Step |
Hyp |
Ref |
Expression |
1 |
|
i1fmulc.2 |
⊢ ( 𝜑 → 𝐹 ∈ dom ∫1 ) |
2 |
|
i1fmulc.3 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
3 |
|
itg10 |
⊢ ( ∫1 ‘ ( ℝ × { 0 } ) ) = 0 |
4 |
|
reex |
⊢ ℝ ∈ V |
5 |
4
|
a1i |
⊢ ( ( 𝜑 ∧ 𝐴 = 0 ) → ℝ ∈ V ) |
6 |
|
i1ff |
⊢ ( 𝐹 ∈ dom ∫1 → 𝐹 : ℝ ⟶ ℝ ) |
7 |
1 6
|
syl |
⊢ ( 𝜑 → 𝐹 : ℝ ⟶ ℝ ) |
8 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 = 0 ) → 𝐹 : ℝ ⟶ ℝ ) |
9 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 = 0 ) → 𝐴 ∈ ℝ ) |
10 |
|
0red |
⊢ ( ( 𝜑 ∧ 𝐴 = 0 ) → 0 ∈ ℝ ) |
11 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝐴 = 0 ) ∧ 𝑥 ∈ ℝ ) → 𝐴 = 0 ) |
12 |
11
|
oveq1d |
⊢ ( ( ( 𝜑 ∧ 𝐴 = 0 ) ∧ 𝑥 ∈ ℝ ) → ( 𝐴 · 𝑥 ) = ( 0 · 𝑥 ) ) |
13 |
|
mul02lem2 |
⊢ ( 𝑥 ∈ ℝ → ( 0 · 𝑥 ) = 0 ) |
14 |
13
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝐴 = 0 ) ∧ 𝑥 ∈ ℝ ) → ( 0 · 𝑥 ) = 0 ) |
15 |
12 14
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝐴 = 0 ) ∧ 𝑥 ∈ ℝ ) → ( 𝐴 · 𝑥 ) = 0 ) |
16 |
5 8 9 10 15
|
caofid2 |
⊢ ( ( 𝜑 ∧ 𝐴 = 0 ) → ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) = ( ℝ × { 0 } ) ) |
17 |
16
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝐴 = 0 ) → ( ∫1 ‘ ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ) = ( ∫1 ‘ ( ℝ × { 0 } ) ) ) |
18 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐴 = 0 ) → 𝐴 = 0 ) |
19 |
18
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝐴 = 0 ) → ( 𝐴 · ( ∫1 ‘ 𝐹 ) ) = ( 0 · ( ∫1 ‘ 𝐹 ) ) ) |
20 |
|
itg1cl |
⊢ ( 𝐹 ∈ dom ∫1 → ( ∫1 ‘ 𝐹 ) ∈ ℝ ) |
21 |
1 20
|
syl |
⊢ ( 𝜑 → ( ∫1 ‘ 𝐹 ) ∈ ℝ ) |
22 |
21
|
recnd |
⊢ ( 𝜑 → ( ∫1 ‘ 𝐹 ) ∈ ℂ ) |
23 |
22
|
mul02d |
⊢ ( 𝜑 → ( 0 · ( ∫1 ‘ 𝐹 ) ) = 0 ) |
24 |
23
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 = 0 ) → ( 0 · ( ∫1 ‘ 𝐹 ) ) = 0 ) |
25 |
19 24
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝐴 = 0 ) → ( 𝐴 · ( ∫1 ‘ 𝐹 ) ) = 0 ) |
26 |
3 17 25
|
3eqtr4a |
⊢ ( ( 𝜑 ∧ 𝐴 = 0 ) → ( ∫1 ‘ ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ) = ( 𝐴 · ( ∫1 ‘ 𝐹 ) ) ) |
27 |
1 2
|
i1fmulc |
⊢ ( 𝜑 → ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∈ dom ∫1 ) |
28 |
27
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ 0 ) → ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∈ dom ∫1 ) |
29 |
|
i1ff |
⊢ ( ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∈ dom ∫1 → ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) : ℝ ⟶ ℝ ) |
30 |
28 29
|
syl |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ 0 ) → ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) : ℝ ⟶ ℝ ) |
31 |
30
|
frnd |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ 0 ) → ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ⊆ ℝ ) |
32 |
31
|
ssdifssd |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ 0 ) → ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ⊆ ℝ ) |
33 |
32
|
sselda |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑚 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) → 𝑚 ∈ ℝ ) |
34 |
33
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑚 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) → 𝑚 ∈ ℂ ) |
35 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ 0 ) → 𝐴 ∈ ℝ ) |
36 |
35
|
recnd |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ 0 ) → 𝐴 ∈ ℂ ) |
37 |
36
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑚 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) → 𝐴 ∈ ℂ ) |
