Metamath Proof Explorer


Theorem binomcxplemnn0

Description: Lemma for binomcxp . When C is a nonnegative integer, the binomial's finite sum value by the standard binomial theorem binom equals this generalized infinite sum: the generalized binomial coefficient and exponentiation operators give exactly the same values in the standard index set ( 0 ... C ) , and when the index set is widened beyond C the additional values are just zeroes. (Contributed by Steve Rodriguez, 22-Apr-2020)

Ref Expression
Hypotheses binomcxp.a ( 𝜑𝐴 ∈ ℝ+ )
binomcxp.b ( 𝜑𝐵 ∈ ℝ )
binomcxp.lt ( 𝜑 → ( abs ‘ 𝐵 ) < ( abs ‘ 𝐴 ) )
binomcxp.c ( 𝜑𝐶 ∈ ℂ )
Assertion binomcxplemnn0 ( ( 𝜑𝐶 ∈ ℕ0 ) → ( ( 𝐴 + 𝐵 ) ↑𝑐 𝐶 ) = Σ 𝑘 ∈ ℕ0 ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴𝑐 ( 𝐶𝑘 ) ) · ( 𝐵𝑘 ) ) ) )

Proof

Step Hyp Ref Expression
1 binomcxp.a ( 𝜑𝐴 ∈ ℝ+ )
2 binomcxp.b ( 𝜑𝐵 ∈ ℝ )
3 binomcxp.lt ( 𝜑 → ( abs ‘ 𝐵 ) < ( abs ‘ 𝐴 ) )
4 binomcxp.c ( 𝜑𝐶 ∈ ℂ )
5 1 rpcnd ( 𝜑𝐴 ∈ ℂ )
6 2 recnd ( 𝜑𝐵 ∈ ℂ )
7 binom ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℕ0 ) → ( ( 𝐴 + 𝐵 ) ↑ 𝐶 ) = Σ 𝑘 ∈ ( 0 ... 𝐶 ) ( ( 𝐶 C 𝑘 ) · ( ( 𝐴 ↑ ( 𝐶𝑘 ) ) · ( 𝐵𝑘 ) ) ) )
8 7 3expia ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐶 ∈ ℕ0 → ( ( 𝐴 + 𝐵 ) ↑ 𝐶 ) = Σ 𝑘 ∈ ( 0 ... 𝐶 ) ( ( 𝐶 C 𝑘 ) · ( ( 𝐴 ↑ ( 𝐶𝑘 ) ) · ( 𝐵𝑘 ) ) ) ) )
9 5 6 8 syl2anc ( 𝜑 → ( 𝐶 ∈ ℕ0 → ( ( 𝐴 + 𝐵 ) ↑ 𝐶 ) = Σ 𝑘 ∈ ( 0 ... 𝐶 ) ( ( 𝐶 C 𝑘 ) · ( ( 𝐴 ↑ ( 𝐶𝑘 ) ) · ( 𝐵𝑘 ) ) ) ) )
10 9 imp ( ( 𝜑𝐶 ∈ ℕ0 ) → ( ( 𝐴 + 𝐵 ) ↑ 𝐶 ) = Σ 𝑘 ∈ ( 0 ... 𝐶 ) ( ( 𝐶 C 𝑘 ) · ( ( 𝐴 ↑ ( 𝐶𝑘 ) ) · ( 𝐵𝑘 ) ) ) )
11 5 adantr ( ( 𝜑𝐶 ∈ ℕ0 ) → 𝐴 ∈ ℂ )
12 6 adantr ( ( 𝜑𝐶 ∈ ℕ0 ) → 𝐵 ∈ ℂ )
13 11 12 addcld ( ( 𝜑𝐶 ∈ ℕ0 ) → ( 𝐴 + 𝐵 ) ∈ ℂ )
14 simpr ( ( 𝜑𝐶 ∈ ℕ0 ) → 𝐶 ∈ ℕ0 )
15 cxpexp ( ( ( 𝐴 + 𝐵 ) ∈ ℂ ∧ 𝐶 ∈ ℕ0 ) → ( ( 𝐴 + 𝐵 ) ↑𝑐 𝐶 ) = ( ( 𝐴 + 𝐵 ) ↑ 𝐶 ) )
16 13 14 15 syl2anc ( ( 𝜑𝐶 ∈ ℕ0 ) → ( ( 𝐴 + 𝐵 ) ↑𝑐 𝐶 ) = ( ( 𝐴 + 𝐵 ) ↑ 𝐶 ) )
17 elfznn0 ( 𝑘 ∈ ( 0 ... 𝐶 ) → 𝑘 ∈ ℕ0 )
18 simplr ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → 𝐶 ∈ ℕ0 )
