Step |
Hyp |
Ref |
Expression |
1 |
|
itg1climres.1 |
⊢ ( 𝜑 → 𝐴 : ℕ ⟶ dom vol ) |
2 |
|
itg1climres.2 |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐴 ‘ 𝑛 ) ⊆ ( 𝐴 ‘ ( 𝑛 + 1 ) ) ) |
3 |
|
itg1climres.3 |
⊢ ( 𝜑 → ∪ ran 𝐴 = ℝ ) |
4 |
|
itg1climres.4 |
⊢ ( 𝜑 → 𝐹 ∈ dom ∫1 ) |
5 |
|
itg1climres.5 |
⊢ 𝐺 = ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ ( 𝐴 ‘ 𝑛 ) , ( 𝐹 ‘ 𝑥 ) , 0 ) ) |
6 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
7 |
|
1zzd |
⊢ ( 𝜑 → 1 ∈ ℤ ) |
8 |
|
i1frn |
⊢ ( 𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin ) |
9 |
4 8
|
syl |
⊢ ( 𝜑 → ran 𝐹 ∈ Fin ) |
10 |
|
difss |
⊢ ( ran 𝐹 ∖ { 0 } ) ⊆ ran 𝐹 |
11 |
|
ssfi |
⊢ ( ( ran 𝐹 ∈ Fin ∧ ( ran 𝐹 ∖ { 0 } ) ⊆ ran 𝐹 ) → ( ran 𝐹 ∖ { 0 } ) ∈ Fin ) |
12 |
9 10 11
|
sylancl |
⊢ ( 𝜑 → ( ran 𝐹 ∖ { 0 } ) ∈ Fin ) |
13 |
|
1zzd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → 1 ∈ ℤ ) |
14 |
|
i1fima |
⊢ ( 𝐹 ∈ dom ∫1 → ( ◡ 𝐹 “ { 𝑘 } ) ∈ dom vol ) |
15 |
4 14
|
syl |
⊢ ( 𝜑 → ( ◡ 𝐹 “ { 𝑘 } ) ∈ dom vol ) |
16 |
15
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ∧ 𝑛 ∈ ℕ ) → ( ◡ 𝐹 “ { 𝑘 } ) ∈ dom vol ) |
17 |
1
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐴 ‘ 𝑛 ) ∈ dom vol ) |
18 |
17
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ∧ 𝑛 ∈ ℕ ) → ( 𝐴 ‘ 𝑛 ) ∈ dom vol ) |
19 |
|
inmbl |
⊢ ( ( ( ◡ 𝐹 “ { 𝑘 } ) ∈ dom vol ∧ ( 𝐴 ‘ 𝑛 ) ∈ dom vol ) → ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ∈ dom vol ) |
20 |
16 18 19
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ∧ 𝑛 ∈ ℕ ) → ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ∈ dom vol ) |
21 |
|
mblvol |
⊢ ( ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ∈ dom vol → ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) = ( vol* ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) |
22 |
20 21
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ∧ 𝑛 ∈ ℕ ) → ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) = ( vol* ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) |
23 |
|
inss1 |
⊢ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ⊆ ( ◡ 𝐹 “ { 𝑘 } ) |
24 |
23
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ∧ 𝑛 ∈ ℕ ) → ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ⊆ ( ◡ 𝐹 “ { 𝑘 } ) ) |
25 |
|
mblss |
⊢ ( ( ◡ 𝐹 “ { 𝑘 } ) ∈ dom vol → ( ◡ 𝐹 “ { 𝑘 } ) ⊆ ℝ ) |
26 |
16 25
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ∧ 𝑛 ∈ ℕ ) → ( ◡ 𝐹 “ { 𝑘 } ) ⊆ ℝ ) |
27 |
|
mblvol |
⊢ ( ( ◡ 𝐹 “ { 𝑘 } ) ∈ dom vol → ( vol ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) = ( vol* ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) ) |
28 |
16 27
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ∧ 𝑛 ∈ ℕ ) → ( vol ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) = ( vol* ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) ) |
29 |
|
i1fima2sn |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( vol ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) ∈ ℝ ) |
30 |
4 29
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( vol ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) ∈ ℝ ) |
31 |
30
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ∧ 𝑛 ∈ ℕ ) → ( vol ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) ∈ ℝ ) |
32 |
28 31
|
eqeltrrd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ∧ 𝑛 ∈ ℕ ) → ( vol* ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) ∈ ℝ ) |
33 |
|
ovolsscl |
⊢ ( ( ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ⊆ ( ◡ 𝐹 “ { 𝑘 } ) ∧ ( ◡ 𝐹 “ { 𝑘 } ) ⊆ ℝ ∧ ( vol* ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) ∈ ℝ ) → ( vol* ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ∈ ℝ ) |
34 |
24 26 32 33
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ∧ 𝑛 ∈ ℕ ) → ( vol* ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ∈ ℝ ) |
35 |
22 34
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ∧ 𝑛 ∈ ℕ ) → ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ∈ ℝ ) |
36 |
35
|
fmpttd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) : ℕ ⟶ ℝ ) |