38 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑚 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) → 𝐴 ≠ 0 ) |
39 |
34 37 38
|
divcan2d |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑚 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) → ( 𝐴 · ( 𝑚 / 𝐴 ) ) = 𝑚 ) |
40 |
1 2
|
i1fmulclem |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑚 ∈ ℝ ) → ( ◡ ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) “ { 𝑚 } ) = ( ◡ 𝐹 “ { ( 𝑚 / 𝐴 ) } ) ) |
41 |
33 40
|
syldan |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑚 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) → ( ◡ ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) “ { 𝑚 } ) = ( ◡ 𝐹 “ { ( 𝑚 / 𝐴 ) } ) ) |
42 |
41
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑚 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) → ( vol ‘ ( ◡ ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) “ { 𝑚 } ) ) = ( vol ‘ ( ◡ 𝐹 “ { ( 𝑚 / 𝐴 ) } ) ) ) |
43 |
42
|
eqcomd |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑚 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) → ( vol ‘ ( ◡ 𝐹 “ { ( 𝑚 / 𝐴 ) } ) ) = ( vol ‘ ( ◡ ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) “ { 𝑚 } ) ) ) |
44 |
39 43
|
oveq12d |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑚 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) → ( ( 𝐴 · ( 𝑚 / 𝐴 ) ) · ( vol ‘ ( ◡ 𝐹 “ { ( 𝑚 / 𝐴 ) } ) ) ) = ( 𝑚 · ( vol ‘ ( ◡ ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) “ { 𝑚 } ) ) ) ) |
45 |
2
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑚 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) → 𝐴 ∈ ℝ ) |
46 |
33 45 38
|
redivcld |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑚 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) → ( 𝑚 / 𝐴 ) ∈ ℝ ) |
47 |
46
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑚 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) → ( 𝑚 / 𝐴 ) ∈ ℂ ) |
48 |
1
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑚 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) → 𝐹 ∈ dom ∫1 ) |
49 |
45
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑚 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) → 𝐴 ∈ ℂ ) |
50 |
|
eldifsni |
⊢ ( 𝑚 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) → 𝑚 ≠ 0 ) |
51 |
50
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑚 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) → 𝑚 ≠ 0 ) |
52 |
34 49 51 38
|
divne0d |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑚 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) → ( 𝑚 / 𝐴 ) ≠ 0 ) |
53 |
|
eldifsn |
⊢ ( ( 𝑚 / 𝐴 ) ∈ ( ℝ ∖ { 0 } ) ↔ ( ( 𝑚 / 𝐴 ) ∈ ℝ ∧ ( 𝑚 / 𝐴 ) ≠ 0 ) ) |
54 |
46 52 53
|
sylanbrc |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑚 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) → ( 𝑚 / 𝐴 ) ∈ ( ℝ ∖ { 0 } ) ) |
55 |
|
i1fima2sn |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ ( 𝑚 / 𝐴 ) ∈ ( ℝ ∖ { 0 } ) ) → ( vol ‘ ( ◡ 𝐹 “ { ( 𝑚 / 𝐴 ) } ) ) ∈ ℝ ) |
56 |
48 54 55
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑚 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) → ( vol ‘ ( ◡ 𝐹 “ { ( 𝑚 / 𝐴 ) } ) ) ∈ ℝ ) |
57 |
56
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑚 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) → ( vol ‘ ( ◡ 𝐹 “ { ( 𝑚 / 𝐴 ) } ) ) ∈ ℂ ) |
