Step |
Hyp |
Ref |
Expression |
1 |
|
wallispilem3.1 |
⊢ 𝐼 = ( 𝑛 ∈ ℕ0 ↦ ∫ ( 0 (,) π ) ( ( sin ‘ 𝑥 ) ↑ 𝑛 ) d 𝑥 ) |
2 |
|
breq2 |
⊢ ( 𝑤 = 0 → ( 𝑚 ≤ 𝑤 ↔ 𝑚 ≤ 0 ) ) |
3 |
2
|
imbi1d |
⊢ ( 𝑤 = 0 → ( ( 𝑚 ≤ 𝑤 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ↔ ( 𝑚 ≤ 0 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) ) |
4 |
3
|
ralbidv |
⊢ ( 𝑤 = 0 → ( ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ 𝑤 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ↔ ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ 0 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) ) |
5 |
|
breq2 |
⊢ ( 𝑤 = 𝑦 → ( 𝑚 ≤ 𝑤 ↔ 𝑚 ≤ 𝑦 ) ) |
6 |
5
|
imbi1d |
⊢ ( 𝑤 = 𝑦 → ( ( 𝑚 ≤ 𝑤 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ↔ ( 𝑚 ≤ 𝑦 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) ) |
7 |
6
|
ralbidv |
⊢ ( 𝑤 = 𝑦 → ( ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ 𝑤 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ↔ ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ 𝑦 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) ) |
8 |
|
breq2 |
⊢ ( 𝑤 = ( 𝑦 + 1 ) → ( 𝑚 ≤ 𝑤 ↔ 𝑚 ≤ ( 𝑦 + 1 ) ) ) |
9 |
8
|
imbi1d |
⊢ ( 𝑤 = ( 𝑦 + 1 ) → ( ( 𝑚 ≤ 𝑤 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ↔ ( 𝑚 ≤ ( 𝑦 + 1 ) → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) ) |
10 |
9
|
ralbidv |
⊢ ( 𝑤 = ( 𝑦 + 1 ) → ( ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ 𝑤 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ↔ ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ ( 𝑦 + 1 ) → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) ) |
11 |
|
breq2 |
⊢ ( 𝑤 = 𝑁 → ( 𝑚 ≤ 𝑤 ↔ 𝑚 ≤ 𝑁 ) ) |
12 |
11
|
imbi1d |
⊢ ( 𝑤 = 𝑁 → ( ( 𝑚 ≤ 𝑤 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ↔ ( 𝑚 ≤ 𝑁 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) ) |
13 |
12
|
ralbidv |
⊢ ( 𝑤 = 𝑁 → ( ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ 𝑤 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ↔ ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ 𝑁 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) ) |
14 |
|
simpr |
⊢ ( ( 𝑚 ∈ ℕ0 ∧ 𝑚 ≤ 0 ) → 𝑚 ≤ 0 ) |
15 |
|
nn0ge0 |
⊢ ( 𝑚 ∈ ℕ0 → 0 ≤ 𝑚 ) |
16 |
15
|
adantr |
⊢ ( ( 𝑚 ∈ ℕ0 ∧ 𝑚 ≤ 0 ) → 0 ≤ 𝑚 ) |
17 |
|
nn0re |
⊢ ( 𝑚 ∈ ℕ0 → 𝑚 ∈ ℝ ) |
18 |
17
|
adantr |
⊢ ( ( 𝑚 ∈ ℕ0 ∧ 𝑚 ≤ 0 ) → 𝑚 ∈ ℝ ) |
19 |
|
0red |
⊢ ( ( 𝑚 ∈ ℕ0 ∧ 𝑚 ≤ 0 ) → 0 ∈ ℝ ) |
20 |
18 19
|
letri3d |
⊢ ( ( 𝑚 ∈ ℕ0 ∧ 𝑚 ≤ 0 ) → ( 𝑚 = 0 ↔ ( 𝑚 ≤ 0 ∧ 0 ≤ 𝑚 ) ) ) |
21 |
14 16 20
|
mpbir2and |
⊢ ( ( 𝑚 ∈ ℕ0 ∧ 𝑚 ≤ 0 ) → 𝑚 = 0 ) |
22 |
21
|
fveq2d |
⊢ ( ( 𝑚 ∈ ℕ0 ∧ 𝑚 ≤ 0 ) → ( 𝐼 ‘ 𝑚 ) = ( 𝐼 ‘ 0 ) ) |
23 |
1
|
wallispilem2 |
⊢ ( ( 𝐼 ‘ 0 ) = π ∧ ( 𝐼 ‘ 1 ) = 2 ∧ ( 𝑚 ∈ ( ℤ≥ ‘ 2 ) → ( 𝐼 ‘ 𝑚 ) = ( ( ( 𝑚 − 1 ) / 𝑚 ) · ( 𝐼 ‘ ( 𝑚 − 2 ) ) ) ) ) |
24 |
23
|
simp1i |
⊢ ( 𝐼 ‘ 0 ) = π |
25 |
|
pirp |
⊢ π ∈ ℝ+ |
26 |
24 25
|
eqeltri |
⊢ ( 𝐼 ‘ 0 ) ∈ ℝ+ |
27 |
22 26
|
eqeltrdi |
⊢ ( ( 𝑚 ∈ ℕ0 ∧ 𝑚 ≤ 0 ) → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) |
