Step |
Hyp |
Ref |
Expression |
1 |
|
itg2i1fseq.1 |
⊢ ( 𝜑 → 𝐹 ∈ MblFn ) |
2 |
|
itg2i1fseq.2 |
⊢ ( 𝜑 → 𝐹 : ℝ ⟶ ( 0 [,) +∞ ) ) |
3 |
|
itg2i1fseq.3 |
⊢ ( 𝜑 → 𝑃 : ℕ ⟶ dom ∫1 ) |
4 |
|
itg2i1fseq.4 |
⊢ ( 𝜑 → ∀ 𝑛 ∈ ℕ ( 0𝑝 ∘r ≤ ( 𝑃 ‘ 𝑛 ) ∧ ( 𝑃 ‘ 𝑛 ) ∘r ≤ ( 𝑃 ‘ ( 𝑛 + 1 ) ) ) ) |
5 |
|
itg2i1fseq.5 |
⊢ ( 𝜑 → ∀ 𝑥 ∈ ℝ ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑥 ) ) ⇝ ( 𝐹 ‘ 𝑥 ) ) |
6 |
|
itg2i1fseq.6 |
⊢ 𝑆 = ( 𝑚 ∈ ℕ ↦ ( ∫1 ‘ ( 𝑃 ‘ 𝑚 ) ) ) |
7 |
|
fveq2 |
⊢ ( 𝑛 = 𝑚 → ( 𝑃 ‘ 𝑛 ) = ( 𝑃 ‘ 𝑚 ) ) |
8 |
7
|
fveq1d |
⊢ ( 𝑛 = 𝑚 → ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑥 ) = ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑥 ) ) |
9 |
8
|
cbvmptv |
⊢ ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑥 ) ) = ( 𝑚 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑥 ) ) |
10 |
|
fveq2 |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑥 ) = ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) ) |
11 |
10
|
mpteq2dv |
⊢ ( 𝑥 = 𝑦 → ( 𝑚 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑥 ) ) = ( 𝑚 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) ) ) |
12 |
9 11
|
syl5eq |
⊢ ( 𝑥 = 𝑦 → ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑥 ) ) = ( 𝑚 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) ) ) |
13 |
12
|
rneqd |
⊢ ( 𝑥 = 𝑦 → ran ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑥 ) ) = ran ( 𝑚 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) ) ) |
14 |
13
|
supeq1d |
⊢ ( 𝑥 = 𝑦 → sup ( ran ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑥 ) ) , ℝ , < ) = sup ( ran ( 𝑚 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) ) , ℝ , < ) ) |
15 |
14
|
cbvmptv |
⊢ ( 𝑥 ∈ ℝ ↦ sup ( ran ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑥 ) ) , ℝ , < ) ) = ( 𝑦 ∈ ℝ ↦ sup ( ran ( 𝑚 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) ) , ℝ , < ) ) |
16 |
3
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 𝑃 ‘ 𝑚 ) ∈ dom ∫1 ) |
17 |
|
i1fmbf |
⊢ ( ( 𝑃 ‘ 𝑚 ) ∈ dom ∫1 → ( 𝑃 ‘ 𝑚 ) ∈ MblFn ) |
18 |
16 17
|
syl |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 𝑃 ‘ 𝑚 ) ∈ MblFn ) |
19 |
|
i1ff |
⊢ ( ( 𝑃 ‘ 𝑚 ) ∈ dom ∫1 → ( 𝑃 ‘ 𝑚 ) : ℝ ⟶ ℝ ) |
20 |
16 19
|
syl |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 𝑃 ‘ 𝑚 ) : ℝ ⟶ ℝ ) |
21 |
7
|
breq2d |
⊢ ( 𝑛 = 𝑚 → ( 0𝑝 ∘r ≤ ( 𝑃 ‘ 𝑛 ) ↔ 0𝑝 ∘r ≤ ( 𝑃 ‘ 𝑚 ) ) ) |
22 |
|
fvoveq1 |
⊢ ( 𝑛 = 𝑚 → ( 𝑃 ‘ ( 𝑛 + 1 ) ) = ( 𝑃 ‘ ( 𝑚 + 1 ) ) ) |
23 |
7 22
|
breq12d |
⊢ ( 𝑛 = 𝑚 → ( ( 𝑃 ‘ 𝑛 ) ∘r ≤ ( 𝑃 ‘ ( 𝑛 + 1 ) ) ↔ ( 𝑃 ‘ 𝑚 ) ∘r ≤ ( 𝑃 ‘ ( 𝑚 + 1 ) ) ) ) |
24 |
21 23
|
anbi12d |
⊢ ( 𝑛 = 𝑚 → ( ( 0𝑝 ∘r ≤ ( 𝑃 ‘ 𝑛 ) ∧ ( 𝑃 ‘ 𝑛 ) ∘r ≤ ( 𝑃 ‘ ( 𝑛 + 1 ) ) ) ↔ ( 0𝑝 ∘r ≤ ( 𝑃 ‘ 𝑚 ) ∧ ( 𝑃 ‘ 𝑚 ) ∘r ≤ ( 𝑃 ‘ ( 𝑚 + 1 ) ) ) ) ) |
25 |
24
|
rspccva |
⊢ ( ( ∀ 𝑛 ∈ ℕ ( 0𝑝 ∘r ≤ ( 𝑃 ‘ 𝑛 ) ∧ ( 𝑃 ‘ 𝑛 ) ∘r ≤ ( 𝑃 ‘ ( 𝑛 + 1 ) ) ) ∧ 𝑚 ∈ ℕ ) → ( 0𝑝 ∘r ≤ ( 𝑃 ‘ 𝑚 ) ∧ ( 𝑃 ‘ 𝑚 ) ∘r ≤ ( 𝑃 ‘ ( 𝑚 + 1 ) ) ) ) |
