Step |
Hyp |
Ref |
Expression |
1 |
|
itg2add.f1 |
⊢ ( 𝜑 → 𝐹 ∈ MblFn ) |
2 |
|
itg2add.f2 |
⊢ ( 𝜑 → 𝐹 : ℝ ⟶ ( 0 [,) +∞ ) ) |
3 |
|
itg2add.f3 |
⊢ ( 𝜑 → ( ∫2 ‘ 𝐹 ) ∈ ℝ ) |
4 |
|
itg2add.g1 |
⊢ ( 𝜑 → 𝐺 ∈ MblFn ) |
5 |
|
itg2add.g2 |
⊢ ( 𝜑 → 𝐺 : ℝ ⟶ ( 0 [,) +∞ ) ) |
6 |
|
itg2add.g3 |
⊢ ( 𝜑 → ( ∫2 ‘ 𝐺 ) ∈ ℝ ) |
7 |
|
itg2add.p1 |
⊢ ( 𝜑 → 𝑃 : ℕ ⟶ dom ∫1 ) |
8 |
|
itg2add.p2 |
⊢ ( 𝜑 → ∀ 𝑛 ∈ ℕ ( 0𝑝 ∘r ≤ ( 𝑃 ‘ 𝑛 ) ∧ ( 𝑃 ‘ 𝑛 ) ∘r ≤ ( 𝑃 ‘ ( 𝑛 + 1 ) ) ) ) |
9 |
|
itg2add.p3 |
⊢ ( 𝜑 → ∀ 𝑥 ∈ ℝ ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑥 ) ) ⇝ ( 𝐹 ‘ 𝑥 ) ) |
10 |
|
itg2add.q1 |
⊢ ( 𝜑 → 𝑄 : ℕ ⟶ dom ∫1 ) |
11 |
|
itg2add.q2 |
⊢ ( 𝜑 → ∀ 𝑛 ∈ ℕ ( 0𝑝 ∘r ≤ ( 𝑄 ‘ 𝑛 ) ∧ ( 𝑄 ‘ 𝑛 ) ∘r ≤ ( 𝑄 ‘ ( 𝑛 + 1 ) ) ) ) |
12 |
|
itg2add.q3 |
⊢ ( 𝜑 → ∀ 𝑥 ∈ ℝ ( 𝑛 ∈ ℕ ↦ ( ( 𝑄 ‘ 𝑛 ) ‘ 𝑥 ) ) ⇝ ( 𝐺 ‘ 𝑥 ) ) |
13 |
1 4
|
mbfadd |
⊢ ( 𝜑 → ( 𝐹 ∘f + 𝐺 ) ∈ MblFn ) |
14 |
|
ge0addcl |
⊢ ( ( 𝑦 ∈ ( 0 [,) +∞ ) ∧ 𝑧 ∈ ( 0 [,) +∞ ) ) → ( 𝑦 + 𝑧 ) ∈ ( 0 [,) +∞ ) ) |
15 |
14
|
adantl |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ( 0 [,) +∞ ) ∧ 𝑧 ∈ ( 0 [,) +∞ ) ) ) → ( 𝑦 + 𝑧 ) ∈ ( 0 [,) +∞ ) ) |
16 |
|
reex |
⊢ ℝ ∈ V |
17 |
16
|
a1i |
⊢ ( 𝜑 → ℝ ∈ V ) |
18 |
|
inidm |
⊢ ( ℝ ∩ ℝ ) = ℝ |
19 |
15 2 5 17 17 18
|
off |
⊢ ( 𝜑 → ( 𝐹 ∘f + 𝐺 ) : ℝ ⟶ ( 0 [,) +∞ ) ) |
20 |
|
simpl |
⊢ ( ( 𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1 ) → 𝑓 ∈ dom ∫1 ) |
21 |
|
simpr |
⊢ ( ( 𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1 ) → 𝑔 ∈ dom ∫1 ) |
22 |
20 21
|
i1fadd |
⊢ ( ( 𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1 ) → ( 𝑓 ∘f + 𝑔 ) ∈ dom ∫1 ) |
23 |
22
|
adantl |
⊢ ( ( 𝜑 ∧ ( 𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1 ) ) → ( 𝑓 ∘f + 𝑔 ) ∈ dom ∫1 ) |
24 |
|
nnex |
⊢ ℕ ∈ V |
25 |
24
|
a1i |
⊢ ( 𝜑 → ℕ ∈ V ) |
26 |
|
inidm |
⊢ ( ℕ ∩ ℕ ) = ℕ |
27 |
23 7 10 25 25 26
|
off |
⊢ ( 𝜑 → ( 𝑃 ∘f ∘f + 𝑄 ) : ℕ ⟶ dom ∫1 ) |
28 |
|
ge0addcl |
⊢ ( ( 𝑓 ∈ ( 0 [,) +∞ ) ∧ 𝑔 ∈ ( 0 [,) +∞ ) ) → ( 𝑓 + 𝑔 ) ∈ ( 0 [,) +∞ ) ) |
29 |
28
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) ∧ ( 𝑓 ∈ ( 0 [,) +∞ ) ∧ 𝑔 ∈ ( 0 [,) +∞ ) ) ) → ( 𝑓 + 𝑔 ) ∈ ( 0 [,) +∞ ) ) |
30 |
7
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 𝑃 ‘ 𝑚 ) ∈ dom ∫1 ) |
31 |
|
fveq2 |
⊢ ( 𝑛 = 𝑚 → ( 𝑃 ‘ 𝑛 ) = ( 𝑃 ‘ 𝑚 ) ) |
32 |
31
|
breq2d |
⊢ ( 𝑛 = 𝑚 → ( 0𝑝 ∘r ≤ ( 𝑃 ‘ 𝑛 ) ↔ 0𝑝 ∘r ≤ ( 𝑃 ‘ 𝑚 ) ) ) |
33 |
|
fvoveq1 |
⊢ ( 𝑛 = 𝑚 → ( 𝑃 ‘ ( 𝑛 + 1 ) ) = ( 𝑃 ‘ ( 𝑚 + 1 ) ) ) |
34 |
31 33
|
breq12d |
⊢ ( 𝑛 = 𝑚 → ( ( 𝑃 ‘ 𝑛 ) ∘r ≤ ( 𝑃 ‘ ( 𝑛 + 1 ) ) ↔ ( 𝑃 ‘ 𝑚 ) ∘r ≤ ( 𝑃 ‘ ( 𝑚 + 1 ) ) ) ) |
35 |
32 34
|
anbi12d |
⊢ ( 𝑛 = 𝑚 → ( ( 0𝑝 ∘r ≤ ( 𝑃 ‘ 𝑛 ) ∧ ( 𝑃 ‘ 𝑛 ) ∘r ≤ ( 𝑃 ‘ ( 𝑛 + 1 ) ) ) ↔ ( 0𝑝 ∘r ≤ ( 𝑃 ‘ 𝑚 ) ∧ ( 𝑃 ‘ 𝑚 ) ∘r ≤ ( 𝑃 ‘ ( 𝑚 + 1 ) ) ) ) ) |
36 |
35
|
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 ) ) ) ) |
37 |
8 36
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 0𝑝 ∘r ≤ ( 𝑃 ‘ 𝑚 ) ∧ ( 𝑃 ‘ 𝑚 ) ∘r ≤ ( 𝑃 ‘ ( 𝑚 + 1 ) ) ) ) |
38 |
37
|
simpld |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → 0𝑝 ∘r ≤ ( 𝑃 ‘ 𝑚 ) ) |
39 |
|
breq2 |
⊢ ( 𝑓 = ( 𝑃 ‘ 𝑚 ) → ( 0𝑝 ∘r ≤ 𝑓 ↔ 0𝑝 ∘r ≤ ( 𝑃 ‘ 𝑚 ) ) ) |
40 |
|
feq1 |
⊢ ( 𝑓 = ( 𝑃 ‘ 𝑚 ) → ( 𝑓 : ℝ ⟶ ( 0 [,) +∞ ) ↔ ( 𝑃 ‘ 𝑚 ) : ℝ ⟶ ( 0 [,) +∞ ) ) ) |
41 |
39 40
|
imbi12d |
⊢ ( 𝑓 = ( 𝑃 ‘ 𝑚 ) → ( ( 0𝑝 ∘r ≤ 𝑓 → 𝑓 : ℝ ⟶ ( 0 [,) +∞ ) ) ↔ ( 0𝑝 ∘r ≤ ( 𝑃 ‘ 𝑚 ) → ( 𝑃 ‘ 𝑚 ) : ℝ ⟶ ( 0 [,) +∞ ) ) ) ) |
42 |
|
i1ff |
⊢ ( 𝑓 ∈ dom ∫1 → 𝑓 : ℝ ⟶ ℝ ) |
43 |
42
|
ffnd |
⊢ ( 𝑓 ∈ dom ∫1 → 𝑓 Fn ℝ ) |
44 |
43
|
adantr |
⊢ ( ( 𝑓 ∈ dom ∫1 ∧ 0𝑝 ∘r ≤ 𝑓 ) → 𝑓 Fn ℝ ) |
45 |
|
0cn |
⊢ 0 ∈ ℂ |
46 |
|
fnconstg |
⊢ ( 0 ∈ ℂ → ( ℂ × { 0 } ) Fn ℂ ) |
47 |
45 46
|
ax-mp |
⊢ ( ℂ × { 0 } ) Fn ℂ |
48 |
|
df-0p |
⊢ 0𝑝 = ( ℂ × { 0 } ) |
49 |
48
|
fneq1i |
⊢ ( 0𝑝 Fn ℂ ↔ ( ℂ × { 0 } ) Fn ℂ ) |
50 |
47 49
|
mpbir |
⊢ 0𝑝 Fn ℂ |
51 |
50
|
a1i |
⊢ ( 𝑓 ∈ dom ∫1 → 0𝑝 Fn ℂ ) |
52 |
|
cnex |
⊢ ℂ ∈ V |
53 |
52
|
a1i |
⊢ ( 𝑓 ∈ dom ∫1 → ℂ ∈ V ) |
54 |
16
|
a1i |
⊢ ( 𝑓 ∈ dom ∫1 → ℝ ∈ V ) |
55 |
|
ax-resscn |
⊢ ℝ ⊆ ℂ |
56 |
|
sseqin2 |
