Step | Hyp | Ref
| Expression |
1 | | oveq2 5463 |
. . . . . . . 8
⊢ (𝑗 = 0 → (A · 𝑗) = (A
· 0)) |
2 | 1 | oveq2d 5471 |
. . . . . . 7
⊢ (𝑗 = 0 → (1 + (A · 𝑗)) = (1 + (A · 0))) |
3 | | oveq2 5463 |
. . . . . . 7
⊢ (𝑗 = 0 → ((1 + A)↑𝑗) = ((1 + A)↑0)) |
4 | 2, 3 | breq12d 3768 |
. . . . . 6
⊢ (𝑗 = 0 → ((1 + (A · 𝑗)) ≤ ((1 + A)↑𝑗) ↔ (1 + (A · 0)) ≤ ((1 + A)↑0))) |
5 | 4 | imbi2d 219 |
. . . . 5
⊢ (𝑗 = 0 → (((A ∈ ℝ ∧ -1 ≤ A)
→ (1 + (A · 𝑗)) ≤ ((1 + A)↑𝑗)) ↔ ((A ∈ ℝ ∧ -1 ≤ A)
→ (1 + (A · 0)) ≤ ((1 +
A)↑0)))) |
6 | | oveq2 5463 |
. . . . . . . 8
⊢ (𝑗 = 𝑘 → (A · 𝑗) = (A
· 𝑘)) |
7 | 6 | oveq2d 5471 |
. . . . . . 7
⊢ (𝑗 = 𝑘 → (1 + (A · 𝑗)) = (1 + (A · 𝑘))) |
8 | | oveq2 5463 |
. . . . . . 7
⊢ (𝑗 = 𝑘 → ((1 + A)↑𝑗) = ((1 + A)↑𝑘)) |
9 | 7, 8 | breq12d 3768 |
. . . . . 6
⊢ (𝑗 = 𝑘 → ((1 + (A · 𝑗)) ≤ ((1 + A)↑𝑗) ↔ (1 + (A · 𝑘)) ≤ ((1 + A)↑𝑘))) |
10 | 9 | imbi2d 219 |
. . . . 5
⊢ (𝑗 = 𝑘 → (((A ∈ ℝ ∧ -1 ≤ A)
→ (1 + (A · 𝑗)) ≤ ((1 + A)↑𝑗)) ↔ ((A ∈ ℝ ∧ -1 ≤ A)
→ (1 + (A · 𝑘)) ≤ ((1 + A)↑𝑘)))) |
11 | | oveq2 5463 |
. . . . . . . 8
⊢ (𝑗 = (𝑘 + 1) → (A · 𝑗) = (A
· (𝑘 +
1))) |
12 | 11 | oveq2d 5471 |
. . . . . . 7
⊢ (𝑗 = (𝑘 + 1) → (1 + (A · 𝑗)) = (1 + (A · (𝑘 + 1)))) |
13 | | oveq2 5463 |
. . . . . . 7
⊢ (𝑗 = (𝑘 + 1) → ((1 + A)↑𝑗) = ((1 + A)↑(𝑘 + 1))) |
14 | 12, 13 | breq12d 3768 |
. . . . . 6
⊢ (𝑗 = (𝑘 + 1) → ((1 + (A · 𝑗)) ≤ ((1 + A)↑𝑗) ↔ (1 + (A · (𝑘 + 1))) ≤ ((1 + A)↑(𝑘 + 1)))) |
15 | 14 | imbi2d 219 |
. . . . 5
⊢ (𝑗 = (𝑘 + 1) → (((A ∈ ℝ ∧ -1 ≤ A)
→ (1 + (A · 𝑗)) ≤ ((1 + A)↑𝑗)) ↔ ((A ∈ ℝ ∧ -1 ≤ A)
→ (1 + (A · (𝑘 + 1))) ≤ ((1 + A)↑(𝑘 + 1))))) |
16 | | oveq2 5463 |
. . . . . . . 8
⊢ (𝑗 = 𝑁 → (A · 𝑗) = (A
· 𝑁)) |
17 | 16 | oveq2d 5471 |
. . . . . . 7
