Theorem binomcxplemdvbinom 38552

Description: Lemma for binomcxp 38556. 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 38554 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 2764	. . . . . 6 |- F/_ b `' abs
3		nfcv 2764	. . . . . . 7 |- F/_ b 0
4		nfcv 2764	. . . . . . 7 |- F/_ b [,)
5		binomcxplem.r	. . . . . . . 8 |- R = sup ( { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } , RR* , < )
6		nfcv 2764	. . . . . . . . . . . 12 |- F/_ b +
7		binomcxplem.s	. . . . . . . . . . . . . 14 |- S = ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) )
8		nfmpt1 4747	. . . . . . . . . . . . . 14 |- F/_ b ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) )
9	7, 8	nfcxfr 2762	. . . . . . . . . . . . 13 |- F/_ b S
10		nfcv 2764	. . . . . . . . . . . . 13 |- F/_ b r
11	9, 10	nffv 6198	. . . . . . . . . . . 12 |- F/_ b ( S ` r )
12	3, 6, 11	nfseq 12811	. . . . . . . . . . 11 |- F/_ b seq 0 ( + , ( S ` r ) )
13	12	nfel1 2779	. . . . . . . . . 10 |- F/ b seq 0 ( + , ( S ` r ) ) e. dom ~~>
14		nfcv 2764	. . . . . . . . . 10 |- F/_ b RR
15	13, 14	nfrab 3123	. . . . . . . . 9 |- F/_ b { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> }
16		nfcv 2764	. . . . . . . . 9 |- F/_ b RR*
17		nfcv 2764	. . . . . . . . 9 |- F/_ b <
18	15, 16, 17	nfsup 8357	. . . . . . . 8 |- F/_ b sup ( { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } , RR* , < )
19	5, 18	nfcxfr 2762	. . . . . . 7 |- F/_ b R
20	3, 4, 19	nfov 6676	. . . . . 6 |- F/_ b ( 0 [,) R )
21	2, 20	nfima 5474	. . . . 5 |- F/_ b ( `' abs " ( 0 [,) R ) )
22	1, 21	nfcxfr 2762	. . . 4 |- F/_ b D
23		nfcv 2764	. . . 4 |- F/_ y D
24		nfcv 2764	. . . 4 |- F/_ y ( ( 1 + b ) ^c -u C )
25		nfcv 2764	. . . 4 |- F/_ b ( ( 1 + y ) ^c -u C )
26		oveq2 6658	. . . . 5 |- ( b = y -> ( 1 + b ) = ( 1 + y ) )
27	26	oveq1d 6665	. . . 4 |- ( b = y -> ( ( 1 + b ) ^c -u C ) = ( ( 1 + y ) ^c -u C ) )
28	22, 23, 24, 25, 27	cbvmptf 4748	. . 3 |- ( b e. D |-> ( ( 1 + b ) ^c -u C ) ) = ( y e. D |-> ( ( 1 + y ) ^c -u C ) )
29	28	oveq2i 6661	. 2 |- ( CC _D ( b e. D |-> ( ( 1 + b ) ^c -u C ) ) ) = ( CC _D ( y e. D |-> ( ( 1 + y ) ^c -u C ) ) )
30		cnelprrecn 10029	. . . . 5 |- CC e. { RR , CC }
31	30	a1i 11	. . . 4 |- ( ( ph /\ -. C e. NN0 ) -> CC e. { RR , CC } )
32		1cnd 10056	. . . . . 6 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> 1 e. CC )
33		cnvimass 5485	. . . . . . . . . 10 |- ( `' abs " ( 0 [,) R ) ) C_ dom abs
34	1, 33	eqsstri 3635	. . . . . . . . 9 |- D C_ dom abs
35		absf 14077	. . . . . . . . . 10 |- abs : CC --> RR
36	35	fdmi 6052	. . . . . . . . 9 |- dom abs = CC
37	34, 36	sseqtri 3637	. . . . . . . 8 |- D C_ CC
38	37	a1i 11	. . . . . . 7 |- ( ( ph /\ -. C e. NN0 ) -> D C_ CC )
39	38	sselda 3603	. . . . . 6 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> y e. CC )
40	32, 39	addcld 10059	. . . . 5 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> ( 1 + y ) e. CC )
41		simpr 477	. . . . . . 7 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ ( 1 + y ) e. RR ) -> ( 1 + y ) e. RR )
