c1 9542   + caddc 9544   x. cmul 9546   -oocmnf 9675   RR*cxr 9676   < clt 9677   <_ cle 9678   - cmin 9862   -ucneg 9863   NNcn 10611   NN0cn0 10871   RR+crp 11304   (,]cioc 11638   [,)cico 11639   seqcseq 12214   ^cexp 12273   abscabs 13291   ~~> cli 13541   ↾_t crest 15312   TopOpenctopn 15313   *Metcxmt 18948   ballcbl 18950  ℂ_fldccnfld 18963   Topctop 19909   TopSpctps 19911   intcnt 20024   _D cdv 22810   ^c ccxp 23497  C_𝑐cbcc 36587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-inf2 8150  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618  ax-pre-sup 9619  ax-addf 9620  ax-mulf 9621

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-fal 1444  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-int 4254  df-iun 4299  df-iin 4300  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-se 4811  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-isom 5608  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-of 6543  df-om 6705  df-1st 6805  df-2nd 6806  df-supp 6924  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-1o 7188  df-2o 7189  df-oadd 7192  df-er 7369  df-map 7480  df-pm 7481  df-ixp 7529  df-en 7576  df-dom 7577  df-sdom 7578  df-fin 7579  df-fsupp 7888  df-fi 7929  df-sup 7960  df-inf 7961  df-oi 8029  df-card 8376  df-cda 8600  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-div 10272  df-nn 10612  df-2 10670  df-3 10671  df-4 10672  df-5 10673  df-6 10674  df-7 10675  df-8 10676  df-9 10677  df-10 10678  df-n0 10872  df-z 10940  df-dec 11054  df-uz 11162  df-q 11267  df-rp 11305  df-xneg 11411  df-xadd 11412  df-xmul 11413  df-ioo 11641  df-ioc 11642  df-ico 11643  df-icc 11644  df-fz 11787  df-fzo 11918  df-fl 12029  df-mod 12098  df-seq 12215  df-exp 12274  df-fac 12461  df-bc 12489  df-hash 12517  df-shft 13124  df-cj 13156  df-re 13157  df-im 13158  df-sqrt 13292  df-abs 13293  df-limsup 13519  df-clim 13545  df-rlim 13546  df-sum 13746  df-prod 13953  df-fallfac 14053  df-ef 14114  df-sin 14116  df-cos 14117  df-tan 14118  df-pi 14119  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-mulr 15197  df-starv 15198  df-sca 15199  df-vsca 15200  df-ip 15201  df-tset 15202  df-ple 15203  df-ds 15205  df-unif 15206  df-hom 15207  df-cco 15208  df-rest 15314  df-topn 15315  df-0g 15333  df-gsum 15334  df-topgen 15335  df-pt 15336  df-prds 15339  df-xrs 15393  df-qtop 15399  df-imas 15400  df-xps 15403  df-mre 15485  df-mrc 15486  df-acs 15488  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-submnd 16576  df-mulg 16669  df-cntz 16964  df-cmn 17425  df-psmet 18955  df-xmet 18956  df-met 18957  df-bl 18958  df-mopn 18959  df-fbas 18960  df-fg 18961  df-cnfld 18964  df-top 19913  df-bases 19914  df-topon 19915  df-topsp 19916  df-cld 20026  df-ntr 20027  df-cls 20028  df-nei 20106  df-lp 20144  df-perf 20145  df-cn 20235  df-cnp 20236  df-haus 20323  df-cmp 20394  df-tx 20569  df-hmeo 20762  df-fil 20853  df-fm 20945  df-flim 20946  df-flf 20947  df-xms 21327  df-ms 21328  df-tms 21329  df-cncf 21902  df-limc 22813  df-dv 22814  df-log 23498  df-cxp 23499  df-bcc 36588

This theorem is referenced by:  binomcxplemnotnn0  36607