Step |
Hyp |
Ref |
Expression |
1 |
|
i1ff |
⊢ ( 𝐹 ∈ dom ∫1 → 𝐹 : ℝ ⟶ ℝ ) |
2 |
1
|
ffvelrnda |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ ) → ( 𝐹 ‘ 𝑥 ) ∈ ℝ ) |
3 |
|
i1ff |
⊢ ( 𝐺 ∈ dom ∫1 → 𝐺 : ℝ ⟶ ℝ ) |
4 |
3
|
ffvelrnda |
⊢ ( ( 𝐺 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ ) → ( 𝐺 ‘ 𝑥 ) ∈ ℝ ) |
5 |
|
absreim |
⊢ ( ( ( 𝐹 ‘ 𝑥 ) ∈ ℝ ∧ ( 𝐺 ‘ 𝑥 ) ∈ ℝ ) → ( abs ‘ ( ( 𝐹 ‘ 𝑥 ) + ( i · ( 𝐺 ‘ 𝑥 ) ) ) ) = ( √ ‘ ( ( ( 𝐹 ‘ 𝑥 ) ↑ 2 ) + ( ( 𝐺 ‘ 𝑥 ) ↑ 2 ) ) ) ) |
6 |
2 4 5
|
syl2an |
⊢ ( ( ( 𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ ) ∧ ( 𝐺 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ ) ) → ( abs ‘ ( ( 𝐹 ‘ 𝑥 ) + ( i · ( 𝐺 ‘ 𝑥 ) ) ) ) = ( √ ‘ ( ( ( 𝐹 ‘ 𝑥 ) ↑ 2 ) + ( ( 𝐺 ‘ 𝑥 ) ↑ 2 ) ) ) ) |
7 |
6
|
anandirs |
⊢ ( ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) ∧ 𝑥 ∈ ℝ ) → ( abs ‘ ( ( 𝐹 ‘ 𝑥 ) + ( i · ( 𝐺 ‘ 𝑥 ) ) ) ) = ( √ ‘ ( ( ( 𝐹 ‘ 𝑥 ) ↑ 2 ) + ( ( 𝐺 ‘ 𝑥 ) ↑ 2 ) ) ) ) |
8 |
2
|
recnd |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ ) → ( 𝐹 ‘ 𝑥 ) ∈ ℂ ) |
9 |
8
|
sqvald |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ ) → ( ( 𝐹 ‘ 𝑥 ) ↑ 2 ) = ( ( 𝐹 ‘ 𝑥 ) · ( 𝐹 ‘ 𝑥 ) ) ) |
10 |
4
|
recnd |
⊢ ( ( 𝐺 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ ) → ( 𝐺 ‘ 𝑥 ) ∈ ℂ ) |
11 |
10
|
sqvald |
⊢ ( ( 𝐺 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ ) → ( ( 𝐺 ‘ 𝑥 ) ↑ 2 ) = ( ( 𝐺 ‘ 𝑥 ) · ( 𝐺 ‘ 𝑥 ) ) ) |
12 |
9 11
|
oveqan12d |
⊢ ( ( ( 𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ ) ∧ ( 𝐺 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ ) ) → ( ( ( 𝐹 ‘ 𝑥 ) ↑ 2 ) + ( ( 𝐺 ‘ 𝑥 ) ↑ 2 ) ) = ( ( ( 𝐹 ‘ 𝑥 ) · ( 𝐹 ‘ 𝑥 ) ) + ( ( 𝐺 ‘ 𝑥 ) · ( 𝐺 ‘ 𝑥 ) ) ) ) |
13 |
12
|
anandirs |
⊢ ( ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) ∧ 𝑥 ∈ ℝ ) → ( ( ( 𝐹 ‘ 𝑥 ) ↑ 2 ) + ( ( 𝐺 ‘ 𝑥 ) ↑ 2 ) ) = ( ( ( 𝐹 ‘ 𝑥 ) · ( 𝐹 ‘ 𝑥 ) ) + ( ( 𝐺 ‘ 𝑥 ) · ( 𝐺 ‘ 𝑥 ) ) ) ) |
14 |
13
|
fveq2d |
⊢ ( ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) ∧ 𝑥 ∈ ℝ ) → ( √ ‘ ( ( ( 𝐹 ‘ 𝑥 ) ↑ 2 ) + ( ( 𝐺 ‘ 𝑥 ) ↑ 2 ) ) ) = ( √ ‘ ( ( ( 𝐹 ‘ 𝑥 ) · ( 𝐹 ‘ 𝑥 ) ) + ( ( 𝐺 ‘ 𝑥 ) · ( 𝐺 ‘ 𝑥 ) ) ) ) ) |
15 |
7 14
|
eqtrd |
⊢ ( ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) ∧ 𝑥 ∈ ℝ ) → ( abs ‘ ( ( 𝐹 ‘ 𝑥 ) + ( i · ( 𝐺 ‘ 𝑥 ) ) ) ) = ( √ ‘ ( ( ( 𝐹 ‘ 𝑥 ) · ( 𝐹 ‘ 𝑥 ) ) + ( ( 𝐺 ‘ 𝑥 ) · ( 𝐺 ‘ 𝑥 ) ) ) ) ) |
16 |
15
|
mpteq2dva |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) → ( 𝑥 ∈ ℝ ↦ ( abs ‘ ( ( 𝐹 ‘ 𝑥 ) + ( i · ( 𝐺 ‘ 𝑥 ) ) ) ) ) = ( 𝑥 ∈ ℝ ↦ ( √ ‘ ( ( ( 𝐹 ‘ 𝑥 ) · ( 𝐹 ‘ 𝑥 ) ) + ( ( 𝐺 ‘ 𝑥 ) · ( 𝐺 ‘ 𝑥 ) ) ) ) ) ) |