37 |
2
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ∧ 𝑛 ∈ ℕ ) → ( 𝐴 ‘ 𝑛 ) ⊆ ( 𝐴 ‘ ( 𝑛 + 1 ) ) ) |
38 |
|
sslin |
⊢ ( ( 𝐴 ‘ 𝑛 ) ⊆ ( 𝐴 ‘ ( 𝑛 + 1 ) ) → ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ⊆ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ ( 𝑛 + 1 ) ) ) ) |
39 |
37 38
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ∧ 𝑛 ∈ ℕ ) → ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ⊆ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ ( 𝑛 + 1 ) ) ) ) |
40 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → 𝐴 : ℕ ⟶ dom vol ) |
41 |
|
peano2nn |
⊢ ( 𝑛 ∈ ℕ → ( 𝑛 + 1 ) ∈ ℕ ) |
42 |
|
ffvelrn |
⊢ ( ( 𝐴 : ℕ ⟶ dom vol ∧ ( 𝑛 + 1 ) ∈ ℕ ) → ( 𝐴 ‘ ( 𝑛 + 1 ) ) ∈ dom vol ) |
43 |
40 41 42
|
syl2an |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ∧ 𝑛 ∈ ℕ ) → ( 𝐴 ‘ ( 𝑛 + 1 ) ) ∈ dom vol ) |
44 |
|
inmbl |
⊢ ( ( ( ◡ 𝐹 “ { 𝑘 } ) ∈ dom vol ∧ ( 𝐴 ‘ ( 𝑛 + 1 ) ) ∈ dom vol ) → ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ ( 𝑛 + 1 ) ) ) ∈ dom vol ) |
45 |
16 43 44
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ∧ 𝑛 ∈ ℕ ) → ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ ( 𝑛 + 1 ) ) ) ∈ dom vol ) |
46 |
|
mblss |
⊢ ( ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ ( 𝑛 + 1 ) ) ) ∈ dom vol → ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ ( 𝑛 + 1 ) ) ) ⊆ ℝ ) |
47 |
45 46
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ∧ 𝑛 ∈ ℕ ) → ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ ( 𝑛 + 1 ) ) ) ⊆ ℝ ) |
48 |
|
ovolss |
⊢ ( ( ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ⊆ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ ( 𝑛 + 1 ) ) ) ∧ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ ( 𝑛 + 1 ) ) ) ⊆ ℝ ) → ( vol* ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ≤ ( vol* ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ ( 𝑛 + 1 ) ) ) ) ) |
49 |
39 47 48
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ∧ 𝑛 ∈ ℕ ) → ( vol* ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ≤ ( vol* ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ ( 𝑛 + 1 ) ) ) ) ) |
50 |
|
mblvol |
⊢ ( ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ ( 𝑛 + 1 ) ) ) ∈ dom vol → ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ ( 𝑛 + 1 ) ) ) ) = ( vol* ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ ( 𝑛 + 1 ) ) ) ) ) |
51 |
45 50
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ∧ 𝑛 ∈ ℕ ) → ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ ( 𝑛 + 1 ) ) ) ) = ( vol* ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ ( 𝑛 + 1 ) ) ) ) ) |
52 |
49 22 51
|
3brtr4d |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ∧ 𝑛 ∈ ℕ ) → ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ≤ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ ( 𝑛 + 1 ) ) ) ) ) |
53 |
52
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ∀ 𝑛 ∈ ℕ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ≤ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ ( 𝑛 + 1 ) ) ) ) ) |
54 |
|
fveq2 |
⊢ ( 𝑛 = 𝑗 → ( 𝐴 ‘ 𝑛 ) = ( 𝐴 ‘ 𝑗 ) ) |
55 |
54
|
ineq2d |
⊢ ( 𝑛 = 𝑗 → ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) = ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑗 ) ) ) |
56 |
55
|
fveq2d |
⊢ ( 𝑛 = 𝑗 → ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) = ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑗 ) ) ) ) |
57 |
|
eqid |
⊢ ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) = ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) |
58 |
|
fvex |
⊢ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑗 ) ) ) ∈ V |
59 |
56 57 58
|
fvmpt |
⊢ ( 𝑗 ∈ ℕ → ( ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ‘ 𝑗 ) = ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑗 ) ) ) ) |
60 |
|
peano2nn |
⊢ ( 𝑗 ∈ ℕ → ( 𝑗 + 1 ) ∈ ℕ ) |
61 |
|
fveq2 |
⊢ ( 𝑛 = ( 𝑗 + 1 ) → ( 𝐴 ‘ 𝑛 ) = ( 𝐴 ‘ ( 𝑗 + 1 ) ) ) |
62 |
61
|
ineq2d |
⊢ ( 𝑛 = ( 𝑗 + 1 ) → ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) = ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ ( 𝑗 + 1 ) ) ) ) |