⊢ ( ℝ ⊆ ℂ ↔ ( ℂ ∩ ℝ ) = ℝ ) |
57 |
55 56
|
mpbi |
⊢ ( ℂ ∩ ℝ ) = ℝ |
58 |
|
0pval |
⊢ ( 𝑥 ∈ ℂ → ( 0𝑝 ‘ 𝑥 ) = 0 ) |
59 |
58
|
adantl |
⊢ ( ( 𝑓 ∈ dom ∫1 ∧ 𝑥 ∈ ℂ ) → ( 0𝑝 ‘ 𝑥 ) = 0 ) |
60 |
|
eqidd |
⊢ ( ( 𝑓 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ ) → ( 𝑓 ‘ 𝑥 ) = ( 𝑓 ‘ 𝑥 ) ) |
61 |
51 43 53 54 57 59 60
|
ofrfval |
⊢ ( 𝑓 ∈ dom ∫1 → ( 0𝑝 ∘r ≤ 𝑓 ↔ ∀ 𝑥 ∈ ℝ 0 ≤ ( 𝑓 ‘ 𝑥 ) ) ) |
62 |
61
|
biimpa |
⊢ ( ( 𝑓 ∈ dom ∫1 ∧ 0𝑝 ∘r ≤ 𝑓 ) → ∀ 𝑥 ∈ ℝ 0 ≤ ( 𝑓 ‘ 𝑥 ) ) |
63 |
42
|
ffvelrnda |
⊢ ( ( 𝑓 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ ) → ( 𝑓 ‘ 𝑥 ) ∈ ℝ ) |
64 |
|
elrege0 |
⊢ ( ( 𝑓 ‘ 𝑥 ) ∈ ( 0 [,) +∞ ) ↔ ( ( 𝑓 ‘ 𝑥 ) ∈ ℝ ∧ 0 ≤ ( 𝑓 ‘ 𝑥 ) ) ) |
65 |
64
|
simplbi2 |
⊢ ( ( 𝑓 ‘ 𝑥 ) ∈ ℝ → ( 0 ≤ ( 𝑓 ‘ 𝑥 ) → ( 𝑓 ‘ 𝑥 ) ∈ ( 0 [,) +∞ ) ) ) |
66 |
63 65
|
syl |
⊢ ( ( 𝑓 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ ) → ( 0 ≤ ( 𝑓 ‘ 𝑥 ) → ( 𝑓 ‘ 𝑥 ) ∈ ( 0 [,) +∞ ) ) ) |
67 |
66
|
ralimdva |
⊢ ( 𝑓 ∈ dom ∫1 → ( ∀ 𝑥 ∈ ℝ 0 ≤ ( 𝑓 ‘ 𝑥 ) → ∀ 𝑥 ∈ ℝ ( 𝑓 ‘ 𝑥 ) ∈ ( 0 [,) +∞ ) ) ) |
68 |
67
|
imp |
⊢ ( ( 𝑓 ∈ dom ∫1 ∧ ∀ 𝑥 ∈ ℝ 0 ≤ ( 𝑓 ‘ 𝑥 ) ) → ∀ 𝑥 ∈ ℝ ( 𝑓 ‘ 𝑥 ) ∈ ( 0 [,) +∞ ) ) |
69 |
62 68
|
syldan |
⊢ ( ( 𝑓 ∈ dom ∫1 ∧ 0𝑝 ∘r ≤ 𝑓 ) → ∀ 𝑥 ∈ ℝ ( 𝑓 ‘ 𝑥 ) ∈ ( 0 [,) +∞ ) ) |
70 |
|
ffnfv |
⊢ ( 𝑓 : ℝ ⟶ ( 0 [,) +∞ ) ↔ ( 𝑓 Fn ℝ ∧ ∀ 𝑥 ∈ ℝ ( 𝑓 ‘ 𝑥 ) ∈ ( 0 [,) +∞ ) ) ) |
71 |
44 69 70
|
sylanbrc |
⊢ ( ( 𝑓 ∈ dom ∫1 ∧ 0𝑝 ∘r ≤ 𝑓 ) → 𝑓 : ℝ ⟶ ( 0 [,) +∞ ) ) |
72 |
71
|
ex |
⊢ ( 𝑓 ∈ dom ∫1 → ( 0𝑝 ∘r ≤ 𝑓 → 𝑓 : ℝ ⟶ ( 0 [,) +∞ ) ) ) |
73 |
41 72
|
vtoclga |
⊢ ( ( 𝑃 ‘ 𝑚 ) ∈ dom ∫1 → ( 0𝑝 ∘r ≤ ( 𝑃 ‘ 𝑚 ) → ( 𝑃 ‘ 𝑚 ) : ℝ ⟶ ( 0 [,) +∞ ) ) ) |
74 |
30 38 73
|
sylc |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 𝑃 ‘ 𝑚 ) : ℝ ⟶ ( 0 [,) +∞ ) ) |
75 |
10
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 𝑄 ‘ 𝑚 ) ∈ dom ∫1 ) |
76 |
|
fveq2 |
⊢ ( 𝑛 = 𝑚 → ( 𝑄 ‘ 𝑛 ) = ( 𝑄 ‘ 𝑚 ) ) |
77 |
76
|
breq2d |
⊢ ( 𝑛 = 𝑚 → ( 0𝑝 ∘r ≤ ( 𝑄 ‘ 𝑛 ) ↔ 0𝑝 ∘r ≤ ( 𝑄 ‘ 𝑚 ) ) ) |
78 |
|
fvoveq1 |
⊢ ( 𝑛 = 𝑚 → ( 𝑄 ‘ ( 𝑛 + 1 ) ) = ( 𝑄 ‘ ( 𝑚 + 1 ) ) ) |
79 |
76 78
|
breq12d |
⊢ ( 𝑛 = 𝑚 → ( ( 𝑄 ‘ 𝑛 ) ∘r ≤ ( 𝑄 ‘ ( 𝑛 + 1 ) ) ↔ ( 𝑄 ‘ 𝑚 ) ∘r ≤ ( 𝑄 ‘ ( 𝑚 + 1 ) ) ) ) |
80 |
77 79
|
anbi12d |
⊢ ( 𝑛 = 𝑚 → ( ( 0𝑝 ∘r ≤ ( 𝑄 ‘ 𝑛 ) ∧ ( 𝑄 ‘ 𝑛 ) ∘r ≤ ( 𝑄 ‘ ( 𝑛 + 1 ) ) ) ↔ ( 0𝑝 ∘r ≤ ( 𝑄 ‘ 𝑚 ) ∧ ( 𝑄 ‘ 𝑚 ) ∘r ≤ ( 𝑄 ‘ ( 𝑚 + 1 ) ) ) ) ) |
81 |
80
|
rspccva |
⊢ ( ( ∀ 𝑛 ∈ ℕ ( 0𝑝 ∘r ≤ ( 𝑄 ‘ 𝑛 ) ∧ ( 𝑄 ‘ 𝑛 ) ∘r ≤ ( 𝑄 ‘ ( 𝑛 + 1 ) ) ) ∧ 𝑚 ∈ ℕ ) → ( 0𝑝 ∘r ≤ ( 𝑄 ‘ 𝑚 ) ∧ ( 𝑄 ‘ 𝑚 ) ∘r ≤ ( 𝑄 ‘ ( 𝑚 + 1 ) ) ) ) |
82 |
11 81
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 0𝑝 ∘r ≤ ( 𝑄 ‘ 𝑚 ) ∧ ( 𝑄 ‘ 𝑚 ) ∘r ≤ ( 𝑄 ‘ ( 𝑚 + 1 ) ) ) ) |
83 |
82
|
simpld |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → 0𝑝 ∘r ≤ ( 𝑄 ‘ 𝑚 ) ) |
84 |
|
breq2 |
⊢ ( 𝑓 = ( 𝑄 ‘ 𝑚 ) → ( 0𝑝 ∘r ≤ 𝑓 ↔ 0𝑝 ∘r ≤ ( 𝑄 ‘ 𝑚 ) ) ) |
85 |
|
feq1 |
⊢ ( 𝑓 = ( 𝑄 ‘ 𝑚 ) → ( 𝑓 : ℝ ⟶ ( 0 [,) +∞ ) ↔ ( 𝑄 ‘ 𝑚 ) : ℝ ⟶ ( 0 [,) +∞ ) ) ) |
86 |
84 85
|
imbi12d |
⊢ ( 𝑓 = ( 𝑄 ‘ 𝑚 ) → ( ( 0𝑝 ∘r ≤ 𝑓 → 𝑓 : ℝ ⟶ ( 0 [,) +∞ ) ) ↔ ( 0𝑝 ∘r ≤ ( 𝑄 ‘ 𝑚 ) → ( 𝑄 ‘ 𝑚 ) : ℝ ⟶ ( 0 [,) +∞ ) ) ) ) |
87 |
86 72
|
vtoclga |
⊢ ( ( 𝑄 ‘ 𝑚 ) ∈ dom ∫1 → ( 0𝑝 ∘r ≤ ( 𝑄 ‘ 𝑚 ) → ( 𝑄 ‘ 𝑚 ) : ℝ ⟶ ( 0 [,) +∞ ) ) ) |
88 |
75 83 87
|
sylc |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 𝑄 ‘ 𝑚 ) : ℝ ⟶ ( 0 [,) +∞ ) ) |
89 |
16
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ℝ ∈ V ) |
90 |
29 74 88 89 89 18
|
off |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( ( 𝑃 ‘ 𝑚 ) ∘f + ( 𝑄 ‘ 𝑚 ) ) : ℝ ⟶ ( 0 [,) +∞ ) ) |
91 |
|
0plef |
⊢ ( ( ( 𝑃 ‘ 𝑚 ) ∘f + ( 𝑄 ‘ 𝑚 ) ) : ℝ ⟶ ( 0 [,) +∞ ) ↔ ( ( ( 𝑃 ‘ 𝑚 ) ∘f + ( 𝑄 ‘ 𝑚 ) ) : ℝ ⟶ ℝ ∧ 0𝑝 ∘r ≤ ( ( 𝑃 ‘ 𝑚 ) ∘f + ( 𝑄 ‘ 𝑚 ) ) ) ) |
92 |
90 91
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( ( ( 𝑃 ‘ 𝑚 ) ∘f + ( 𝑄 ‘ 𝑚 ) ) : ℝ ⟶ ℝ ∧ 0𝑝 ∘r ≤ ( ( 𝑃 ‘ 𝑚 ) ∘f + ( 𝑄 ‘ 𝑚 ) ) ) ) |
93 |
92
|
simprd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → 0𝑝 ∘r ≤ ( ( 𝑃 ‘ 𝑚 ) ∘f + ( 𝑄 ‘ 𝑚 ) ) ) |
94 |
7
|
ffnd |
⊢ ( 𝜑 → 𝑃 Fn ℕ ) |
95 |
10
|
ffnd |
⊢ ( 𝜑 → 𝑄 Fn ℕ ) |
