Theorem binomcxplemdvbinom 36702

Description: Lemma for binomcxp 36706. 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 36704 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 2592	. . . . . 6 |- F/_ b `' abs
3		nfcv 2592	. . . . . . 7 |- F/_ b 0
4		nfcv 2592	. . . . . . 7 |- F/_ b [,)
5		binomcxplem.r	. . . . . . . 8 |- R = sup ( { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } , RR* , < )
6		nfcv 2592	. . . . . . . . . . . 12 |- F/_ b +
7		binomcxplem.s	. . . . . . . . . . . . . 14 |- S = ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) )
8		nfmpt1 4492	. . . . . . . . . . . . . 14 |- F/_ b ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) )
9	7, 8	nfcxfr 2590	. . . . . . . . . . . . 13 |- F/_ b S
10		nfcv 2592	. . . . . . . . . . . . 13 |- F/_ b r
11	9, 10	nffv 5872	. . . . . . . . . . . 12 |- F/_ b ( S ` r )
12	3, 6, 11	nfseq 12223	. . . . . . . . . . 11 |- F/_ b seq 0 ( + , ( S ` r ) )
13	12	nfel1 2606	. . . . . . . . . 10 |- F/ b seq 0 ( + , ( S ` r ) ) e. dom ~~>
14		nfcv 2592	. . . . . . . . . 10 |- F/_ b RR
15	13, 14	nfrab 2972	. . . . . . . . 9 |- F/_ b { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> }
16		nfcv 2592	. . . . . . . . 9 |- F/_ b RR*
17		nfcv 2592	. . . . . . . . 9 |- F/_ b <
18	15, 16, 17	nfsup 7965	. . . . . . . 8 |- F/_ b sup ( { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } , RR* , < )
19	5, 18	nfcxfr 2590	. . . . . . 7 |- F/_ b R
20	3, 4, 19	nfov 6316	. . . . . 6 |- F/_ b ( 0 [,) R )
21	2, 20	nfima 5176	. . . . 5 |- F/_ b ( `' abs " ( 0 [,) R ) )
22	1, 21	nfcxfr 2590	. . . 4 |- F/_ b D
23		nfcv 2592	. . . 4 |- F/_ y D
24		nfcv 2592	. . . 4 |- F/_ y ( ( 1 + b ) ^c -u C )
25		nfcv 2592	. . . 4 |- F/_ b ( ( 1 + y ) ^c -u C )
26		oveq2 6298	. . . . 5 |- ( b = y -> ( 1 + b ) = ( 1 + y ) )
27	26	oveq1d 6305	. . . 4 |- ( b = y -> ( ( 1 + b ) ^c -u C ) = ( ( 1 + y ) ^c -u C ) )
28	22, 23, 24, 25, 27	cbvmptf 4493	. . 3 |- ( b e. D |-> ( ( 1 + b ) ^c -u C ) ) = ( y e. D |-> ( ( 1 + y ) ^c -u C ) )
29	28	oveq2i 6301	. 2 |- ( CC _D ( b e. D |-> ( ( 1 + b ) ^c -u C ) ) ) = ( CC _D ( y e. D |-> ( ( 1 + y ) ^c -u C ) ) )
30		cnelprrecn 9632	. . . . 5 |- CC e. { RR , CC }
31	30	a1i 11	. . . 4 |- ( ( ph /\ -. C e. NN0 ) -> CC e. { RR , CC } )
32		1cnd 9659	. . . . . 6 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> 1 e. CC )
33		cnvimass 5188	. . . . . . . . . 10 |- ( `' abs " ( 0 [,) R ) ) C_ dom abs
34	1, 33	eqsstri 3462	. . . . . . . . 9 |- D C_ dom abs
35		absf 13400	. . . . . . . . . 10 |- abs : CC --> RR
36	35	fdmi 5734	. . . . . . . . 9 |- dom abs = CC
37	34, 36	sseqtri 3464	. . . . . . . 8 |- D C_ CC
38	37	a1i 11	. . . . . . 7 |- ( ( ph /\ -. C e. NN0 ) -> D C_ CC )
39	38	sselda 3432	. . . . . 6 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> y e. CC )
40	32, 39	addcld 9662	. . . . 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 9659	. . . . . . . . . 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 9988	. . . . . . . . 9 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ ( 1 + y ) e. RR ) -> ( ( 1 + y ) - 1 ) = y )
45		1red 9658	. . . . . . . . . 10 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ ( 1 + y ) e. RR ) -> 1 e. RR )
46	41, 45	resubcld 10047	. . . . . . . . 9 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ ( 1 + y ) e. RR ) -> ( ( 1 + y ) - 1 ) e. RR )
