Step |
Hyp |
Ref |
Expression |
1 |
|
aaliou3lem.c |
⊢ 𝐹 = ( 𝑎 ∈ ℕ ↦ ( 2 ↑ - ( ! ‘ 𝑎 ) ) ) |
2 |
|
aaliou3lem.d |
⊢ 𝐿 = Σ 𝑏 ∈ ℕ ( 𝐹 ‘ 𝑏 ) |
3 |
|
aaliou3lem.e |
⊢ 𝐻 = ( 𝑐 ∈ ℕ ↦ Σ 𝑏 ∈ ( 1 ... 𝑐 ) ( 𝐹 ‘ 𝑏 ) ) |
4 |
|
peano2nn |
⊢ ( 𝐴 ∈ ℕ → ( 𝐴 + 1 ) ∈ ℕ ) |
5 |
|
eqid |
⊢ ( 𝑐 ∈ ( ℤ≥ ‘ ( 𝐴 + 1 ) ) ↦ ( ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) · ( ( 1 / 2 ) ↑ ( 𝑐 − ( 𝐴 + 1 ) ) ) ) ) = ( 𝑐 ∈ ( ℤ≥ ‘ ( 𝐴 + 1 ) ) ↦ ( ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) · ( ( 1 / 2 ) ↑ ( 𝑐 − ( 𝐴 + 1 ) ) ) ) ) |
6 |
5 1
|
aaliou3lem3 |
⊢ ( ( 𝐴 + 1 ) ∈ ℕ → ( seq ( 𝐴 + 1 ) ( + , 𝐹 ) ∈ dom ⇝ ∧ Σ 𝑏 ∈ ( ℤ≥ ‘ ( 𝐴 + 1 ) ) ( 𝐹 ‘ 𝑏 ) ∈ ℝ+ ∧ Σ 𝑏 ∈ ( ℤ≥ ‘ ( 𝐴 + 1 ) ) ( 𝐹 ‘ 𝑏 ) ≤ ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ) ) |
7 |
|
3simpc |
⊢ ( ( seq ( 𝐴 + 1 ) ( + , 𝐹 ) ∈ dom ⇝ ∧ Σ 𝑏 ∈ ( ℤ≥ ‘ ( 𝐴 + 1 ) ) ( 𝐹 ‘ 𝑏 ) ∈ ℝ+ ∧ Σ 𝑏 ∈ ( ℤ≥ ‘ ( 𝐴 + 1 ) ) ( 𝐹 ‘ 𝑏 ) ≤ ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ) → ( Σ 𝑏 ∈ ( ℤ≥ ‘ ( 𝐴 + 1 ) ) ( 𝐹 ‘ 𝑏 ) ∈ ℝ+ ∧ Σ 𝑏 ∈ ( ℤ≥ ‘ ( 𝐴 + 1 ) ) ( 𝐹 ‘ 𝑏 ) ≤ ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ) ) |
8 |
4 6 7
|
3syl |
⊢ ( 𝐴 ∈ ℕ → ( Σ 𝑏 ∈ ( ℤ≥ ‘ ( 𝐴 + 1 ) ) ( 𝐹 ‘ 𝑏 ) ∈ ℝ+ ∧ Σ 𝑏 ∈ ( ℤ≥ ‘ ( 𝐴 + 1 ) ) ( 𝐹 ‘ 𝑏 ) ≤ ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ) ) |
9 |
|
nncn |
⊢ ( 𝐴 ∈ ℕ → 𝐴 ∈ ℂ ) |
10 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
11 |
|
pncan |
⊢ ( ( 𝐴 ∈ ℂ ∧ 1 ∈ ℂ ) → ( ( 𝐴 + 1 ) − 1 ) = 𝐴 ) |
12 |
9 10 11
|
sylancl |
⊢ ( 𝐴 ∈ ℕ → ( ( 𝐴 + 1 ) − 1 ) = 𝐴 ) |
13 |
12
|
oveq2d |
⊢ ( 𝐴 ∈ ℕ → ( 1 ... ( ( 𝐴 + 1 ) − 1 ) ) = ( 1 ... 𝐴 ) ) |
14 |
13
|
sumeq1d |
⊢ ( 𝐴 ∈ ℕ → Σ 𝑏 ∈ ( 1 ... ( ( 𝐴 + 1 ) − 1 ) ) ( 𝐹 ‘ 𝑏 ) = Σ 𝑏 ∈ ( 1 ... 𝐴 ) ( 𝐹 ‘ 𝑏 ) ) |
15 |
14
|
oveq1d |
⊢ ( 𝐴 ∈ ℕ → ( Σ 𝑏 ∈ ( 1 ... ( ( 𝐴 + 1 ) − 1 ) ) ( 𝐹 ‘ 𝑏 ) + Σ 𝑏 ∈ ( ℤ≥ ‘ ( 𝐴 + 1 ) ) ( 𝐹 ‘ 𝑏 ) ) = ( Σ 𝑏 ∈ ( 1 ... 𝐴 ) ( 𝐹 ‘ 𝑏 ) + Σ 𝑏 ∈ ( ℤ≥ ‘ ( 𝐴 + 1 ) ) ( 𝐹 ‘ 𝑏 ) ) ) |
16 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
17 |
|
eqid |
⊢ ( ℤ≥ ‘ ( 𝐴 + 1 ) ) = ( ℤ≥ ‘ ( 𝐴 + 1 ) ) |
18 |
|
eqidd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑏 ∈ ℕ ) → ( 𝐹 ‘ 𝑏 ) = ( 𝐹 ‘ 𝑏 ) ) |
19 |
|
fveq2 |
⊢ ( 𝑎 = 𝑏 → ( ! ‘ 𝑎 ) = ( ! ‘ 𝑏 ) ) |
20 |
19
|
negeqd |
⊢ ( 𝑎 = 𝑏 → - ( ! ‘ 𝑎 ) = - ( ! ‘ 𝑏 ) ) |
21 |
20
|
oveq2d |
⊢ ( 𝑎 = 𝑏 → ( 2 ↑ - ( ! ‘ 𝑎 ) ) = ( 2 ↑ - ( ! ‘ 𝑏 ) ) ) |
22 |
|