96 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 𝑃 ‘ 𝑚 ) = ( 𝑃 ‘ 𝑚 ) ) |
97 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 𝑄 ‘ 𝑚 ) = ( 𝑄 ‘ 𝑚 ) ) |
98 |
94 95 25 25 26 96 97
|
ofval |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑚 ) = ( ( 𝑃 ‘ 𝑚 ) ∘f + ( 𝑄 ‘ 𝑚 ) ) ) |
99 |
93 98
|
breqtrrd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → 0𝑝 ∘r ≤ ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑚 ) ) |
100 |
|
i1ff |
⊢ ( ( 𝑃 ‘ 𝑚 ) ∈ dom ∫1 → ( 𝑃 ‘ 𝑚 ) : ℝ ⟶ ℝ ) |
101 |
30 100
|
syl |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 𝑃 ‘ 𝑚 ) : ℝ ⟶ ℝ ) |
102 |
101
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) ∧ 𝑦 ∈ ℝ ) → ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) ∈ ℝ ) |
103 |
|
i1ff |
⊢ ( ( 𝑄 ‘ 𝑚 ) ∈ dom ∫1 → ( 𝑄 ‘ 𝑚 ) : ℝ ⟶ ℝ ) |
104 |
75 103
|
syl |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 𝑄 ‘ 𝑚 ) : ℝ ⟶ ℝ ) |
105 |
104
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) ∧ 𝑦 ∈ ℝ ) → ( ( 𝑄 ‘ 𝑚 ) ‘ 𝑦 ) ∈ ℝ ) |
106 |
|
peano2nn |
⊢ ( 𝑚 ∈ ℕ → ( 𝑚 + 1 ) ∈ ℕ ) |
107 |
|
ffvelrn |
⊢ ( ( 𝑃 : ℕ ⟶ dom ∫1 ∧ ( 𝑚 + 1 ) ∈ ℕ ) → ( 𝑃 ‘ ( 𝑚 + 1 ) ) ∈ dom ∫1 ) |
108 |
7 106 107
|
syl2an |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 𝑃 ‘ ( 𝑚 + 1 ) ) ∈ dom ∫1 ) |
109 |
|
i1ff |
⊢ ( ( 𝑃 ‘ ( 𝑚 + 1 ) ) ∈ dom ∫1 → ( 𝑃 ‘ ( 𝑚 + 1 ) ) : ℝ ⟶ ℝ ) |
110 |
108 109
|
syl |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 𝑃 ‘ ( 𝑚 + 1 ) ) : ℝ ⟶ ℝ ) |
111 |
110
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) ∧ 𝑦 ∈ ℝ ) → ( ( 𝑃 ‘ ( 𝑚 + 1 ) ) ‘ 𝑦 ) ∈ ℝ ) |
112 |
|
ffvelrn |
⊢ ( ( 𝑄 : ℕ ⟶ dom ∫1 ∧ ( 𝑚 + 1 ) ∈ ℕ ) → ( 𝑄 ‘ ( 𝑚 + 1 ) ) ∈ dom ∫1 ) |
113 |
10 106 112
|
syl2an |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 𝑄 ‘ ( 𝑚 + 1 ) ) ∈ dom ∫1 ) |
114 |
|
i1ff |
⊢ ( ( 𝑄 ‘ ( 𝑚 + 1 ) ) ∈ dom ∫1 → ( 𝑄 ‘ ( 𝑚 + 1 ) ) : ℝ ⟶ ℝ ) |
115 |
113 114
|
syl |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 𝑄 ‘ ( 𝑚 + 1 ) ) : ℝ ⟶ ℝ ) |
116 |
115
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) ∧ 𝑦 ∈ ℝ ) → ( ( 𝑄 ‘ ( 𝑚 + 1 ) ) ‘ 𝑦 ) ∈ ℝ ) |
117 |
37
|
simprd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 𝑃 ‘ 𝑚 ) ∘r ≤ ( 𝑃 ‘ ( 𝑚 + 1 ) ) ) |
118 |
101
|
ffnd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 𝑃 ‘ 𝑚 ) Fn ℝ ) |
119 |
110
|
ffnd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 𝑃 ‘ ( 𝑚 + 1 ) ) Fn ℝ ) |
120 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) ∧ 𝑦 ∈ ℝ ) → ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) = ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) ) |
121 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) ∧ 𝑦 ∈ ℝ ) → ( ( 𝑃 ‘ ( 𝑚 + 1 ) ) ‘ 𝑦 ) = ( ( 𝑃 ‘ ( 𝑚 + 1 ) ) ‘ 𝑦 ) ) |
122 |
118 119 89 89 18 120 121
|
ofrfval |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( ( 𝑃 ‘ 𝑚 ) ∘r ≤ ( 𝑃 ‘ ( 𝑚 + 1 ) ) ↔ ∀ 𝑦 ∈ ℝ ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) ≤ ( ( 𝑃 ‘ ( 𝑚 + 1 ) ) ‘ 𝑦 ) ) ) |
123 |
117 122
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ∀ 𝑦 ∈ ℝ ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) ≤ ( ( 𝑃 ‘ ( 𝑚 + 1 ) ) ‘ 𝑦 ) ) |
124 |
123
|
r19.21bi |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) ∧ 𝑦 ∈ ℝ ) → ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) ≤ ( ( 𝑃 ‘ ( 𝑚 + 1 ) ) ‘ 𝑦 ) ) |
125 |
82
|
simprd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 𝑄 ‘ 𝑚 ) ∘r ≤ ( 𝑄 ‘ ( 𝑚 + 1 ) ) ) |
126 |
104
|
ffnd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 𝑄 ‘ 𝑚 ) Fn ℝ ) |
127 |
115
|
ffnd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 𝑄 ‘ ( 𝑚 + 1 ) ) Fn ℝ ) |
128 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) ∧ 𝑦 ∈ ℝ ) → ( ( 𝑄 ‘ 𝑚 ) ‘ 𝑦 ) = ( ( 𝑄 ‘ 𝑚 ) ‘ 𝑦 ) ) |
129 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) ∧ 𝑦 ∈ ℝ ) → ( ( 𝑄 ‘ ( 𝑚 + 1 ) ) ‘ 𝑦 ) = ( ( 𝑄 ‘ ( 𝑚 + 1 ) ) ‘ 𝑦 ) ) |
130 |
126 127 89 89 18 128 129
|