42		1cnd 10056	. . . . . . . . . 10 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ ( 1 + y ) e. RR ) -> 1 e. CC )
43	39	adantr 481	. . . . . . . . . 10 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ ( 1 + y ) e. RR ) -> y e. CC )
44	42, 43	pncan2d 10394	. . . . . . . . 9 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ ( 1 + y ) e. RR ) -> ( ( 1 + y ) - 1 ) = y )
45		1red 10055	. . . . . . . . . 10 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ ( 1 + y ) e. RR ) -> 1 e. RR )
46	41, 45	resubcld 10458	. . . . . . . . 9 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ ( 1 + y ) e. RR ) -> ( ( 1 + y ) - 1 ) e. RR )
47	44, 46	eqeltrrd 2702	. . . . . . . 8 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ ( 1 + y ) e. RR ) -> y e. RR )
48		1pneg1e0 11129	. . . . . . . . 9 |- ( 1 + -u 1 ) = 0
49		1red 10055	. . . . . . . . . . 11 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ y e. RR ) -> 1 e. RR )
50	49	renegcld 10457	. . . . . . . . . 10 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ y e. RR ) -> -u 1 e. RR )
51		simpr 477	. . . . . . . . . 10 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ y e. RR ) -> y e. RR )
52		ffn 6045	. . . . . . . . . . . . . . . . . . . 20 |- ( abs : CC --> RR -> abs Fn CC )
53		elpreima 6337	. . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . 18 |- ( y e. ( `' abs " ( 0 [,) R ) ) -> ( abs ` y ) e. ( 0 [,) R ) )
56	55, 1	eleq2s 2719	. . . . . . . . . . . . . . . . 17 |- ( y e. D -> ( abs ` y ) e. ( 0 [,) R ) )
57		0re 10040	. . . . . . . . . . . . . . . . . 18 |- 0 e. RR
58		ssrab2 3687	. . . . . . . . . . . . . . . . . . . . 21 |- { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } C_ RR
59		ressxr 10083	. . . . . . . . . . . . . . . . . . . . 21 |- RR C_ RR*
60	58, 59	sstri 3612	. . . . . . . . . . . . . . . . . . . 20 |- { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } C_ RR*
61		supxrcl 12145	. . . . . . . . . . . . . . . . . . . 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 2697	. . . . . . . . . . . . . . . . . 18 |- R e. RR*
64		elico2 12237	. . . . . . . . . . . . . . . . . 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 708	. . . . . . . . . . . . . . . . 17 |- ( ( abs ` y ) e. ( 0 [,) R ) <-> ( ( abs ` y ) e. RR /\ 0 <_ ( abs ` y ) /\ ( abs ` y ) < R ) )
66	56, 65	sylib 208	. . . . . . . . . . . . . . . 16 |- ( y e. D -> ( ( abs ` y ) e. RR /\ 0 <_ ( abs ` y ) /\ ( abs ` y ) < R ) )
67	66	simp3d 1075	. . . . . . . . . . . . . . 15 |- ( y e. D -> ( abs ` y ) < R )
68	67	adantl 482	. . . . . . . . . . . . . 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 38551	. . . . . . . . . . . . . . 15 |- ( ( ph /\ -. C e. NN0 ) -> R = 1 )
75	74	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> R = 1 )
76	68, 75	breqtrd 4679	. . . . . . . . . . . . 13 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> ( abs ` y ) < 1 )
77	76	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ y e. RR ) -> ( abs ` y ) < 1 )
78	51, 49	absltd 14168	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ y e. RR ) -> ( ( abs ` y ) < 1 <-> ( -u 1 < y /\ y < 1 ) ) )