58 |
37 47 57
|
mulassd |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑚 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) → ( ( 𝐴 · ( 𝑚 / 𝐴 ) ) · ( vol ‘ ( ◡ 𝐹 “ { ( 𝑚 / 𝐴 ) } ) ) ) = ( 𝐴 · ( ( 𝑚 / 𝐴 ) · ( vol ‘ ( ◡ 𝐹 “ { ( 𝑚 / 𝐴 ) } ) ) ) ) ) |
59 |
44 58
|
eqtr3d |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑚 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) → ( 𝑚 · ( vol ‘ ( ◡ ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) “ { 𝑚 } ) ) ) = ( 𝐴 · ( ( 𝑚 / 𝐴 ) · ( vol ‘ ( ◡ 𝐹 “ { ( 𝑚 / 𝐴 ) } ) ) ) ) ) |
60 |
59
|
sumeq2dv |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ 0 ) → Σ 𝑚 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ( 𝑚 · ( vol ‘ ( ◡ ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) “ { 𝑚 } ) ) ) = Σ 𝑚 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ( 𝐴 · ( ( 𝑚 / 𝐴 ) · ( vol ‘ ( ◡ 𝐹 “ { ( 𝑚 / 𝐴 ) } ) ) ) ) ) |
61 |
|
i1frn |
⊢ ( ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∈ dom ∫1 → ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∈ Fin ) |
62 |
28 61
|
syl |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ 0 ) → ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∈ Fin ) |
63 |
|
difss |
⊢ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ⊆ ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) |
64 |
|
ssfi |
⊢ ( ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∈ Fin ∧ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ⊆ ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ) → ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ∈ Fin ) |
65 |
62 63 64
|
sylancl |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ 0 ) → ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ∈ Fin ) |
66 |
47 57
|
mulcld |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑚 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) → ( ( 𝑚 / 𝐴 ) · ( vol ‘ ( ◡ 𝐹 “ { ( 𝑚 / 𝐴 ) } ) ) ) ∈ ℂ ) |
67 |
65 36 66
|
fsummulc2 |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ 0 ) → ( 𝐴 · Σ 𝑚 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ( ( 𝑚 / 𝐴 ) · ( vol ‘ ( ◡ 𝐹 “ { ( 𝑚 / 𝐴 ) } ) ) ) ) = Σ 𝑚 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ( 𝐴 · ( ( 𝑚 / 𝐴 ) · ( vol ‘ ( ◡ 𝐹 “ { ( 𝑚 / 𝐴 ) } ) ) ) ) ) |
68 |
60 67
|
eqtr4d |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ 0 ) → Σ 𝑚 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ( 𝑚 · ( vol ‘ ( ◡ ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) “ { 𝑚 } ) ) ) = ( 𝐴 · Σ 𝑚 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ( ( 𝑚 / 𝐴 ) · ( vol ‘ ( ◡ 𝐹 “ { ( 𝑚 / 𝐴 ) } ) ) ) ) ) |
69 |
|
itg1val |
⊢ ( ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∈ dom ∫1 → ( ∫1 ‘ ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ) = Σ 𝑚 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ( 𝑚 · ( vol ‘ ( ◡ ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) “ { 𝑚 } ) ) ) ) |
70 |
28 69
|
syl |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ 0 ) → ( ∫1 ‘ ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ) = Σ 𝑚 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ( 𝑚 · ( vol ‘ ( ◡ ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) “ { 𝑚 } ) ) ) ) |
71 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ 0 ) → 𝐹 ∈ dom ∫1 ) |