26 |
4 25
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 0𝑝 ∘r ≤ ( 𝑃 ‘ 𝑚 ) ∧ ( 𝑃 ‘ 𝑚 ) ∘r ≤ ( 𝑃 ‘ ( 𝑚 + 1 ) ) ) ) |
27 |
26
|
simpld |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → 0𝑝 ∘r ≤ ( 𝑃 ‘ 𝑚 ) ) |
28 |
|
0plef |
⊢ ( ( 𝑃 ‘ 𝑚 ) : ℝ ⟶ ( 0 [,) +∞ ) ↔ ( ( 𝑃 ‘ 𝑚 ) : ℝ ⟶ ℝ ∧ 0𝑝 ∘r ≤ ( 𝑃 ‘ 𝑚 ) ) ) |
29 |
20 27 28
|
sylanbrc |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 𝑃 ‘ 𝑚 ) : ℝ ⟶ ( 0 [,) +∞ ) ) |
30 |
26
|
simprd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 𝑃 ‘ 𝑚 ) ∘r ≤ ( 𝑃 ‘ ( 𝑚 + 1 ) ) ) |
31 |
|
rge0ssre |
⊢ ( 0 [,) +∞ ) ⊆ ℝ |
32 |
2
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( 𝐹 ‘ 𝑦 ) ∈ ( 0 [,) +∞ ) ) |
33 |
31 32
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( 𝐹 ‘ 𝑦 ) ∈ ℝ ) |
34 |
1 2 3 4 5
|
itg2i1fseqle |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 𝑃 ‘ 𝑚 ) ∘r ≤ 𝐹 ) |
35 |
20
|
ffnd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 𝑃 ‘ 𝑚 ) Fn ℝ ) |
36 |
2
|
ffnd |
⊢ ( 𝜑 → 𝐹 Fn ℝ ) |
37 |
36
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → 𝐹 Fn ℝ ) |
38 |
|
reex |
⊢ ℝ ∈ V |
39 |
38
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ℝ ∈ V ) |
40 |
|
inidm |
⊢ ( ℝ ∩ ℝ ) = ℝ |
41 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) ∧ 𝑦 ∈ ℝ ) → ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) = ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) ) |
42 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) ∧ 𝑦 ∈ ℝ ) → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) ) |
43 |
35 37 39 39 40 41 42
|
ofrfval |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( ( 𝑃 ‘ 𝑚 ) ∘r ≤ 𝐹 ↔ ∀ 𝑦 ∈ ℝ ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) ≤ ( 𝐹 ‘ 𝑦 ) ) ) |
44 |
34 43
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ∀ 𝑦 ∈ ℝ ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) ≤ ( 𝐹 ‘ 𝑦 ) ) |
45 |
44
|
r19.21bi |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) ∧ 𝑦 ∈ ℝ ) → ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) ≤ ( 𝐹 ‘ 𝑦 ) ) |
46 |
45
|
an32s |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑚 ∈ ℕ ) → ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) ≤ ( 𝐹 ‘ 𝑦 ) ) |
47 |
46
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ∀ 𝑚 ∈ ℕ ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) ≤ ( 𝐹 ‘ 𝑦 ) ) |
48 |
|
brralrspcev |
⊢ ( ( ( 𝐹 ‘ 𝑦 ) ∈ ℝ ∧ ∀ 𝑚 ∈ ℕ ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) ≤ ( 𝐹 ‘ 𝑦 ) ) → ∃ 𝑧 ∈ ℝ ∀ 𝑚 ∈ ℕ ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) ≤ 𝑧 ) |
49 |
33 47 48
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ∃ 𝑧 ∈ ℝ ∀ 𝑚 ∈ ℕ ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) ≤ 𝑧 ) |
50 |
7
|
fveq2d |
⊢ ( 𝑛 = 𝑚 → ( ∫2 ‘ ( 𝑃 ‘ 𝑛 ) ) = ( ∫2 ‘ ( 𝑃 ‘ 𝑚 ) ) ) |
51 |
50
|
cbvmptv |
⊢ ( 𝑛 ∈ ℕ ↦ ( ∫2 ‘ ( 𝑃 ‘ 𝑛 ) ) ) = ( 𝑚 ∈ ℕ ↦ ( ∫2 ‘ ( 𝑃 ‘ 𝑚 ) ) ) |
52 |
51