19 simpr ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → 𝑘 ∈ ℕ0 )
20 18 19 bccbc ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( 𝐶 C𝑐 𝑘 ) = ( 𝐶 C 𝑘 ) )
21 17 20 sylan2 ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( 0 ... 𝐶 ) ) → ( 𝐶 C𝑐 𝑘 ) = ( 𝐶 C 𝑘 ) )
22 5 ad2antrr ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( 0 ... 𝐶 ) ) → 𝐴 ∈ ℂ )
23 elfzle2 ( 𝑘 ∈ ( 0 ... 𝐶 ) → 𝑘𝐶 )
24 23 adantl ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( 0 ... 𝐶 ) ) → 𝑘𝐶 )
25 nn0sub ( ( 𝑘 ∈ ℕ0𝐶 ∈ ℕ0 ) → ( 𝑘𝐶 ↔ ( 𝐶𝑘 ) ∈ ℕ0 ) )
26 25 ancoms ( ( 𝐶 ∈ ℕ0𝑘 ∈ ℕ0 ) → ( 𝑘𝐶 ↔ ( 𝐶𝑘 ) ∈ ℕ0 ) )
27 26 adantll ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( 𝑘𝐶 ↔ ( 𝐶𝑘 ) ∈ ℕ0 ) )
28 17 27 sylan2 ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( 0 ... 𝐶 ) ) → ( 𝑘𝐶 ↔ ( 𝐶𝑘 ) ∈ ℕ0 ) )
29 24 28 mpbid ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( 0 ... 𝐶 ) ) → ( 𝐶𝑘 ) ∈ ℕ0 )
30 cxpexp ( ( 𝐴 ∈ ℂ ∧ ( 𝐶𝑘 ) ∈ ℕ0 ) → ( 𝐴𝑐 ( 𝐶𝑘 ) ) = ( 𝐴 ↑ ( 𝐶𝑘 ) ) )
31 22 29 30 syl2anc ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( 0 ... 𝐶 ) ) → ( 𝐴𝑐 ( 𝐶𝑘 ) ) = ( 𝐴 ↑ ( 𝐶𝑘 ) ) )
32 31 oveq1d ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( 0 ... 𝐶 ) ) → ( ( 𝐴𝑐 ( 𝐶𝑘 ) ) · ( 𝐵𝑘 ) ) = ( ( 𝐴 ↑ ( 𝐶𝑘 ) ) · ( 𝐵𝑘 ) ) )
33 21 32 oveq12d ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( 0 ... 𝐶 ) ) → ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴𝑐 ( 𝐶𝑘 ) ) · ( 𝐵𝑘 ) ) ) = ( ( 𝐶 C 𝑘 ) · ( ( 𝐴 ↑ ( 𝐶𝑘 ) ) · ( 𝐵𝑘 ) ) ) )
34 33 sumeq2dv ( ( 𝜑𝐶 ∈ ℕ0 ) → Σ 𝑘 ∈ ( 0 ... 𝐶 ) ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴𝑐 ( 𝐶𝑘 ) ) · ( 𝐵𝑘 ) ) ) = Σ 𝑘 ∈ ( 0 ... 𝐶 ) ( ( 𝐶 C 𝑘 ) · ( ( 𝐴 ↑ ( 𝐶𝑘 ) ) · ( 𝐵𝑘 ) ) ) )
35 10 16 34 3eqtr4d ( ( 𝜑𝐶 ∈ ℕ0 ) → ( ( 𝐴 + 𝐵 ) ↑𝑐 𝐶 ) = Σ 𝑘 ∈ ( 0 ... 𝐶 ) ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴𝑐 ( 𝐶𝑘 ) ) · ( 𝐵𝑘 ) ) ) )
36 4 adantr ( ( 𝜑𝐶 ∈ ℕ0 ) → 𝐶 ∈ ℂ )
37 13 36 cxpcld ( ( 𝜑𝐶 ∈ ℕ0 ) → ( ( 𝐴 + 𝐵 ) ↑𝑐 𝐶 ) ∈ ℂ )