28 |
27
|
ex |
⊢ ( 𝑚 ∈ ℕ0 → ( 𝑚 ≤ 0 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) |
29 |
28
|
rgen |
⊢ ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ 0 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) |
30 |
|
nfv |
⊢ Ⅎ 𝑚 𝑦 ∈ ℕ0 |
31 |
|
nfra1 |
⊢ Ⅎ 𝑚 ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ 𝑦 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) |
32 |
30 31
|
nfan |
⊢ Ⅎ 𝑚 ( 𝑦 ∈ ℕ0 ∧ ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ 𝑦 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) |
33 |
|
simpllr |
⊢ ( ( ( ( 𝑦 ∈ ℕ0 ∧ ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ 𝑦 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) ∧ 𝑚 ∈ ℕ0 ) ∧ 𝑚 ≤ ( 𝑦 + 1 ) ) → ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ 𝑦 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) |
34 |
|
simplr |
⊢ ( ( ( ( 𝑦 ∈ ℕ0 ∧ ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ 𝑦 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) ∧ 𝑚 ∈ ℕ0 ) ∧ 𝑚 ≤ ( 𝑦 + 1 ) ) → 𝑚 ∈ ℕ0 ) |
35 |
|
rsp |
⊢ ( ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ 𝑦 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) → ( 𝑚 ∈ ℕ0 → ( 𝑚 ≤ 𝑦 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) ) |
36 |
33 34 35
|
sylc |
⊢ ( ( ( ( 𝑦 ∈ ℕ0 ∧ ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ 𝑦 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) ∧ 𝑚 ∈ ℕ0 ) ∧ 𝑚 ≤ ( 𝑦 + 1 ) ) → ( 𝑚 ≤ 𝑦 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) |
37 |
|
fveq2 |
⊢ ( 𝑚 = 1 → ( 𝐼 ‘ 𝑚 ) = ( 𝐼 ‘ 1 ) ) |
38 |
23
|
simp2i |
⊢ ( 𝐼 ‘ 1 ) = 2 |
39 |
|
2rp |
⊢ 2 ∈ ℝ+ |
40 |
38 39
|
eqeltri |
⊢ ( 𝐼 ‘ 1 ) ∈ ℝ+ |
41 |
37 40
|
eqeltrdi |
⊢ ( 𝑚 = 1 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) |
42 |
41
|
a1i |
⊢ ( ( ( ( 𝑦 ∈ ℕ0 ∧ ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ 𝑦 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) ∧ 𝑚 ∈ ℕ0 ) ∧ 𝑚 = ( 𝑦 + 1 ) ) → ( 𝑚 = 1 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) |
43 |
23
|
simp3i |
⊢ ( 𝑚 ∈ ( ℤ≥ ‘ 2 ) → ( 𝐼 ‘ 𝑚 ) = ( ( ( 𝑚 − 1 ) / 𝑚 ) · ( 𝐼 ‘ ( 𝑚 − 2 ) ) ) ) |
44 |
43
|
adantl |
⊢ ( ( ( ( 𝑦 ∈ ℕ0 ∧ ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ 𝑦 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) ∧ 𝑚 = ( 𝑦 + 1 ) ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 2 ) ) → ( 𝐼 ‘ 𝑚 ) = ( ( ( 𝑚 − 1 ) / 𝑚 ) · ( 𝐼 ‘ ( 𝑚 − 2 ) ) ) ) |
45 |
|
eluz2nn |
⊢ ( 𝑚 ∈ ( ℤ≥ ‘ 2 ) → 𝑚 ∈ ℕ ) |
46 |
|
nnre |
⊢ ( 𝑚 ∈ ℕ → 𝑚 ∈ ℝ ) |
47 |
|
1red |
⊢ ( 𝑚 ∈ ℕ → 1 ∈ ℝ ) |
48 |
46 47
|
resubcld |
⊢ ( 𝑚 ∈ ℕ → ( 𝑚 − 1 ) ∈ ℝ ) |
49 |
45 48
|
syl |
⊢ ( 𝑚 ∈ ( ℤ≥ ‘ 2 ) → ( 𝑚 − 1 ) ∈ ℝ ) |
50 |
|
1m1e0 |
⊢ ( 1 − 1 ) = 0 |
51 |
|
1red |
⊢ ( 𝑚 ∈ ( ℤ≥ ‘ 2 ) → 1 ∈ ℝ ) |
52 |
|
eluzelre |
⊢ ( 𝑚 ∈ ( ℤ≥ ‘ 2 ) → 𝑚 ∈ ℝ ) |
53 |
|
eluz2b2 |
⊢ ( 𝑚 ∈ ( ℤ≥ ‘ 2 ) ↔ ( 𝑚 ∈ ℕ ∧ 1 < 𝑚 ) ) |
54 |
53
|
simprbi |
⊢ ( 𝑚 ∈ ( ℤ≥ ‘ 2 ) → 1 < 𝑚 ) |