ofrfval |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( ( 𝑄 ‘ 𝑚 ) ∘r ≤ ( 𝑄 ‘ ( 𝑚 + 1 ) ) ↔ ∀ 𝑦 ∈ ℝ ( ( 𝑄 ‘ 𝑚 ) ‘ 𝑦 ) ≤ ( ( 𝑄 ‘ ( 𝑚 + 1 ) ) ‘ 𝑦 ) ) ) |
131 |
125 130
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ∀ 𝑦 ∈ ℝ ( ( 𝑄 ‘ 𝑚 ) ‘ 𝑦 ) ≤ ( ( 𝑄 ‘ ( 𝑚 + 1 ) ) ‘ 𝑦 ) ) |
132 |
131
|
r19.21bi |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) ∧ 𝑦 ∈ ℝ ) → ( ( 𝑄 ‘ 𝑚 ) ‘ 𝑦 ) ≤ ( ( 𝑄 ‘ ( 𝑚 + 1 ) ) ‘ 𝑦 ) ) |
133 |
102 105 111 116 124 132
|
le2addd |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) ∧ 𝑦 ∈ ℝ ) → ( ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) + ( ( 𝑄 ‘ 𝑚 ) ‘ 𝑦 ) ) ≤ ( ( ( 𝑃 ‘ ( 𝑚 + 1 ) ) ‘ 𝑦 ) + ( ( 𝑄 ‘ ( 𝑚 + 1 ) ) ‘ 𝑦 ) ) ) |
134 |
133
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ∀ 𝑦 ∈ ℝ ( ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) + ( ( 𝑄 ‘ 𝑚 ) ‘ 𝑦 ) ) ≤ ( ( ( 𝑃 ‘ ( 𝑚 + 1 ) ) ‘ 𝑦 ) + ( ( 𝑄 ‘ ( 𝑚 + 1 ) ) ‘ 𝑦 ) ) ) |
135 |
30 75
|
i1fadd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( ( 𝑃 ‘ 𝑚 ) ∘f + ( 𝑄 ‘ 𝑚 ) ) ∈ dom ∫1 ) |
136 |
|
i1ff |
⊢ ( ( ( 𝑃 ‘ 𝑚 ) ∘f + ( 𝑄 ‘ 𝑚 ) ) ∈ dom ∫1 → ( ( 𝑃 ‘ 𝑚 ) ∘f + ( 𝑄 ‘ 𝑚 ) ) : ℝ ⟶ ℝ ) |
137 |
|
ffn |
⊢ ( ( ( 𝑃 ‘ 𝑚 ) ∘f + ( 𝑄 ‘ 𝑚 ) ) : ℝ ⟶ ℝ → ( ( 𝑃 ‘ 𝑚 ) ∘f + ( 𝑄 ‘ 𝑚 ) ) Fn ℝ ) |
138 |
135 136 137
|
3syl |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( ( 𝑃 ‘ 𝑚 ) ∘f + ( 𝑄 ‘ 𝑚 ) ) Fn ℝ ) |
139 |
108 113
|
i1fadd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( ( 𝑃 ‘ ( 𝑚 + 1 ) ) ∘f + ( 𝑄 ‘ ( 𝑚 + 1 ) ) ) ∈ dom ∫1 ) |
140 |
|
i1ff |
⊢ ( ( ( 𝑃 ‘ ( 𝑚 + 1 ) ) ∘f + ( 𝑄 ‘ ( 𝑚 + 1 ) ) ) ∈ dom ∫1 → ( ( 𝑃 ‘ ( 𝑚 + 1 ) ) ∘f + ( 𝑄 ‘ ( 𝑚 + 1 ) ) ) : ℝ ⟶ ℝ ) |
141 |
139 140
|
syl |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( ( 𝑃 ‘ ( 𝑚 + 1 ) ) ∘f + ( 𝑄 ‘ ( 𝑚 + 1 ) ) ) : ℝ ⟶ ℝ ) |
142 |
141
|
ffnd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( ( 𝑃 ‘ ( 𝑚 + 1 ) ) ∘f + ( 𝑄 ‘ ( 𝑚 + 1 ) ) ) Fn ℝ ) |
143 |
118 126 89 89 18 120 128
|
ofval |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) ∧ 𝑦 ∈ ℝ ) → ( ( ( 𝑃 ‘ 𝑚 ) ∘f + ( 𝑄 ‘ 𝑚 ) ) ‘ 𝑦 ) = ( ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) + ( ( 𝑄 ‘ 𝑚 ) ‘ 𝑦 ) ) ) |
144 |
119 127 89 89 18 121 129
|
ofval |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) ∧ 𝑦 ∈ ℝ ) → ( ( ( 𝑃 ‘ ( 𝑚 + 1 ) ) ∘f + ( 𝑄 ‘ ( 𝑚 + 1 ) ) ) ‘ 𝑦 ) = ( ( ( 𝑃 ‘ ( 𝑚 + 1 ) ) ‘ 𝑦 ) + ( ( 𝑄 ‘ ( 𝑚 + 1 ) ) ‘ 𝑦 ) ) ) |
145 |
138 142 89 89 18 143 144
|
ofrfval |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( ( ( 𝑃 ‘ 𝑚 ) ∘f + ( 𝑄 ‘ 𝑚 ) ) ∘r ≤ ( ( 𝑃 ‘ ( 𝑚 + 1 ) ) ∘f + ( 𝑄 ‘ ( 𝑚 + 1 ) ) ) ↔ ∀ 𝑦 ∈ ℝ ( ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) + ( ( 𝑄 ‘ 𝑚 ) ‘ 𝑦 ) ) ≤ ( ( ( 𝑃 ‘ ( 𝑚 + 1 ) ) ‘ 𝑦 ) + ( ( 𝑄 ‘ ( 𝑚 + 1 ) ) ‘ 𝑦 ) ) ) ) |
146 |
134 145
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( ( 𝑃 ‘ 𝑚 ) ∘f + ( 𝑄 ‘ 𝑚 ) ) ∘r ≤ ( ( 𝑃 ‘ ( 𝑚 + 1 ) ) ∘f + ( 𝑄 ‘ ( 𝑚 + 1 ) ) ) ) |
147 |
|
eqidd |
⊢ ( ( 𝜑 ∧ ( 𝑚 + 1 ) ∈ ℕ ) → ( 𝑃 ‘ ( 𝑚 + 1 ) ) = ( 𝑃 ‘ ( 𝑚 + 1 ) ) ) |
148 |
|
eqidd |
⊢ ( ( 𝜑 ∧ ( 𝑚 + 1 ) ∈ ℕ ) → ( 𝑄 ‘ ( 𝑚 + 1 ) ) = ( 𝑄 ‘ ( 𝑚 + 1 ) ) ) |
149 |
94 95 25 25 26 147 148
|
ofval |
⊢ ( ( 𝜑 ∧ ( 𝑚 + 1 ) ∈ ℕ ) → ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ ( 𝑚 + 1 ) ) = ( ( 𝑃 ‘ ( 𝑚 + 1 ) ) ∘f + ( 𝑄 ‘ ( 𝑚 + 1 ) ) ) ) |
150 |
106 149
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ ( 𝑚 + 1 ) ) = ( ( 𝑃 ‘ ( 𝑚 + 1 ) ) ∘f + ( 𝑄 ‘ ( 𝑚 + 1 ) ) ) ) |
151 |
146 98 150
|
3brtr4d |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑚 ) ∘r ≤ ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ ( 𝑚 + 1 ) ) ) |
152 |
99 151
|
jca |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 0𝑝 ∘r ≤ ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑚 ) ∧ ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑚 ) ∘r ≤ ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ ( 𝑚 + 1 ) ) ) ) |
153 |
152
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑚 ∈ ℕ ( 0𝑝 ∘r ≤ ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑚 ) ∧ ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑚 ) ∘r ≤ ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ ( 𝑚 + 1 ) ) ) ) |