⊢ (𝑗 = 𝑁 → (1 + (A · 𝑗)) = (1 + (A · 𝑁))) |
18 | | oveq2 5463 |
. . . . . . 7
⊢ (𝑗 = 𝑁 → ((1 + A)↑𝑗) = ((1 + A)↑𝑁)) |
19 | 17, 18 | breq12d 3768 |
. . . . . 6
⊢ (𝑗 = 𝑁 → ((1 + (A · 𝑗)) ≤ ((1 + A)↑𝑗) ↔ (1 + (A · 𝑁)) ≤ ((1 + A)↑𝑁))) |
20 | 19 | imbi2d 219 |
. . . . 5
⊢ (𝑗 = 𝑁 → (((A ∈ ℝ ∧ -1 ≤ A)
→ (1 + (A · 𝑗)) ≤ ((1 + A)↑𝑗)) ↔ ((A ∈ ℝ ∧ -1 ≤ A)
→ (1 + (A · 𝑁)) ≤ ((1 + A)↑𝑁)))) |
21 | | recn 6812 |
. . . . . . 7
⊢ (A ∈ ℝ →
A ∈
ℂ) |
22 | | mul01 7182 |
. . . . . . . . . 10
⊢ (A ∈ ℂ →
(A · 0) = 0) |
23 | 22 | oveq2d 5471 |
. . . . . . . . 9
⊢ (A ∈ ℂ →
(1 + (A · 0)) = (1 +
0)) |
24 | | 1p0e1 7810 |
. . . . . . . . 9
⊢ (1 + 0) =
1 |
25 | 23, 24 | syl6eq 2085 |
. . . . . . . 8
⊢ (A ∈ ℂ →
(1 + (A · 0)) = 1) |
26 | | 1le1 7356 |
. . . . . . . . 9
⊢ 1 ≤
1 |
27 | | ax-1cn 6776 |
. . . . . . . . . . 11
⊢ 1 ∈ ℂ |
28 | | addcl 6804 |
. . . . . . . . . . 11
⊢ ((1 ∈ ℂ ∧
A ∈
ℂ) → (1 + A) ∈ ℂ) |
29 | 27, 28 | mpan 400 |
. . . . . . . . . 10
⊢ (A ∈ ℂ →
(1 + A) ∈
ℂ) |
30 | | exp0 8913 |
. . . . . . . . . 10
⊢ ((1 +
A) ∈
ℂ → ((1 + A)↑0) =
1) |
31 | 29, 30 | syl 14 |
. . . . . . . . 9
⊢ (A ∈ ℂ →
((1 + A)↑0) = 1) |
32 | 26, 31 | syl5breqr 3791 |
. . . . . . . 8
⊢ (A ∈ ℂ →
1 ≤ ((1 + A)↑0)) |
33 | 25, 32 | eqbrtrd 3775 |
. . . . . . 7
⊢ (A ∈ ℂ →
(1 + (A · 0)) ≤ ((1 + A)↑0)) |
34 | 21, 33 | syl 14 |
. . . . . 6
⊢ (A ∈ ℝ →
(1 + (A · 0)) ≤ ((1 + A)↑0)) |
35 | 34 | adantr 261 |
. . . . 5
⊢
((A ∈ ℝ ∧ -1
≤ A) → (1 + (A · 0)) ≤ ((1 + A)↑0)) |
36 | | 1re 6824 |
. . . . . . . . . . . . . 14
⊢ 1 ∈ ℝ |
37 | | nn0re 7966 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈
ℝ) |
38 | | remulcl 6807 |
. . . . . . . . . . . . . . 15
⊢
((A ∈ ℝ ∧ 𝑘 ∈ ℝ) → (A · 𝑘) ∈
ℝ) |