38 35 37 eqeltrrd ( ( 𝜑𝐶 ∈ ℕ0 ) → Σ 𝑘 ∈ ( 0 ... 𝐶 ) ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴𝑐 ( 𝐶𝑘 ) ) · ( 𝐵𝑘 ) ) ) ∈ ℂ )
39 38 addid1d ( ( 𝜑𝐶 ∈ ℕ0 ) → ( Σ 𝑘 ∈ ( 0 ... 𝐶 ) ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴𝑐 ( 𝐶𝑘 ) ) · ( 𝐵𝑘 ) ) ) + 0 ) = Σ 𝑘 ∈ ( 0 ... 𝐶 ) ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴𝑐 ( 𝐶𝑘 ) ) · ( 𝐵𝑘 ) ) ) )
40 nn0uz 0 = ( ℤ ‘ 0 )
41 eqid ( ℤ ‘ ( 𝐶 + 1 ) ) = ( ℤ ‘ ( 𝐶 + 1 ) )
42 1nn0 1 ∈ ℕ0
43 42 a1i ( ( 𝜑𝐶 ∈ ℕ0 ) → 1 ∈ ℕ0 )
44 14 43 nn0addcld ( ( 𝜑𝐶 ∈ ℕ0 ) → ( 𝐶 + 1 ) ∈ ℕ0 )
45 eqidd ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( 𝑗 ∈ ℕ0 ↦ ( ( 𝐶 C𝑐 𝑗 ) · ( ( 𝐴𝑐 ( 𝐶𝑗 ) ) · ( 𝐵𝑗 ) ) ) ) = ( 𝑗 ∈ ℕ0 ↦ ( ( 𝐶 C𝑐 𝑗 ) · ( ( 𝐴𝑐 ( 𝐶𝑗 ) ) · ( 𝐵𝑗 ) ) ) ) )
46 simpr ( ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑗 = 𝑘 ) → 𝑗 = 𝑘 )
47 46 oveq2d ( ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑗 = 𝑘 ) → ( 𝐶 C𝑐 𝑗 ) = ( 𝐶 C𝑐 𝑘 ) )
48 46 oveq2d ( ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑗 = 𝑘 ) → ( 𝐶𝑗 ) = ( 𝐶𝑘 ) )
49 48 oveq2d ( ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑗 = 𝑘 ) → ( 𝐴𝑐 ( 𝐶𝑗 ) ) = ( 𝐴𝑐 ( 𝐶𝑘 ) ) )
50 46 oveq2d ( ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑗 = 𝑘 ) → ( 𝐵𝑗 ) = ( 𝐵𝑘 ) )
51 49 50 oveq12d ( ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑗 = 𝑘 ) → ( ( 𝐴𝑐 ( 𝐶𝑗 ) ) · ( 𝐵𝑗 ) ) = ( ( 𝐴𝑐 ( 𝐶𝑘 ) ) · ( 𝐵𝑘 ) ) )
52 47 51 oveq12d ( ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑗 = 𝑘 ) → ( ( 𝐶 C𝑐 𝑗 ) · ( ( 𝐴𝑐 ( 𝐶𝑗 ) ) · ( 𝐵𝑗 ) ) ) = ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴𝑐 ( 𝐶𝑘 ) ) · ( 𝐵𝑘 ) ) ) )
53 4 ad2antrr ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → 𝐶 ∈ ℂ )
54 53 19 bcccl ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( 𝐶 C𝑐 𝑘 ) ∈ ℂ )
55 5 ad2antrr ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → 𝐴 ∈ ℂ )
56 19 nn0cnd ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → 𝑘 ∈ ℂ )
57 53 56 subcld ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( 𝐶𝑘 ) ∈ ℂ )