72 |
|
itg1val |
⊢ ( 𝐹 ∈ dom ∫1 → ( ∫1 ‘ 𝐹 ) = Σ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ( 𝑘 · ( vol ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) ) ) |
73 |
71 72
|
syl |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ 0 ) → ( ∫1 ‘ 𝐹 ) = Σ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ( 𝑘 · ( vol ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) ) ) |
74 |
|
id |
⊢ ( 𝑘 = ( 𝑚 / 𝐴 ) → 𝑘 = ( 𝑚 / 𝐴 ) ) |
75 |
|
sneq |
⊢ ( 𝑘 = ( 𝑚 / 𝐴 ) → { 𝑘 } = { ( 𝑚 / 𝐴 ) } ) |
76 |
75
|
imaeq2d |
⊢ ( 𝑘 = ( 𝑚 / 𝐴 ) → ( ◡ 𝐹 “ { 𝑘 } ) = ( ◡ 𝐹 “ { ( 𝑚 / 𝐴 ) } ) ) |
77 |
76
|
fveq2d |
⊢ ( 𝑘 = ( 𝑚 / 𝐴 ) → ( vol ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) = ( vol ‘ ( ◡ 𝐹 “ { ( 𝑚 / 𝐴 ) } ) ) ) |
78 |
74 77
|
oveq12d |
⊢ ( 𝑘 = ( 𝑚 / 𝐴 ) → ( 𝑘 · ( vol ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) ) = ( ( 𝑚 / 𝐴 ) · ( vol ‘ ( ◡ 𝐹 “ { ( 𝑚 / 𝐴 ) } ) ) ) ) |
79 |
|
eqid |
⊢ ( 𝑛 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ↦ ( 𝑛 / 𝐴 ) ) = ( 𝑛 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ↦ ( 𝑛 / 𝐴 ) ) |
80 |
|
eldifi |
⊢ ( 𝑛 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) → 𝑛 ∈ ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ) |
81 |
4
|
a1i |
⊢ ( 𝜑 → ℝ ∈ V ) |
82 |
7
|
ffnd |
⊢ ( 𝜑 → 𝐹 Fn ℝ ) |
83 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) ) |
84 |
81 2 82 83
|
ofc1 |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ‘ 𝑦 ) = ( 𝐴 · ( 𝐹 ‘ 𝑦 ) ) ) |
85 |
84
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑦 ∈ ℝ ) → ( ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ‘ 𝑦 ) = ( 𝐴 · ( 𝐹 ‘ 𝑦 ) ) ) |
86 |
85
|
oveq1d |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑦 ∈ ℝ ) → ( ( ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ‘ 𝑦 ) / 𝐴 ) = ( ( 𝐴 · ( 𝐹 ‘ 𝑦 ) ) / 𝐴 ) ) |
87 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ 0 ) → 𝐹 : ℝ ⟶ ℝ ) |
88 |
87
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑦 ∈ ℝ ) → ( 𝐹 ‘ 𝑦 ) ∈ ℝ ) |
89 |
88
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑦 ∈ ℝ ) → ( 𝐹 ‘ 𝑦 ) ∈ ℂ ) |
90 |
36
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑦 ∈ ℝ ) → 𝐴 ∈ ℂ ) |
91 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑦 ∈ ℝ ) → 𝐴 ≠ 0 ) |
92 |
89 90 91
|
divcan3d |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑦 ∈ ℝ ) → ( ( 𝐴 · ( 𝐹 ‘ 𝑦 ) ) / 𝐴 ) = ( 𝐹 ‘ 𝑦 ) ) |
93 |
86 92
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑦 ∈ ℝ ) → ( ( ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ‘ 𝑦 ) / 𝐴 ) = ( 𝐹 ‘ 𝑦 ) ) |
94 |
87
|
ffnd |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ 0 ) → 𝐹 Fn ℝ ) |
95 |
|
fnfvelrn |
⊢ ( ( 𝐹 Fn ℝ ∧ 𝑦 ∈ ℝ ) → ( 𝐹 ‘ 𝑦 ) ∈ ran 𝐹 ) |
96 |
94 95
|
sylan |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑦 ∈ ℝ ) → ( 𝐹 ‘ 𝑦 ) ∈ ran 𝐹 ) |
97 |
93 96
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑦 ∈ ℝ ) → ( ( ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ‘ 𝑦 ) / 𝐴 ) ∈ ran 𝐹 ) |
98 |
97
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ 0 ) → ∀ 𝑦 ∈ ℝ ( ( ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ‘ 𝑦 ) / 𝐴 ) ∈ ran 𝐹 ) |
99 |
30
|
ffnd |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ 0 ) → ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) Fn ℝ ) |
100 |
|
oveq1 |