55 |
51 52 51 54
|
ltsub1dd |
⊢ ( 𝑚 ∈ ( ℤ≥ ‘ 2 ) → ( 1 − 1 ) < ( 𝑚 − 1 ) ) |
56 |
50 55
|
eqbrtrrid |
⊢ ( 𝑚 ∈ ( ℤ≥ ‘ 2 ) → 0 < ( 𝑚 − 1 ) ) |
57 |
49 56
|
elrpd |
⊢ ( 𝑚 ∈ ( ℤ≥ ‘ 2 ) → ( 𝑚 − 1 ) ∈ ℝ+ ) |
58 |
45
|
nnrpd |
⊢ ( 𝑚 ∈ ( ℤ≥ ‘ 2 ) → 𝑚 ∈ ℝ+ ) |
59 |
57 58
|
rpdivcld |
⊢ ( 𝑚 ∈ ( ℤ≥ ‘ 2 ) → ( ( 𝑚 − 1 ) / 𝑚 ) ∈ ℝ+ ) |
60 |
59
|
adantl |
⊢ ( ( ( ( 𝑦 ∈ ℕ0 ∧ ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ 𝑦 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) ∧ 𝑚 = ( 𝑦 + 1 ) ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 2 ) ) → ( ( 𝑚 − 1 ) / 𝑚 ) ∈ ℝ+ ) |
61 |
|
breq1 |
⊢ ( 𝑚 = 𝑘 → ( 𝑚 ≤ 𝑦 ↔ 𝑘 ≤ 𝑦 ) ) |
62 |
|
fveq2 |
⊢ ( 𝑚 = 𝑘 → ( 𝐼 ‘ 𝑚 ) = ( 𝐼 ‘ 𝑘 ) ) |
63 |
62
|
eleq1d |
⊢ ( 𝑚 = 𝑘 → ( ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ↔ ( 𝐼 ‘ 𝑘 ) ∈ ℝ+ ) ) |
64 |
61 63
|
imbi12d |
⊢ ( 𝑚 = 𝑘 → ( ( 𝑚 ≤ 𝑦 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ↔ ( 𝑘 ≤ 𝑦 → ( 𝐼 ‘ 𝑘 ) ∈ ℝ+ ) ) ) |
65 |
64
|
cbvralvw |
⊢ ( ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ 𝑦 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ↔ ∀ 𝑘 ∈ ℕ0 ( 𝑘 ≤ 𝑦 → ( 𝐼 ‘ 𝑘 ) ∈ ℝ+ ) ) |
66 |
65
|
biimpi |
⊢ ( ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ 𝑦 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) → ∀ 𝑘 ∈ ℕ0 ( 𝑘 ≤ 𝑦 → ( 𝐼 ‘ 𝑘 ) ∈ ℝ+ ) ) |
67 |
66
|
ad3antlr |
⊢ ( ( ( ( 𝑦 ∈ ℕ0 ∧ ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ 𝑦 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) ∧ 𝑚 = ( 𝑦 + 1 ) ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 2 ) ) → ∀ 𝑘 ∈ ℕ0 ( 𝑘 ≤ 𝑦 → ( 𝐼 ‘ 𝑘 ) ∈ ℝ+ ) ) |
68 |
|
uznn0sub |
⊢ ( 𝑚 ∈ ( ℤ≥ ‘ 2 ) → ( 𝑚 − 2 ) ∈ ℕ0 ) |
69 |
68
|
adantl |
⊢ ( ( ( ( 𝑦 ∈ ℕ0 ∧ ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ 𝑦 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) ∧ 𝑚 = ( 𝑦 + 1 ) ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 2 ) ) → ( 𝑚 − 2 ) ∈ ℕ0 ) |
70 |
67 69
|
jca |
⊢ ( ( ( ( 𝑦 ∈ ℕ0 ∧ ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ 𝑦 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) ∧ 𝑚 = ( 𝑦 + 1 ) ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 2 ) ) → ( ∀ 𝑘 ∈ ℕ0 ( 𝑘 ≤ 𝑦 → ( 𝐼 ‘ 𝑘 ) ∈ ℝ+ ) ∧ ( 𝑚 − 2 ) ∈ ℕ0 ) ) |
71 |
|
simplll |
⊢ ( ( ( ( 𝑦 ∈ ℕ0 ∧ ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ 𝑦 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) ∧ 𝑚 = ( 𝑦 + 1 ) ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 2 ) ) → 𝑦 ∈ ℕ0 ) |
72 |
|
simplr |
⊢ ( ( ( ( 𝑦 ∈ ℕ0 ∧ ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ 𝑦 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) ∧ 𝑚 = ( 𝑦 + 1 ) ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 2 ) ) → 𝑚 = ( 𝑦 + 1 ) ) |
73 |
|
simpr |
⊢ ( ( ( ( 𝑦 ∈ ℕ0 ∧ ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ 𝑦 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) ∧ 𝑚 = ( 𝑦 + 1 ) ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 2 ) ) → 𝑚 ∈ ( ℤ≥ ‘ 2 ) ) |