79	77, 78	mpbid 222	. . . . . . . . . . 11 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ y e. RR ) -> ( -u 1 < y /\ y < 1 ) )
80	79	simpld 475	. . . . . . . . . 10 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ y e. RR ) -> -u 1 < y )
81	50, 51, 49, 80	ltadd2dd 10196	. . . . . . . . 9 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ y e. RR ) -> ( 1 + -u 1 ) < ( 1 + y ) )
82	48, 81	syl5eqbrr 4689	. . . . . . . 8 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ y e. RR ) -> 0 < ( 1 + y ) )
83	47, 82	syldan 487	. . . . . . 7 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ ( 1 + y ) e. RR ) -> 0 < ( 1 + y ) )
84	41, 83	elrpd 11869	. . . . . 6 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ ( 1 + y ) e. RR ) -> ( 1 + y ) e. RR+ )
85	84	ex 450	. . . . 5 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> ( ( 1 + y ) e. RR -> ( 1 + y ) e. RR+ ) )
86		eqid 2622	. . . . . 6 |- ( CC \ ( -oo (,] 0 ) ) = ( CC \ ( -oo (,] 0 ) )
87	86	ellogdm 24385	. . . . 5 |- ( ( 1 + y ) e. ( CC \ ( -oo (,] 0 ) ) <-> ( ( 1 + y ) e. CC /\ ( ( 1 + y ) e. RR -> ( 1 + y ) e. RR+ ) ) )
88	40, 85, 87	sylanbrc 698	. . . 4 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> ( 1 + y ) e. ( CC \ ( -oo (,] 0 ) ) )
89		eldifi 3732	. . . . . 6 |- ( x e. ( CC \ ( -oo (,] 0 ) ) -> x e. CC )
90	89	adantl 482	. . . . 5 |- ( ( ( ph /\ -. C e. NN0 ) /\ x e. ( CC \ ( -oo (,] 0 ) ) ) -> x e. CC )
91	72	adantr 481	. . . . . . 7 |- ( ( ph /\ -. C e. NN0 ) -> C e. CC )
92	91	negcld 10379	. . . . . 6 |- ( ( ph /\ -. C e. NN0 ) -> -u C e. CC )
93	92	adantr 481	. . . . 5 |- ( ( ( ph /\ -. C e. NN0 ) /\ x e. ( CC \ ( -oo (,] 0 ) ) ) -> -u C e. CC )
94	90, 93	cxpcld 24454	. . . 4 |- ( ( ( ph /\ -. C e. NN0 ) /\ x e. ( CC \ ( -oo (,] 0 ) ) ) -> ( x ^c -u C ) e. CC )
95		ovexd 6680	. . . 4 |- ( ( ( ph /\ -. C e. NN0 ) /\ x e. ( CC \ ( -oo (,] 0 ) ) ) -> ( -u C x. ( x ^c ( -u C - 1 ) ) ) e. _V )
96		1cnd 10056	. . . . . . 7 |- ( ( ( ph /\ -. C e. NN0 ) /\ x e. CC ) -> 1 e. CC )
97		simpr 477	. . . . . . 7 |- ( ( ( ph /\ -. C e. NN0 ) /\ x e. CC ) -> x e. CC )
98	96, 97	addcld 10059	. . . . . 6 |- ( ( ( ph /\ -. C e. NN0 ) /\ x e. CC ) -> ( 1 + x ) e. CC )
99		c0ex 10034	. . . . . . . . 9 |- 0 e. _V
100	99	a1i 11	. . . . . . . 8 |- ( ( ( ph /\ -. C e. NN0 ) /\ x e. CC ) -> 0 e. _V )
101		1cnd 10056	. . . . . . . . 9 |- ( ( ph /\ -. C e. NN0 ) -> 1 e. CC )
102	31, 101	dvmptc 23721	. . . . . . . 8 |- ( ( ph /\ -. C e. NN0 ) -> ( CC _D ( x e. CC |-> 1 ) ) = ( x e. CC |-> 0 ) )
103	31	dvmptid 23720	. . . . . . . 8 |- ( ( ph /\ -. C e. NN0 ) -> ( CC _D ( x e. CC |-> x ) ) = ( x e. CC |-> 1 ) )
104	31, 96, 100, 102, 97, 96, 103	dvmptadd 23723	. . . . . . 7 |- ( ( ph /\ -. C e. NN0 ) -> ( CC _D ( x e. CC |-> ( 1 + x ) ) ) = ( x e. CC |-> ( 0 + 1 ) ) )
105		0p1e1 11132	. . . . . . . 8 |- ( 0 + 1 ) = 1
106	105	mpteq2i 4741	. . . . . . 7 |- ( x e. CC |-> ( 0 + 1 ) ) = ( x e. CC |-> 1 )
107	104, 106	syl6eq 2672	. . . . . 6 |- ( ( ph /\ -. C e. NN0 ) -> ( CC _D ( x e. CC |-> ( 1 + x ) ) ) = ( x e. CC |-> 1 ) )