39 | 37, 38 | sylan2 270 |
. . . . . . . . . . . . . 14
⊢
((A ∈ ℝ ∧ 𝑘 ∈ ℕ0) → (A · 𝑘) ∈
ℝ) |
40 | | readdcl 6805 |
. . . . . . . . . . . . . 14
⊢ ((1 ∈ ℝ ∧
(A · 𝑘) ∈
ℝ) → (1 + (A · 𝑘)) ∈ ℝ) |
41 | 36, 39, 40 | sylancr 393 |
. . . . . . . . . . . . 13
⊢
((A ∈ ℝ ∧ 𝑘 ∈ ℕ0) → (1 + (A · 𝑘)) ∈
ℝ) |
42 | | simpl 102 |
. . . . . . . . . . . . 13
⊢
((A ∈ ℝ ∧ 𝑘 ∈ ℕ0) → A ∈
ℝ) |
43 | | readdcl 6805 |
. . . . . . . . . . . . 13
⊢ (((1 +
(A · 𝑘)) ∈
ℝ ∧ A ∈ ℝ)
→ ((1 + (A · 𝑘)) + A)
∈ ℝ) |
44 | 41, 42, 43 | syl2anc 391 |
. . . . . . . . . . . 12
⊢
((A ∈ ℝ ∧ 𝑘 ∈ ℕ0) → ((1 + (A · 𝑘)) + A)
∈ ℝ) |
45 | 44 | adantr 261 |
. . . . . . . . . . 11
⊢
(((A ∈ ℝ ∧ 𝑘 ∈ ℕ0) ∧ (-1 ≤ A
∧ (1 + (A
· 𝑘)) ≤ ((1 +
A)↑𝑘))) → ((1 + (A · 𝑘)) + A)
∈ ℝ) |
46 | | readdcl 6805 |
. . . . . . . . . . . . . . 15
⊢ ((1 ∈ ℝ ∧
A ∈
ℝ) → (1 + A) ∈ ℝ) |
47 | 36, 46 | mpan 400 |
. . . . . . . . . . . . . 14
⊢ (A ∈ ℝ →
(1 + A) ∈
ℝ) |
48 | 47 | adantr 261 |
. . . . . . . . . . . . 13
⊢
((A ∈ ℝ ∧ 𝑘 ∈ ℕ0) → (1 + A) ∈
ℝ) |
49 | 41, 48 | remulcld 6853 |
. . . . . . . . . . . 12
⊢
((A ∈ ℝ ∧ 𝑘 ∈ ℕ0) → ((1 + (A · 𝑘)) · (1 + A)) ∈
ℝ) |
50 | 49 | adantr 261 |
. . . . . . . . . . 11
⊢
(((A ∈ ℝ ∧ 𝑘 ∈ ℕ0) ∧ (-1 ≤ A
∧ (1 + (A
· 𝑘)) ≤ ((1 +
A)↑𝑘))) → ((1 + (A · 𝑘)) · (1 + A)) ∈
ℝ) |
51 | | reexpcl 8926 |
. . . . . . . . . . . . . 14
⊢ (((1 +
A) ∈
ℝ ∧ 𝑘 ∈
ℕ0) → ((1 + A)↑𝑘) ∈
ℝ) |
52 | 47, 51 | sylan 267 |
. . . . . . . . . . . . 13
⊢
((A ∈ ℝ ∧ 𝑘 ∈ ℕ0) → ((1 + A)↑𝑘) ∈
ℝ) |
53 | 52, 48 | remulcld 6853 |
. . . . . . . . . . . 12
⊢
((A ∈ ℝ ∧ 𝑘 ∈ ℕ0) → (((1 + A)↑𝑘) · (1 + A)) ∈
ℝ) |
54 | 53 | adantr 261 |
. . . . . . . . . . 11
⊢
(((A ∈ ℝ ∧ 𝑘 ∈ ℕ0) ∧ (-1 ≤ A
∧ (1 + (A
· 𝑘)) ≤ ((1 +
A)↑𝑘))) → (((1 + A)↑𝑘) · (1 + A)) ∈
ℝ) |