63 |
62
|
fveq2d |
⊢ ( 𝑛 = ( 𝑗 + 1 ) → ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) = ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ ( 𝑗 + 1 ) ) ) ) ) |
64 |
|
fvex |
⊢ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ ( 𝑗 + 1 ) ) ) ) ∈ V |
65 |
63 57 64
|
fvmpt |
⊢ ( ( 𝑗 + 1 ) ∈ ℕ → ( ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ‘ ( 𝑗 + 1 ) ) = ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ ( 𝑗 + 1 ) ) ) ) ) |
66 |
60 65
|
syl |
⊢ ( 𝑗 ∈ ℕ → ( ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ‘ ( 𝑗 + 1 ) ) = ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ ( 𝑗 + 1 ) ) ) ) ) |
67 |
59 66
|
breq12d |
⊢ ( 𝑗 ∈ ℕ → ( ( ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ‘ 𝑗 ) ≤ ( ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ‘ ( 𝑗 + 1 ) ) ↔ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑗 ) ) ) ≤ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ ( 𝑗 + 1 ) ) ) ) ) ) |
68 |
67
|
ralbiia |
⊢ ( ∀ 𝑗 ∈ ℕ ( ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ‘ 𝑗 ) ≤ ( ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ‘ ( 𝑗 + 1 ) ) ↔ ∀ 𝑗 ∈ ℕ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑗 ) ) ) ≤ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ ( 𝑗 + 1 ) ) ) ) ) |
69 |
|
fvoveq1 |
⊢ ( 𝑛 = 𝑗 → ( 𝐴 ‘ ( 𝑛 + 1 ) ) = ( 𝐴 ‘ ( 𝑗 + 1 ) ) ) |
70 |
69
|
ineq2d |
⊢ ( 𝑛 = 𝑗 → ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ ( 𝑛 + 1 ) ) ) = ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ ( 𝑗 + 1 ) ) ) ) |
71 |
70
|
fveq2d |
⊢ ( 𝑛 = 𝑗 → ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ ( 𝑛 + 1 ) ) ) ) = ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ ( 𝑗 + 1 ) ) ) ) ) |
72 |
56 71
|
breq12d |
⊢ ( 𝑛 = 𝑗 → ( ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ≤ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ ( 𝑛 + 1 ) ) ) ) ↔ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑗 ) ) ) ≤ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ ( 𝑗 + 1 ) ) ) ) ) ) |
73 |
72
|
cbvralvw |
⊢ ( ∀ 𝑛 ∈ ℕ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ≤ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ ( 𝑛 + 1 ) ) ) ) ↔ ∀ 𝑗 ∈ ℕ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑗 ) ) ) ≤ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ ( 𝑗 + 1 ) ) ) ) ) |
74 |
68 73
|
bitr4i |
⊢ ( ∀ 𝑗 ∈ ℕ ( ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ‘ 𝑗 ) ≤ ( ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ‘ ( 𝑗 + 1 ) ) ↔ ∀ 𝑛 ∈ ℕ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ≤ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ ( 𝑛 + 1 ) ) ) ) ) |
75 |
53 74
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ∀ 𝑗 ∈ ℕ ( ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ‘ 𝑗 ) ≤ ( ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ‘ ( 𝑗 + 1 ) ) ) |
76 |
75
|
r19.21bi |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ∧ 𝑗 ∈ ℕ ) → ( ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ‘ 𝑗 ) ≤ ( ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ‘ ( 𝑗 + 1 ) ) ) |
77 |
|
ovolss |
⊢ ( ( ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ⊆ ( ◡ 𝐹 “ { 𝑘 } ) ∧ ( ◡ 𝐹 “ { 𝑘 } ) ⊆ ℝ ) → ( vol* ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ≤ ( vol* ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) ) |
78 |
23 26 77
|
sylancr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ∧ 𝑛 ∈ ℕ ) → ( vol* ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ≤ ( vol* ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) ) |
79 |
78 22 28
|
3brtr4d |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ∧ 𝑛 ∈ ℕ ) → ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ≤ ( vol ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) ) |
80 |
79
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ∀ 𝑛 ∈ ℕ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ≤ ( vol ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) ) |
81 |
59
|
breq1d |