74 |
|
simp2 |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 = ( 𝑦 + 1 ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 2 ) ) → 𝑚 = ( 𝑦 + 1 ) ) |
75 |
74
|
oveq1d |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 = ( 𝑦 + 1 ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 2 ) ) → ( 𝑚 − 2 ) = ( ( 𝑦 + 1 ) − 2 ) ) |
76 |
|
nn0re |
⊢ ( 𝑦 ∈ ℕ0 → 𝑦 ∈ ℝ ) |
77 |
76
|
3ad2ant1 |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 = ( 𝑦 + 1 ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 2 ) ) → 𝑦 ∈ ℝ ) |
78 |
77
|
recnd |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 = ( 𝑦 + 1 ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 2 ) ) → 𝑦 ∈ ℂ ) |
79 |
|
df-2 |
⊢ 2 = ( 1 + 1 ) |
80 |
79
|
a1i |
⊢ ( 𝑦 ∈ ℂ → 2 = ( 1 + 1 ) ) |
81 |
80
|
oveq2d |
⊢ ( 𝑦 ∈ ℂ → ( ( 𝑦 + 1 ) − 2 ) = ( ( 𝑦 + 1 ) − ( 1 + 1 ) ) ) |
82 |
|
id |
⊢ ( 𝑦 ∈ ℂ → 𝑦 ∈ ℂ ) |
83 |
|
1cnd |
⊢ ( 𝑦 ∈ ℂ → 1 ∈ ℂ ) |
84 |
82 83 83
|
pnpcan2d |
⊢ ( 𝑦 ∈ ℂ → ( ( 𝑦 + 1 ) − ( 1 + 1 ) ) = ( 𝑦 − 1 ) ) |
85 |
81 84
|
eqtrd |
⊢ ( 𝑦 ∈ ℂ → ( ( 𝑦 + 1 ) − 2 ) = ( 𝑦 − 1 ) ) |
86 |
78 85
|
syl |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 = ( 𝑦 + 1 ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 2 ) ) → ( ( 𝑦 + 1 ) − 2 ) = ( 𝑦 − 1 ) ) |
87 |
75 86
|
eqtrd |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 = ( 𝑦 + 1 ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 2 ) ) → ( 𝑚 − 2 ) = ( 𝑦 − 1 ) ) |
88 |
77
|
lem1d |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 = ( 𝑦 + 1 ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 2 ) ) → ( 𝑦 − 1 ) ≤ 𝑦 ) |
89 |
87 88
|
eqbrtrd |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 = ( 𝑦 + 1 ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 2 ) ) → ( 𝑚 − 2 ) ≤ 𝑦 ) |
90 |
71 72 73 89
|
syl3anc |
⊢ ( ( ( ( 𝑦 ∈ ℕ0 ∧ ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ 𝑦 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) ∧ 𝑚 = ( 𝑦 + 1 ) ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 2 ) ) → ( 𝑚 − 2 ) ≤ 𝑦 ) |
91 |
|
breq1 |
⊢ ( 𝑘 = ( 𝑚 − 2 ) → ( 𝑘 ≤ 𝑦 ↔ ( 𝑚 − 2 ) ≤ 𝑦 ) ) |
92 |
|
fveq2 |
⊢ ( 𝑘 = ( 𝑚 − 2 ) → ( 𝐼 ‘ 𝑘 ) = ( 𝐼 ‘ ( 𝑚 − 2 ) ) ) |
93 |
92
|
eleq1d |
⊢ ( 𝑘 = ( 𝑚 − 2 ) → ( ( 𝐼 ‘ 𝑘 ) ∈ ℝ+ ↔ ( 𝐼 ‘ ( 𝑚 − 2 ) ) ∈ ℝ+ ) ) |
94 |
91 93
|
imbi12d |
⊢ ( 𝑘 = ( 𝑚 − 2 ) → ( ( 𝑘 ≤ 𝑦 → ( 𝐼 ‘ 𝑘 ) ∈ ℝ+ ) ↔ ( ( 𝑚 − 2 ) ≤ 𝑦 → ( 𝐼 ‘ ( 𝑚 − 2 ) ) ∈ ℝ+ ) ) ) |
95 |
94
|
rspccva |
⊢ ( ( ∀ 𝑘 ∈ ℕ0 ( 𝑘 ≤ 𝑦 → ( 𝐼 ‘ 𝑘 ) ∈ ℝ+ ) ∧ ( 𝑚 − 2 ) ∈ ℕ0 ) → ( ( 𝑚 − 2 ) ≤ 𝑦 → ( 𝐼 ‘ ( 𝑚 − 2 ) ) ∈ ℝ+ ) ) |
96 |
70 90 95
|
sylc |
⊢ ( ( ( ( 𝑦 ∈ ℕ0 ∧ ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ 𝑦 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) ∧ 𝑚 = ( 𝑦 + 1 ) ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 2 ) ) → ( 𝐼 ‘ ( 𝑚 − 2 ) ) ∈ ℝ+ ) |
97 |
60 96
|
rpmulcld |