55 | | remulcl 6807 |
. . . . . . . . . . . . . . . . . 18
⊢
((A ∈ ℝ ∧
A ∈
ℝ) → (A · A) ∈
ℝ) |
56 | 55 | anidms 377 |
. . . . . . . . . . . . . . . . 17
⊢ (A ∈ ℝ →
(A · A) ∈
ℝ) |
57 | | msqge0 7400 |
. . . . . . . . . . . . . . . . 17
⊢ (A ∈ ℝ →
0 ≤ (A · A)) |
58 | 56, 57 | jca 290 |
. . . . . . . . . . . . . . . 16
⊢ (A ∈ ℝ →
((A · A) ∈ ℝ ∧ 0 ≤ (A
· A))) |
59 | | nn0ge0 7983 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈ ℕ0 → 0 ≤ 𝑘) |
60 | 37, 59 | jca 290 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 ∈ ℕ0 → (𝑘 ∈ ℝ
∧ 0 ≤ 𝑘)) |
61 | | mulge0 7403 |
. . . . . . . . . . . . . . . 16
⊢
((((A · A) ∈ ℝ ∧ 0 ≤ (A
· A)) ∧ (𝑘 ∈ ℝ
∧ 0 ≤ 𝑘)) → 0 ≤ ((A · A)
· 𝑘)) |
62 | 58, 60, 61 | syl2an 273 |
. . . . . . . . . . . . . . 15
⊢
((A ∈ ℝ ∧ 𝑘 ∈ ℕ0) → 0 ≤ ((A · A)
· 𝑘)) |
63 | 21 | adantr 261 |
. . . . . . . . . . . . . . . 16
⊢
((A ∈ ℝ ∧ 𝑘 ∈ ℕ0) → A ∈
ℂ) |
64 | | nn0cn 7967 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈
ℂ) |
65 | 64 | adantl 262 |
. . . . . . . . . . . . . . . 16
⊢
((A ∈ ℝ ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈
ℂ) |
66 | 63, 63, 65 | mul32d 6963 |
. . . . . . . . . . . . . . 15
⊢
((A ∈ ℝ ∧ 𝑘 ∈ ℕ0) → ((A · A)
· 𝑘) = ((A · 𝑘) · A)) |
67 | 62, 66 | breqtrd 3779 |
. . . . . . . . . . . . . 14
⊢
((A ∈ ℝ ∧ 𝑘 ∈ ℕ0) → 0 ≤ ((A · 𝑘) · A)) |
68 | | simpl 102 |
. . . . . . . . . . . . . . . . 17
⊢
((A ∈ ℝ ∧ 𝑘 ∈ ℝ) → A ∈
ℝ) |
69 | 38, 68 | remulcld 6853 |
. . . . . . . . . . . . . . . 16
⊢
((A ∈ ℝ ∧ 𝑘 ∈ ℝ) → ((A · 𝑘) · A) ∈
ℝ) |
70 | 37, 69 | sylan2 270 |
. . . . . . . . . . . . . . 15
⊢
((A ∈ ℝ ∧ 𝑘 ∈ ℕ0) → ((A · 𝑘) · A) ∈
ℝ) |
71 | 44, 70 | addge01d 7319 |
. . . . . . . . . . . . . 14
⊢