17 |
|
ax-icn |
⊢ i ∈ ℂ |
18 |
|
mulcl |
⊢ ( ( i ∈ ℂ ∧ ( 𝐺 ‘ 𝑥 ) ∈ ℂ ) → ( i · ( 𝐺 ‘ 𝑥 ) ) ∈ ℂ ) |
19 |
17 10 18
|
sylancr |
⊢ ( ( 𝐺 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ ) → ( i · ( 𝐺 ‘ 𝑥 ) ) ∈ ℂ ) |
20 |
|
addcl |
⊢ ( ( ( 𝐹 ‘ 𝑥 ) ∈ ℂ ∧ ( i · ( 𝐺 ‘ 𝑥 ) ) ∈ ℂ ) → ( ( 𝐹 ‘ 𝑥 ) + ( i · ( 𝐺 ‘ 𝑥 ) ) ) ∈ ℂ ) |
21 |
8 19 20
|
syl2an |
⊢ ( ( ( 𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ ) ∧ ( 𝐺 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ ) ) → ( ( 𝐹 ‘ 𝑥 ) + ( i · ( 𝐺 ‘ 𝑥 ) ) ) ∈ ℂ ) |
22 |
21
|
anandirs |
⊢ ( ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) ∧ 𝑥 ∈ ℝ ) → ( ( 𝐹 ‘ 𝑥 ) + ( i · ( 𝐺 ‘ 𝑥 ) ) ) ∈ ℂ ) |
23 |
|
reex |
⊢ ℝ ∈ V |
24 |
23
|
a1i |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) → ℝ ∈ V ) |
25 |
2
|
adantlr |
⊢ ( ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) ∧ 𝑥 ∈ ℝ ) → ( 𝐹 ‘ 𝑥 ) ∈ ℝ ) |
26 |
|
ovexd |
⊢ ( ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) ∧ 𝑥 ∈ ℝ ) → ( i · ( 𝐺 ‘ 𝑥 ) ) ∈ V ) |
27 |
1
|
feqmptd |
⊢ ( 𝐹 ∈ dom ∫1 → 𝐹 = ( 𝑥 ∈ ℝ ↦ ( 𝐹 ‘ 𝑥 ) ) ) |
28 |
27
|
adantr |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) → 𝐹 = ( 𝑥 ∈ ℝ ↦ ( 𝐹 ‘ 𝑥 ) ) ) |
29 |
23
|
a1i |
⊢ ( 𝐺 ∈ dom ∫1 → ℝ ∈ V ) |
30 |
17
|
a1i |
⊢ ( ( 𝐺 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ ) → i ∈ ℂ ) |
31 |
|
fconstmpt |
⊢ ( ℝ × { i } ) = ( 𝑥 ∈ ℝ ↦ i ) |
32 |
31
|
a1i |
⊢ ( 𝐺 ∈ dom ∫1 → ( ℝ × { i } ) = ( 𝑥 ∈ ℝ ↦ i ) ) |
33 |
3
|
feqmptd |
⊢ ( 𝐺 ∈ dom ∫1 → 𝐺 = ( 𝑥 ∈ ℝ ↦ ( 𝐺 ‘ 𝑥 ) ) ) |
34 |
29 30 4 32 33
|
offval2 |
⊢ ( 𝐺 ∈ dom ∫1 → ( ( ℝ × { i } ) ∘f · 𝐺 ) = ( 𝑥 ∈ ℝ ↦ ( i · ( 𝐺 ‘ 𝑥 ) ) ) ) |
35 |
34
|
adantl |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) → ( ( ℝ × { i } ) ∘f · 𝐺 ) = ( 𝑥 ∈ ℝ ↦ ( i · ( 𝐺 ‘ 𝑥 ) ) ) ) |
36 |
24 25 26 28 35
|
offval2 |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) → ( 𝐹 ∘f + ( ( ℝ × { i } ) ∘f · 𝐺 ) ) = ( 𝑥 ∈ ℝ ↦ ( ( 𝐹 ‘ 𝑥 ) + ( i · ( 𝐺 ‘ 𝑥 ) ) ) ) ) |
37 |
|
absf |
⊢ abs : ℂ ⟶ ℝ |
38 |
37
|
a1i |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) → abs : ℂ ⟶ ℝ ) |
39 |
38
|
feqmptd |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) → abs = ( 𝑦 ∈ ℂ ↦ ( abs ‘ 𝑦 ) ) ) |
40 |
|
fveq2 |
⊢ ( 𝑦 = ( ( 𝐹 ‘ 𝑥 ) + ( i · ( 𝐺 ‘ 𝑥 ) ) ) → ( abs ‘ 𝑦 ) = ( abs ‘ ( ( 𝐹 ‘ 𝑥 ) + ( i · ( 𝐺 ‘ 𝑥 ) ) ) ) ) |
41 |
22 36 39 40
|
fmptco |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) → ( abs ∘ ( 𝐹 ∘f + ( ( ℝ × { i } ) ∘f · 𝐺 ) ) ) = ( 𝑥 ∈ ℝ ↦ ( abs ‘ ( ( 𝐹 ‘ 𝑥 ) + ( i · ( 𝐺 ‘ 𝑥 ) ) ) ) ) ) |