58 55 57 cxpcld ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( 𝐴𝑐 ( 𝐶𝑘 ) ) ∈ ℂ )
59 6 ad2antrr ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → 𝐵 ∈ ℂ )
60 59 19 expcld ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( 𝐵𝑘 ) ∈ ℂ )
61 58 60 mulcld ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝐴𝑐 ( 𝐶𝑘 ) ) · ( 𝐵𝑘 ) ) ∈ ℂ )
62 54 61 mulcld ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴𝑐 ( 𝐶𝑘 ) ) · ( 𝐵𝑘 ) ) ) ∈ ℂ )
63 45 52 19 62 fvmptd ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝑗 ∈ ℕ0 ↦ ( ( 𝐶 C𝑐 𝑗 ) · ( ( 𝐴𝑐 ( 𝐶𝑗 ) ) · ( 𝐵𝑗 ) ) ) ) ‘ 𝑘 ) = ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴𝑐 ( 𝐶𝑘 ) ) · ( 𝐵𝑘 ) ) ) )
64 peano2nn0 ( 𝐶 ∈ ℕ0 → ( 𝐶 + 1 ) ∈ ℕ0 )
65 64 adantl ( ( 𝜑𝐶 ∈ ℕ0 ) → ( 𝐶 + 1 ) ∈ ℕ0 )
66 c0ex 0 ∈ V
67 66 fconst ( ℕ0 × { 0 } ) : ℕ0 ⟶ { 0 }
68 67 a1i ( ( 𝜑𝐶 ∈ ℕ0 ) → ( ℕ0 × { 0 } ) : ℕ0 ⟶ { 0 } )
69 0red ( ( 𝜑𝐶 ∈ ℕ0 ) → 0 ∈ ℝ )
70 69 snssd ( ( 𝜑𝐶 ∈ ℕ0 ) → { 0 } ⊆ ℝ )
71 68 70 fssd ( ( 𝜑𝐶 ∈ ℕ0 ) → ( ℕ0 × { 0 } ) : ℕ0 ⟶ ℝ )
72 71 ffvelrnda ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( ( ℕ0 × { 0 } ) ‘ 𝑘 ) ∈ ℝ )
73 63 62 eqeltrd ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝑗 ∈ ℕ0 ↦ ( ( 𝐶 C𝑐 𝑗 ) · ( ( 𝐴𝑐 ( 𝐶𝑗 ) ) · ( 𝐵𝑗 ) ) ) ) ‘ 𝑘 ) ∈ ℂ )
74 climrel Rel ⇝
75 40 xpeq1i ( ℕ0 × { 0 } ) = ( ( ℤ ‘ 0 ) × { 0 } )
76 seqeq3 ( ( ℕ0 × { 0 } ) = ( ( ℤ ‘ 0 ) × { 0 } ) → seq 0 ( + , ( ℕ0 × { 0 } ) ) = seq 0 ( + , ( ( ℤ ‘ 0 ) × { 0 } ) ) )
77 75 76 ax-mp seq 0 ( + , ( ℕ0 × { 0 } ) ) = seq 0 ( + , ( ( ℤ ‘ 0 ) × { 0 } ) )
78 0z 0 ∈ ℤ
79 serclim0 ( 0 ∈ ℤ → seq 0 ( + , ( ( ℤ ‘ 0 ) × { 0 } ) ) ⇝ 0 )
80 78 79 ax-mp seq 0 ( + , ( ( ℤ ‘ 0 ) × { 0 } ) ) ⇝ 0
81 77 80 eqbrtri seq 0 ( + , ( ℕ0 × { 0 } ) ) ⇝ 0
82 releldm ( ( Rel ⇝ ∧ seq 0 ( + , ( ℕ0 × { 0 } ) ) ⇝ 0 ) → seq 0 ( + , ( ℕ0 × { 0 } ) ) ∈ dom ⇝ )
83 74 81 82 mp2an seq 0 ( + , ( ℕ0 × { 0 } ) ) ∈ dom ⇝
84 83 a1i ( ( 𝜑𝐶 ∈ ℕ0 ) → seq 0 ( + , ( ℕ0 × { 0 } ) ) ∈ dom ⇝ )
85 eluznn0 ( ( ( 𝐶 + 1 ) ∈ ℕ0𝑘 ∈ ( ℤ ‘ ( 𝐶 + 1 ) ) ) → 𝑘 ∈ ℕ0 )