⊢ ( ( ( ( 𝑦 ∈ ℕ0 ∧ ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ 𝑦 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) ∧ 𝑚 = ( 𝑦 + 1 ) ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 2 ) ) → ( ( ( 𝑚 − 1 ) / 𝑚 ) · ( 𝐼 ‘ ( 𝑚 − 2 ) ) ) ∈ ℝ+ ) |
98 |
44 97
|
eqeltrd |
⊢ ( ( ( ( 𝑦 ∈ ℕ0 ∧ ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ 𝑦 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) ∧ 𝑚 = ( 𝑦 + 1 ) ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 2 ) ) → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) |
99 |
98
|
adantllr |
⊢ ( ( ( ( ( 𝑦 ∈ ℕ0 ∧ ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ 𝑦 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) ∧ 𝑚 ∈ ℕ0 ) ∧ 𝑚 = ( 𝑦 + 1 ) ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 2 ) ) → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) |
100 |
99
|
ex |
⊢ ( ( ( ( 𝑦 ∈ ℕ0 ∧ ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ 𝑦 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) ∧ 𝑚 ∈ ℕ0 ) ∧ 𝑚 = ( 𝑦 + 1 ) ) → ( 𝑚 ∈ ( ℤ≥ ‘ 2 ) → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) |
101 |
|
simplll |
⊢ ( ( ( ( 𝑦 ∈ ℕ0 ∧ ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ 𝑦 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) ∧ 𝑚 ∈ ℕ0 ) ∧ 𝑚 = ( 𝑦 + 1 ) ) → 𝑦 ∈ ℕ0 ) |
102 |
|
simplr |
⊢ ( ( ( ( 𝑦 ∈ ℕ0 ∧ ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ 𝑦 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) ∧ 𝑚 ∈ ℕ0 ) ∧ 𝑚 = ( 𝑦 + 1 ) ) → 𝑚 ∈ ℕ0 ) |
103 |
|
simpr |
⊢ ( ( ( ( 𝑦 ∈ ℕ0 ∧ ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ 𝑦 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) ∧ 𝑚 ∈ ℕ0 ) ∧ 𝑚 = ( 𝑦 + 1 ) ) → 𝑚 = ( 𝑦 + 1 ) ) |
104 |
|
simp3 |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0 ∧ 𝑚 = ( 𝑦 + 1 ) ) → 𝑚 = ( 𝑦 + 1 ) ) |
105 |
|
nn0p1nn |
⊢ ( 𝑦 ∈ ℕ0 → ( 𝑦 + 1 ) ∈ ℕ ) |
106 |
105
|
3ad2ant1 |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0 ∧ 𝑚 = ( 𝑦 + 1 ) ) → ( 𝑦 + 1 ) ∈ ℕ ) |
107 |
104 106
|
eqeltrd |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0 ∧ 𝑚 = ( 𝑦 + 1 ) ) → 𝑚 ∈ ℕ ) |
108 |
|
elnnuz |
⊢ ( 𝑚 ∈ ℕ ↔ 𝑚 ∈ ( ℤ≥ ‘ 1 ) ) |
109 |
107 108
|
sylib |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0 ∧ 𝑚 = ( 𝑦 + 1 ) ) → 𝑚 ∈ ( ℤ≥ ‘ 1 ) ) |
110 |
|
uzp1 |
⊢ ( 𝑚 ∈ ( ℤ≥ ‘ 1 ) → ( 𝑚 = 1 ∨ 𝑚 ∈ ( ℤ≥ ‘ ( 1 + 1 ) ) ) ) |
111 |
|
1p1e2 |
⊢ ( 1 + 1 ) = 2 |
112 |
111
|
fveq2i |
⊢ ( ℤ≥ ‘ ( 1 + 1 ) ) = ( ℤ≥ ‘ 2 ) |
113 |
112
|
eleq2i |
⊢ ( 𝑚 ∈ ( ℤ≥ ‘ ( 1 + 1 ) ) ↔ 𝑚 ∈ ( ℤ≥ ‘ 2 ) ) |
114 |
113
|
orbi2i |
⊢ ( ( 𝑚 = 1 ∨ 𝑚 ∈ ( ℤ≥ ‘ ( 1 + 1 ) ) ) ↔ ( 𝑚 = 1 ∨ 𝑚 ∈ ( ℤ≥ ‘ 2 ) ) ) |
115 |
110 114
|
sylib |
⊢ ( 𝑚 ∈ ( ℤ≥ ‘ 1 ) → ( 𝑚 = 1 ∨ 𝑚 ∈ ( ℤ≥ ‘ 2 ) ) ) |
116 |
109 115
|
syl |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0 ∧ 𝑚 = ( 𝑦 + 1 ) ) → ( 𝑚 = 1 ∨ 𝑚 ∈ ( ℤ≥ ‘ 2 ) ) ) |
117 |
101 102 103 116
|
syl3anc |
⊢ ( ( ( ( 𝑦 ∈ ℕ0 ∧ ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ 𝑦 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) ∧ 𝑚 ∈ ℕ0 ) ∧ 𝑚 = ( 𝑦 + 1 ) ) → ( 𝑚 = 1 ∨ 𝑚 ∈ ( ℤ≥ ‘ 2 ) ) ) |