ovex |
⊢ ( 2 ↑ - ( ! ‘ 𝑏 ) ) ∈ V |
23 |
21 1 22
|
fvmpt |
⊢ ( 𝑏 ∈ ℕ → ( 𝐹 ‘ 𝑏 ) = ( 2 ↑ - ( ! ‘ 𝑏 ) ) ) |
24 |
|
2rp |
⊢ 2 ∈ ℝ+ |
25 |
|
nnnn0 |
⊢ ( 𝑏 ∈ ℕ → 𝑏 ∈ ℕ0 ) |
26 |
|
faccl |
⊢ ( 𝑏 ∈ ℕ0 → ( ! ‘ 𝑏 ) ∈ ℕ ) |
27 |
25 26
|
syl |
⊢ ( 𝑏 ∈ ℕ → ( ! ‘ 𝑏 ) ∈ ℕ ) |
28 |
27
|
nnzd |
⊢ ( 𝑏 ∈ ℕ → ( ! ‘ 𝑏 ) ∈ ℤ ) |
29 |
28
|
znegcld |
⊢ ( 𝑏 ∈ ℕ → - ( ! ‘ 𝑏 ) ∈ ℤ ) |
30 |
|
rpexpcl |
⊢ ( ( 2 ∈ ℝ+ ∧ - ( ! ‘ 𝑏 ) ∈ ℤ ) → ( 2 ↑ - ( ! ‘ 𝑏 ) ) ∈ ℝ+ ) |
31 |
24 29 30
|
sylancr |
⊢ ( 𝑏 ∈ ℕ → ( 2 ↑ - ( ! ‘ 𝑏 ) ) ∈ ℝ+ ) |
32 |
31
|
rpcnd |
⊢ ( 𝑏 ∈ ℕ → ( 2 ↑ - ( ! ‘ 𝑏 ) ) ∈ ℂ ) |
33 |
23 32
|
eqeltrd |
⊢ ( 𝑏 ∈ ℕ → ( 𝐹 ‘ 𝑏 ) ∈ ℂ ) |
34 |
33
|
adantl |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑏 ∈ ℕ ) → ( 𝐹 ‘ 𝑏 ) ∈ ℂ ) |
35 |
|
1nn |
⊢ 1 ∈ ℕ |
36 |
|
eqid |
⊢ ( 𝑐 ∈ ( ℤ≥ ‘ 1 ) ↦ ( ( 2 ↑ - ( ! ‘ 1 ) ) · ( ( 1 / 2 ) ↑ ( 𝑐 − 1 ) ) ) ) = ( 𝑐 ∈ ( ℤ≥ ‘ 1 ) ↦ ( ( 2 ↑ - ( ! ‘ 1 ) ) · ( ( 1 / 2 ) ↑ ( 𝑐 − 1 ) ) ) ) |
37 |
36 1
|
aaliou3lem3 |
⊢ ( 1 ∈ ℕ → ( seq 1 ( + , 𝐹 ) ∈ dom ⇝ ∧ Σ 𝑏 ∈ ( ℤ≥ ‘ 1 ) ( 𝐹 ‘ 𝑏 ) ∈ ℝ+ ∧ Σ 𝑏 ∈ ( ℤ≥ ‘ 1 ) ( 𝐹 ‘ 𝑏 ) ≤ ( 2 · ( 2 ↑ - ( ! ‘ 1 ) ) ) ) ) |
38 |
37
|
simp1d |
⊢ ( 1 ∈ ℕ → seq 1 ( + , 𝐹 ) ∈ dom ⇝ ) |
39 |
35 38
|
mp1i |
⊢ ( 𝐴 ∈ ℕ → seq 1 ( + , 𝐹 ) ∈ dom ⇝ ) |
40 |
16 17 4 18 34 39
|
isumsplit |
⊢ ( 𝐴 ∈ ℕ → Σ 𝑏 ∈ ℕ ( 𝐹 ‘ 𝑏 ) = ( Σ 𝑏 ∈ ( 1 ... ( ( 𝐴 + 1 ) − 1 ) ) ( 𝐹 ‘ 𝑏 ) + Σ 𝑏 ∈ ( ℤ≥ ‘ ( 𝐴 + 1 ) ) ( 𝐹 ‘ 𝑏 ) ) ) |
41 |
|
oveq2 |
⊢ ( 𝑐 = 𝐴 → ( 1 ... 𝑐 ) = ( 1 ... 𝐴 ) ) |
42 |
41
|
sumeq1d |
⊢ ( 𝑐 = 𝐴 → Σ 𝑏 ∈ ( 1 ... 𝑐 ) ( 𝐹 ‘ 𝑏 ) = Σ 𝑏 ∈ ( 1 ... 𝐴 ) ( 𝐹 ‘ 𝑏 ) ) |
43 |
|
sumex |
⊢ Σ 𝑏 ∈ ( 1 ... 𝐴 ) ( 𝐹 ‘ 𝑏 ) ∈ V |
44 |
42 3 43
|
fvmpt |
⊢ ( 𝐴 ∈ ℕ → ( 𝐻 ‘ 𝐴 ) = Σ 𝑏 ∈ ( 1 ... 𝐴 ) ( 𝐹 ‘ 𝑏 ) ) |
45 |
44
|
oveq1d |
⊢ ( 𝐴 ∈ ℕ → ( ( 𝐻 ‘ 𝐴 ) + Σ 𝑏 ∈ ( ℤ≥ ‘ ( 𝐴 + 1 ) ) ( 𝐹 ‘ 𝑏 ) ) = ( Σ 𝑏 ∈ ( 1 ... 𝐴 ) ( 𝐹 ‘ 𝑏 ) + Σ 𝑏 ∈ ( ℤ≥ ‘ ( 𝐴 + 1 ) ) ( 𝐹 ‘ 𝑏 ) ) ) |
46 |
15 40 45
|
3eqtr4rd |
⊢ ( 𝐴 ∈ ℕ → ( ( 𝐻 ‘ 𝐴 ) + Σ 𝑏 ∈ ( ℤ≥ ‘ ( 𝐴 + 1 ) ) ( 𝐹 ‘ 𝑏 ) ) = Σ 𝑏 ∈ ℕ ( 𝐹 ‘ 𝑏 ) ) |
47 |
46 2
|
eqtr4di |
⊢ ( 𝐴 ∈ ℕ → ( ( 𝐻 ‘ 𝐴 ) + Σ 𝑏 ∈ ( ℤ≥ ‘ ( 𝐴 + 1 ) ) ( 𝐹 ‘ 𝑏 ) ) = 𝐿 ) |
48 |
1 2 3
|
aaliou3lem4 |
⊢ 𝐿 ∈ ℝ |
49 |
48
|
recni |
⊢ 𝐿 ∈ ℂ |
50 |
49
|
a1i |
⊢ ( 𝐴 ∈ ℕ → 𝐿 ∈ ℂ ) |
51 |
1 2 3
|
aaliou3lem5 |
⊢ ( 𝐴 ∈ ℕ → ( 𝐻 ‘ 𝐴 ) ∈ ℝ ) |
52 |
51
|
recnd |
⊢ ( 𝐴 ∈ ℕ → ( 𝐻 ‘ 𝐴 ) ∈ ℂ ) |
53 |
6
|
simp2d |
⊢ ( ( 𝐴 + 1 ) ∈ ℕ → Σ 𝑏 ∈ ( ℤ≥ ‘ ( 𝐴 + 1 ) ) ( 𝐹 ‘ 𝑏 ) ∈ ℝ+ ) |