86 65 85 sylan ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ ‘ ( 𝐶 + 1 ) ) ) → 𝑘 ∈ ℕ0 )
87 86 63 syldan ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ ‘ ( 𝐶 + 1 ) ) ) → ( ( 𝑗 ∈ ℕ0 ↦ ( ( 𝐶 C𝑐 𝑗 ) · ( ( 𝐴𝑐 ( 𝐶𝑗 ) ) · ( 𝐵𝑗 ) ) ) ) ‘ 𝑘 ) = ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴𝑐 ( 𝐶𝑘 ) ) · ( 𝐵𝑘 ) ) ) )
88 0zd ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ ‘ ( 𝐶 + 1 ) ) ) → 0 ∈ ℤ )
89 86 nn0zd ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ ‘ ( 𝐶 + 1 ) ) ) → 𝑘 ∈ ℤ )
90 1zzd ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ ‘ ( 𝐶 + 1 ) ) ) → 1 ∈ ℤ )
91 89 90 zsubcld ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ ‘ ( 𝐶 + 1 ) ) ) → ( 𝑘 − 1 ) ∈ ℤ )
92 14 nn0zd ( ( 𝜑𝐶 ∈ ℕ0 ) → 𝐶 ∈ ℤ )
93 92 adantr ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ ‘ ( 𝐶 + 1 ) ) ) → 𝐶 ∈ ℤ )
94 14 nn0ge0d ( ( 𝜑𝐶 ∈ ℕ0 ) → 0 ≤ 𝐶 )
95 94 adantr ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ ‘ ( 𝐶 + 1 ) ) ) → 0 ≤ 𝐶 )
96 eluzle ( 𝑘 ∈ ( ℤ ‘ ( 𝐶 + 1 ) ) → ( 𝐶 + 1 ) ≤ 𝑘 )
97 96 adantl ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ ‘ ( 𝐶 + 1 ) ) ) → ( 𝐶 + 1 ) ≤ 𝑘 )
98 93 zred ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ ‘ ( 𝐶 + 1 ) ) ) → 𝐶 ∈ ℝ )
99 1red ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ ‘ ( 𝐶 + 1 ) ) ) → 1 ∈ ℝ )
100 86 nn0red ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ ‘ ( 𝐶 + 1 ) ) ) → 𝑘 ∈ ℝ )
101 leaddsub ( ( 𝐶 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑘 ∈ ℝ ) → ( ( 𝐶 + 1 ) ≤ 𝑘𝐶 ≤ ( 𝑘 − 1 ) ) )
102 98 99 100 101 syl3anc ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ ‘ ( 𝐶 + 1 ) ) ) → ( ( 𝐶 + 1 ) ≤ 𝑘𝐶 ≤ ( 𝑘 − 1 ) ) )
103 97 102 mpbid ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ ‘ ( 𝐶 + 1 ) ) ) → 𝐶 ≤ ( 𝑘 − 1 ) )
104 88 91 93 95 103 elfzd ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ ‘ ( 𝐶 + 1 ) ) ) → 𝐶 ∈ ( 0 ... ( 𝑘 − 1 ) ) )