|
rneqi |
⊢ ran ( 𝑛 ∈ ℕ ↦ ( ∫2 ‘ ( 𝑃 ‘ 𝑛 ) ) ) = ran ( 𝑚 ∈ ℕ ↦ ( ∫2 ‘ ( 𝑃 ‘ 𝑚 ) ) ) |
53 |
52
|
supeq1i |
⊢ sup ( ran ( 𝑛 ∈ ℕ ↦ ( ∫2 ‘ ( 𝑃 ‘ 𝑛 ) ) ) , ℝ* , < ) = sup ( ran ( 𝑚 ∈ ℕ ↦ ( ∫2 ‘ ( 𝑃 ‘ 𝑚 ) ) ) , ℝ* , < ) |
54 |
15 18 29 30 49 53
|
itg2mono |
⊢ ( 𝜑 → ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ sup ( ran ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑥 ) ) , ℝ , < ) ) ) = sup ( ran ( 𝑛 ∈ ℕ ↦ ( ∫2 ‘ ( 𝑃 ‘ 𝑛 ) ) ) , ℝ* , < ) ) |
55 |
2
|
feqmptd |
⊢ ( 𝜑 → 𝐹 = ( 𝑦 ∈ ℝ ↦ ( 𝐹 ‘ 𝑦 ) ) ) |
56 |
7
|
fveq1d |
⊢ ( 𝑛 = 𝑚 → ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑦 ) = ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) ) |
57 |
56
|
cbvmptv |
⊢ ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑦 ) ) = ( 𝑚 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) ) |
58 |
57
|
rneqi |
⊢ ran ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑦 ) ) = ran ( 𝑚 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) ) |
59 |
58
|
supeq1i |
⊢ sup ( ran ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑦 ) ) , ℝ , < ) = sup ( ran ( 𝑚 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) ) , ℝ , < ) |
60 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
61 |
|
1zzd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → 1 ∈ ℤ ) |
62 |
20
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) ∧ 𝑦 ∈ ℝ ) → ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) ∈ ℝ ) |
63 |
62
|
an32s |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑚 ∈ ℕ ) → ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) ∈ ℝ ) |
64 |
63 57
|
fmptd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑦 ) ) : ℕ ⟶ ℝ ) |
65 |
|
peano2nn |
⊢ ( 𝑚 ∈ ℕ → ( 𝑚 + 1 ) ∈ ℕ ) |
66 |
|
ffvelrn |
⊢ ( ( 𝑃 : ℕ ⟶ dom ∫1 ∧ ( 𝑚 + 1 ) ∈ ℕ ) → ( 𝑃 ‘ ( 𝑚 + 1 ) ) ∈ dom ∫1 ) |
67 |
3 65 66
|
syl2an |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 𝑃 ‘ ( 𝑚 + 1 ) ) ∈ dom ∫1 ) |
68 |
|
i1ff |
⊢ ( ( 𝑃 ‘ ( 𝑚 + 1 ) ) ∈ dom ∫1 → ( 𝑃 ‘ ( 𝑚 + 1 ) ) : ℝ ⟶ ℝ ) |
69 |
67 68
|
syl |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 𝑃 ‘ ( 𝑚 + 1 ) ) : ℝ ⟶ ℝ ) |
70 |
69
|
ffnd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 𝑃 ‘ ( 𝑚 + 1 ) ) Fn ℝ ) |
71 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) ∧ 𝑦 ∈ ℝ ) → ( ( 𝑃 ‘ ( 𝑚 + 1 ) ) ‘ 𝑦 ) = ( ( 𝑃 ‘ ( 𝑚 + 1 ) ) ‘ 𝑦 ) ) |
72 |
35 70 39 39 40 41 71
|
ofrfval |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( ( 𝑃 ‘ 𝑚 ) ∘r ≤ ( 𝑃 ‘ ( 𝑚 + 1 ) ) ↔ ∀ 𝑦 ∈ ℝ ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) ≤ ( ( 𝑃 ‘ ( 𝑚 + 1 ) ) ‘ 𝑦 ) ) ) |
73 |
30 72
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ∀ 𝑦 ∈ ℝ ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) ≤ ( ( 𝑃 ‘ ( 𝑚 + 1 ) ) ‘ 𝑦 ) ) |
74 |
73
|
r19.21bi |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) ∧ 𝑦 ∈ ℝ ) → ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) ≤ ( ( 𝑃 ‘ ( 𝑚 + 1 ) ) ‘ 𝑦 ) ) |