42 |
8 8
|
mulcld |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ ) → ( ( 𝐹 ‘ 𝑥 ) · ( 𝐹 ‘ 𝑥 ) ) ∈ ℂ ) |
43 |
10 10
|
mulcld |
⊢ ( ( 𝐺 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ ) → ( ( 𝐺 ‘ 𝑥 ) · ( 𝐺 ‘ 𝑥 ) ) ∈ ℂ ) |
44 |
|
addcl |
⊢ ( ( ( ( 𝐹 ‘ 𝑥 ) · ( 𝐹 ‘ 𝑥 ) ) ∈ ℂ ∧ ( ( 𝐺 ‘ 𝑥 ) · ( 𝐺 ‘ 𝑥 ) ) ∈ ℂ ) → ( ( ( 𝐹 ‘ 𝑥 ) · ( 𝐹 ‘ 𝑥 ) ) + ( ( 𝐺 ‘ 𝑥 ) · ( 𝐺 ‘ 𝑥 ) ) ) ∈ ℂ ) |
45 |
42 43 44
|
syl2an |
⊢ ( ( ( 𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ ) ∧ ( 𝐺 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ ) ) → ( ( ( 𝐹 ‘ 𝑥 ) · ( 𝐹 ‘ 𝑥 ) ) + ( ( 𝐺 ‘ 𝑥 ) · ( 𝐺 ‘ 𝑥 ) ) ) ∈ ℂ ) |
46 |
45
|
anandirs |
⊢ ( ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) ∧ 𝑥 ∈ ℝ ) → ( ( ( 𝐹 ‘ 𝑥 ) · ( 𝐹 ‘ 𝑥 ) ) + ( ( 𝐺 ‘ 𝑥 ) · ( 𝐺 ‘ 𝑥 ) ) ) ∈ ℂ ) |
47 |
42
|
adantlr |
⊢ ( ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) ∧ 𝑥 ∈ ℝ ) → ( ( 𝐹 ‘ 𝑥 ) · ( 𝐹 ‘ 𝑥 ) ) ∈ ℂ ) |
48 |
43
|
adantll |
⊢ ( ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) ∧ 𝑥 ∈ ℝ ) → ( ( 𝐺 ‘ 𝑥 ) · ( 𝐺 ‘ 𝑥 ) ) ∈ ℂ ) |
49 |
23
|
a1i |
⊢ ( 𝐹 ∈ dom ∫1 → ℝ ∈ V ) |
50 |
49 2 2 27 27
|
offval2 |
⊢ ( 𝐹 ∈ dom ∫1 → ( 𝐹 ∘f · 𝐹 ) = ( 𝑥 ∈ ℝ ↦ ( ( 𝐹 ‘ 𝑥 ) · ( 𝐹 ‘ 𝑥 ) ) ) ) |
51 |
50
|
adantr |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) → ( 𝐹 ∘f · 𝐹 ) = ( 𝑥 ∈ ℝ ↦ ( ( 𝐹 ‘ 𝑥 ) · ( 𝐹 ‘ 𝑥 ) ) ) ) |
52 |
29 4 4 33 33
|
offval2 |
⊢ ( 𝐺 ∈ dom ∫1 → ( 𝐺 ∘f · 𝐺 ) = ( 𝑥 ∈ ℝ ↦ ( ( 𝐺 ‘ 𝑥 ) · ( 𝐺 ‘ 𝑥 ) ) ) ) |
53 |
52
|
adantl |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) → ( 𝐺 ∘f · 𝐺 ) = ( 𝑥 ∈ ℝ ↦ ( ( 𝐺 ‘ 𝑥 ) · ( 𝐺 ‘ 𝑥 ) ) ) ) |
54 |
24 47 48 51 53
|
offval2 |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) → ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) = ( 𝑥 ∈ ℝ ↦ ( ( ( 𝐹 ‘ 𝑥 ) · ( 𝐹 ‘ 𝑥 ) ) + ( ( 𝐺 ‘ 𝑥 ) · ( 𝐺 ‘ 𝑥 ) ) ) ) ) |
55 |
|
sqrtf |
⊢ √ : ℂ ⟶ ℂ |
56 |
55
|
a1i |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) → √ : ℂ ⟶ ℂ ) |
57 |
56
|
feqmptd |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) → √ = ( 𝑦 ∈ ℂ ↦ ( √ ‘ 𝑦 ) ) ) |
58 |
|
fveq2 |
⊢ ( 𝑦 = ( ( ( 𝐹 ‘ 𝑥 ) · ( 𝐹 ‘ 𝑥 ) ) + ( ( 𝐺 ‘ 𝑥 ) · ( 𝐺 ‘ 𝑥 ) ) ) → ( √ ‘ 𝑦 ) = ( √ ‘ ( ( ( 𝐹 ‘ 𝑥 ) · ( 𝐹 ‘ 𝑥 ) ) + ( ( 𝐺 ‘ 𝑥 ) · ( 𝐺 ‘ 𝑥 ) ) ) ) ) |
59 |
46 54 57 58
|
fmptco |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) → ( √ ∘ ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) ) = ( 𝑥 ∈ ℝ ↦ ( √ ‘ ( ( ( 𝐹 ‘ 𝑥 ) · ( 𝐹 ‘ 𝑥 ) ) + ( ( 𝐺 ‘ 𝑥 ) · ( 𝐺 ‘ 𝑥 ) ) ) ) ) ) |