105 4 ad2antrr ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ ‘ ( 𝐶 + 1 ) ) ) → 𝐶 ∈ ℂ )
106 105 86 bcc0 ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ ‘ ( 𝐶 + 1 ) ) ) → ( ( 𝐶 C𝑐 𝑘 ) = 0 ↔ 𝐶 ∈ ( 0 ... ( 𝑘 − 1 ) ) ) )
107 104 106 mpbird ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ ‘ ( 𝐶 + 1 ) ) ) → ( 𝐶 C𝑐 𝑘 ) = 0 )
108 107 oveq1d ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ ‘ ( 𝐶 + 1 ) ) ) → ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴𝑐 ( 𝐶𝑘 ) ) · ( 𝐵𝑘 ) ) ) = ( 0 · ( ( 𝐴𝑐 ( 𝐶𝑘 ) ) · ( 𝐵𝑘 ) ) ) )
109 5 ad2antrr ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ ‘ ( 𝐶 + 1 ) ) ) → 𝐴 ∈ ℂ )
110 eluzelcn ( 𝑘 ∈ ( ℤ ‘ ( 𝐶 + 1 ) ) → 𝑘 ∈ ℂ )
111 110 adantl ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ ‘ ( 𝐶 + 1 ) ) ) → 𝑘 ∈ ℂ )
112 105 111 subcld ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ ‘ ( 𝐶 + 1 ) ) ) → ( 𝐶𝑘 ) ∈ ℂ )
113 109 112 cxpcld ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ ‘ ( 𝐶 + 1 ) ) ) → ( 𝐴𝑐 ( 𝐶𝑘 ) ) ∈ ℂ )
114 6 ad2antrr ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ ‘ ( 𝐶 + 1 ) ) ) → 𝐵 ∈ ℂ )
115 114 86 expcld ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ ‘ ( 𝐶 + 1 ) ) ) → ( 𝐵𝑘 ) ∈ ℂ )
116 113 115 mulcld ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ ‘ ( 𝐶 + 1 ) ) ) → ( ( 𝐴𝑐 ( 𝐶𝑘 ) ) · ( 𝐵𝑘 ) ) ∈ ℂ )
117 116 mul02d ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ ‘ ( 𝐶 + 1 ) ) ) → ( 0 · ( ( 𝐴𝑐 ( 𝐶𝑘 ) ) · ( 𝐵𝑘 ) ) ) = 0 )
118 108 117 eqtrd ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ ‘ ( 𝐶 + 1 ) ) ) → ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴𝑐 ( 𝐶𝑘 ) ) · ( 𝐵𝑘 ) ) ) = 0 )
119 87 118 eqtrd ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ ‘ ( 𝐶 + 1 ) ) ) → ( ( 𝑗 ∈ ℕ0 ↦ ( ( 𝐶 C𝑐 𝑗 ) · ( ( 𝐴𝑐 ( 𝐶𝑗 ) ) · ( 𝐵𝑗 ) ) ) ) ‘ 𝑘 ) = 0 )
120 119 abs00bd ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ ‘ ( 𝐶 + 1 ) ) ) → ( abs ‘ ( ( 𝑗 ∈ ℕ0 ↦ ( ( 𝐶 C𝑐 𝑗 ) · ( ( 𝐴𝑐 ( 𝐶𝑗 ) ) · ( 𝐵𝑗 ) ) ) ) ‘ 𝑘 ) ) = 0 )
121 0re 0 ∈ ℝ