75 |
74
|
an32s |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑚 ∈ ℕ ) → ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) ≤ ( ( 𝑃 ‘ ( 𝑚 + 1 ) ) ‘ 𝑦 ) ) |
76 |
|
eqid |
⊢ ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑦 ) ) = ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑦 ) ) |
77 |
|
fvex |
⊢ ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) ∈ V |
78 |
56 76 77
|
fvmpt |
⊢ ( 𝑚 ∈ ℕ → ( ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑦 ) ) ‘ 𝑚 ) = ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) ) |
79 |
78
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑚 ∈ ℕ ) → ( ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑦 ) ) ‘ 𝑚 ) = ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) ) |
80 |
|
fveq2 |
⊢ ( 𝑛 = ( 𝑚 + 1 ) → ( 𝑃 ‘ 𝑛 ) = ( 𝑃 ‘ ( 𝑚 + 1 ) ) ) |
81 |
80
|
fveq1d |
⊢ ( 𝑛 = ( 𝑚 + 1 ) → ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑦 ) = ( ( 𝑃 ‘ ( 𝑚 + 1 ) ) ‘ 𝑦 ) ) |
82 |
|
fvex |
⊢ ( ( 𝑃 ‘ ( 𝑚 + 1 ) ) ‘ 𝑦 ) ∈ V |
83 |
81 76 82
|
fvmpt |
⊢ ( ( 𝑚 + 1 ) ∈ ℕ → ( ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑦 ) ) ‘ ( 𝑚 + 1 ) ) = ( ( 𝑃 ‘ ( 𝑚 + 1 ) ) ‘ 𝑦 ) ) |
84 |
65 83
|
syl |
⊢ ( 𝑚 ∈ ℕ → ( ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑦 ) ) ‘ ( 𝑚 + 1 ) ) = ( ( 𝑃 ‘ ( 𝑚 + 1 ) ) ‘ 𝑦 ) ) |
85 |
84
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑚 ∈ ℕ ) → ( ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑦 ) ) ‘ ( 𝑚 + 1 ) ) = ( ( 𝑃 ‘ ( 𝑚 + 1 ) ) ‘ 𝑦 ) ) |
86 |
75 79 85
|
3brtr4d |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑚 ∈ ℕ ) → ( ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑦 ) ) ‘ 𝑚 ) ≤ ( ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑦 ) ) ‘ ( 𝑚 + 1 ) ) ) |
87 |
78
|
breq1d |
⊢ ( 𝑚 ∈ ℕ → ( ( ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑦 ) ) ‘ 𝑚 ) ≤ 𝑧 ↔ ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) ≤ 𝑧 ) ) |
88 |
87
|
ralbiia |
⊢ ( ∀ 𝑚 ∈ ℕ ( ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑦 ) ) ‘ 𝑚 ) ≤ 𝑧 ↔ ∀ 𝑚 ∈ ℕ ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) ≤ 𝑧 ) |
89 |
88
|
rexbii |
⊢ ( ∃ 𝑧 ∈ ℝ ∀ 𝑚 ∈ ℕ ( ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑦 ) ) ‘ 𝑚 ) ≤ 𝑧 ↔ ∃ 𝑧 ∈ ℝ ∀ 𝑚 ∈ ℕ ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) ≤ 𝑧 ) |
90 |
49 89
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ∃ 𝑧 ∈ ℝ ∀ 𝑚 ∈ ℕ ( ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑦 ) ) ‘ 𝑚 ) ≤ 𝑧 ) |
91 |
60 61 64 86 90
|
climsup |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑦 ) ) ⇝ sup ( ran ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑦 ) ) , ℝ , < ) ) |
92 |
|
fveq2 |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑥 ) = ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑦 ) ) |
93 |
92
|