60 |
16 41 59
|
3eqtr4d |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) → ( abs ∘ ( 𝐹 ∘f + ( ( ℝ × { i } ) ∘f · 𝐺 ) ) ) = ( √ ∘ ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) ) ) |
61 |
|
elrege0 |
⊢ ( 𝑥 ∈ ( 0 [,) +∞ ) ↔ ( 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥 ) ) |
62 |
|
resqrtcl |
⊢ ( ( 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥 ) → ( √ ‘ 𝑥 ) ∈ ℝ ) |
63 |
61 62
|
sylbi |
⊢ ( 𝑥 ∈ ( 0 [,) +∞ ) → ( √ ‘ 𝑥 ) ∈ ℝ ) |
64 |
63
|
adantl |
⊢ ( ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) ∧ 𝑥 ∈ ( 0 [,) +∞ ) ) → ( √ ‘ 𝑥 ) ∈ ℝ ) |
65 |
|
id |
⊢ ( √ : ℂ ⟶ ℂ → √ : ℂ ⟶ ℂ ) |
66 |
65
|
feqmptd |
⊢ ( √ : ℂ ⟶ ℂ → √ = ( 𝑥 ∈ ℂ ↦ ( √ ‘ 𝑥 ) ) ) |
67 |
55 66
|
ax-mp |
⊢ √ = ( 𝑥 ∈ ℂ ↦ ( √ ‘ 𝑥 ) ) |
68 |
67
|
reseq1i |
⊢ ( √ ↾ ( 0 [,) +∞ ) ) = ( ( 𝑥 ∈ ℂ ↦ ( √ ‘ 𝑥 ) ) ↾ ( 0 [,) +∞ ) ) |
69 |
|
rge0ssre |
⊢ ( 0 [,) +∞ ) ⊆ ℝ |
70 |
|
ax-resscn |
⊢ ℝ ⊆ ℂ |
71 |
69 70
|
sstri |
⊢ ( 0 [,) +∞ ) ⊆ ℂ |
72 |
|
resmpt |
⊢ ( ( 0 [,) +∞ ) ⊆ ℂ → ( ( 𝑥 ∈ ℂ ↦ ( √ ‘ 𝑥 ) ) ↾ ( 0 [,) +∞ ) ) = ( 𝑥 ∈ ( 0 [,) +∞ ) ↦ ( √ ‘ 𝑥 ) ) ) |
73 |
71 72
|
ax-mp |
⊢ ( ( 𝑥 ∈ ℂ ↦ ( √ ‘ 𝑥 ) ) ↾ ( 0 [,) +∞ ) ) = ( 𝑥 ∈ ( 0 [,) +∞ ) ↦ ( √ ‘ 𝑥 ) ) |
74 |
68 73
|
eqtri |
⊢ ( √ ↾ ( 0 [,) +∞ ) ) = ( 𝑥 ∈ ( 0 [,) +∞ ) ↦ ( √ ‘ 𝑥 ) ) |
75 |
64 74
|
fmptd |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) → ( √ ↾ ( 0 [,) +∞ ) ) : ( 0 [,) +∞ ) ⟶ ℝ ) |
76 |
|
ge0addcl |
⊢ ( ( 𝑥 ∈ ( 0 [,) +∞ ) ∧ 𝑦 ∈ ( 0 [,) +∞ ) ) → ( 𝑥 + 𝑦 ) ∈ ( 0 [,) +∞ ) ) |
77 |
76
|
adantl |
⊢ ( ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) ∧ ( 𝑥 ∈ ( 0 [,) +∞ ) ∧ 𝑦 ∈ ( 0 [,) +∞ ) ) ) → ( 𝑥 + 𝑦 ) ∈ ( 0 [,) +∞ ) ) |
78 |
|
oveq12 |
⊢ ( ( 𝑧 = 𝐹 ∧ 𝑧 = 𝐹 ) → ( 𝑧 ∘f · 𝑧 ) = ( 𝐹 ∘f · 𝐹 ) ) |
79 |
78
|
anidms |
⊢ ( 𝑧 = 𝐹 → ( 𝑧 ∘f · 𝑧 ) = ( 𝐹 ∘f · 𝐹 ) ) |
80 |
79
|
feq1d |
⊢ ( 𝑧 = 𝐹 → ( ( 𝑧 ∘f · 𝑧 ) : ℝ ⟶ ( 0 [,) +∞ ) ↔ ( 𝐹 ∘f · 𝐹 ) : ℝ ⟶ ( 0 [,) +∞ ) ) ) |
81 |
|
i1ff |
⊢ ( 𝑧 ∈ dom ∫1 → 𝑧 : ℝ ⟶ ℝ ) |
82 |
81
|
ffvelrnda |
⊢ ( ( 𝑧 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ ) → ( 𝑧 ‘ 𝑥 ) ∈ ℝ ) |
83 |
82 82
|
remulcld |
⊢ ( ( 𝑧 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ ) → ( ( 𝑧 ‘ 𝑥 ) · ( 𝑧 ‘ 𝑥 ) ) ∈ ℝ ) |
84 |
82
|
msqge0d |
⊢ ( ( 𝑧 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ ) → 0 ≤ ( ( 𝑧 ‘ 𝑥 ) · ( 𝑧 ‘ 𝑥 ) ) ) |
85 |
|
elrege0 |
⊢ ( ( ( 𝑧 ‘ 𝑥 ) · ( 𝑧 ‘ 𝑥 ) ) ∈ ( 0 [,) +∞ ) ↔ ( ( ( 𝑧 ‘ 𝑥 ) · ( 𝑧 ‘ 𝑥 ) ) ∈ ℝ ∧ 0 ≤ ( ( 𝑧 ‘ 𝑥 ) · ( 𝑧 ‘ 𝑥 ) ) ) ) |
86 |
83 84 85
|
sylanbrc |