122 120 121 eqeltrdi ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ ‘ ( 𝐶 + 1 ) ) ) → ( abs ‘ ( ( 𝑗 ∈ ℕ0 ↦ ( ( 𝐶 C𝑐 𝑗 ) · ( ( 𝐴𝑐 ( 𝐶𝑗 ) ) · ( 𝐵𝑗 ) ) ) ) ‘ 𝑘 ) ) ∈ ℝ )
123 eqle ( ( ( abs ‘ ( ( 𝑗 ∈ ℕ0 ↦ ( ( 𝐶 C𝑐 𝑗 ) · ( ( 𝐴𝑐 ( 𝐶𝑗 ) ) · ( 𝐵𝑗 ) ) ) ) ‘ 𝑘 ) ) ∈ ℝ ∧ ( abs ‘ ( ( 𝑗 ∈ ℕ0 ↦ ( ( 𝐶 C𝑐 𝑗 ) · ( ( 𝐴𝑐 ( 𝐶𝑗 ) ) · ( 𝐵𝑗 ) ) ) ) ‘ 𝑘 ) ) = 0 ) → ( abs ‘ ( ( 𝑗 ∈ ℕ0 ↦ ( ( 𝐶 C𝑐 𝑗 ) · ( ( 𝐴𝑐 ( 𝐶𝑗 ) ) · ( 𝐵𝑗 ) ) ) ) ‘ 𝑘 ) ) ≤ 0 )
124 122 120 123 syl2anc ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ ‘ ( 𝐶 + 1 ) ) ) → ( abs ‘ ( ( 𝑗 ∈ ℕ0 ↦ ( ( 𝐶 C𝑐 𝑗 ) · ( ( 𝐴𝑐 ( 𝐶𝑗 ) ) · ( 𝐵𝑗 ) ) ) ) ‘ 𝑘 ) ) ≤ 0 )
125 72 recnd ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( ( ℕ0 × { 0 } ) ‘ 𝑘 ) ∈ ℂ )
126 86 125 syldan ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ ‘ ( 𝐶 + 1 ) ) ) → ( ( ℕ0 × { 0 } ) ‘ 𝑘 ) ∈ ℂ )
127 126 mul02d ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ ‘ ( 𝐶 + 1 ) ) ) → ( 0 · ( ( ℕ0 × { 0 } ) ‘ 𝑘 ) ) = 0 )
128 124 127 breqtrrd ( ( ( 𝜑𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ ‘ ( 𝐶 + 1 ) ) ) → ( abs ‘ ( ( 𝑗 ∈ ℕ0 ↦ ( ( 𝐶 C𝑐 𝑗 ) · ( ( 𝐴𝑐 ( 𝐶𝑗 ) ) · ( 𝐵𝑗 ) ) ) ) ‘ 𝑘 ) ) ≤ ( 0 · ( ( ℕ0 × { 0 } ) ‘ 𝑘 ) ) )
129 40 65 72 73 84 69 128 cvgcmpce ( ( 𝜑𝐶 ∈ ℕ0 ) → seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( ( 𝐶 C𝑐 𝑗 ) · ( ( 𝐴𝑐 ( 𝐶𝑗 ) ) · ( 𝐵𝑗 ) ) ) ) ) ∈ dom ⇝ )
130 40 41 44 63 62 129 isumsplit ( ( 𝜑𝐶 ∈ ℕ0 ) → Σ 𝑘 ∈ ℕ0 ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴𝑐 ( 𝐶𝑘 ) ) · ( 𝐵𝑘 ) ) ) = ( Σ 𝑘 ∈ ( 0 ... ( ( 𝐶 + 1 ) − 1 ) ) ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴𝑐 ( 𝐶𝑘 ) ) · ( 𝐵𝑘 ) ) ) + Σ 𝑘 ∈ ( ℤ ‘ ( 𝐶 + 1 ) ) ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴𝑐 ( 𝐶𝑘 ) ) · ( 𝐵𝑘 ) ) ) ) )
131 1cnd ( ( 𝜑𝐶 ∈ ℕ0 ) → 1 ∈ ℂ )
132 36 131 pncand ( ( 𝜑𝐶 ∈ ℕ0 ) → ( ( 𝐶 + 1 ) − 1 ) = 𝐶 )
133 132 oveq2d ( ( 𝜑𝐶 ∈ ℕ0 ) → ( 0 ... ( ( 𝐶 + 1 ) − 1 ) ) = ( 0 ... 𝐶 ) )