mpteq2dv |
⊢ ( 𝑥 = 𝑦 → ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑥 ) ) = ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑦 ) ) ) |
94 |
|
fveq2 |
⊢ ( 𝑥 = 𝑦 → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ) |
95 |
93 94
|
breq12d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑥 ) ) ⇝ ( 𝐹 ‘ 𝑥 ) ↔ ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑦 ) ) ⇝ ( 𝐹 ‘ 𝑦 ) ) ) |
96 |
95
|
rspccva |
⊢ ( ( ∀ 𝑥 ∈ ℝ ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑥 ) ) ⇝ ( 𝐹 ‘ 𝑥 ) ∧ 𝑦 ∈ ℝ ) → ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑦 ) ) ⇝ ( 𝐹 ‘ 𝑦 ) ) |
97 |
5 96
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑦 ) ) ⇝ ( 𝐹 ‘ 𝑦 ) ) |
98 |
|
climuni |
⊢ ( ( ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑦 ) ) ⇝ sup ( ran ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑦 ) ) , ℝ , < ) ∧ ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑦 ) ) ⇝ ( 𝐹 ‘ 𝑦 ) ) → sup ( ran ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑦 ) ) , ℝ , < ) = ( 𝐹 ‘ 𝑦 ) ) |
99 |
91 97 98
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → sup ( ran ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑦 ) ) , ℝ , < ) = ( 𝐹 ‘ 𝑦 ) ) |
100 |
59 99
|
eqtr3id |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → sup ( ran ( 𝑚 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) ) , ℝ , < ) = ( 𝐹 ‘ 𝑦 ) ) |
101 |
100
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑦 ∈ ℝ ↦ sup ( ran ( 𝑚 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) ) , ℝ , < ) ) = ( 𝑦 ∈ ℝ ↦ ( 𝐹 ‘ 𝑦 ) ) ) |
102 |
55 101
|
eqtr4d |
⊢ ( 𝜑 → 𝐹 = ( 𝑦 ∈ ℝ ↦ sup ( ran ( 𝑚 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) ) , ℝ , < ) ) ) |
103 |
102 15
|
eqtr4di |
⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ ℝ ↦ sup ( ran ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑥 ) ) , ℝ , < ) ) ) |
104 |
103
|
fveq2d |
⊢ ( 𝜑 → ( ∫2 ‘ 𝐹 ) = ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ sup ( ran ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑥 ) ) , ℝ , < ) ) ) ) |
105 |
|
itg2itg1 |
⊢ ( ( ( 𝑃 ‘ 𝑚 ) ∈ dom ∫1 ∧ 0𝑝 ∘r ≤ ( 𝑃 ‘ 𝑚 ) ) → ( ∫2 ‘ ( 𝑃 ‘ 𝑚 ) ) = ( ∫1 ‘ ( 𝑃 ‘ 𝑚 ) ) ) |
106 |
16 27 105
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( ∫2 ‘ ( 𝑃 ‘ 𝑚 ) ) = ( ∫1 ‘ ( 𝑃 ‘ 𝑚 ) ) ) |
107 |
106
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑚 ∈ ℕ ↦ ( ∫2 ‘ ( 𝑃 ‘ 𝑚 ) ) ) = ( 𝑚 ∈ ℕ ↦ ( ∫1 ‘ ( 𝑃 ‘ 𝑚 ) ) ) ) |
108 |
6 107
|
eqtr4id |
⊢ ( 𝜑 → 𝑆 = ( 𝑚 ∈ ℕ ↦ ( ∫2 ‘ ( 𝑃 ‘ 𝑚 ) ) ) ) |
109 |
108 51
|
eqtr4di |
⊢ ( 𝜑 → 𝑆 = ( 𝑛 ∈ ℕ ↦ ( ∫2 ‘ ( 𝑃 ‘ 𝑛 ) ) ) ) |
110 |
109
|
rneqd |
⊢ ( 𝜑 → ran 𝑆 = ran ( 𝑛 ∈ ℕ ↦ ( ∫2 ‘ ( 𝑃 ‘ 𝑛 ) ) ) ) |
111 |
110
|
supeq1d |
⊢ ( 𝜑 → sup ( ran 𝑆 , ℝ* , < ) = sup ( ran ( 𝑛 ∈ ℕ ↦ ( ∫2 ‘ ( 𝑃 ‘ 𝑛 ) ) ) , ℝ* , < ) ) |
112 |
54 104 111
|
3eqtr4d |
⊢ ( 𝜑 → ( ∫2 ‘ 𝐹 ) = sup ( ran 𝑆 , ℝ* , < ) ) |