((A ∈ ℝ ∧ 𝑘 ∈ ℕ0) → (0 ≤ ((A · 𝑘) · A) ↔ ((1 + (A · 𝑘)) + A)
≤ (((1 + (A · 𝑘)) + A) +
((A · 𝑘) · A)))) |
72 | 67, 71 | mpbid 135 |
. . . . . . . . . . . . 13
⊢
((A ∈ ℝ ∧ 𝑘 ∈ ℕ0) → ((1 + (A · 𝑘)) + A)
≤ (((1 + (A · 𝑘)) + A) +
((A · 𝑘) · A))) |
73 | | mulcl 6806 |
. . . . . . . . . . . . . . . . 17
⊢
((A ∈ ℂ ∧ 𝑘 ∈ ℂ) → (A · 𝑘) ∈
ℂ) |
74 | | addcl 6804 |
. . . . . . . . . . . . . . . . 17
⊢ ((1 ∈ ℂ ∧
(A · 𝑘) ∈
ℂ) → (1 + (A · 𝑘)) ∈ ℂ) |
75 | 27, 73, 74 | sylancr 393 |
. . . . . . . . . . . . . . . 16
⊢
((A ∈ ℂ ∧ 𝑘 ∈ ℂ) → (1 + (A · 𝑘)) ∈
ℂ) |
76 | | simpl 102 |
. . . . . . . . . . . . . . . 16
⊢
((A ∈ ℂ ∧ 𝑘 ∈ ℂ) → A ∈
ℂ) |
77 | 73, 76 | mulcld 6845 |
. . . . . . . . . . . . . . . 16
⊢
((A ∈ ℂ ∧ 𝑘 ∈ ℂ) → ((A · 𝑘) · A) ∈
ℂ) |
78 | 75, 76, 77 | addassd 6847 |
. . . . . . . . . . . . . . 15
⊢
((A ∈ ℂ ∧ 𝑘 ∈ ℂ) → (((1 + (A · 𝑘)) + A) +
((A · 𝑘) · A)) = ((1 + (A
· 𝑘)) + (A + ((A ·
𝑘) · A)))) |
79 | | muladd11 6943 |
. . . . . . . . . . . . . . . 16
⊢
(((A · 𝑘) ∈ ℂ
∧ A ∈ ℂ) → ((1 + (A · 𝑘)) · (1 + A)) = ((1 + (A
· 𝑘)) + (A + ((A ·
𝑘) · A)))) |
80 | 73, 76, 79 | syl2anc 391 |
. . . . . . . . . . . . . . 15
⊢
((A ∈ ℂ ∧ 𝑘 ∈ ℂ) → ((1 + (A · 𝑘)) · (1 + A)) = ((1 + (A
· 𝑘)) + (A + ((A ·
𝑘) · A)))) |
81 | 78, 80 | eqtr4d 2072 |
. . . . . . . . . . . . . 14
⊢
((A ∈ ℂ ∧ 𝑘 ∈ ℂ) → (((1 + (A · 𝑘)) + A) +
((A · 𝑘) · A)) = ((1 + (A
· 𝑘)) · (1 +
A))) |
82 | 21, 64, 81 | syl2an 273 |
. . . . . . . . . . . . 13
⊢
((A ∈ ℝ ∧ 𝑘 ∈ ℕ0) → (((1 + (A · 𝑘)) + A) +
((A · 𝑘) · A)) = ((1 + (A
· 𝑘)) · (1 +
A))) |
83 | 72, 82 | breqtrd 3779 |
. . . . . . . . . . . 12
⊢
((A ∈ ℝ ∧ 𝑘 ∈ ℕ0) → ((1 + (A · 𝑘)) + A)
≤ ((1 + (A · 𝑘)) · (1 + A))) |
84 | 83 | adantr 261 |
. . . . . . . . . . 11
⊢