134 133 sumeq1d ( ( 𝜑𝐶 ∈ ℕ0 ) → Σ 𝑘 ∈ ( 0 ... ( ( 𝐶 + 1 ) − 1 ) ) ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴𝑐 ( 𝐶𝑘 ) ) · ( 𝐵𝑘 ) ) ) = Σ 𝑘 ∈ ( 0 ... 𝐶 ) ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴𝑐 ( 𝐶𝑘 ) ) · ( 𝐵𝑘 ) ) ) )
135 134 oveq1d ( ( 𝜑𝐶 ∈ ℕ0 ) → ( Σ 𝑘 ∈ ( 0 ... ( ( 𝐶 + 1 ) − 1 ) ) ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴𝑐 ( 𝐶𝑘 ) ) · ( 𝐵𝑘 ) ) ) + Σ 𝑘 ∈ ( ℤ ‘ ( 𝐶 + 1 ) ) ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴𝑐 ( 𝐶𝑘 ) ) · ( 𝐵𝑘 ) ) ) ) = ( Σ 𝑘 ∈ ( 0 ... 𝐶 ) ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴𝑐 ( 𝐶𝑘 ) ) · ( 𝐵𝑘 ) ) ) + Σ 𝑘 ∈ ( ℤ ‘ ( 𝐶 + 1 ) ) ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴𝑐 ( 𝐶𝑘 ) ) · ( 𝐵𝑘 ) ) ) ) )
136 118 sumeq2dv ( ( 𝜑𝐶 ∈ ℕ0 ) → Σ 𝑘 ∈ ( ℤ ‘ ( 𝐶 + 1 ) ) ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴𝑐 ( 𝐶𝑘 ) ) · ( 𝐵𝑘 ) ) ) = Σ 𝑘 ∈ ( ℤ ‘ ( 𝐶 + 1 ) ) 0 )
137 ssid ( ℤ ‘ ( 𝐶 + 1 ) ) ⊆ ( ℤ ‘ ( 𝐶 + 1 ) )
138 137 orci ( ( ℤ ‘ ( 𝐶 + 1 ) ) ⊆ ( ℤ ‘ ( 𝐶 + 1 ) ) ∨ ( ℤ ‘ ( 𝐶 + 1 ) ) ∈ Fin )
139 sumz ( ( ( ℤ ‘ ( 𝐶 + 1 ) ) ⊆ ( ℤ ‘ ( 𝐶 + 1 ) ) ∨ ( ℤ ‘ ( 𝐶 + 1 ) ) ∈ Fin ) → Σ 𝑘 ∈ ( ℤ ‘ ( 𝐶 + 1 ) ) 0 = 0 )
140 138 139 ax-mp Σ 𝑘 ∈ ( ℤ ‘ ( 𝐶 + 1 ) ) 0 = 0
141 136 140 eqtrdi ( ( 𝜑𝐶 ∈ ℕ0 ) → Σ 𝑘 ∈ ( ℤ ‘ ( 𝐶 + 1 ) ) ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴𝑐 ( 𝐶𝑘 ) ) · ( 𝐵𝑘 ) ) ) = 0 )
142 141 oveq2d ( ( 𝜑𝐶 ∈ ℕ0 ) → ( Σ 𝑘 ∈ ( 0 ... 𝐶 ) ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴𝑐 ( 𝐶𝑘 ) ) · ( 𝐵𝑘 ) ) ) + Σ 𝑘 ∈ ( ℤ ‘ ( 𝐶 + 1 ) ) ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴𝑐 ( 𝐶𝑘 ) ) · ( 𝐵𝑘 ) ) ) ) = ( Σ 𝑘 ∈ ( 0 ... 𝐶 ) ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴𝑐 ( 𝐶𝑘 ) ) · ( 𝐵𝑘 ) ) ) + 0 ) )
143 130 135 142 3eqtrd ( ( 𝜑𝐶 ∈ ℕ0 ) → Σ 𝑘 ∈ ℕ0 ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴𝑐 ( 𝐶𝑘 ) ) · ( 𝐵𝑘 ) ) ) = ( Σ 𝑘 ∈ ( 0 ... 𝐶 ) ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴𝑐 ( 𝐶𝑘 ) ) · ( 𝐵𝑘 ) ) ) + 0 ) )
144 39 143 35 3eqtr4rd ( ( 𝜑𝐶 ∈ ℕ0 ) → ( ( 𝐴 + 𝐵 ) ↑𝑐 𝐶 ) = Σ 𝑘 ∈ ℕ0 ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴𝑐 ( 𝐶𝑘 ) ) · ( 𝐵𝑘 ) ) ) )