⊢ ( ( 𝑧 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ ) → ( ( 𝑧 ‘ 𝑥 ) · ( 𝑧 ‘ 𝑥 ) ) ∈ ( 0 [,) +∞ ) ) |
87 |
86
|
fmpttd |
⊢ ( 𝑧 ∈ dom ∫1 → ( 𝑥 ∈ ℝ ↦ ( ( 𝑧 ‘ 𝑥 ) · ( 𝑧 ‘ 𝑥 ) ) ) : ℝ ⟶ ( 0 [,) +∞ ) ) |
88 |
23
|
a1i |
⊢ ( 𝑧 ∈ dom ∫1 → ℝ ∈ V ) |
89 |
81
|
feqmptd |
⊢ ( 𝑧 ∈ dom ∫1 → 𝑧 = ( 𝑥 ∈ ℝ ↦ ( 𝑧 ‘ 𝑥 ) ) ) |
90 |
88 82 82 89 89
|
offval2 |
⊢ ( 𝑧 ∈ dom ∫1 → ( 𝑧 ∘f · 𝑧 ) = ( 𝑥 ∈ ℝ ↦ ( ( 𝑧 ‘ 𝑥 ) · ( 𝑧 ‘ 𝑥 ) ) ) ) |
91 |
90
|
feq1d |
⊢ ( 𝑧 ∈ dom ∫1 → ( ( 𝑧 ∘f · 𝑧 ) : ℝ ⟶ ( 0 [,) +∞ ) ↔ ( 𝑥 ∈ ℝ ↦ ( ( 𝑧 ‘ 𝑥 ) · ( 𝑧 ‘ 𝑥 ) ) ) : ℝ ⟶ ( 0 [,) +∞ ) ) ) |
92 |
87 91
|
mpbird |
⊢ ( 𝑧 ∈ dom ∫1 → ( 𝑧 ∘f · 𝑧 ) : ℝ ⟶ ( 0 [,) +∞ ) ) |
93 |
80 92
|
vtoclga |
⊢ ( 𝐹 ∈ dom ∫1 → ( 𝐹 ∘f · 𝐹 ) : ℝ ⟶ ( 0 [,) +∞ ) ) |
94 |
93
|
adantr |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) → ( 𝐹 ∘f · 𝐹 ) : ℝ ⟶ ( 0 [,) +∞ ) ) |
95 |
|
oveq12 |
⊢ ( ( 𝑧 = 𝐺 ∧ 𝑧 = 𝐺 ) → ( 𝑧 ∘f · 𝑧 ) = ( 𝐺 ∘f · 𝐺 ) ) |
96 |
95
|
anidms |
⊢ ( 𝑧 = 𝐺 → ( 𝑧 ∘f · 𝑧 ) = ( 𝐺 ∘f · 𝐺 ) ) |
97 |
96
|
feq1d |
⊢ ( 𝑧 = 𝐺 → ( ( 𝑧 ∘f · 𝑧 ) : ℝ ⟶ ( 0 [,) +∞ ) ↔ ( 𝐺 ∘f · 𝐺 ) : ℝ ⟶ ( 0 [,) +∞ ) ) ) |
98 |
97 92
|
vtoclga |
⊢ ( 𝐺 ∈ dom ∫1 → ( 𝐺 ∘f · 𝐺 ) : ℝ ⟶ ( 0 [,) +∞ ) ) |
99 |
98
|
adantl |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) → ( 𝐺 ∘f · 𝐺 ) : ℝ ⟶ ( 0 [,) +∞ ) ) |
100 |
|
inidm |
⊢ ( ℝ ∩ ℝ ) = ℝ |
101 |
77 94 99 24 24 100
|
off |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) → ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) : ℝ ⟶ ( 0 [,) +∞ ) ) |
102 |
|
fco2 |
⊢ ( ( ( √ ↾ ( 0 [,) +∞ ) ) : ( 0 [,) +∞ ) ⟶ ℝ ∧ ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) : ℝ ⟶ ( 0 [,) +∞ ) ) → ( √ ∘ ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) ) : ℝ ⟶ ℝ ) |
103 |
75 101 102
|
syl2anc |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) → ( √ ∘ ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) ) : ℝ ⟶ ℝ ) |
104 |
|
rnco |
⊢ ran ( √ ∘ ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) ) = ran ( √ ↾ ran ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) ) |
105 |
|
ffn |
⊢ ( √ : ℂ ⟶ ℂ → √ Fn ℂ ) |
106 |
55 105
|
ax-mp |
⊢ √ Fn ℂ |
107 |
|
readdcl |
⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( 𝑥 + 𝑦 ) ∈ ℝ ) |
108 |
107
|
adantl |
⊢ ( ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) ∧ ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ) → ( 𝑥 + 𝑦 ) ∈ ℝ ) |
109 |
|
remulcl |
⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( 𝑥 · 𝑦 ) ∈ ℝ ) |
110 |
109
|