154 |
|
fveq2 |
⊢ ( 𝑛 = 𝑚 → ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑛 ) = ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑚 ) ) |
155 |
154
|
fveq1d |
⊢ ( 𝑛 = 𝑚 → ( ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑛 ) ‘ 𝑦 ) = ( ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑚 ) ‘ 𝑦 ) ) |
156 |
155
|
cbvmptv |
⊢ ( 𝑛 ∈ ℕ ↦ ( ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑛 ) ‘ 𝑦 ) ) = ( 𝑚 ∈ ℕ ↦ ( ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑚 ) ‘ 𝑦 ) ) |
157 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
158 |
|
1zzd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → 1 ∈ ℤ ) |
159 |
|
fveq2 |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑥 ) = ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑦 ) ) |
160 |
159
|
mpteq2dv |
⊢ ( 𝑥 = 𝑦 → ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑥 ) ) = ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑦 ) ) ) |
161 |
|
fveq2 |
⊢ ( 𝑥 = 𝑦 → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ) |
162 |
160 161
|
breq12d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑥 ) ) ⇝ ( 𝐹 ‘ 𝑥 ) ↔ ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑦 ) ) ⇝ ( 𝐹 ‘ 𝑦 ) ) ) |
163 |
162
|
rspccva |
⊢ ( ( ∀ 𝑥 ∈ ℝ ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑥 ) ) ⇝ ( 𝐹 ‘ 𝑥 ) ∧ 𝑦 ∈ ℝ ) → ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑦 ) ) ⇝ ( 𝐹 ‘ 𝑦 ) ) |
164 |
9 163
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑦 ) ) ⇝ ( 𝐹 ‘ 𝑦 ) ) |
165 |
24
|
mptex |
⊢ ( 𝑛 ∈ ℕ ↦ ( ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑛 ) ‘ 𝑦 ) ) ∈ V |
166 |
165
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( 𝑛 ∈ ℕ ↦ ( ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑛 ) ‘ 𝑦 ) ) ∈ V ) |
167 |
|
fveq2 |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑄 ‘ 𝑛 ) ‘ 𝑥 ) = ( ( 𝑄 ‘ 𝑛 ) ‘ 𝑦 ) ) |
168 |
167
|
mpteq2dv |
⊢ ( 𝑥 = 𝑦 → ( 𝑛 ∈ ℕ ↦ ( ( 𝑄 ‘ 𝑛 ) ‘ 𝑥 ) ) = ( 𝑛 ∈ ℕ ↦ ( ( 𝑄 ‘ 𝑛 ) ‘ 𝑦 ) ) ) |
169 |
|
fveq2 |
⊢ ( 𝑥 = 𝑦 → ( 𝐺 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑦 ) ) |
170 |
168 169
|
breq12d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑛 ∈ ℕ ↦ ( ( 𝑄 ‘ 𝑛 ) ‘ 𝑥 ) ) ⇝ ( 𝐺 ‘ 𝑥 ) ↔ ( 𝑛 ∈ ℕ ↦ ( ( 𝑄 ‘ 𝑛 ) ‘ 𝑦 ) ) ⇝ ( 𝐺 ‘ 𝑦 ) ) ) |
171 |
170
|
rspccva |
⊢ ( ( ∀ 𝑥 ∈ ℝ ( 𝑛 ∈ ℕ ↦ ( ( 𝑄 ‘ 𝑛 ) ‘ 𝑥 ) ) ⇝ ( 𝐺 ‘ 𝑥 ) ∧ 𝑦 ∈ ℝ ) → ( 𝑛 ∈ ℕ ↦ ( ( 𝑄 ‘ 𝑛 ) ‘ 𝑦 ) ) ⇝ ( 𝐺 ‘ 𝑦 ) ) |
172 |
12 171
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( 𝑛 ∈ ℕ ↦ ( ( 𝑄 ‘ 𝑛 ) ‘ 𝑦 ) ) ⇝ ( 𝐺 ‘ 𝑦 ) ) |
173 |
31
|
fveq1d |
⊢ ( 𝑛 = 𝑚 → ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑦 ) = ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) ) |
174 |
|
eqid |
⊢ ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑦 ) ) = ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑦 ) ) |
175 |
|
fvex |
⊢ ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) ∈ V |
176 |
173 174 175
|
fvmpt |
⊢ ( 𝑚 ∈ ℕ → ( ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑦 ) ) ‘ 𝑚 ) = ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) ) |
177 |
176
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑚 ∈ ℕ ) → ( ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑦 ) ) ‘ 𝑚 ) = ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) ) |
178 |
102
|
an32s |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑚 ∈ ℕ ) → ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) ∈ ℝ ) |
179 |
177 178
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑚 ∈ ℕ ) → ( ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑦 ) ) ‘ 𝑚 ) ∈ ℝ ) |
180 |
179
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑚 ∈ ℕ ) → ( ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑦 ) ) ‘ 𝑚 ) ∈ ℂ ) |
181 |
76
|
fveq1d |
⊢ ( 𝑛 = 𝑚 → ( ( 𝑄 ‘ 𝑛 ) ‘ 𝑦 ) = ( ( 𝑄 ‘ 𝑚 ) ‘ 𝑦 ) ) |
182 |
|
eqid |