⊢ ( 𝑗 ∈ ℕ → ( ( ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ‘ 𝑗 ) ≤ ( vol ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) ↔ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑗 ) ) ) ≤ ( vol ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) ) ) |
82 |
81
|
ralbiia |
⊢ ( ∀ 𝑗 ∈ ℕ ( ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ‘ 𝑗 ) ≤ ( vol ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) ↔ ∀ 𝑗 ∈ ℕ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑗 ) ) ) ≤ ( vol ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) ) |
83 |
56
|
breq1d |
⊢ ( 𝑛 = 𝑗 → ( ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ≤ ( vol ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) ↔ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑗 ) ) ) ≤ ( vol ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) ) ) |
84 |
83
|
cbvralvw |
⊢ ( ∀ 𝑛 ∈ ℕ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ≤ ( vol ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) ↔ ∀ 𝑗 ∈ ℕ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑗 ) ) ) ≤ ( vol ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) ) |
85 |
82 84
|
bitr4i |
⊢ ( ∀ 𝑗 ∈ ℕ ( ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ‘ 𝑗 ) ≤ ( vol ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) ↔ ∀ 𝑛 ∈ ℕ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ≤ ( vol ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) ) |
86 |
80 85
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ∀ 𝑗 ∈ ℕ ( ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ‘ 𝑗 ) ≤ ( vol ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) ) |
87 |
|
brralrspcev |
⊢ ( ( ( vol ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) ∈ ℝ ∧ ∀ 𝑗 ∈ ℕ ( ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ‘ 𝑗 ) ≤ ( vol ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) ) → ∃ 𝑥 ∈ ℝ ∀ 𝑗 ∈ ℕ ( ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ‘ 𝑗 ) ≤ 𝑥 ) |
88 |
30 86 87
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ∃ 𝑥 ∈ ℝ ∀ 𝑗 ∈ ℕ ( ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ‘ 𝑗 ) ≤ 𝑥 ) |
89 |
6 13 36 76 88
|
climsup |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ⇝ sup ( ran ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) , ℝ , < ) ) |
90 |
20
|
fmpttd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( 𝑛 ∈ ℕ ↦ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) : ℕ ⟶ dom vol ) |
91 |
39
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ∀ 𝑛 ∈ ℕ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ⊆ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ ( 𝑛 + 1 ) ) ) ) |
92 |
|
eqid |
⊢ ( 𝑛 ∈ ℕ ↦ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) = ( 𝑛 ∈ ℕ ↦ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) |
93 |
|
fvex |
⊢ ( 𝐴 ‘ 𝑗 ) ∈ V |
94 |
93
|
inex2 |
⊢ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑗 ) ) ∈ V |
95 |
55 92 94
|
fvmpt |
⊢ ( 𝑗 ∈ ℕ → ( ( 𝑛 ∈ ℕ ↦ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ‘ 𝑗 ) = ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑗 ) ) ) |
96 |
|
fvex |
⊢ ( 𝐴 ‘ ( 𝑗 + 1 ) ) ∈ V |
97 |
96
|
inex2 |
⊢ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ ( 𝑗 + 1 ) ) ) ∈ V |
98 |
62 92 97
|
fvmpt |
⊢ ( ( 𝑗 + 1 ) ∈ ℕ → ( ( 𝑛 ∈ ℕ ↦ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ‘ ( 𝑗 + 1 ) ) = ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ ( 𝑗 + 1 ) ) ) ) |
99 |
60 98
|
syl |
⊢ ( 𝑗 ∈ ℕ → ( ( 𝑛 ∈ ℕ ↦ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ‘ ( 𝑗 + 1 ) ) = ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ ( 𝑗 + 1 ) ) ) ) |
100 |
95 99
|
sseq12d |
⊢ ( 𝑗 ∈ ℕ → ( ( ( 𝑛 ∈ ℕ ↦ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ‘ 𝑗 ) ⊆ ( ( 𝑛 ∈ ℕ ↦ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ‘ ( 𝑗 + 1 ) ) ↔ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑗 ) ) ⊆ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ ( 𝑗 + 1 ) ) ) ) ) |
101 |
100
|
ralbiia |