47	44, 46	eqeltrrd 2530	. . . . . . . 8 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ ( 1 + y ) e. RR ) -> y e. RR )
48		1pneg1e0 10718	. . . . . . . . 9 |- ( 1 + -u 1 ) = 0
49		1red 9658	. . . . . . . . . . 11 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ y e. RR ) -> 1 e. RR )
50	49	renegcld 10046	. . . . . . . . . 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 5728	. . . . . . . . . . . . . . . . . . . 20 |- ( abs : CC --> RR -> abs Fn CC )
53		elpreima 6002	. . . . . . . . . . . . . . . . . . . 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 2547	. . . . . . . . . . . . . . . . 17 |- ( y e. D -> ( abs ` y ) e. ( 0 [,) R ) )
57		0re 9643	. . . . . . . . . . . . . . . . . 18 |- 0 e. RR
58		ssrab2 3514	. . . . . . . . . . . . . . . . . . . . 21 |- { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } C_ RR
59		ressxr 9684	. . . . . . . . . . . . . . . . . . . . 21 |- RR C_ RR*
60	58, 59	sstri 3441	. . . . . . . . . . . . . . . . . . . 20 |- { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } C_ RR*
61		supxrcl 11600	. . . . . . . . . . . . . . . . . . . 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 2525	. . . . . . . . . . . . . . . . . 18 |- R e. RR*
64		elico2 11698	. . . . . . . . . . . . . . . . . 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 678	. . . . . . . . . . . . . . . . 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 1022	. . . . . . . . . . . . . . 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 36701	. . . . . . . . . . . . . . 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 4427	. . . . . . . . . . . . 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 13491	. . . . . . . . . . . 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 9794	. . . . . . . . 9 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ y e. RR ) -> ( 1 + -u 1 ) < ( 1 + y ) )
82	48, 81	syl5eqbrr 4437	. . . . . . . 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 11338	. . . . . 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 2451	. . . . . 6 |- ( CC \ ( -oo (,] 0 ) ) = ( CC \ ( -oo (,] 0 ) )
87	86	ellogdm 23584	. . . . 5 |- ( ( 1 + y ) e. ( CC \ ( -oo (,] 0 ) ) <-> ( ( 1 + y ) e. CC /\ ( ( 1 + y ) e. RR -> ( 1 + y ) e. RR+ ) ) )
88	40, 85, 87	sylanbrc 670	. . . 4 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> ( 1 + y ) e. ( CC \ ( -oo (,] 0 ) ) )
89		eldifi 3555	. . . . . 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 9973	. . . . . 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 23653	. . . 4 |- ( ( ( ph /\ -. C e. NN0 ) /\ x e. ( CC \ ( -oo (,] 0 ) ) ) -> ( x ^c -u C ) e. CC )
95		ovex 6318	. . . . 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 9659	. . . . . . 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 9662	. . . . . 6 |- ( ( ( ph /\ -. C e. NN0 ) /\ x e. CC ) -> ( 1 + x ) e. CC )
100		c0ex 9637	. . . . . . . . 9 |- 0 e. _V
101	100	a1i 11	. . . . . . . 8 |- ( ( ( ph /\ -. C e. NN0 ) /\ x e. CC ) -> 0 e. _V )
102		1cnd 9659	. . . . . . . . 9 |- ( ( ph /\ -. C e. NN0 ) -> 1 e. CC )
103	31, 102	dvmptc 22912	. . . . . . . 8 |- ( ( ph /\ -. C e. NN0 ) -> ( CC _D ( x e. CC |-> 1 ) ) = ( x e. CC |-> 0 ) )
104	31	dvmptid 22911	. . . . . . . 8 |- ( ( ph /\ -. C e. NN0 ) -> ( CC _D ( x e. CC |-> x ) ) = ( x e. CC |-> 1 ) )