118 |
42 100 117
|
mpjaod |
⊢ ( ( ( ( 𝑦 ∈ ℕ0 ∧ ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ 𝑦 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) ∧ 𝑚 ∈ ℕ0 ) ∧ 𝑚 = ( 𝑦 + 1 ) ) → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) |
119 |
118
|
adantlr |
⊢ ( ( ( ( ( 𝑦 ∈ ℕ0 ∧ ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ 𝑦 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) ∧ 𝑚 ∈ ℕ0 ) ∧ 𝑚 ≤ ( 𝑦 + 1 ) ) ∧ 𝑚 = ( 𝑦 + 1 ) ) → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) |
120 |
119
|
ex |
⊢ ( ( ( ( 𝑦 ∈ ℕ0 ∧ ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ 𝑦 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) ∧ 𝑚 ∈ ℕ0 ) ∧ 𝑚 ≤ ( 𝑦 + 1 ) ) → ( 𝑚 = ( 𝑦 + 1 ) → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) |
121 |
|
simplll |
⊢ ( ( ( ( 𝑦 ∈ ℕ0 ∧ ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ 𝑦 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) ∧ 𝑚 ∈ ℕ0 ) ∧ 𝑚 ≤ ( 𝑦 + 1 ) ) → 𝑦 ∈ ℕ0 ) |
122 |
|
simpr |
⊢ ( ( ( ( 𝑦 ∈ ℕ0 ∧ ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ 𝑦 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) ∧ 𝑚 ∈ ℕ0 ) ∧ 𝑚 ≤ ( 𝑦 + 1 ) ) → 𝑚 ≤ ( 𝑦 + 1 ) ) |
123 |
|
simpl1 |
⊢ ( ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0 ∧ 𝑚 ≤ ( 𝑦 + 1 ) ) ∧ 𝑚 < ( 𝑦 + 1 ) ) → 𝑦 ∈ ℕ0 ) |
124 |
|
simpl2 |
⊢ ( ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0 ∧ 𝑚 ≤ ( 𝑦 + 1 ) ) ∧ 𝑚 < ( 𝑦 + 1 ) ) → 𝑚 ∈ ℕ0 ) |
125 |
|
simpr |
⊢ ( ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0 ∧ 𝑚 ≤ ( 𝑦 + 1 ) ) ∧ 𝑚 < ( 𝑦 + 1 ) ) → 𝑚 < ( 𝑦 + 1 ) ) |
126 |
|
simpr |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 = 0 ) → 𝑚 = 0 ) |
127 |
|
nn0ge0 |
⊢ ( 𝑦 ∈ ℕ0 → 0 ≤ 𝑦 ) |
128 |
127
|
adantr |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 = 0 ) → 0 ≤ 𝑦 ) |
129 |
126 128
|
eqbrtrd |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 = 0 ) → 𝑚 ≤ 𝑦 ) |
130 |
129
|
3ad2antl1 |
⊢ ( ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0 ∧ 𝑚 < ( 𝑦 + 1 ) ) ∧ 𝑚 = 0 ) → 𝑚 ≤ 𝑦 ) |
131 |
|
simpl1 |
⊢ ( ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0 ∧ 𝑚 < ( 𝑦 + 1 ) ) ∧ 𝑚 ∈ ℕ ) → 𝑦 ∈ ℕ0 ) |
132 |
|
simpr |
⊢ ( ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0 ∧ 𝑚 < ( 𝑦 + 1 ) ) ∧ 𝑚 ∈ ℕ ) → 𝑚 ∈ ℕ ) |
133 |
|
simpl3 |
⊢ ( ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0 ∧ 𝑚 < ( 𝑦 + 1 ) ) ∧ 𝑚 ∈ ℕ ) → 𝑚 < ( 𝑦 + 1 ) ) |
134 |
|
simp3 |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ℕ ∧ 𝑚 < ( 𝑦 + 1 ) ) → 𝑚 < ( 𝑦 + 1 ) ) |
135 |
|
simp2 |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ℕ ∧ 𝑚 < ( 𝑦 + 1 ) ) → 𝑚 ∈ ℕ ) |
136 |
|
simp1 |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ℕ ∧ 𝑚 < ( 𝑦 + 1 ) ) → 𝑦 ∈ ℕ0 ) |
137 |
|
0red |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ℕ ∧ 𝑚 < ( 𝑦 + 1 ) ) → 0 ∈ ℝ ) |
138 |
48
|
3ad2ant2 |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ℕ ∧ 𝑚 < ( 𝑦 + 1 ) ) → ( 𝑚 − 1 ) ∈ ℝ ) |