⊢ ( 𝑛 = ( ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ‘ 𝑦 ) → ( 𝑛 / 𝐴 ) = ( ( ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ‘ 𝑦 ) / 𝐴 ) ) |
101 |
100
|
eleq1d |
⊢ ( 𝑛 = ( ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ‘ 𝑦 ) → ( ( 𝑛 / 𝐴 ) ∈ ran 𝐹 ↔ ( ( ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ‘ 𝑦 ) / 𝐴 ) ∈ ran 𝐹 ) ) |
102 |
101
|
ralrn |
⊢ ( ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) Fn ℝ → ( ∀ 𝑛 ∈ ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ( 𝑛 / 𝐴 ) ∈ ran 𝐹 ↔ ∀ 𝑦 ∈ ℝ ( ( ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ‘ 𝑦 ) / 𝐴 ) ∈ ran 𝐹 ) ) |
103 |
99 102
|
syl |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ 0 ) → ( ∀ 𝑛 ∈ ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ( 𝑛 / 𝐴 ) ∈ ran 𝐹 ↔ ∀ 𝑦 ∈ ℝ ( ( ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ‘ 𝑦 ) / 𝐴 ) ∈ ran 𝐹 ) ) |
104 |
98 103
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ 0 ) → ∀ 𝑛 ∈ ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ( 𝑛 / 𝐴 ) ∈ ran 𝐹 ) |
105 |
104
|
r19.21bi |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑛 ∈ ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ) → ( 𝑛 / 𝐴 ) ∈ ran 𝐹 ) |
106 |
80 105
|
sylan2 |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑛 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) → ( 𝑛 / 𝐴 ) ∈ ran 𝐹 ) |
107 |
32
|
sselda |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑛 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) → 𝑛 ∈ ℝ ) |
108 |
107
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑛 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) → 𝑛 ∈ ℂ ) |
109 |
36
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑛 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) → 𝐴 ∈ ℂ ) |
110 |
|
eldifsni |
⊢ ( 𝑛 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) → 𝑛 ≠ 0 ) |
111 |
110
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑛 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) → 𝑛 ≠ 0 ) |
112 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑛 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) → 𝐴 ≠ 0 ) |
113 |
108 109 111 112
|
divne0d |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑛 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) → ( 𝑛 / 𝐴 ) ≠ 0 ) |
114 |
|
eldifsn |
⊢ ( ( 𝑛 / 𝐴 ) ∈ ( ran 𝐹 ∖ { 0 } ) ↔ ( ( 𝑛 / 𝐴 ) ∈ ran 𝐹 ∧ ( 𝑛 / 𝐴 ) ≠ 0 ) ) |
115 |
106 113 114
|
sylanbrc |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑛 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) → ( 𝑛 / 𝐴 ) ∈ ( ran 𝐹 ∖ { 0 } ) ) |
116 |
|
eldifi |
⊢ ( 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) → 𝑘 ∈ ran 𝐹 ) |
117 |
|
fnfvelrn |
⊢ ( ( ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) Fn ℝ ∧ 𝑦 ∈ ℝ ) → ( ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ‘ 𝑦 ) ∈ ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ) |
118 |
99 117
|
sylan |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑦 ∈ ℝ ) → ( ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ‘ 𝑦 ) ∈ ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ) |
119 |
85 118
|
eqeltrrd |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑦 ∈ ℝ ) → ( 𝐴 · ( 𝐹 ‘ 𝑦 ) ) ∈ ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ) |
120 |
119
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ 0 ) → ∀ 𝑦 ∈ ℝ ( 𝐴 · ( 𝐹 ‘ 𝑦 ) ) ∈ ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ) |