54 |
4 53
|
syl |
⊢ ( 𝐴 ∈ ℕ → Σ 𝑏 ∈ ( ℤ≥ ‘ ( 𝐴 + 1 ) ) ( 𝐹 ‘ 𝑏 ) ∈ ℝ+ ) |
55 |
54
|
rpcnd |
⊢ ( 𝐴 ∈ ℕ → Σ 𝑏 ∈ ( ℤ≥ ‘ ( 𝐴 + 1 ) ) ( 𝐹 ‘ 𝑏 ) ∈ ℂ ) |
56 |
50 52 55
|
subaddd |
⊢ ( 𝐴 ∈ ℕ → ( ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) = Σ 𝑏 ∈ ( ℤ≥ ‘ ( 𝐴 + 1 ) ) ( 𝐹 ‘ 𝑏 ) ↔ ( ( 𝐻 ‘ 𝐴 ) + Σ 𝑏 ∈ ( ℤ≥ ‘ ( 𝐴 + 1 ) ) ( 𝐹 ‘ 𝑏 ) ) = 𝐿 ) ) |
57 |
47 56
|
mpbird |
⊢ ( 𝐴 ∈ ℕ → ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) = Σ 𝑏 ∈ ( ℤ≥ ‘ ( 𝐴 + 1 ) ) ( 𝐹 ‘ 𝑏 ) ) |
58 |
57
|
eqcomd |
⊢ ( 𝐴 ∈ ℕ → Σ 𝑏 ∈ ( ℤ≥ ‘ ( 𝐴 + 1 ) ) ( 𝐹 ‘ 𝑏 ) = ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ) |
59 |
|
eleq1 |
⊢ ( Σ 𝑏 ∈ ( ℤ≥ ‘ ( 𝐴 + 1 ) ) ( 𝐹 ‘ 𝑏 ) = ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) → ( Σ 𝑏 ∈ ( ℤ≥ ‘ ( 𝐴 + 1 ) ) ( 𝐹 ‘ 𝑏 ) ∈ ℝ+ ↔ ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ∈ ℝ+ ) ) |
60 |
|
breq1 |
⊢ ( Σ 𝑏 ∈ ( ℤ≥ ‘ ( 𝐴 + 1 ) ) ( 𝐹 ‘ 𝑏 ) = ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) → ( Σ 𝑏 ∈ ( ℤ≥ ‘ ( 𝐴 + 1 ) ) ( 𝐹 ‘ 𝑏 ) ≤ ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ↔ ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ≤ ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ) ) |
61 |
59 60
|
anbi12d |
⊢ ( Σ 𝑏 ∈ ( ℤ≥ ‘ ( 𝐴 + 1 ) ) ( 𝐹 ‘ 𝑏 ) = ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) → ( ( Σ 𝑏 ∈ ( ℤ≥ ‘ ( 𝐴 + 1 ) ) ( 𝐹 ‘ 𝑏 ) ∈ ℝ+ ∧ Σ 𝑏 ∈ ( ℤ≥ ‘ ( 𝐴 + 1 ) ) ( 𝐹 ‘ 𝑏 ) ≤ ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ) ↔ ( ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ∈ ℝ+ ∧ ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ≤ ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ) ) ) |
62 |
58 61
|
syl |
⊢ ( 𝐴 ∈ ℕ → ( ( Σ 𝑏 ∈ ( ℤ≥ ‘ ( 𝐴 + 1 ) ) ( 𝐹 ‘ 𝑏 ) ∈ ℝ+ ∧ Σ 𝑏 ∈ ( ℤ≥ ‘ ( 𝐴 + 1 ) ) ( 𝐹 ‘ 𝑏 ) ≤ ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ) ↔ ( ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ∈ ℝ+ ∧ ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ≤ ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ) ) ) |
63 |
51
|
adantr |
⊢ ( ( 𝐴 ∈ ℕ ∧ ( ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ∈ ℝ+ ∧ ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ≤ ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ) ) → ( 𝐻 ‘ 𝐴 ) ∈ ℝ ) |
64 |
|
simprl |
