Metamath Proof Explorer


Theorem binomcxplemdvbinom

Description: Lemma for binomcxp . 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 by this derivative to show that ( ( 1 + b ) ^c C ) (with a nonnegated C ) and the later sum, since both at b = 0 equal one, are the same. (Contributed by Steve Rodriguez, 22-Apr-2020)

Ref Expression
Hypotheses binomcxp.a φ A +
binomcxp.b φ B
binomcxp.lt φ B < A
binomcxp.c φ C
binomcxplem.f F = j 0 C C 𝑐 j
binomcxplem.s S = b k 0 F k b k
binomcxplem.r R = sup r | seq 0 + S r dom * <
binomcxplem.e E = b k k F k b k 1
binomcxplem.d D = abs -1 0 R
Assertion binomcxplemdvbinom φ ¬ C 0 db D 1 + b C d b = b D C 1 + b - C - 1

Proof

Step Hyp Ref Expression
1 binomcxp.a φ A +
2 binomcxp.b φ B
3 binomcxp.lt φ B < A
4 binomcxp.c φ C
5 binomcxplem.f F = j 0 C C 𝑐 j
6 binomcxplem.s S = b k 0 F k b k
7 binomcxplem.r R = sup r | seq 0 + S r dom * <
8 binomcxplem.e E = b k k F k b k 1
9 binomcxplem.d D = abs -1 0 R
10 nfcv _ b abs -1
11 nfcv _ b 0
12 nfcv _ b .
13 nfcv _ b +
14 nfmpt1 _ b b k 0 F k b k
15 6 14 nfcxfr _ b S
16 nfcv _ b r
17 15 16 nffv _ b S r
18 11 13 17 nfseq _ b seq 0 + S r
19 18 nfel1 b seq 0 + S r dom
20 nfcv _ b
21 19 20 nfrabw _ b r | seq 0 + S r dom
22 nfcv _ b *
23 nfcv _ b <
24 21 22 23 nfsup _ b sup r | seq 0 + S r dom * <
25 7 24 nfcxfr _ b R
26 11 12 25 nfov _ b 0 R
27 10 26 nfima _ b abs -1 0 R
28 9 27 nfcxfr _ b D
29 nfcv _ y D
30 nfcv _ y 1 + b C
31 nfcv _ b 1 + y C
32 oveq2 b = y 1 + b = 1 + y
33 32 oveq1d b = y 1 + b C = 1 + y C
34 28 29 30 31 33 cbvmptf b D 1 + b C = y D 1 + y C
35 34 oveq2i db D 1 + b C d b = dy D 1 + y C d y
36 cnelprrecn
37 36 a1i φ ¬ C 0
38 1cnd φ ¬ C 0 y D 1
39 cnvimass abs -1 0 R dom abs
40 9 39 eqsstri D dom abs
41 absf abs :
42 41 fdmi dom abs =
43 40 42 sseqtri D
44 43 a1i φ ¬ C 0 D
45 44 sselda φ ¬ C 0 y D y
46 38 45 addcld φ ¬ C 0 y D 1 + y
47 simpr φ ¬ C 0 y D 1 + y 1 + y
48 1cnd φ ¬ C 0 y D 1 + y 1
49 45 adantr φ ¬ C 0 y D 1 + y y
50 48 49 pncan2d φ ¬ C 0 y D 1 + y 1 + y - 1 = y
51 1red φ ¬ C 0 y D 1 + y 1
52 47 51 resubcld φ ¬ C 0 y D 1 + y 1 + y - 1
53 50 52 eqeltrrd φ ¬ C 0 y D 1 + y y
54 1pneg1e0 1 + -1 = 0
55 1red φ ¬ C 0 y D y 1
56 55 renegcld φ ¬ C 0 y D y 1
57 simpr φ ¬ C 0 y D y y
58 ffn abs : abs Fn
59 elpreima abs Fn y abs -1 0 R y y 0 R
60 41 58 59 mp2b y abs -1 0 R y y 0 R
61 60 simprbi y abs -1 0 R y 0 R
62 61 9 eleq2s y D y 0 R
63 0re 0
64 ssrab2 r | seq 0 + S r dom
65 ressxr *
66 64 65 sstri r | seq 0 + S r dom *
67 supxrcl r | seq 0 + S r dom * sup r | seq 0 + S r dom * < *
68 66 67 ax-mp sup r | seq 0 + S r dom * < *
69 7 68 eqeltri R *
70 elico2 0 R * y 0 R y 0 y y < R
71 63 69 70 mp2an y 0 R y 0 y y < R
72 62 71 sylib y D y 0 y y < R
73 72 simp3d y D y < R
74 73 adantl φ ¬ C 0 y D y < R
75 1 2 3 4 5 6 7 binomcxplemradcnv φ ¬ C 0 R = 1
76 75 adantr φ ¬ C 0 y D R = 1
77 74 76 breqtrd φ ¬ C 0 y D y < 1
78 77 adantr φ ¬ C 0 y D y y < 1
79 57 55 absltd φ ¬ C 0 y D y y < 1 1 < y y < 1
80 78 79 mpbid φ ¬ C 0 y D y 1 < y y < 1
81 80 simpld φ ¬ C 0 y D y 1 < y
82 56 57 55 81 ltadd2dd φ ¬ C 0 y D y 1 + -1 < 1 + y
83 54 82 eqbrtrrid φ ¬ C 0 y D y 0 < 1 + y