⊢ ( 𝑛 ∈ ℕ ↦ ( ( 𝑄 ‘ 𝑛 ) ‘ 𝑦 ) ) = ( 𝑛 ∈ ℕ ↦ ( ( 𝑄 ‘ 𝑛 ) ‘ 𝑦 ) ) |
183 |
|
fvex |
⊢ ( ( 𝑄 ‘ 𝑚 ) ‘ 𝑦 ) ∈ V |
184 |
181 182 183
|
fvmpt |
⊢ ( 𝑚 ∈ ℕ → ( ( 𝑛 ∈ ℕ ↦ ( ( 𝑄 ‘ 𝑛 ) ‘ 𝑦 ) ) ‘ 𝑚 ) = ( ( 𝑄 ‘ 𝑚 ) ‘ 𝑦 ) ) |
185 |
184
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑚 ∈ ℕ ) → ( ( 𝑛 ∈ ℕ ↦ ( ( 𝑄 ‘ 𝑛 ) ‘ 𝑦 ) ) ‘ 𝑚 ) = ( ( 𝑄 ‘ 𝑚 ) ‘ 𝑦 ) ) |
186 |
105
|
an32s |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑚 ∈ ℕ ) → ( ( 𝑄 ‘ 𝑚 ) ‘ 𝑦 ) ∈ ℝ ) |
187 |
185 186
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑚 ∈ ℕ ) → ( ( 𝑛 ∈ ℕ ↦ ( ( 𝑄 ‘ 𝑛 ) ‘ 𝑦 ) ) ‘ 𝑚 ) ∈ ℝ ) |
188 |
187
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑚 ∈ ℕ ) → ( ( 𝑛 ∈ ℕ ↦ ( ( 𝑄 ‘ 𝑛 ) ‘ 𝑦 ) ) ‘ 𝑚 ) ∈ ℂ ) |
189 |
98
|
fveq1d |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑚 ) ‘ 𝑦 ) = ( ( ( 𝑃 ‘ 𝑚 ) ∘f + ( 𝑄 ‘ 𝑚 ) ) ‘ 𝑦 ) ) |
190 |
189
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) ∧ 𝑦 ∈ ℝ ) → ( ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑚 ) ‘ 𝑦 ) = ( ( ( 𝑃 ‘ 𝑚 ) ∘f + ( 𝑄 ‘ 𝑚 ) ) ‘ 𝑦 ) ) |
191 |
190 143
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) ∧ 𝑦 ∈ ℝ ) → ( ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑚 ) ‘ 𝑦 ) = ( ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) + ( ( 𝑄 ‘ 𝑚 ) ‘ 𝑦 ) ) ) |
192 |
191
|
an32s |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑚 ∈ ℕ ) → ( ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑚 ) ‘ 𝑦 ) = ( ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) + ( ( 𝑄 ‘ 𝑚 ) ‘ 𝑦 ) ) ) |
193 |
|
eqid |
⊢ ( 𝑛 ∈ ℕ ↦ ( ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑛 ) ‘ 𝑦 ) ) = ( 𝑛 ∈ ℕ ↦ ( ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑛 ) ‘ 𝑦 ) ) |
194 |
|
fvex |
⊢ ( ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑚 ) ‘ 𝑦 ) ∈ V |
195 |
155 193 194
|
fvmpt |
⊢ ( 𝑚 ∈ ℕ → ( ( 𝑛 ∈ ℕ ↦ ( ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑛 ) ‘ 𝑦 ) ) ‘ 𝑚 ) = ( ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑚 ) ‘ 𝑦 ) ) |
196 |
195
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑚 ∈ ℕ ) → ( ( 𝑛 ∈ ℕ ↦ ( ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑛 ) ‘ 𝑦 ) ) ‘ 𝑚 ) = ( ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑚 ) ‘ 𝑦 ) ) |
197 |
177 185
|
oveq12d |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑚 ∈ ℕ ) → ( ( ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑦 ) ) ‘ 𝑚 ) + ( ( 𝑛 ∈ ℕ ↦ ( ( 𝑄 ‘ 𝑛 ) ‘ 𝑦 ) ) ‘ 𝑚 ) ) = ( ( ( 𝑃 ‘ 𝑚 ) ‘ 𝑦 ) + ( ( 𝑄 ‘ 𝑚 ) ‘ 𝑦 ) ) ) |
198 |
192 196 197
|
3eqtr4d |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑚 ∈ ℕ ) → ( ( 𝑛 ∈ ℕ ↦ ( ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑛 ) ‘ 𝑦 ) ) ‘ 𝑚 ) = ( ( ( 𝑛 ∈ ℕ ↦ ( ( 𝑃 ‘ 𝑛 ) ‘ 𝑦 ) ) ‘ 𝑚 ) + ( ( 𝑛 ∈ ℕ ↦ ( ( 𝑄 ‘ 𝑛 ) ‘ 𝑦 ) ) ‘ 𝑚 ) ) ) |
199 |
157 158 164 166 172 180 188 198
|
climadd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( 𝑛 ∈ ℕ ↦ ( ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑛 ) ‘ 𝑦 ) ) ⇝ ( ( 𝐹 ‘ 𝑦 ) + ( 𝐺 ‘ 𝑦 ) ) ) |
200 |
156 199
|
eqbrtrrid |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( 𝑚 ∈ ℕ ↦ ( ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑚 ) ‘ 𝑦 ) ) ⇝ ( ( 𝐹 ‘ 𝑦 ) + ( 𝐺 ‘ 𝑦 ) ) ) |
201 |
2
|
ffnd |
⊢ ( 𝜑 → 𝐹 Fn ℝ ) |
202 |
5
|
ffnd |
⊢ ( 𝜑 → 𝐺 Fn ℝ ) |
203 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) ) |
204 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐺 ‘ 𝑦 ) ) |
205 |
201 202 17 17 18 203 204
|
ofval |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( ( 𝐹 ∘f + 𝐺 ) ‘ 𝑦 ) = ( ( 𝐹 ‘ 𝑦 ) + ( 𝐺 ‘ 𝑦 ) ) ) |
206 |
200 205
|
breqtrrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( 𝑚 ∈ ℕ ↦ ( ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑚 ) ‘ 𝑦 ) ) ⇝ ( ( 𝐹 ∘f + 𝐺 ) ‘ 𝑦 ) ) |
207 |
206
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑦 ∈ ℝ ( 𝑚 ∈ ℕ ↦ ( ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑚 ) ‘ 𝑦 ) ) ⇝ ( ( 𝐹 ∘f + 𝐺 ) ‘ 𝑦 ) ) |
208 |
|