108		fvex 6201	. . . . . . . 8 |- ( TopOpen ` ℂfld ) e. _V
109		cnfldtps 22581	. . . . . . . . . 10 |- ℂfld e. TopSp
110		cnfldbas 19750	. . . . . . . . . . 11 |- CC = ( Base ` ℂfld )
111		eqid 2622	. . . . . . . . . . 11 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
112	110, 111	tpsuni 20740	. . . . . . . . . 10 |- (ℂfld e. TopSp -> CC = U. ( TopOpen ` ℂfld ) )
113	109, 112	ax-mp 5	. . . . . . . . 9 |- CC = U. ( TopOpen ` ℂfld )
114	113	restid 16094	. . . . . . . 8 |- ( ( TopOpen ` ℂfld ) e. _V -> ( ( TopOpen ` ℂfld ) ↾t CC ) = ( TopOpen ` ℂfld ) )
115	108, 114	ax-mp 5	. . . . . . 7 |- ( ( TopOpen ` ℂfld ) ↾t CC ) = ( TopOpen ` ℂfld )
116	115	eqcomi 2631	. . . . . 6 |- ( TopOpen ` ℂfld ) = ( ( TopOpen ` ℂfld ) ↾t CC )
117	111	cnfldtop 22587	. . . . . . . 8 |- ( TopOpen ` ℂfld ) e. Top
118		eqid 2622	. . . . . . . . . . . 12 |- ( abs o. - ) = ( abs o. - )
119	118	cnbl0 22577	. . . . . . . . . . 11 |- ( R e. RR* -> ( `' abs " ( 0 [,) R ) ) = ( 0 ( ball ` ( abs o. - ) ) R ) )
120	63, 119	ax-mp 5	. . . . . . . . . 10 |- ( `' abs " ( 0 [,) R ) ) = ( 0 ( ball ` ( abs o. - ) ) R )
121	1, 120	eqtri 2644	. . . . . . . . 9 |- D = ( 0 ( ball ` ( abs o. - ) ) R )
122		cnxmet 22576	. . . . . . . . . 10 |- ( abs o. - ) e. ( *Met ` CC )
123		0cn 10032	. . . . . . . . . 10 |- 0 e. CC
124	111	cnfldtopn 22585	. . . . . . . . . . 11 |- ( TopOpen ` ℂfld ) = ( MetOpen ` ( abs o. - ) )
125	124	blopn 22305	. . . . . . . . . 10 |- ( ( ( abs o. - ) e. ( *Met ` CC ) /\ 0 e. CC /\ R e. RR* ) -> ( 0 ( ball ` ( abs o. - ) ) R ) e. ( TopOpen ` ℂfld ) )
126	122, 123, 63, 125	mp3an 1424	. . . . . . . . 9 |- ( 0 ( ball ` ( abs o. - ) ) R ) e. ( TopOpen ` ℂfld )
127	121, 126	eqeltri 2697	. . . . . . . 8 |- D e. ( TopOpen ` ℂfld )
128		isopn3i 20886	. . . . . . . 8 |- ( ( ( TopOpen ` ℂfld ) e. Top /\ D e. ( TopOpen ` ℂfld ) ) -> ( ( int ` ( TopOpen ` ℂfld ) ) ` D ) = D )
129	117, 127, 128	mp2an 708	. . . . . . 7 |- ( ( int ` ( TopOpen ` ℂfld ) ) ` D ) = D
130	129	a1i 11	. . . . . 6 |- ( ( ph /\ -. C e. NN0 ) -> ( ( int ` ( TopOpen ` ℂfld ) ) ` D ) = D )
131	31, 98, 96, 107, 38, 116, 111, 130	dvmptres2 23725	. . . . 5 |- ( ( ph /\ -. C e. NN0 ) -> ( CC _D ( x e. D |-> ( 1 + x ) ) ) = ( x e. D |-> 1 ) )
132		oveq2 6658	. . . . . . 7 |- ( x = y -> ( 1 + x ) = ( 1 + y ) )
133	132	cbvmptv 4750	. . . . . 6 |- ( x e. D |-> ( 1 + x ) ) = ( y e. D |-> ( 1 + y ) )
134	133	oveq2i 6661	. . . . 5 |- ( CC _D ( x e. D |-> ( 1 + x ) ) ) = ( CC _D ( y e. D |-> ( 1 + y ) ) )
135		eqidd 2623	. . . . . 6 |- ( x = y -> 1 = 1 )
136	135	cbvmptv 4750	. . . . 5 |- ( x e. D |-> 1 ) = ( y e. D |-> 1 )
137	131, 134, 136	3eqtr3g 2679	. . . 4 |- ( ( ph /\ -. C e. NN0 ) -> ( CC _D ( y e. D |-> ( 1 + y ) ) ) = ( y e. D |-> 1 ) )
138	86	dvcncxp1 24484	. . . . 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 ) ) ) ) )
139	92, 138	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 ) ) ) ) )
140		oveq1 6657	. . . 4 |- ( x = ( 1 + y ) -> ( x ^c -u C ) = ( ( 1 + y ) ^c -u C ) )