⊢ ( ( 𝐴 ∈ ℕ ∧ ( ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ∈ ℝ+ ∧ ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ≤ ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ) ) → ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ∈ ℝ+ ) |
65 |
|
difrp |
⊢ ( ( ( 𝐻 ‘ 𝐴 ) ∈ ℝ ∧ 𝐿 ∈ ℝ ) → ( ( 𝐻 ‘ 𝐴 ) < 𝐿 ↔ ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ∈ ℝ+ ) ) |
66 |
63 48 65
|
sylancl |
⊢ ( ( 𝐴 ∈ ℕ ∧ ( ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ∈ ℝ+ ∧ ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ≤ ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ) ) → ( ( 𝐻 ‘ 𝐴 ) < 𝐿 ↔ ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ∈ ℝ+ ) ) |
67 |
64 66
|
mpbird |
⊢ ( ( 𝐴 ∈ ℕ ∧ ( ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ∈ ℝ+ ∧ ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ≤ ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ) ) → ( 𝐻 ‘ 𝐴 ) < 𝐿 ) |
68 |
63 67
|
ltned |
⊢ ( ( 𝐴 ∈ ℕ ∧ ( ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ∈ ℝ+ ∧ ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ≤ ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ) ) → ( 𝐻 ‘ 𝐴 ) ≠ 𝐿 ) |
69 |
|
nnnn0 |
⊢ ( ( 𝐴 + 1 ) ∈ ℕ → ( 𝐴 + 1 ) ∈ ℕ0 ) |
70 |
|
faccl |
⊢ ( ( 𝐴 + 1 ) ∈ ℕ0 → ( ! ‘ ( 𝐴 + 1 ) ) ∈ ℕ ) |
71 |
4 69 70
|
3syl |
⊢ ( 𝐴 ∈ ℕ → ( ! ‘ ( 𝐴 + 1 ) ) ∈ ℕ ) |
72 |
71
|
nnzd |
⊢ ( 𝐴 ∈ ℕ → ( ! ‘ ( 𝐴 + 1 ) ) ∈ ℤ ) |
73 |
72
|
znegcld |
⊢ ( 𝐴 ∈ ℕ → - ( ! ‘ ( 𝐴 + 1 ) ) ∈ ℤ ) |
74 |
|
rpexpcl |
⊢ ( ( 2 ∈ ℝ+ ∧ - ( ! ‘ ( 𝐴 + 1 ) ) ∈ ℤ ) → ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ∈ ℝ+ ) |
75 |
24 73 74
|
sylancr |
⊢ ( 𝐴 ∈ ℕ → ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ∈ ℝ+ ) |
76 |
|
rpmulcl |
⊢ ( ( 2 ∈ ℝ+ ∧ ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ∈ ℝ+ ) → ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ∈ ℝ+ ) |
77 |
24 75 76
|
sylancr |
⊢ ( 𝐴 ∈ ℕ → ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ∈ ℝ+ ) |
78 |
77
|
adantr |
⊢ ( ( 𝐴 ∈ ℕ ∧ ( ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ∈ ℝ+ ∧ ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ≤ ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ) ) → ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ∈ ℝ+ ) |
79 |
78
|
rpred |