2fveq3 |
⊢ ( 𝑛 = 𝑗 → ( ∫1 ‘ ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑛 ) ) = ( ∫1 ‘ ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑗 ) ) ) |
209 |
208
|
cbvmptv |
⊢ ( 𝑛 ∈ ℕ ↦ ( ∫1 ‘ ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑛 ) ) ) = ( 𝑗 ∈ ℕ ↦ ( ∫1 ‘ ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑗 ) ) ) |
210 |
3 6
|
readdcld |
⊢ ( 𝜑 → ( ( ∫2 ‘ 𝐹 ) + ( ∫2 ‘ 𝐺 ) ) ∈ ℝ ) |
211 |
98
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( ∫1 ‘ ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑚 ) ) = ( ∫1 ‘ ( ( 𝑃 ‘ 𝑚 ) ∘f + ( 𝑄 ‘ 𝑚 ) ) ) ) |
212 |
30 75
|
itg1add |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( ∫1 ‘ ( ( 𝑃 ‘ 𝑚 ) ∘f + ( 𝑄 ‘ 𝑚 ) ) ) = ( ( ∫1 ‘ ( 𝑃 ‘ 𝑚 ) ) + ( ∫1 ‘ ( 𝑄 ‘ 𝑚 ) ) ) ) |
213 |
211 212
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( ∫1 ‘ ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑚 ) ) = ( ( ∫1 ‘ ( 𝑃 ‘ 𝑚 ) ) + ( ∫1 ‘ ( 𝑄 ‘ 𝑚 ) ) ) ) |
214 |
|
itg1cl |
⊢ ( ( 𝑃 ‘ 𝑚 ) ∈ dom ∫1 → ( ∫1 ‘ ( 𝑃 ‘ 𝑚 ) ) ∈ ℝ ) |
215 |
30 214
|
syl |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( ∫1 ‘ ( 𝑃 ‘ 𝑚 ) ) ∈ ℝ ) |
216 |
|
itg1cl |
⊢ ( ( 𝑄 ‘ 𝑚 ) ∈ dom ∫1 → ( ∫1 ‘ ( 𝑄 ‘ 𝑚 ) ) ∈ ℝ ) |
217 |
75 216
|
syl |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( ∫1 ‘ ( 𝑄 ‘ 𝑚 ) ) ∈ ℝ ) |
218 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( ∫2 ‘ 𝐹 ) ∈ ℝ ) |
219 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( ∫2 ‘ 𝐺 ) ∈ ℝ ) |
220 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → 𝐹 : ℝ ⟶ ( 0 [,) +∞ ) ) |
221 |
|
icossicc |
⊢ ( 0 [,) +∞ ) ⊆ ( 0 [,] +∞ ) |
222 |
|
fss |
⊢ ( ( 𝐹 : ℝ ⟶ ( 0 [,) +∞ ) ∧ ( 0 [,) +∞ ) ⊆ ( 0 [,] +∞ ) ) → 𝐹 : ℝ ⟶ ( 0 [,] +∞ ) ) |
223 |
220 221 222
|
sylancl |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → 𝐹 : ℝ ⟶ ( 0 [,] +∞ ) ) |
224 |
1 2 7 8 9
|
itg2i1fseqle |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 𝑃 ‘ 𝑚 ) ∘r ≤ 𝐹 ) |
225 |
|
itg2ub |
⊢ ( ( 𝐹 : ℝ ⟶ ( 0 [,] +∞ ) ∧ ( 𝑃 ‘ 𝑚 ) ∈ dom ∫1 ∧ ( 𝑃 ‘ 𝑚 ) ∘r ≤ 𝐹 ) → ( ∫1 ‘ ( 𝑃 ‘ 𝑚 ) ) ≤ ( ∫2 ‘ 𝐹 ) ) |
226 |
223 30 224 225
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( ∫1 ‘ ( 𝑃 ‘ 𝑚 ) ) ≤ ( ∫2 ‘ 𝐹 ) ) |
227 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → 𝐺 : ℝ ⟶ ( 0 [,) +∞ ) ) |
228 |
|
fss |
⊢ ( ( 𝐺 : ℝ ⟶ ( 0 [,) +∞ ) ∧ ( 0 [,) +∞ ) ⊆ ( 0 [,] +∞ ) ) → 𝐺 : ℝ ⟶ ( 0 [,] +∞ ) ) |
229 |
227 221 228
|
sylancl |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → 𝐺 : ℝ ⟶ ( 0 [,] +∞ ) ) |
230 |
4 5 10 11 12
|
itg2i1fseqle |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 𝑄 ‘ 𝑚 ) ∘r ≤ 𝐺 ) |
231 |
|
itg2ub |
⊢ ( ( 𝐺 : ℝ ⟶ ( 0 [,] +∞ ) ∧ ( 𝑄 ‘ 𝑚 ) ∈ dom ∫1 ∧ ( 𝑄 ‘ 𝑚 ) ∘r ≤ 𝐺 ) → ( ∫1 ‘ ( 𝑄 ‘ 𝑚 ) ) ≤ ( ∫2 ‘ 𝐺 ) ) |
232 |
229 75 230 231
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( ∫1 ‘ ( 𝑄 ‘ 𝑚 ) ) ≤ ( ∫2 ‘ 𝐺 ) ) |
233 |
215 217 218 219 226 232
|
le2addd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( ( ∫1 ‘ ( 𝑃 ‘ 𝑚 ) ) + ( ∫1 ‘ ( 𝑄 ‘ 𝑚 ) ) ) ≤ ( ( ∫2 ‘ 𝐹 ) + ( ∫2 ‘ 𝐺 ) ) ) |
234 |
213 233
|
eqbrtrd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( ∫1 ‘ ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑚 ) ) ≤ ( ( ∫2 ‘ 𝐹 ) + ( ∫2 ‘ 𝐺 ) ) ) |
235 |
234
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑚 ∈ ℕ ( ∫1 ‘ ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑚 ) ) ≤ ( ( ∫2 ‘ 𝐹 ) + ( ∫2 ‘ 𝐺 ) ) ) |
236 |
|
2fveq3 |
⊢ ( 𝑚 = 𝑘 → ( ∫1 ‘ ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑚 ) ) = ( ∫1 ‘ ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑘 ) ) ) |
237 |
236
|
breq1d |
⊢ ( 𝑚 = 𝑘 → ( ( ∫1 ‘ ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑚 ) ) ≤ ( ( ∫2 ‘ 𝐹 ) + ( ∫2 ‘ 𝐺 ) ) ↔ ( ∫1 ‘ ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑘 ) ) ≤ ( ( ∫2 ‘ 𝐹 ) + ( ∫2 ‘ 𝐺 ) ) ) ) |
238 |
237
|
rspccva |
⊢ ( ( ∀ 𝑚 ∈ ℕ ( ∫1 ‘ ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑚 ) ) ≤ ( ( ∫2 ‘ 𝐹 ) + ( ∫2 ‘ 𝐺 ) ) ∧ 𝑘 ∈ ℕ ) → ( ∫1 ‘ ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑘 ) ) ≤ ( ( ∫2 ‘ 𝐹 ) + ( ∫2 ‘ 𝐺 ) ) ) |