adantl |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ) → ( 𝑥 · 𝑦 ) ∈ ℝ ) |
111 |
110 1 1 49 49 100
|
off |
⊢ ( 𝐹 ∈ dom ∫1 → ( 𝐹 ∘f · 𝐹 ) : ℝ ⟶ ℝ ) |
112 |
111
|
adantr |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) → ( 𝐹 ∘f · 𝐹 ) : ℝ ⟶ ℝ ) |
113 |
109
|
adantl |
⊢ ( ( 𝐺 ∈ dom ∫1 ∧ ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ) → ( 𝑥 · 𝑦 ) ∈ ℝ ) |
114 |
113 3 3 29 29 100
|
off |
⊢ ( 𝐺 ∈ dom ∫1 → ( 𝐺 ∘f · 𝐺 ) : ℝ ⟶ ℝ ) |
115 |
114
|
adantl |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) → ( 𝐺 ∘f · 𝐺 ) : ℝ ⟶ ℝ ) |
116 |
108 112 115 24 24 100
|
off |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) → ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) : ℝ ⟶ ℝ ) |
117 |
116
|
frnd |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) → ran ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) ⊆ ℝ ) |
118 |
117 70
|
sstrdi |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) → ran ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) ⊆ ℂ ) |
119 |
|
fnssres |
⊢ ( ( √ Fn ℂ ∧ ran ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) ⊆ ℂ ) → ( √ ↾ ran ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) ) Fn ran ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) ) |
120 |
106 118 119
|
sylancr |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) → ( √ ↾ ran ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) ) Fn ran ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) ) |
121 |
|
id |
⊢ ( 𝐹 ∈ dom ∫1 → 𝐹 ∈ dom ∫1 ) |
122 |
121 121
|
i1fmul |
⊢ ( 𝐹 ∈ dom ∫1 → ( 𝐹 ∘f · 𝐹 ) ∈ dom ∫1 ) |
123 |
122
|
adantr |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) → ( 𝐹 ∘f · 𝐹 ) ∈ dom ∫1 ) |
124 |
|
id |
⊢ ( 𝐺 ∈ dom ∫1 → 𝐺 ∈ dom ∫1 ) |
125 |
124 124
|
i1fmul |
⊢ ( 𝐺 ∈ dom ∫1 → ( 𝐺 ∘f · 𝐺 ) ∈ dom ∫1 ) |
126 |
125
|
adantl |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) → ( 𝐺 ∘f · 𝐺 ) ∈ dom ∫1 ) |
127 |
123 126
|
i1fadd |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) → ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) ∈ dom ∫1 ) |
128 |
|
i1frn |
⊢ ( ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) ∈ dom ∫1 → ran ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) ∈ Fin ) |
129 |
127 128
|
syl |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) → ran ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) ∈ Fin ) |
130 |
|
fnfi |
