Metamath Proof Explorer


Theorem binomcxplemradcnv

Description: Lemma for binomcxp . By binomcxplemfrat and radcnvrat the radius of convergence of power series sum_ k e. NN0 ( ( Fk ) x. ( b ^ k ) ) is one. (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 * <
Assertion binomcxplemradcnv φ ¬ C 0 R = 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 simpl b = x k 0 b = x
9 8 oveq1d b = x k 0 b k = x k
10 9 oveq2d b = x k 0 F k b k = F k x k
11 10 mpteq2dva b = x k 0 F k b k = k 0 F k x k
12 fveq2 k = y F k = F y
13 oveq2 k = y x k = x y
14 12 13 oveq12d k = y F k x k = F y x y
15 14 cbvmptv k 0 F k x k = y 0 F y x y
16 11 15 eqtrdi b = x k 0 F k b k = y 0 F y x y
17 16 cbvmptv b k 0 F k b k = x y 0 F y x y
18 6 17 eqtri S = x y 0 F y x y
19 4 ad2antrr φ ¬ C 0 j 0 C
20 simpr φ ¬ C 0 j 0 j 0
21 19 20 bcccl φ ¬ C 0 j 0 C C 𝑐 j
22 21 5 fmptd φ ¬ C 0 F : 0
23 fvoveq1 k = i F k + 1 = F i + 1
24 fveq2 k = i F k = F i
25 23 24 oveq12d k = i F k + 1 F k = F i + 1 F i
26 25 fveq2d k = i F k + 1 F k = F i + 1 F i
27 26 cbvmptv k 0 F k + 1 F k = i 0 F i + 1 F i
28 nn0uz 0 = 0
29 0nn0 0 0
30 29 a1i φ ¬ C 0 0 0
31 5 a1i φ ¬ C 0 i 0 F = j 0 C C 𝑐 j
32 simpr φ ¬ C 0 i 0 j = i j = i
33 32 oveq2d φ ¬ C 0 i 0 j = i C C 𝑐 j = C C 𝑐 i
34 simpr φ ¬ C 0 i 0 i 0
35 ovexd φ ¬ C 0 i 0 C C 𝑐 i V
36 31 33 34 35 fvmptd φ ¬ C 0 i 0 F i = C C 𝑐 i
37 elfznn0 C 0 i 1 C 0
38 37 con3i ¬ C 0 ¬ C 0 i 1
39 38 ad2antlr φ ¬ C 0 i 0 ¬ C 0 i 1
40 4 adantr φ i 0 C
41 simpr φ i 0 i 0
42 40 41 bcc0 φ i 0 C C 𝑐 i = 0 C 0 i 1
43 42 necon3abid φ i 0 C C 𝑐 i 0 ¬ C 0 i 1
44 43 adantlr φ ¬ C 0 i 0 C C 𝑐 i 0 ¬ C 0 i 1
45 39 44 mpbird φ ¬ C 0 i 0 C C 𝑐 i 0
46 36 45 eqnetrd φ ¬ C 0 i 0 F i 0
47 1 2 3 4 5 binomcxplemfrat φ ¬ C 0 k 0 F k + 1 F k 1
48 ax-1ne0 1 0
49 48 a1i φ ¬ C 0 1 0
50 18 22 7 27 28 30 46 47 49 radcnvrat φ ¬ C 0 R = 1 1
51 1div1e1 1 1 = 1
52 50 51 eqtrdi φ ¬ C 0 R = 1