⊢ ( ∀ 𝑗 ∈ ℕ ( ( 𝑛 ∈ ℕ ↦ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ‘ 𝑗 ) ⊆ ( ( 𝑛 ∈ ℕ ↦ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ‘ ( 𝑗 + 1 ) ) ↔ ∀ 𝑗 ∈ ℕ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑗 ) ) ⊆ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ ( 𝑗 + 1 ) ) ) ) |
102 |
55 70
|
sseq12d |
⊢ ( 𝑛 = 𝑗 → ( ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ⊆ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ ( 𝑛 + 1 ) ) ) ↔ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑗 ) ) ⊆ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ ( 𝑗 + 1 ) ) ) ) ) |
103 |
102
|
cbvralvw |
⊢ ( ∀ 𝑛 ∈ ℕ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ⊆ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ ( 𝑛 + 1 ) ) ) ↔ ∀ 𝑗 ∈ ℕ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑗 ) ) ⊆ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ ( 𝑗 + 1 ) ) ) ) |
104 |
101 103
|
bitr4i |
⊢ ( ∀ 𝑗 ∈ ℕ ( ( 𝑛 ∈ ℕ ↦ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ‘ 𝑗 ) ⊆ ( ( 𝑛 ∈ ℕ ↦ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ‘ ( 𝑗 + 1 ) ) ↔ ∀ 𝑛 ∈ ℕ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ⊆ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ ( 𝑛 + 1 ) ) ) ) |
105 |
91 104
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ∀ 𝑗 ∈ ℕ ( ( 𝑛 ∈ ℕ ↦ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ‘ 𝑗 ) ⊆ ( ( 𝑛 ∈ ℕ ↦ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ‘ ( 𝑗 + 1 ) ) ) |
106 |
|
volsup |
⊢ ( ( ( 𝑛 ∈ ℕ ↦ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) : ℕ ⟶ dom vol ∧ ∀ 𝑗 ∈ ℕ ( ( 𝑛 ∈ ℕ ↦ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ‘ 𝑗 ) ⊆ ( ( 𝑛 ∈ ℕ ↦ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ‘ ( 𝑗 + 1 ) ) ) → ( vol ‘ ∪ ran ( 𝑛 ∈ ℕ ↦ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) = sup ( ( vol “ ran ( 𝑛 ∈ ℕ ↦ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) , ℝ* , < ) ) |
107 |
90 105 106
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( vol ‘ ∪ ran ( 𝑛 ∈ ℕ ↦ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) = sup ( ( vol “ ran ( 𝑛 ∈ ℕ ↦ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) , ℝ* , < ) ) |
108 |
95
|
iuneq2i |
⊢ ∪ 𝑗 ∈ ℕ ( ( 𝑛 ∈ ℕ ↦ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ‘ 𝑗 ) = ∪ 𝑗 ∈ ℕ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑗 ) ) |
109 |
55
|
cbviunv |
⊢ ∪ 𝑛 ∈ ℕ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) = ∪ 𝑗 ∈ ℕ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑗 ) ) |
110 |
|
iunin2 |
⊢ ∪ 𝑛 ∈ ℕ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) = ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ∪ 𝑛 ∈ ℕ ( 𝐴 ‘ 𝑛 ) ) |
111 |
108 109 110
|
3eqtr2i |
⊢ ∪ 𝑗 ∈ ℕ ( ( 𝑛 ∈ ℕ ↦ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ‘ 𝑗 ) = ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ∪ 𝑛 ∈ ℕ ( 𝐴 ‘ 𝑛 ) ) |
112 |
|
ffn |
⊢ ( 𝐴 : ℕ ⟶ dom vol → 𝐴 Fn ℕ ) |
113 |
|
fniunfv |
⊢ ( 𝐴 Fn ℕ → ∪ 𝑛 ∈ ℕ ( 𝐴 ‘ 𝑛 ) = ∪ ran 𝐴 ) |
114 |
1 112 113
|
3syl |
⊢ ( 𝜑 → ∪ 𝑛 ∈ ℕ ( 𝐴 ‘ 𝑛 ) = ∪ ran 𝐴 ) |
115 |
114 3
|
eqtrd |
⊢ ( 𝜑 → ∪ 𝑛 ∈ ℕ ( 𝐴 ‘ 𝑛 ) = ℝ ) |
116 |
115
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ∪ 𝑛 ∈ ℕ ( 𝐴 ‘ 𝑛 ) = ℝ ) |
117 |
116
|
ineq2d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ∪ 𝑛 ∈ ℕ ( 𝐴 ‘ 𝑛 ) ) = ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ℝ ) ) |
118 |
15
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( ◡ 𝐹 “ { 𝑘 } ) ∈ dom vol ) |
119 |
118 25
|
syl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( ◡ 𝐹 “ { 𝑘 } ) ⊆ ℝ ) |
120 |
|
df-ss |
⊢ ( ( ◡ 𝐹 “ { 𝑘 } ) ⊆ ℝ ↔ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ℝ ) = ( ◡ 𝐹 “ { 𝑘 } ) ) |
121 |
119 120
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ℝ ) = ( ◡ 𝐹 “ { 𝑘 } ) ) |
122 |
117 121
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ∪ 𝑛 ∈ ℕ ( 𝐴 ‘ 𝑛 ) ) = ( ◡ 𝐹 “ { 𝑘 } ) ) |
123 |
111 122
|
syl5eq |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ∪ 𝑗 ∈ ℕ ( ( 𝑛 ∈ ℕ ↦ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ‘ 𝑗 ) = ( ◡ 𝐹 “ { 𝑘 } ) ) |
124 |
|
ffn |