⊢ ( ( ( √ ↾ ran ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) ) Fn ran ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) ∧ ran ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) ∈ Fin ) → ( √ ↾ ran ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) ) ∈ Fin ) |
131 |
120 129 130
|
syl2anc |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) → ( √ ↾ ran ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) ) ∈ Fin ) |
132 |
|
rnfi |
⊢ ( ( √ ↾ ran ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) ) ∈ Fin → ran ( √ ↾ ran ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) ) ∈ Fin ) |
133 |
131 132
|
syl |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) → ran ( √ ↾ ran ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) ) ∈ Fin ) |
134 |
104 133
|
eqeltrid |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) → ran ( √ ∘ ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) ) ∈ Fin ) |
135 |
|
cnvco |
⊢ ◡ ( √ ∘ ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) ) = ( ◡ ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) ∘ ◡ √ ) |
136 |
135
|
imaeq1i |
⊢ ( ◡ ( √ ∘ ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) ) “ { 𝑥 } ) = ( ( ◡ ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) ∘ ◡ √ ) “ { 𝑥 } ) |
137 |
|
imaco |
⊢ ( ( ◡ ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) ∘ ◡ √ ) “ { 𝑥 } ) = ( ◡ ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) “ ( ◡ √ “ { 𝑥 } ) ) |
138 |
136 137
|
eqtri |
⊢ ( ◡ ( √ ∘ ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) ) “ { 𝑥 } ) = ( ◡ ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) “ ( ◡ √ “ { 𝑥 } ) ) |
139 |
|
i1fima |
⊢ ( ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) ∈ dom ∫1 → ( ◡ ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) “ ( ◡ √ “ { 𝑥 } ) ) ∈ dom vol ) |
140 |
127 139
|
syl |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) → ( ◡ ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) “ ( ◡ √ “ { 𝑥 } ) ) ∈ dom vol ) |
141 |
138 140
|
eqeltrid |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) → ( ◡ ( √ ∘ ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) ) “ { 𝑥 } ) ∈ dom vol ) |
142 |
141
|
adantr |
⊢ ( ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) ∧ 𝑥 ∈ ( ran ( √ ∘ ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) ) ∖ { 0 } ) ) → ( ◡ ( √ ∘ ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) ) “ { 𝑥 } ) ∈ dom vol ) |
143 |
138
|
fveq2i |