84 53 83 syldan φ ¬ C 0 y D 1 + y 0 < 1 + y
85 47 84 elrpd φ ¬ C 0 y D 1 + y 1 + y +
86 85 ex φ ¬ C 0 y D 1 + y 1 + y +
87 eqid −∞ 0 = −∞ 0
88 87 ellogdm 1 + y −∞ 0 1 + y 1 + y 1 + y +
89 46 86 88 sylanbrc φ ¬ C 0 y D 1 + y −∞ 0
90 eldifi x −∞ 0 x
91 90 adantl φ ¬ C 0 x −∞ 0 x
92 4 adantr φ ¬ C 0 C
93 92 negcld φ ¬ C 0 C
94 93 adantr φ ¬ C 0 x −∞ 0 C
95 91 94 cxpcld φ ¬ C 0 x −∞ 0 x C
96 ovexd φ ¬ C 0 x −∞ 0 C x - C - 1 V
97 1cnd φ ¬ C 0 x 1
98 simpr φ ¬ C 0 x x
99 97 98 addcld φ ¬ C 0 x 1 + x
100 c0ex 0 V
101 100 a1i φ ¬ C 0 x 0 V
102 1cnd φ ¬ C 0 1
103 37 102 dvmptc φ ¬ C 0 dx 1 d x = x 0
104 37 dvmptid φ ¬ C 0 dx x d x = x 1
105 37 97 101 103 98 97 104 dvmptadd φ ¬ C 0 dx 1 + x d x = x 0 + 1
106 0p1e1 0 + 1 = 1
107 106 mpteq2i x 0 + 1 = x 1
108 105 107 eqtrdi φ ¬ C 0 dx 1 + x d x = x 1
109 fvex TopOpen fld V
110 cnfldtps fld TopSp
111 cnfldbas = Base fld
112 eqid TopOpen fld = TopOpen fld
113 111 112 tpsuni fld TopSp = TopOpen fld
114 110 113 ax-mp = TopOpen fld
115 114 restid TopOpen fld V TopOpen fld 𝑡 = TopOpen fld
116 109 115 ax-mp TopOpen fld 𝑡 = TopOpen fld
117 116 eqcomi TopOpen fld = TopOpen fld 𝑡
118 112 cnfldtop TopOpen fld Top
119 eqid abs = abs
120 119 cnbl0 R * abs -1 0 R = 0 ball abs R
121 69 120 ax-mp abs -1 0 R = 0 ball abs R
122 9 121 eqtri D = 0 ball abs R
123 cnxmet abs ∞Met
124 0cn 0
125 112 cnfldtopn TopOpen fld = MetOpen abs
126 125 blopn abs ∞Met 0 R * 0 ball abs R TopOpen fld
127 123 124 69 126 mp3an 0 ball abs R TopOpen fld
128 122 127 eqeltri D TopOpen fld
129 isopn3i TopOpen fld Top D TopOpen fld int TopOpen fld D = D
130 118 128 129 mp2an int TopOpen fld D = D
131 130 a1i φ ¬ C 0 int TopOpen fld D = D
132 37 99 97 108 44 117 112 131 dvmptres2 φ ¬ C 0 dx D 1 + x d x = x D 1
133 oveq2 x = y 1 + x = 1 + y
134 133 cbvmptv x D 1 + x = y D 1 + y
135 134 oveq2i dx D 1 + x d x = dy D 1 + y d y
136 eqidd x = y 1 = 1
137 136 cbvmptv x D 1 = y D 1
138 132 135 137 3eqtr3g φ ¬ C 0 dy D 1 + y d y = y D 1
139 87 dvcncxp1 C dx −∞ 0 x C d x = x −∞ 0 C x - C - 1
140 93 139 syl φ ¬ C 0 dx −∞ 0 x C d x = x −∞ 0 C x - C - 1
141 oveq1 x = 1 + y x C = 1 + y C
142 oveq1 x = 1 + y x - C - 1 = 1 + y - C - 1
143 142 oveq2d x = 1 + y C x - C - 1 = C 1 + y - C - 1
144 37 37 89 38 95 96 138 140 141 143 dvmptco φ ¬ C 0 dy D 1 + y C d y = y D C 1 + y - C - 1 1
145 92 adantr φ ¬ C 0 y D C
146 145 negcld φ ¬ C 0 y D C
147 146 38 subcld φ ¬ C 0 y D - C - 1
148 46 147 cxpcld φ ¬ C 0 y D 1 + y - C - 1
149 146 148 mulcld φ ¬ C 0 y D C 1 + y - C - 1
150 149 mulid1d φ ¬ C 0 y D C 1 + y - C - 1 1 = C 1 + y - C - 1
151 150 mpteq2dva φ ¬ C 0 y D C 1 + y - C - 1 1 = y D C 1 + y - C - 1
152 nfcv _ b C 1 + y - C - 1
153 nfcv _ y C 1 + b - C - 1
154 oveq2 y = b 1 + y = 1 + b
155 154 oveq1d y = b 1 + y - C - 1 = 1 + b - C - 1
156 155 oveq2d y = b C 1 + y - C - 1 = C 1 + b - C - 1
157 29 28 152 153 156 cbvmptf y D C 1 + y - C - 1 = b D C 1 + b - C - 1
158 157 a1i φ ¬ C 0 y D C 1 + y - C - 1 = b D C 1 + b - C - 1
159 144 151 158 3eqtrd φ ¬ C 0 dy D 1 + y C d y = b D C 1 + b - C - 1
160 35 159 syl5eq φ ¬ C 0 db D 1 + b C d b = b D C 1 + b - C - 1