121 |
|
oveq2 |
⊢ ( 𝑘 = ( 𝐹 ‘ 𝑦 ) → ( 𝐴 · 𝑘 ) = ( 𝐴 · ( 𝐹 ‘ 𝑦 ) ) ) |
122 |
121
|
eleq1d |
⊢ ( 𝑘 = ( 𝐹 ‘ 𝑦 ) → ( ( 𝐴 · 𝑘 ) ∈ ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ↔ ( 𝐴 · ( 𝐹 ‘ 𝑦 ) ) ∈ ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ) ) |
123 |
122
|
ralrn |
⊢ ( 𝐹 Fn ℝ → ( ∀ 𝑘 ∈ ran 𝐹 ( 𝐴 · 𝑘 ) ∈ ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ↔ ∀ 𝑦 ∈ ℝ ( 𝐴 · ( 𝐹 ‘ 𝑦 ) ) ∈ ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ) ) |
124 |
94 123
|
syl |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ 0 ) → ( ∀ 𝑘 ∈ ran 𝐹 ( 𝐴 · 𝑘 ) ∈ ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ↔ ∀ 𝑦 ∈ ℝ ( 𝐴 · ( 𝐹 ‘ 𝑦 ) ) ∈ ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ) ) |
125 |
120 124
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ 0 ) → ∀ 𝑘 ∈ ran 𝐹 ( 𝐴 · 𝑘 ) ∈ ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ) |
126 |
125
|
r19.21bi |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑘 ∈ ran 𝐹 ) → ( 𝐴 · 𝑘 ) ∈ ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ) |
127 |
116 126
|
sylan2 |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( 𝐴 · 𝑘 ) ∈ ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ) |
128 |
36
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → 𝐴 ∈ ℂ ) |
129 |
87
|
frnd |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ 0 ) → ran 𝐹 ⊆ ℝ ) |
130 |
129
|
ssdifssd |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ 0 ) → ( ran 𝐹 ∖ { 0 } ) ⊆ ℝ ) |
131 |
130
|
sselda |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → 𝑘 ∈ ℝ ) |
132 |
131
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → 𝑘 ∈ ℂ ) |
133 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → 𝐴 ≠ 0 ) |
134 |
|
eldifsni |
⊢ ( 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) → 𝑘 ≠ 0 ) |
135 |
134
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → 𝑘 ≠ 0 ) |
136 |
128 132 133 135
|
mulne0d |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( 𝐴 · 𝑘 ) ≠ 0 ) |
137 |
|
eldifsn |
⊢ ( ( 𝐴 · 𝑘 ) ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ↔ ( ( 𝐴 · 𝑘 ) ∈ ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∧ ( 𝐴 · 𝑘 ) ≠ 0 ) ) |
138 |
127 136 137
|
sylanbrc |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( 𝐴 · 𝑘 ) ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) |
139 |
|
simpl |
⊢ ( ( 𝑛 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → 𝑛 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) |
140 |
|
ssel2 |
⊢ ( ( ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ⊆ ℝ ∧ 𝑛 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) → 𝑛 ∈ ℝ ) |
141 |
32 139 140
|
syl2an |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ ( 𝑛 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ) → 𝑛 ∈ ℝ ) |
142 |
141
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ ( 𝑛 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ) → 𝑛 ∈ ℂ ) |
143 |
2
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ ( 𝑛 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ) → 𝐴 ∈ ℝ ) |
144 |
143
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ ( 𝑛 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ) → 𝐴 ∈ ℂ ) |
145 |
131
|
adantrl |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ ( 𝑛 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ) → 𝑘 ∈ ℝ ) |