(((A ∈ ℝ ∧ 𝑘 ∈ ℕ0) ∧ (-1 ≤ A
∧ (1 + (A
· 𝑘)) ≤ ((1 +
A)↑𝑘))) → ((1 + (A · 𝑘)) + A)
≤ ((1 + (A · 𝑘)) · (1 + A))) |
85 | 41 | adantr 261 |
. . . . . . . . . . . 12
⊢
(((A ∈ ℝ ∧ 𝑘 ∈ ℕ0) ∧ (-1 ≤ A
∧ (1 + (A
· 𝑘)) ≤ ((1 +
A)↑𝑘))) → (1 + (A · 𝑘)) ∈
ℝ) |
86 | 52 | adantr 261 |
. . . . . . . . . . . 12
⊢
(((A ∈ ℝ ∧ 𝑘 ∈ ℕ0) ∧ (-1 ≤ A
∧ (1 + (A
· 𝑘)) ≤ ((1 +
A)↑𝑘))) → ((1 + A)↑𝑘) ∈
ℝ) |
87 | 48 | adantr 261 |
. . . . . . . . . . . 12
⊢
(((A ∈ ℝ ∧ 𝑘 ∈ ℕ0) ∧ (-1 ≤ A
∧ (1 + (A
· 𝑘)) ≤ ((1 +
A)↑𝑘))) → (1 + A) ∈
ℝ) |
88 | | neg1rr 7801 |
. . . . . . . . . . . . . . . 16
⊢ -1 ∈ ℝ |
89 | | leadd2 7221 |
. . . . . . . . . . . . . . . 16
⊢ ((-1
∈ ℝ ∧
A ∈
ℝ ∧ 1 ∈ ℝ) → (-1 ≤ A ↔ (1 + -1) ≤ (1 + A))) |
90 | 88, 36, 89 | mp3an13 1222 |
. . . . . . . . . . . . . . 15
⊢ (A ∈ ℝ →
(-1 ≤ A ↔ (1 + -1) ≤ (1 +
A))) |
91 | | 1pneg1e0 7806 |
. . . . . . . . . . . . . . . 16
⊢ (1 + -1)
= 0 |
92 | 91 | breq1i 3762 |
. . . . . . . . . . . . . . 15
⊢ ((1 + -1)
≤ (1 + A) ↔ 0 ≤ (1 + A)) |
93 | 90, 92 | syl6bb 185 |
. . . . . . . . . . . . . 14
⊢ (A ∈ ℝ →
(-1 ≤ A ↔ 0 ≤ (1 + A))) |
94 | 93 | biimpa 280 |
. . . . . . . . . . . . 13
⊢
((A ∈ ℝ ∧ -1
≤ A) → 0 ≤ (1 + A)) |
95 | 94 | ad2ant2r 478 |
. . . . . . . . . . . 12
⊢
(((A ∈ ℝ ∧ 𝑘 ∈ ℕ0) ∧ (-1 ≤ A
∧ (1 + (A
· 𝑘)) ≤ ((1 +
A)↑𝑘))) → 0 ≤ (1 + A)) |
96 | | simprr 484 |
. . . . . . . . . . . 12
⊢
(((A ∈ ℝ ∧ 𝑘 ∈ ℕ0) ∧ (-1 ≤ A
∧ (1 + (A
· 𝑘)) ≤ ((1 +
A)↑𝑘))) → (1 + (A · 𝑘)) ≤ ((1 + A)↑𝑘)) |
97 | 85, 86, 87, 95, 96 | lemul1ad 7686 |
. . . . . . . . . . 11
⊢
(((A ∈ ℝ ∧ 𝑘 ∈ ℕ0) ∧ (-1 ≤ A
∧ (1 + (A
· 𝑘)) ≤ ((1 +
A)↑𝑘))) → ((1 + (A · 𝑘)) · (1 + A)) ≤ (((1 + A)↑𝑘) · (1 + A))) |
98 | 45, 50, 54, 84, 97 | letrd 6935 |
. . . . . . . . . 10
⊢