⊢ ( ( 𝐴 ∈ ℕ ∧ ( ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ∈ ℝ+ ∧ ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ≤ ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ) ) → ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ∈ ℝ ) |
80 |
63 79
|
resubcld |
⊢ ( ( 𝐴 ∈ ℕ ∧ ( ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ∈ ℝ+ ∧ ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ≤ ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ) ) → ( ( 𝐻 ‘ 𝐴 ) − ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ) ∈ ℝ ) |
81 |
48
|
a1i |
⊢ ( ( 𝐴 ∈ ℕ ∧ ( ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ∈ ℝ+ ∧ ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ≤ ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ) ) → 𝐿 ∈ ℝ ) |
82 |
63 78
|
ltsubrpd |
⊢ ( ( 𝐴 ∈ ℕ ∧ ( ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ∈ ℝ+ ∧ ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ≤ ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ) ) → ( ( 𝐻 ‘ 𝐴 ) − ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ) < ( 𝐻 ‘ 𝐴 ) ) |
83 |
80 63 81 82 67
|
lttrd |
⊢ ( ( 𝐴 ∈ ℕ ∧ ( ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ∈ ℝ+ ∧ ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ≤ ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ) ) → ( ( 𝐻 ‘ 𝐴 ) − ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ) < 𝐿 ) |
84 |
80 81 83
|
ltled |
⊢ ( ( 𝐴 ∈ ℕ ∧ ( ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ∈ ℝ+ ∧ ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ≤ ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ) ) → ( ( 𝐻 ‘ 𝐴 ) − ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ) ≤ 𝐿 ) |
85 |
|
simprr |
⊢ ( ( 𝐴 ∈ ℕ ∧ ( ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ∈ ℝ+ ∧ ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ≤ ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ) ) → ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ≤ ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ) |
86 |
81 63 79
|
lesubadd2d |
⊢ ( ( 𝐴 ∈ ℕ ∧ ( ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ∈ ℝ+ ∧ ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ≤ ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ) ) → ( ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ≤ ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ↔ 𝐿 ≤ ( ( 𝐻 ‘ 𝐴 ) + ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ) ) ) |
87 |