146 |
145
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ ( 𝑛 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ) → 𝑘 ∈ ℂ ) |
147 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ ( 𝑛 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ) → 𝐴 ≠ 0 ) |
148 |
142 144 146 147
|
divmuld |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ ( 𝑛 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ) → ( ( 𝑛 / 𝐴 ) = 𝑘 ↔ ( 𝐴 · 𝑘 ) = 𝑛 ) ) |
149 |
148
|
bicomd |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ ( 𝑛 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ) → ( ( 𝐴 · 𝑘 ) = 𝑛 ↔ ( 𝑛 / 𝐴 ) = 𝑘 ) ) |
150 |
|
eqcom |
⊢ ( 𝑛 = ( 𝐴 · 𝑘 ) ↔ ( 𝐴 · 𝑘 ) = 𝑛 ) |
151 |
|
eqcom |
⊢ ( 𝑘 = ( 𝑛 / 𝐴 ) ↔ ( 𝑛 / 𝐴 ) = 𝑘 ) |
152 |
149 150 151
|
3bitr4g |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ ( 𝑛 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ) → ( 𝑛 = ( 𝐴 · 𝑘 ) ↔ 𝑘 = ( 𝑛 / 𝐴 ) ) ) |
153 |
79 115 138 152
|
f1o2d |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ 0 ) → ( 𝑛 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ↦ ( 𝑛 / 𝐴 ) ) : ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) –1-1-onto→ ( ran 𝐹 ∖ { 0 } ) ) |
154 |
|
oveq1 |
⊢ ( 𝑛 = 𝑚 → ( 𝑛 / 𝐴 ) = ( 𝑚 / 𝐴 ) ) |
155 |
|
ovex |
⊢ ( 𝑚 / 𝐴 ) ∈ V |
156 |
154 79 155
|
fvmpt |
⊢ ( 𝑚 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) → ( ( 𝑛 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ↦ ( 𝑛 / 𝐴 ) ) ‘ 𝑚 ) = ( 𝑚 / 𝐴 ) ) |
157 |
156
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑚 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) → ( ( 𝑛 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ↦ ( 𝑛 / 𝐴 ) ) ‘ 𝑚 ) = ( 𝑚 / 𝐴 ) ) |
158 |
|
i1fima2sn |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( vol ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) ∈ ℝ ) |
159 |
71 158
|
sylan |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( vol ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) ∈ ℝ ) |
160 |
131 159
|
remulcld |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( 𝑘 · ( vol ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) ) ∈ ℝ ) |
161 |
160
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( 𝑘 · ( vol ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) ) ∈ ℂ ) |
162 |
78 65 153 157 161
|
fsumf1o |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ 0 ) → Σ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ( 𝑘 · ( vol ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) ) = Σ 𝑚 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ( ( 𝑚 / 𝐴 ) · ( vol ‘ ( ◡ 𝐹 “ { ( 𝑚 / 𝐴 ) } ) ) ) ) |
163 |
73 162
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ 0 ) → ( ∫1 ‘ 𝐹 ) = Σ 𝑚 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ( ( 𝑚 / 𝐴 ) · ( vol ‘ ( ◡ 𝐹 “ { ( 𝑚 / 𝐴 ) } ) ) ) ) |
164 |
163
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ 0 ) → ( 𝐴 · ( ∫1 ‘ 𝐹 ) ) = ( 𝐴 · Σ 𝑚 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ( ( 𝑚 / 𝐴 ) · ( vol ‘ ( ◡ 𝐹 “ { ( 𝑚 / 𝐴 ) } ) ) ) ) ) |
165 |
68 70 164
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ 0 ) → ( ∫1 ‘ ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ) = ( 𝐴 · ( ∫1 ‘ 𝐹 ) ) ) |
166 |
26 165
|
pm2.61dane |
⊢ ( 𝜑 → ( ∫1 ‘ ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ) = ( 𝐴 · ( ∫1 ‘ 𝐹 ) ) ) |