239 |
235 238
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ∫1 ‘ ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑘 ) ) ≤ ( ( ∫2 ‘ 𝐹 ) + ( ∫2 ‘ 𝐺 ) ) ) |
240 |
13 19 27 153 207 209 210 239
|
itg2i1fseq2 |
⊢ ( 𝜑 → ( 𝑛 ∈ ℕ ↦ ( ∫1 ‘ ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑛 ) ) ) ⇝ ( ∫2 ‘ ( 𝐹 ∘f + 𝐺 ) ) ) |
241 |
|
1zzd |
⊢ ( 𝜑 → 1 ∈ ℤ ) |
242 |
|
eqid |
⊢ ( 𝑘 ∈ ℕ ↦ ( ∫1 ‘ ( 𝑃 ‘ 𝑘 ) ) ) = ( 𝑘 ∈ ℕ ↦ ( ∫1 ‘ ( 𝑃 ‘ 𝑘 ) ) ) |
243 |
1 2 7 8 9 242 3
|
itg2i1fseq3 |
⊢ ( 𝜑 → ( 𝑘 ∈ ℕ ↦ ( ∫1 ‘ ( 𝑃 ‘ 𝑘 ) ) ) ⇝ ( ∫2 ‘ 𝐹 ) ) |
244 |
24
|
mptex |
⊢ ( 𝑛 ∈ ℕ ↦ ( ∫1 ‘ ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑛 ) ) ) ∈ V |
245 |
244
|
a1i |
⊢ ( 𝜑 → ( 𝑛 ∈ ℕ ↦ ( ∫1 ‘ ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑛 ) ) ) ∈ V ) |
246 |
|
eqid |
⊢ ( 𝑘 ∈ ℕ ↦ ( ∫1 ‘ ( 𝑄 ‘ 𝑘 ) ) ) = ( 𝑘 ∈ ℕ ↦ ( ∫1 ‘ ( 𝑄 ‘ 𝑘 ) ) ) |
247 |
4 5 10 11 12 246 6
|
itg2i1fseq3 |
⊢ ( 𝜑 → ( 𝑘 ∈ ℕ ↦ ( ∫1 ‘ ( 𝑄 ‘ 𝑘 ) ) ) ⇝ ( ∫2 ‘ 𝐺 ) ) |
248 |
|
2fveq3 |
⊢ ( 𝑘 = 𝑚 → ( ∫1 ‘ ( 𝑃 ‘ 𝑘 ) ) = ( ∫1 ‘ ( 𝑃 ‘ 𝑚 ) ) ) |
249 |
|
fvex |
⊢ ( ∫1 ‘ ( 𝑃 ‘ 𝑚 ) ) ∈ V |
250 |
248 242 249
|
fvmpt |
⊢ ( 𝑚 ∈ ℕ → ( ( 𝑘 ∈ ℕ ↦ ( ∫1 ‘ ( 𝑃 ‘ 𝑘 ) ) ) ‘ 𝑚 ) = ( ∫1 ‘ ( 𝑃 ‘ 𝑚 ) ) ) |
251 |
250
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( ( 𝑘 ∈ ℕ ↦ ( ∫1 ‘ ( 𝑃 ‘ 𝑘 ) ) ) ‘ 𝑚 ) = ( ∫1 ‘ ( 𝑃 ‘ 𝑚 ) ) ) |
252 |
215
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( ∫1 ‘ ( 𝑃 ‘ 𝑚 ) ) ∈ ℂ ) |
253 |
251 252
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( ( 𝑘 ∈ ℕ ↦ ( ∫1 ‘ ( 𝑃 ‘ 𝑘 ) ) ) ‘ 𝑚 ) ∈ ℂ ) |
254 |
|
2fveq3 |
⊢ ( 𝑘 = 𝑚 → ( ∫1 ‘ ( 𝑄 ‘ 𝑘 ) ) = ( ∫1 ‘ ( 𝑄 ‘ 𝑚 ) ) ) |
255 |
|
fvex |
⊢ ( ∫1 ‘ ( 𝑄 ‘ 𝑚 ) ) ∈ V |
256 |
254 246 255
|
fvmpt |
⊢ ( 𝑚 ∈ ℕ → ( ( 𝑘 ∈ ℕ ↦ ( ∫1 ‘ ( 𝑄 ‘ 𝑘 ) ) ) ‘ 𝑚 ) = ( ∫1 ‘ ( 𝑄 ‘ 𝑚 ) ) ) |
257 |
256
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( ( 𝑘 ∈ ℕ ↦ ( ∫1 ‘ ( 𝑄 ‘ 𝑘 ) ) ) ‘ 𝑚 ) = ( ∫1 ‘ ( 𝑄 ‘ 𝑚 ) ) ) |
258 |
217
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( ∫1 ‘ ( 𝑄 ‘ 𝑚 ) ) ∈ ℂ ) |
259 |
257 258
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( ( 𝑘 ∈ ℕ ↦ ( ∫1 ‘ ( 𝑄 ‘ 𝑘 ) ) ) ‘ 𝑚 ) ∈ ℂ ) |
260 |
|
2fveq3 |
⊢ ( 𝑗 = 𝑚 → ( ∫1 ‘ ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑗 ) ) = ( ∫1 ‘ ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑚 ) ) ) |
261 |
|
fvex |
⊢ ( ∫1 ‘ ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑚 ) ) ∈ V |
262 |
260 209 261
|
fvmpt |
⊢ ( 𝑚 ∈ ℕ → ( ( 𝑛 ∈ ℕ ↦ ( ∫1 ‘ ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑛 ) ) ) ‘ 𝑚 ) = ( ∫1 ‘ ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑚 ) ) ) |
263 |
262
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( ( 𝑛 ∈ ℕ ↦ ( ∫1 ‘ ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑛 ) ) ) ‘ 𝑚 ) = ( ∫1 ‘ ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑚 ) ) ) |
264 |
251 257
|
oveq12d |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( ( ( 𝑘 ∈ ℕ ↦ ( ∫1 ‘ ( 𝑃 ‘ 𝑘 ) ) ) ‘ 𝑚 ) + ( ( 𝑘 ∈ ℕ ↦ ( ∫1 ‘ ( 𝑄 ‘ 𝑘 ) ) ) ‘ 𝑚 ) ) = ( ( ∫1 ‘ ( 𝑃 ‘ 𝑚 ) ) + ( ∫1 ‘ ( 𝑄 ‘ 𝑚 ) ) ) ) |
265 |
213 263 264
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( ( 𝑛 ∈ ℕ ↦ ( ∫1 ‘ ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑛 ) ) ) ‘ 𝑚 ) = ( ( ( 𝑘 ∈ ℕ ↦ ( ∫1 ‘ ( 𝑃 ‘ 𝑘 ) ) ) ‘ 𝑚 ) + ( ( 𝑘 ∈ ℕ ↦ ( ∫1 ‘ ( 𝑄 ‘ 𝑘 ) ) ) ‘ 𝑚 ) ) ) |
266 |
157 241 243 245 247 253 259 265
|
climadd |
⊢ ( 𝜑 → ( 𝑛 ∈ ℕ ↦ ( ∫1 ‘ ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑛 ) ) ) ⇝ ( ( ∫2 ‘ 𝐹 ) + ( ∫2 ‘ 𝐺 ) ) ) |
267 |
|
climuni |
⊢ ( ( ( 𝑛 ∈ ℕ ↦ ( ∫1 ‘ ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑛 ) ) ) ⇝ ( ∫2 ‘ ( 𝐹 ∘f + 𝐺 ) ) ∧ ( 𝑛 ∈ ℕ ↦ ( ∫1 ‘ ( ( 𝑃 ∘f ∘f + 𝑄 ) ‘ 𝑛 ) ) ) ⇝ ( ( ∫2 ‘ 𝐹 ) + ( ∫2 ‘ 𝐺 ) ) ) → ( ∫2 ‘ ( 𝐹 ∘f + 𝐺 ) ) = ( ( ∫2 ‘ 𝐹 ) + ( ∫2 ‘ 𝐺 ) ) ) |
268 |
240 266 267
|
syl2anc |
⊢ ( 𝜑 → ( ∫2 ‘ ( 𝐹 ∘f + 𝐺 ) ) = ( ( ∫2 ‘ 𝐹 ) + ( ∫2 ‘ 𝐺 ) ) ) |