139 |
76
|
3ad2ant1 |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ℕ ∧ 𝑚 < ( 𝑦 + 1 ) ) → 𝑦 ∈ ℝ ) |
140 |
|
nnm1ge0 |
⊢ ( 𝑚 ∈ ℕ → 0 ≤ ( 𝑚 − 1 ) ) |
141 |
140
|
3ad2ant2 |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ℕ ∧ 𝑚 < ( 𝑦 + 1 ) ) → 0 ≤ ( 𝑚 − 1 ) ) |
142 |
46
|
3ad2ant2 |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ℕ ∧ 𝑚 < ( 𝑦 + 1 ) ) → 𝑚 ∈ ℝ ) |
143 |
|
1red |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ℕ ∧ 𝑚 < ( 𝑦 + 1 ) ) → 1 ∈ ℝ ) |
144 |
142 143 139
|
ltsubaddd |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ℕ ∧ 𝑚 < ( 𝑦 + 1 ) ) → ( ( 𝑚 − 1 ) < 𝑦 ↔ 𝑚 < ( 𝑦 + 1 ) ) ) |
145 |
134 144
|
mpbird |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ℕ ∧ 𝑚 < ( 𝑦 + 1 ) ) → ( 𝑚 − 1 ) < 𝑦 ) |
146 |
137 138 139 141 145
|
lelttrd |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ℕ ∧ 𝑚 < ( 𝑦 + 1 ) ) → 0 < 𝑦 ) |
147 |
146
|
gt0ne0d |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ℕ ∧ 𝑚 < ( 𝑦 + 1 ) ) → 𝑦 ≠ 0 ) |
148 |
|
elnnne0 |
⊢ ( 𝑦 ∈ ℕ ↔ ( 𝑦 ∈ ℕ0 ∧ 𝑦 ≠ 0 ) ) |
149 |
136 147 148
|
sylanbrc |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ℕ ∧ 𝑚 < ( 𝑦 + 1 ) ) → 𝑦 ∈ ℕ ) |
150 |
|
nnleltp1 |
⊢ ( ( 𝑚 ∈ ℕ ∧ 𝑦 ∈ ℕ ) → ( 𝑚 ≤ 𝑦 ↔ 𝑚 < ( 𝑦 + 1 ) ) ) |
151 |
135 149 150
|
syl2anc |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ℕ ∧ 𝑚 < ( 𝑦 + 1 ) ) → ( 𝑚 ≤ 𝑦 ↔ 𝑚 < ( 𝑦 + 1 ) ) ) |
152 |
134 151
|
mpbird |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ℕ ∧ 𝑚 < ( 𝑦 + 1 ) ) → 𝑚 ≤ 𝑦 ) |
153 |
131 132 133 152
|
syl3anc |
⊢ ( ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0 ∧ 𝑚 < ( 𝑦 + 1 ) ) ∧ 𝑚 ∈ ℕ ) → 𝑚 ≤ 𝑦 ) |
154 |
|
elnn0 |
⊢ ( 𝑚 ∈ ℕ0 ↔ ( 𝑚 ∈ ℕ ∨ 𝑚 = 0 ) ) |
155 |
154
|
biimpi |
⊢ ( 𝑚 ∈ ℕ0 → ( 𝑚 ∈ ℕ ∨ 𝑚 = 0 ) ) |
156 |
155
|
orcomd |
⊢ ( 𝑚 ∈ ℕ0 → ( 𝑚 = 0 ∨ 𝑚 ∈ ℕ ) ) |
157 |
156
|
3ad2ant2 |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0 ∧ 𝑚 < ( 𝑦 + 1 ) ) → ( 𝑚 = 0 ∨ 𝑚 ∈ ℕ ) ) |
158 |
130 153 157
|
mpjaodan |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0 ∧ 𝑚 < ( 𝑦 + 1 ) ) → 𝑚 ≤ 𝑦 ) |
159 |
158
|
orcd |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0 ∧ 𝑚 < ( 𝑦 + 1 ) ) → ( 𝑚 ≤ 𝑦 ∨ 𝑚 = ( 𝑦 + 1 ) ) ) |
160 |
123 124 125 159
|
syl3anc |
⊢ ( ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0 ∧ 𝑚 ≤ ( 𝑦 + 1 ) ) ∧ 𝑚 < ( 𝑦 + 1 ) ) → ( 𝑚 ≤ 𝑦 ∨ 𝑚 = ( 𝑦 + 1 ) ) ) |
161 |
|
simpr |
⊢ ( ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0 ∧ 𝑚 ≤ ( 𝑦 + 1 ) ) ∧ 𝑚 = ( 𝑦 + 1 ) ) → 𝑚 = ( 𝑦 + 1 ) ) |
162 |
161
|
olcd |
⊢ ( ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0 ∧ 𝑚 ≤ ( 𝑦 + 1 ) ) ∧ 𝑚 = ( 𝑦 + 1 ) ) → ( 𝑚 ≤ 𝑦 ∨ 𝑚 = ( 𝑦 + 1 ) ) ) |
163 |
|
simp3 |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0 ∧ 𝑚 ≤ ( 𝑦 + 1 ) ) → 𝑚 ≤ ( 𝑦 + 1 ) ) |
164 |
17
|
3ad2ant2 |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0 ∧ 𝑚 ≤ ( 𝑦 + 1 ) ) → 𝑚 ∈ ℝ ) |
165 |
76
|
3ad2ant1 |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0 ∧ 𝑚 ≤ ( 𝑦 + 1 ) ) → 𝑦 ∈ ℝ ) |