105	31, 97, 101, 103, 98, 97, 104	dvmptadd 22914	. . . . . . 7 |- ( ( ph /\ -. C e. NN0 ) -> ( CC _D ( x e. CC |-> ( 1 + x ) ) ) = ( x e. CC |-> ( 0 + 1 ) ) )
106		0p1e1 10721	. . . . . . . 8 |- ( 0 + 1 ) = 1
107	106	mpteq2i 4486	. . . . . . 7 |- ( x e. CC |-> ( 0 + 1 ) ) = ( x e. CC |-> 1 )
108	105, 107	syl6eq 2501	. . . . . 6 |- ( ( ph /\ -. C e. NN0 ) -> ( CC _D ( x e. CC |-> ( 1 + x ) ) ) = ( x e. CC |-> 1 ) )
109		fvex 5875	. . . . . . . 8 |- ( TopOpen ` ℂfld ) e. _V
110		cnfldtps 21798	. . . . . . . . . 10 |- ℂfld e. TopSp
111		cnfldbas 18974	. . . . . . . . . . 11 |- CC = ( Base ` ℂfld )
112		eqid 2451	. . . . . . . . . . 11 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
113	111, 112	tpsuni 19953	. . . . . . . . . 10 |- (ℂfld e. TopSp -> CC = U. ( TopOpen ` ℂfld ) )
114	110, 113	ax-mp 5	. . . . . . . . 9 |- CC = U. ( TopOpen ` ℂfld )
115	114	restid 15332	. . . . . . . 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 2460	. . . . . 6 |- ( TopOpen ` ℂfld ) = ( ( TopOpen ` ℂfld ) ↾t CC )
118	112	cnfldtop 21804	. . . . . . . 8 |- ( TopOpen ` ℂfld ) e. Top
119		eqid 2451	. . . . . . . . . . . 12 |- ( abs o. - ) = ( abs o. - )
120	119	cnbl0 21794	. . . . . . . . . . 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 2473	. . . . . . . . 9 |- D = ( 0 ( ball ` ( abs o. - ) ) R )
123		cnxmet 21793	. . . . . . . . . 10 |- ( abs o. - ) e. ( *Met ` CC )
124		0cn 9635	. . . . . . . . . 10 |- 0 e. CC
125	112	cnfldtopn 21802	. . . . . . . . . . 11 |- ( TopOpen ` ℂfld ) = ( MetOpen ` ( abs o. - ) )
126	125	blopn 21515	. . . . . . . . . 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 1364	. . . . . . . . 9 |- ( 0 ( ball ` ( abs o. - ) ) R ) e. ( TopOpen ` ℂfld )
128	122, 127	eqeltri 2525	. . . . . . . 8 |- D e. ( TopOpen ` ℂfld )
129		isopn3i 20098	. . . . . . . 8 |- ( ( ( TopOpen ` ℂfld ) e. Top /\ D e. ( TopOpen ` ℂfld ) ) -> ( ( int ` ( TopOpen ` ℂfld ) ) ` D ) = D )
130	118, 128, 129	mp2an 678	. . . . . . 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 22916	. . . . 5 |- ( ( ph /\ -. C e. NN0 ) -> ( CC _D ( x e. D |-> ( 1 + x ) ) ) = ( x e. D |-> 1 ) )
133		oveq2 6298	. . . . . . 7 |- ( x = y -> ( 1 + x ) = ( 1 + y ) )
134	133	cbvmptv 4495	. . . . . 6 |- ( x e. D |-> ( 1 + x ) ) = ( y e. D |-> ( 1 + y ) )
135	134	oveq2i 6301	. . . . 5 |- ( CC _D ( x e. D |-> ( 1 + x ) ) ) = ( CC _D ( y e. D |-> ( 1 + y ) ) )
136		eqidd 2452	. . . . . 6 |- ( x = y -> 1 = 1 )
137	136	cbvmptv 4495	. . . . 5 |- ( x e. D |-> 1 ) = ( y e. D |-> 1 )
138	132, 135, 137	3eqtr3g 2508	. . . 4 |- ( ( ph /\ -. C e. NN0 ) -> ( CC _D ( y e. D |-> ( 1 + y ) ) ) = ( y e. D |-> 1 ) )
139	86	dvcncxp1 23683	. . . . 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 6297	. . . 4 |- ( x = ( 1 + y ) -> ( x ^c -u C ) = ( ( 1 + y ) ^c -u C ) )
142		oveq1 6297	. . . . 5 |- ( x = ( 1 + y ) -> ( x ^c ( -u C - 1 ) ) = ( ( 1 + y ) ^c ( -u C - 1 ) ) )
143	142	oveq2d 6306	. . . 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 22926	. . 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 9973	. . . . . 6 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> -u C e. CC )
147	146, 32	subcld 9986	. . . . . . 7 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> ( -u C - 1 ) e. CC )
148	40, 147	cxpcld 23653	. . . . . 6 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> ( ( 1 + y ) ^c ( -u C - 1 ) ) e. CC )