⊢ ( vol ‘ ( ◡ ( √ ∘ ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) ) “ { 𝑥 } ) ) = ( vol ‘ ( ◡ ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) “ ( ◡ √ “ { 𝑥 } ) ) ) |
144 |
|
eldifsni |
⊢ ( 𝑥 ∈ ( ran ( √ ∘ ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) ) ∖ { 0 } ) → 𝑥 ≠ 0 ) |
145 |
|
c0ex |
⊢ 0 ∈ V |
146 |
145
|
elsn |
⊢ ( 0 ∈ { 𝑥 } ↔ 0 = 𝑥 ) |
147 |
|
eqcom |
⊢ ( 0 = 𝑥 ↔ 𝑥 = 0 ) |
148 |
146 147
|
bitri |
⊢ ( 0 ∈ { 𝑥 } ↔ 𝑥 = 0 ) |
149 |
148
|
necon3bbii |
⊢ ( ¬ 0 ∈ { 𝑥 } ↔ 𝑥 ≠ 0 ) |
150 |
|
sqrt0 |
⊢ ( √ ‘ 0 ) = 0 |
151 |
150
|
eleq1i |
⊢ ( ( √ ‘ 0 ) ∈ { 𝑥 } ↔ 0 ∈ { 𝑥 } ) |
152 |
149 151
|
xchnxbir |
⊢ ( ¬ ( √ ‘ 0 ) ∈ { 𝑥 } ↔ 𝑥 ≠ 0 ) |
153 |
144 152
|
sylibr |
⊢ ( 𝑥 ∈ ( ran ( √ ∘ ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) ) ∖ { 0 } ) → ¬ ( √ ‘ 0 ) ∈ { 𝑥 } ) |
154 |
153
|
olcd |
⊢ ( 𝑥 ∈ ( ran ( √ ∘ ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) ) ∖ { 0 } ) → ( ¬ 0 ∈ ℂ ∨ ¬ ( √ ‘ 0 ) ∈ { 𝑥 } ) ) |
155 |
|
ianor |
⊢ ( ¬ ( 0 ∈ ℂ ∧ ( √ ‘ 0 ) ∈ { 𝑥 } ) ↔ ( ¬ 0 ∈ ℂ ∨ ¬ ( √ ‘ 0 ) ∈ { 𝑥 } ) ) |
156 |
|
elpreima |
⊢ ( √ Fn ℂ → ( 0 ∈ ( ◡ √ “ { 𝑥 } ) ↔ ( 0 ∈ ℂ ∧ ( √ ‘ 0 ) ∈ { 𝑥 } ) ) ) |
157 |
55 105 156
|
mp2b |
⊢ ( 0 ∈ ( ◡ √ “ { 𝑥 } ) ↔ ( 0 ∈ ℂ ∧ ( √ ‘ 0 ) ∈ { 𝑥 } ) ) |
158 |
155 157
|
xchnxbir |
⊢ ( ¬ 0 ∈ ( ◡ √ “ { 𝑥 } ) ↔ ( ¬ 0 ∈ ℂ ∨ ¬ ( √ ‘ 0 ) ∈ { 𝑥 } ) ) |
159 |
154 158
|
sylibr |
⊢ ( 𝑥 ∈ ( ran ( √ ∘ ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) ) ∖ { 0 } ) → ¬ 0 ∈ ( ◡ √ “ { 𝑥 } ) ) |
160 |
|
i1fima2 |
⊢ ( ( ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) ∈ dom ∫1 ∧ ¬ 0 ∈ ( ◡ √ “ { 𝑥 } ) ) → ( vol ‘ ( ◡ ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) “ ( ◡ √ “ { 𝑥 } ) ) ) ∈ ℝ ) |
161 |
127 159 160
|
syl2an |
⊢ ( ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) ∧ 𝑥 ∈ ( ran ( √ ∘ ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) ) ∖ { 0 } ) ) → ( vol ‘ ( ◡ ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) “ ( ◡ √ “ { 𝑥 } ) ) ) ∈ ℝ ) |
162 |
143 161
|
eqeltrid |
⊢ ( ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) ∧ 𝑥 ∈ ( ran ( √ ∘ ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) ) ∖ { 0 } ) ) → ( vol ‘ ( ◡ ( √ ∘ ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) ) “ { 𝑥 } ) ) ∈ ℝ ) |
163 |
103 134 142 162
|
i1fd |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) → ( √ ∘ ( ( 𝐹 ∘f · 𝐹 ) ∘f + ( 𝐺 ∘f · 𝐺 ) ) ) ∈ dom ∫1 ) |
164 |
60 163
|
eqeltrd |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ) → ( abs ∘ ( 𝐹 ∘f + ( ( ℝ × { i } ) ∘f · 𝐺 ) ) ) ∈ dom ∫1 ) |