⊢ ( ( 𝑛 ∈ ℕ ↦ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) : ℕ ⟶ dom vol → ( 𝑛 ∈ ℕ ↦ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) Fn ℕ ) |
125 |
|
fniunfv |
⊢ ( ( 𝑛 ∈ ℕ ↦ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) Fn ℕ → ∪ 𝑗 ∈ ℕ ( ( 𝑛 ∈ ℕ ↦ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ‘ 𝑗 ) = ∪ ran ( 𝑛 ∈ ℕ ↦ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) |
126 |
90 124 125
|
3syl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ∪ 𝑗 ∈ ℕ ( ( 𝑛 ∈ ℕ ↦ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ‘ 𝑗 ) = ∪ ran ( 𝑛 ∈ ℕ ↦ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) |
127 |
123 126
|
eqtr3d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( ◡ 𝐹 “ { 𝑘 } ) = ∪ ran ( 𝑛 ∈ ℕ ↦ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) |
128 |
127
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( vol ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) = ( vol ‘ ∪ ran ( 𝑛 ∈ ℕ ↦ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ) |
129 |
36
|
frnd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ran ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ⊆ ℝ ) |
130 |
36
|
fdmd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → dom ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) = ℕ ) |
131 |
|
1nn |
⊢ 1 ∈ ℕ |
132 |
|
ne0i |
⊢ ( 1 ∈ ℕ → ℕ ≠ ∅ ) |
133 |
131 132
|
mp1i |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ℕ ≠ ∅ ) |
134 |
130 133
|
eqnetrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → dom ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ≠ ∅ ) |
135 |
|
dm0rn0 |
⊢ ( dom ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) = ∅ ↔ ran ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) = ∅ ) |
136 |
135
|
necon3bii |
⊢ ( dom ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ≠ ∅ ↔ ran ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ≠ ∅ ) |
137 |
134 136
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ran ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ≠ ∅ ) |
138 |
|
ffn |
⊢ ( ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) : ℕ ⟶ ℝ → ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) Fn ℕ ) |
139 |
|
breq1 |
⊢ ( 𝑧 = ( ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ‘ 𝑗 ) → ( 𝑧 ≤ 𝑥 ↔ ( ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ‘ 𝑗 ) ≤ 𝑥 ) ) |
140 |
139
|
ralrn |
⊢ ( ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) Fn ℕ → ( ∀ 𝑧 ∈ ran ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) 𝑧 ≤ 𝑥 ↔ ∀ 𝑗 ∈ ℕ ( ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ‘ 𝑗 ) ≤ 𝑥 ) ) |
141 |
36 138 140
|
3syl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( ∀ 𝑧 ∈ ran ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) 𝑧 ≤ 𝑥 ↔ ∀ 𝑗 ∈ ℕ ( ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ‘ 𝑗 ) ≤ 𝑥 ) ) |
142 |
141
|
rexbidv |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( ∃ 𝑥 ∈ ℝ ∀ 𝑧 ∈ ran ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) 𝑧 ≤ 𝑥 ↔ ∃ 𝑥 ∈ ℝ ∀ 𝑗 ∈ ℕ ( ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ‘ 𝑗 ) ≤ 𝑥 ) ) |
143 |
88 142
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ∃ 𝑥 ∈ ℝ ∀ 𝑧 ∈ ran ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) 𝑧 ≤ 𝑥 ) |
144 |
|
supxrre |
⊢ ( ( ran ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ⊆ ℝ ∧ ran ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑧 ∈ ran ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) 𝑧 ≤ 𝑥 ) → sup ( ran ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) , ℝ* , < ) = sup ( ran ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) , ℝ , < ) ) |
145 |
129 137 143 144
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → sup ( ran ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) , ℝ* , < ) = sup ( ran ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) , ℝ , < ) ) |
146 |
|
volf |
⊢ vol : dom vol ⟶ ( 0 [,] +∞ ) |
147 |
146
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → vol : dom vol ⟶ ( 0 [,] +∞ ) ) |
148 |
147 20
|