(((A ∈ ℝ ∧ 𝑘 ∈ ℕ0) ∧ (-1 ≤ A
∧ (1 + (A
· 𝑘)) ≤ ((1 +
A)↑𝑘))) → ((1 + (A · 𝑘)) + A)
≤ (((1 + A)↑𝑘) · (1 + A))) |
99 | | adddi 6811 |
. . . . . . . . . . . . . . . 16
⊢
((A ∈ ℂ ∧ 𝑘 ∈ ℂ ∧ 1
∈ ℂ) → (A · (𝑘 + 1)) = ((A · 𝑘) + (A
· 1))) |
100 | 27, 99 | mp3an3 1220 |
. . . . . . . . . . . . . . 15
⊢
((A ∈ ℂ ∧ 𝑘 ∈ ℂ) → (A · (𝑘 + 1)) = ((A · 𝑘) + (A
· 1))) |
101 | | mulid1 6822 |
. . . . . . . . . . . . . . . . 17
⊢ (A ∈ ℂ →
(A · 1) = A) |
102 | 101 | adantr 261 |
. . . . . . . . . . . . . . . 16
⊢
((A ∈ ℂ ∧ 𝑘 ∈ ℂ) → (A · 1) = A) |
103 | 102 | oveq2d 5471 |
. . . . . . . . . . . . . . 15
⊢
((A ∈ ℂ ∧ 𝑘 ∈ ℂ) → ((A · 𝑘) + (A
· 1)) = ((A · 𝑘) + A)) |
104 | 100, 103 | eqtrd 2069 |
. . . . . . . . . . . . . 14
⊢
((A ∈ ℂ ∧ 𝑘 ∈ ℂ) → (A · (𝑘 + 1)) = ((A · 𝑘) + A)) |
105 | 104 | oveq2d 5471 |
. . . . . . . . . . . . 13
⊢
((A ∈ ℂ ∧ 𝑘 ∈ ℂ) → (1 + (A · (𝑘 + 1))) = (1 + ((A · 𝑘) + A))) |
106 | | addass 6809 |
. . . . . . . . . . . . . . 15
⊢ ((1 ∈ ℂ ∧
(A · 𝑘) ∈ ℂ
∧ A ∈ ℂ) → ((1 + (A · 𝑘)) + A) =
(1 + ((A · 𝑘) + A))) |
107 | 27, 106 | mp3an1 1218 |
. . . . . . . . . . . . . 14
⊢
(((A · 𝑘) ∈ ℂ
∧ A ∈ ℂ) → ((1 + (A · 𝑘)) + A) =
(1 + ((A · 𝑘) + A))) |
108 | 73, 76, 107 | syl2anc 391 |
. . . . . . . . . . . . 13
⊢
((A ∈ ℂ ∧ 𝑘 ∈ ℂ) → ((1 + (A · 𝑘)) + A) =
(1 + ((A · 𝑘) + A))) |
109 | 105, 108 | eqtr4d 2072 |
. . . . . . . . . . . 12
⊢
((A ∈ ℂ ∧ 𝑘 ∈ ℂ) → (1 + (A · (𝑘 + 1))) = ((1 + (A · 𝑘)) + A)) |
110 | 21, 64, 109 | syl2an 273 |
. . . . . . . . . . 11
⊢
((A ∈ ℝ ∧ 𝑘 ∈ ℕ0) → (1 + (A · (𝑘 + 1))) = ((1 + (A · 𝑘)) + A)) |
111 | 110 | adantr 261 |
. . . . . . . . . 10
⊢
(((A ∈ ℝ ∧ 𝑘 ∈ ℕ0) ∧ (-1 ≤ A
∧ (1 + (A
· 𝑘)) ≤ ((1 +
A)↑𝑘))) → (1 + (A · (𝑘 + 1))) = ((1 + (A · 𝑘)) + A)) |