166 |
|
1red |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0 ∧ 𝑚 ≤ ( 𝑦 + 1 ) ) → 1 ∈ ℝ ) |
167 |
165 166
|
readdcld |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0 ∧ 𝑚 ≤ ( 𝑦 + 1 ) ) → ( 𝑦 + 1 ) ∈ ℝ ) |
168 |
164 167
|
leloed |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0 ∧ 𝑚 ≤ ( 𝑦 + 1 ) ) → ( 𝑚 ≤ ( 𝑦 + 1 ) ↔ ( 𝑚 < ( 𝑦 + 1 ) ∨ 𝑚 = ( 𝑦 + 1 ) ) ) ) |
169 |
163 168
|
mpbid |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0 ∧ 𝑚 ≤ ( 𝑦 + 1 ) ) → ( 𝑚 < ( 𝑦 + 1 ) ∨ 𝑚 = ( 𝑦 + 1 ) ) ) |
170 |
160 162 169
|
mpjaodan |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0 ∧ 𝑚 ≤ ( 𝑦 + 1 ) ) → ( 𝑚 ≤ 𝑦 ∨ 𝑚 = ( 𝑦 + 1 ) ) ) |
171 |
121 34 122 170
|
syl3anc |
⊢ ( ( ( ( 𝑦 ∈ ℕ0 ∧ ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ 𝑦 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) ∧ 𝑚 ∈ ℕ0 ) ∧ 𝑚 ≤ ( 𝑦 + 1 ) ) → ( 𝑚 ≤ 𝑦 ∨ 𝑚 = ( 𝑦 + 1 ) ) ) |
172 |
36 120 171
|
mpjaod |
⊢ ( ( ( ( 𝑦 ∈ ℕ0 ∧ ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ 𝑦 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) ∧ 𝑚 ∈ ℕ0 ) ∧ 𝑚 ≤ ( 𝑦 + 1 ) ) → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) |
173 |
172
|
exp31 |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ 𝑦 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) → ( 𝑚 ∈ ℕ0 → ( 𝑚 ≤ ( 𝑦 + 1 ) → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) ) |
174 |
32 173
|
ralrimi |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ 𝑦 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) → ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ ( 𝑦 + 1 ) → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) |
175 |
174
|
ex |
⊢ ( 𝑦 ∈ ℕ0 → ( ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ 𝑦 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) → ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ ( 𝑦 + 1 ) → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) ) |
176 |
4 7 10 13 29 175
|
nn0ind |
⊢ ( 𝑁 ∈ ℕ0 → ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ 𝑁 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ) |
177 |
176
|
ancri |
⊢ ( 𝑁 ∈ ℕ0 → ( ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ 𝑁 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ∧ 𝑁 ∈ ℕ0 ) ) |
178 |
|
nn0re |
⊢ ( 𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ ) |
179 |
178
|
leidd |
⊢ ( 𝑁 ∈ ℕ0 → 𝑁 ≤ 𝑁 ) |
180 |
|
breq1 |
⊢ ( 𝑚 = 𝑁 → ( 𝑚 ≤ 𝑁 ↔ 𝑁 ≤ 𝑁 ) ) |
181 |
|
fveq2 |
⊢ ( 𝑚 = 𝑁 → ( 𝐼 ‘ 𝑚 ) = ( 𝐼 ‘ 𝑁 ) ) |
182 |
181
|
eleq1d |
⊢ ( 𝑚 = 𝑁 → ( ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ↔ ( 𝐼 ‘ 𝑁 ) ∈ ℝ+ ) ) |
183 |
180 182
|
imbi12d |
⊢ ( 𝑚 = 𝑁 → ( ( 𝑚 ≤ 𝑁 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ↔ ( 𝑁 ≤ 𝑁 → ( 𝐼 ‘ 𝑁 ) ∈ ℝ+ ) ) ) |
184 |
183
|
rspccva |
⊢ ( ( ∀ 𝑚 ∈ ℕ0 ( 𝑚 ≤ 𝑁 → ( 𝐼 ‘ 𝑚 ) ∈ ℝ+ ) ∧ 𝑁 ∈ ℕ0 ) → ( 𝑁 ≤ 𝑁 → ( 𝐼 ‘ 𝑁 ) ∈ ℝ+ ) ) |
185 |
177 179 184
|
sylc |
⊢ ( 𝑁 ∈ ℕ0 → ( 𝐼 ‘ 𝑁 ) ∈ ℝ+ ) |