cofmpt |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( vol ∘ ( 𝑛 ∈ ℕ ↦ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) = ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ) |
149 |
148
|
rneqd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ran ( vol ∘ ( 𝑛 ∈ ℕ ↦ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) = ran ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ) |
150 |
|
rnco2 |
⊢ ran ( vol ∘ ( 𝑛 ∈ ℕ ↦ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) = ( vol “ ran ( 𝑛 ∈ ℕ ↦ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) |
151 |
149 150
|
eqtr3di |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ran ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) = ( vol “ ran ( 𝑛 ∈ ℕ ↦ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ) |
152 |
151
|
supeq1d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → sup ( ran ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) , ℝ* , < ) = sup ( ( vol “ ran ( 𝑛 ∈ ℕ ↦ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) , ℝ* , < ) ) |
153 |
145 152
|
eqtr3d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → sup ( ran ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) , ℝ , < ) = sup ( ( vol “ ran ( 𝑛 ∈ ℕ ↦ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) , ℝ* , < ) ) |
154 |
107 128 153
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( vol ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) = sup ( ran ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) , ℝ , < ) ) |
155 |
89 154
|
breqtrrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ⇝ ( vol ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) ) |
156 |
|
i1ff |
⊢ ( 𝐹 ∈ dom ∫1 → 𝐹 : ℝ ⟶ ℝ ) |
157 |
|
frn |
⊢ ( 𝐹 : ℝ ⟶ ℝ → ran 𝐹 ⊆ ℝ ) |
158 |
4 156 157
|
3syl |
⊢ ( 𝜑 → ran 𝐹 ⊆ ℝ ) |
159 |
158
|
ssdifssd |
⊢ ( 𝜑 → ( ran 𝐹 ∖ { 0 } ) ⊆ ℝ ) |
160 |
159
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → 𝑘 ∈ ℝ ) |
161 |
160
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → 𝑘 ∈ ℂ ) |
162 |
|
nnex |
⊢ ℕ ∈ V |
163 |
162
|
mptex |
⊢ ( 𝑛 ∈ ℕ ↦ ( 𝑘 · ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ) ∈ V |
164 |
163
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( 𝑛 ∈ ℕ ↦ ( 𝑘 · ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ) ∈ V ) |
165 |
36
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ∧ 𝑗 ∈ ℕ ) → ( ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ‘ 𝑗 ) ∈ ℝ ) |
166 |
165
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ∧ 𝑗 ∈ ℕ ) → ( ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ‘ 𝑗 ) ∈ ℂ ) |
167 |
56
|
oveq2d |
⊢ ( 𝑛 = 𝑗 → ( 𝑘 · ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) = ( 𝑘 · ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑗 ) ) ) ) ) |
168 |
|
eqid |
⊢ ( 𝑛 ∈ ℕ ↦ ( 𝑘 · ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ) = ( 𝑛 ∈ ℕ ↦ ( 𝑘 · ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ) |
169 |
|
ovex |
⊢ ( 𝑘 · ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑗 ) ) ) ) ∈ V |
170 |
167 168 169
|
fvmpt |
⊢ ( 𝑗 ∈ ℕ → ( ( 𝑛 ∈ ℕ ↦ ( 𝑘 · ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ) ‘ 𝑗 ) = ( 𝑘 · ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑗 ) ) ) ) ) |
171 |
59
|
oveq2d |
⊢ ( 𝑗 ∈ ℕ → ( 𝑘 · ( ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ‘ 𝑗 ) ) = ( 𝑘 · ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑗 ) ) ) ) ) |
172 |
170 171
|
eqtr4d |
⊢ ( 𝑗 ∈ ℕ → ( ( 𝑛 ∈ ℕ ↦ ( 𝑘 · ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ) ‘ 𝑗 ) = ( 𝑘 · ( ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ‘ 𝑗 ) ) ) |
173 |
172
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ∧ 𝑗 ∈ ℕ ) → ( ( 𝑛 ∈ ℕ ↦ ( 𝑘 · ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ) ‘ 𝑗 ) = ( 𝑘 · ( ( 𝑛 ∈ ℕ ↦ ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ‘ 𝑗 ) ) ) |
174 |
6 13 155 161 164 166 173
|
climmulc2 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( 𝑛 ∈ ℕ ↦ ( 𝑘 · ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ) ⇝ ( 𝑘 · ( vol ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) ) ) |