112 | 27, 21, 28 | sylancr 393 |
. . . . . . . . . . . 12
⊢ (A ∈ ℝ →
(1 + A) ∈
ℂ) |
113 | | expp1 8916 |
. . . . . . . . . . . 12
⊢ (((1 +
A) ∈
ℂ ∧ 𝑘 ∈
ℕ0) → ((1 + A)↑(𝑘 + 1)) = (((1 + A)↑𝑘) · (1 + A))) |
114 | 112, 113 | sylan 267 |
. . . . . . . . . . 11
⊢
((A ∈ ℝ ∧ 𝑘 ∈ ℕ0) → ((1 + A)↑(𝑘 + 1)) = (((1 + A)↑𝑘) · (1 + A))) |
115 | 114 | adantr 261 |
. . . . . . . . . 10
⊢
(((A ∈ ℝ ∧ 𝑘 ∈ ℕ0) ∧ (-1 ≤ A
∧ (1 + (A
· 𝑘)) ≤ ((1 +
A)↑𝑘))) → ((1 + A)↑(𝑘 + 1)) = (((1 + A)↑𝑘) · (1 + A))) |
116 | 98, 111, 115 | 3brtr4d 3785 |
. . . . . . . . 9
⊢
(((A ∈ ℝ ∧ 𝑘 ∈ ℕ0) ∧ (-1 ≤ A
∧ (1 + (A
· 𝑘)) ≤ ((1 +
A)↑𝑘))) → (1 + (A · (𝑘 + 1))) ≤ ((1 + A)↑(𝑘 + 1))) |
117 | 116 | exp43 354 |
. . . . . . . 8
⊢ (A ∈ ℝ →
(𝑘 ∈ ℕ0 → (-1 ≤ A → ((1 + (A · 𝑘)) ≤ ((1 + A)↑𝑘) → (1 + (A · (𝑘 + 1))) ≤ ((1 + A)↑(𝑘 + 1)))))) |
118 | 117 | com12 27 |
. . . . . . 7
⊢ (𝑘 ∈ ℕ0 → (A ∈ ℝ →
(-1 ≤ A → ((1 + (A · 𝑘)) ≤ ((1 + A)↑𝑘) → (1 + (A · (𝑘 + 1))) ≤ ((1 + A)↑(𝑘 + 1)))))) |
119 | 118 | impd 242 |
. . . . . 6
⊢ (𝑘 ∈ ℕ0 → ((A ∈ ℝ ∧ -1 ≤ A)
→ ((1 + (A · 𝑘)) ≤ ((1 + A)↑𝑘) → (1 + (A · (𝑘 + 1))) ≤ ((1 + A)↑(𝑘 + 1))))) |
120 | 119 | a2d 23 |
. . . . 5
⊢ (𝑘 ∈ ℕ0 → (((A ∈ ℝ ∧ -1 ≤ A)
→ (1 + (A · 𝑘)) ≤ ((1 + A)↑𝑘)) → ((A ∈ ℝ ∧ -1 ≤ A)
→ (1 + (A · (𝑘 + 1))) ≤ ((1 + A)↑(𝑘 + 1))))) |
121 | 5, 10, 15, 20, 35, 120 | nn0ind 8128 |
. . . 4
⊢ (𝑁 ∈ ℕ0 → ((A ∈ ℝ ∧ -1 ≤ A)
→ (1 + (A · 𝑁)) ≤ ((1 + A)↑𝑁))) |
122 | 121 | expd 245 |
. . 3
⊢ (𝑁 ∈ ℕ0 → (A ∈ ℝ →
(-1 ≤ A → (1 + (A · 𝑁)) ≤ ((1 + A)↑𝑁)))) |
123 | 122 | com12 27 |
. 2
⊢ (A ∈ ℝ →
(𝑁 ∈ ℕ0 → (-1 ≤ A → (1 + (A
· 𝑁)) ≤ ((1 +
A)↑𝑁)))) |
124 | 123 | 3imp 1097 |
1
⊢
((A ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ -1 ≤ A)
→ (1 + (A · 𝑁)) ≤ ((1 + A)↑𝑁)) |