149	146, 148	mulcld 9663	. . . . 5 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> ( -u C x. ( ( 1 + y ) ^c ( -u C - 1 ) ) ) e. CC )
150	149	mulid1d 9660	. . . 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 4489	. . 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 2592	. . . . 5 |- F/_ b ( -u C x. ( ( 1 + y ) ^c ( -u C - 1 ) ) )
153		nfcv 2592	. . . . 5 |- F/_ y ( -u C x. ( ( 1 + b ) ^c ( -u C - 1 ) ) )
154		oveq2 6298	. . . . . . 7 |- ( y = b -> ( 1 + y ) = ( 1 + b ) )
155	154	oveq1d 6305	. . . . . 6 |- ( y = b -> ( ( 1 + y ) ^c ( -u C - 1 ) ) = ( ( 1 + b ) ^c ( -u C - 1 ) ) )
156	155	oveq2d 6306	. . . . 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 4493	. . . 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 2489	. 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 2497	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 985   = wceq 1444   e. wcel 1887   {crab 2741   _Vcvv 3045   \ cdif 3401   C_ wss 3404   {cpr 3970   U.cuni 4198   class class class wbr 4402   |-> cmpt 4461   `'ccnv 4833   dom cdm 4834   "cima 4837   o. ccom 4838   Fn wfn 5577   -->wf 5578   ` cfv 5582  (class class class)co 6290   supcsup 7954   CCcc 9537   RRcr 9538   0cc0 9539   1c1 9540   + caddc 9542   x. cmul 9544   -oocmnf 9673   RR*cxr 9674   < clt 9675   <_ cle 9676   - cmin 9860   -ucneg 9861   NNcn 10609   NN0cn0 10869   RR+crp 11302   (,]cioc 11636   [,)cico 11637   seqcseq 12213   ^cexp 12272   abscabs 13297   ~~> cli 13548   ↾_t crest 15319   TopOpenctopn 15320   *Metcxmt 18955   ballcbl 18957  ℂ_fldccnfld 18970   Topctop 19917   TopSpctps 19919   intcnt 20032   _D cdv 22818   ^c ccxp 23505  C_𝑐cbcc 36685

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

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-ixp 7523  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-fi 7925  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ioc 11640  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-fl 12028  df-mod 12097  df-seq 12214  df-exp 12273  df-fac 12460  df-bc 12488  df-hash 12516  df-shft 13130  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-limsup 13526  df-clim 13552  df-rlim 13553  df-sum 13753  df-prod 13960  df-fallfac 14060  df-ef 14121  df-sin 14123  df-cos 14124  df-tan 14125  df-pi 14126  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-starv 15205  df-sca 15206  df-vsca 15207  df-ip 15208  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-hom 15214  df-cco 15215  df-rest 15321  df-topn 15322  df-0g 15340  df-gsum 15341  df-topgen 15342  df-pt 15343  df-prds 15346  df-xrs 15400  df-qtop 15406  df-imas 15407  df-xps 15410  df-mre 15492  df-mrc 15493  df-acs 15495  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-submnd 16583  df-mulg 16676  df-cntz 16971  df-cmn 17432  df-psmet 18962  df-xmet 18963  df-met 18964  df-bl 18965  df-mopn 18966  df-fbas 18967  df-fg 18968  df-cnfld 18971  df-top 19921  df-bases 19922  df-topon 19923  df-topsp 19924  df-cld 20034  df-ntr 20035  df-cls 20036  df-nei 20114  df-lp 20152  df-perf 20153  df-cn 20243  df-cnp 20244  df-haus 20331  df-cmp 20402  df-tx 20577  df-hmeo 20770  df-fil 20861  df-fm 20953  df-flim 20954  df-flf 20955  df-xms 21335  df-ms 21336  df-tms 21337  df-cncf 21910  df-limc 22821  df-dv 22822  df-log 23506  df-cxp 23507  df-bcc 36686

This theorem is referenced by:  binomcxplemnotnn0  36705