141		oveq1 6657	. . . . 5 |- ( x = ( 1 + y ) -> ( x ^c ( -u C - 1 ) ) = ( ( 1 + y ) ^c ( -u C - 1 ) ) )
142	141	oveq2d 6666	. . . 4 |- ( x = ( 1 + y ) -> ( -u C x. ( x ^c ( -u C - 1 ) ) ) = ( -u C x. ( ( 1 + y ) ^c ( -u C - 1 ) ) ) )
143	31, 31, 88, 32, 94, 95, 137, 139, 140, 142	dvmptco 23735	. . 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 ) ) )
144	91	adantr 481	. . . . . . 7 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> C e. CC )
145	144	negcld 10379	. . . . . 6 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> -u C e. CC )
146	145, 32	subcld 10392	. . . . . . 7 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> ( -u C - 1 ) e. CC )
147	40, 146	cxpcld 24454	. . . . . 6 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> ( ( 1 + y ) ^c ( -u C - 1 ) ) e. CC )
148	145, 147	mulcld 10060	. . . . 5 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> ( -u C x. ( ( 1 + y ) ^c ( -u C - 1 ) ) ) e. CC )
149	148	mulid1d 10057	. . . 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 ) ) ) )
150	149	mpteq2dva 4744	. . 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 ) ) ) ) )
151		nfcv 2764	. . . . 5 |- F/_ b ( -u C x. ( ( 1 + y ) ^c ( -u C - 1 ) ) )
152		nfcv 2764	. . . . 5 |- F/_ y ( -u C x. ( ( 1 + b ) ^c ( -u C - 1 ) ) )
153		oveq2 6658	. . . . . . 7 |- ( y = b -> ( 1 + y ) = ( 1 + b ) )
154	153	oveq1d 6665	. . . . . 6 |- ( y = b -> ( ( 1 + y ) ^c ( -u C - 1 ) ) = ( ( 1 + b ) ^c ( -u C - 1 ) ) )
155	154	oveq2d 6666	. . . . 5 |- ( y = b -> ( -u C x. ( ( 1 + y ) ^c ( -u C - 1 ) ) ) = ( -u C x. ( ( 1 + b ) ^c ( -u C - 1 ) ) ) )
156	23, 22, 151, 152, 155	cbvmptf 4748	. . . 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 ) ) ) )
157	156	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 ) ) ) ) )
158	143, 150, 157	3eqtrd 2660	. 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 ) ) ) ) )
159	29, 158	syl5eq 2668	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 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   \ cdif 3571   C_ wss 3574   {cpr 4179   U.cuni 4436   class class class wbr 4653   |-> cmpt 4729   `'ccnv 5113   dom cdm 5114   "cima 5117   o. ccom 5118   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   supcsup 8346   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   -oocmnf 10072   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   -ucneg 10267   NNcn 11020   NN0cn0 11292   RR+crp 11832   (,]cioc 12176   [,)cico 12177   seqcseq 12801   ^cexp 12860   abscabs 13974   ~~> cli 14215   ↾_t crest 16081   TopOpenctopn 16082   *Metcxmt 19731   ballcbl 19733  ℂ_fldccnfld 19746   Topctop 20698   TopSpctps 20736   intcnt 20821   _D cdv 23627   ^c ccxp 24302  C_𝑐cbcc 38535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-prod 14636  df-fallfac 14738  df-ef 14798  df-sin 14800  df-cos 14801  df-tan 14802  df-pi 14803  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-cmp 21190  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631  df-log 24303  df-cxp 24304  df-bcc 38536

This theorem is referenced by:  binomcxplemnotnn0  38555