85 86
|
mpbid |
⊢ ( ( 𝐴 ∈ ℕ ∧ ( ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ∈ ℝ+ ∧ ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ≤ ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ) ) → 𝐿 ≤ ( ( 𝐻 ‘ 𝐴 ) + ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ) ) |
88 |
81 63 79
|
absdifled |
⊢ ( ( 𝐴 ∈ ℕ ∧ ( ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ∈ ℝ+ ∧ ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ≤ ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ) ) → ( ( abs ‘ ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ) ≤ ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ↔ ( ( ( 𝐻 ‘ 𝐴 ) − ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ) ≤ 𝐿 ∧ 𝐿 ≤ ( ( 𝐻 ‘ 𝐴 ) + ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ) ) ) ) |
89 |
84 87 88
|
mpbir2and |
⊢ ( ( 𝐴 ∈ ℕ ∧ ( ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ∈ ℝ+ ∧ ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ≤ ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ) ) → ( abs ‘ ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ) ≤ ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ) |
90 |
68 89
|
jca |
⊢ ( ( 𝐴 ∈ ℕ ∧ ( ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ∈ ℝ+ ∧ ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ≤ ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ) ) → ( ( 𝐻 ‘ 𝐴 ) ≠ 𝐿 ∧ ( abs ‘ ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ) ≤ ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ) ) |
91 |
90
|
ex |
⊢ ( 𝐴 ∈ ℕ → ( ( ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ∈ ℝ+ ∧ ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ≤ ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ) → ( ( 𝐻 ‘ 𝐴 ) ≠ 𝐿 ∧ ( abs ‘ ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ) ≤ ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ) ) ) |
92 |
62 91
|
sylbid |
⊢ ( 𝐴 ∈ ℕ → ( ( Σ 𝑏 ∈ ( ℤ≥ ‘ ( 𝐴 + 1 ) ) ( 𝐹 ‘ 𝑏 ) ∈ ℝ+ ∧ Σ 𝑏 ∈ ( ℤ≥ ‘ ( 𝐴 + 1 ) ) ( 𝐹 ‘ 𝑏 ) ≤ ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ) → ( ( 𝐻 ‘ 𝐴 ) ≠ 𝐿 ∧ ( abs ‘ ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ) ≤ ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ) ) ) |
93 |
8 92
|
mpd |
⊢ ( 𝐴 ∈ ℕ → ( ( 𝐻 ‘ 𝐴 ) ≠ 𝐿 ∧ ( abs ‘ ( 𝐿 − ( 𝐻 ‘ 𝐴 ) ) ) ≤ ( 2 · ( 2 ↑ - ( ! ‘ ( 𝐴 + 1 ) ) ) ) ) ) |