175 |
162
|
mptex |
⊢ ( 𝑛 ∈ ℕ ↦ ( ∫1 ‘ 𝐺 ) ) ∈ V |
176 |
175
|
a1i |
⊢ ( 𝜑 → ( 𝑛 ∈ ℕ ↦ ( ∫1 ‘ 𝐺 ) ) ∈ V ) |
177 |
160
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ∧ 𝑛 ∈ ℕ ) → 𝑘 ∈ ℝ ) |
178 |
177 35
|
remulcld |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ∧ 𝑛 ∈ ℕ ) → ( 𝑘 · ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ∈ ℝ ) |
179 |
178
|
fmpttd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( 𝑛 ∈ ℕ ↦ ( 𝑘 · ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ) : ℕ ⟶ ℝ ) |
180 |
179
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ∧ 𝑗 ∈ ℕ ) → ( ( 𝑛 ∈ ℕ ↦ ( 𝑘 · ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ) ‘ 𝑗 ) ∈ ℝ ) |
181 |
180
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ∧ 𝑗 ∈ ℕ ) → ( ( 𝑛 ∈ ℕ ↦ ( 𝑘 · ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ) ‘ 𝑗 ) ∈ ℂ ) |
182 |
181
|
anasss |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ∧ 𝑗 ∈ ℕ ) ) → ( ( 𝑛 ∈ ℕ ↦ ( 𝑘 · ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ) ‘ 𝑗 ) ∈ ℂ ) |
183 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → 𝐹 ∈ dom ∫1 ) |
184 |
5
|
i1fres |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ ( 𝐴 ‘ 𝑛 ) ∈ dom vol ) → 𝐺 ∈ dom ∫1 ) |
185 |
183 17 184
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → 𝐺 ∈ dom ∫1 ) |
186 |
12
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ran 𝐹 ∖ { 0 } ) ∈ Fin ) |
187 |
|
ffn |
⊢ ( 𝐹 : ℝ ⟶ ℝ → 𝐹 Fn ℝ ) |
188 |
4 156 187
|
3syl |
⊢ ( 𝜑 → 𝐹 Fn ℝ ) |
189 |
188
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → 𝐹 Fn ℝ ) |
190 |
|
fnfvelrn |
⊢ ( ( 𝐹 Fn ℝ ∧ 𝑥 ∈ ℝ ) → ( 𝐹 ‘ 𝑥 ) ∈ ran 𝐹 ) |
191 |
189 190
|
sylan |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑥 ∈ ℝ ) → ( 𝐹 ‘ 𝑥 ) ∈ ran 𝐹 ) |
192 |
|
i1f0rn |
⊢ ( 𝐹 ∈ dom ∫1 → 0 ∈ ran 𝐹 ) |
193 |
4 192
|
syl |
⊢ ( 𝜑 → 0 ∈ ran 𝐹 ) |
194 |
193
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑥 ∈ ℝ ) → 0 ∈ ran 𝐹 ) |
195 |
191 194
|
ifcld |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑥 ∈ ℝ ) → if ( 𝑥 ∈ ( 𝐴 ‘ 𝑛 ) , ( 𝐹 ‘ 𝑥 ) , 0 ) ∈ ran 𝐹 ) |
196 |
195 5
|
fmptd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → 𝐺 : ℝ ⟶ ran 𝐹 ) |
197 |
|
frn |
⊢ ( 𝐺 : ℝ ⟶ ran 𝐹 → ran 𝐺 ⊆ ran 𝐹 ) |
198 |
|
ssdif |
⊢ ( ran 𝐺 ⊆ ran 𝐹 → ( ran 𝐺 ∖ { 0 } ) ⊆ ( ran 𝐹 ∖ { 0 } ) ) |
199 |
196 197 198
|
3syl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ran 𝐺 ∖ { 0 } ) ⊆ ( ran 𝐹 ∖ { 0 } ) ) |
200 |
158
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ran 𝐹 ⊆ ℝ ) |
201 |
200
|
ssdifd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ran 𝐹 ∖ { 0 } ) ⊆ ( ℝ ∖ { 0 } ) ) |
202 |
|
itg1val2 |
⊢ ( ( 𝐺 ∈ dom ∫1 ∧ ( ( ran 𝐹 ∖ { 0 } ) ∈ Fin ∧ ( ran 𝐺 ∖ { 0 } ) ⊆ ( ran 𝐹 ∖ { 0 } ) ∧ ( ran 𝐹 ∖ { 0 } ) ⊆ ( ℝ ∖ { 0 } ) ) ) → ( ∫1 ‘ 𝐺 ) = Σ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ( 𝑘 · ( vol ‘ ( ◡ 𝐺 “ { 𝑘 } ) ) ) ) |
203 |
185 186 199 201 202
|
syl13anc |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ∫1 ‘ 𝐺 ) = Σ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ( 𝑘 · ( vol ‘ ( ◡ 𝐺 “ { 𝑘 } ) ) ) ) |
204 |
|
fvex |
⊢ ( 𝐹 ‘ 𝑥 ) ∈ V |
205 |
|
c0ex |
⊢ 0 ∈ V |
206 |
204 205
|
ifex |
⊢ if ( 𝑥 ∈ ( 𝐴 ‘ 𝑛 ) , ( 𝐹 ‘ 𝑥 ) , 0 ) ∈ V |
207 |
5
|
fvmpt2 |
⊢ ( ( 𝑥 ∈ ℝ ∧ if ( 𝑥 ∈ ( 𝐴 ‘ 𝑛 ) , ( 𝐹 ‘ 𝑥 ) , 0 ) ∈ V ) → ( 𝐺 ‘ 𝑥 ) = if ( 𝑥 ∈ ( 𝐴 ‘ 𝑛 ) , ( 𝐹 ‘ 𝑥 ) , 0 ) ) |
208 |
206 207
|
mpan2 |
⊢ ( 𝑥 ∈ ℝ → ( 𝐺 ‘ 𝑥 ) = if ( 𝑥 ∈ ( 𝐴 ‘ 𝑛 ) , ( 𝐹 ‘ 𝑥 ) , 0 ) ) |
209 |
208
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ∧ 𝑥 ∈ ℝ ) → ( 𝐺 ‘ 𝑥 ) = if ( 𝑥 ∈ ( 𝐴 ‘ 𝑛 ) , ( 𝐹 ‘ 𝑥 ) , 0 ) ) |
210 |
209
|
eqeq1d |
⊢ ( ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ∧ 𝑥 ∈ ℝ ) → ( ( 𝐺 ‘ 𝑥 ) = 𝑘 ↔ if ( 𝑥 ∈ ( 𝐴 ‘ 𝑛 ) , ( 𝐹 ‘ 𝑥 ) , 0 ) = 𝑘 ) ) |
211 |
|
eldifsni |
⊢ ( 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) → 𝑘 ≠ 0 ) |
212 |
211
|
ad2antlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ∧ 𝑥 ∈ ℝ ) → 𝑘 ≠ 0 ) |
213 |
|
neeq1 |
⊢ ( if ( 𝑥 ∈ ( 𝐴 ‘ 𝑛 ) , ( 𝐹 ‘ 𝑥 ) , 0 ) = 𝑘 → ( if ( 𝑥 ∈ ( 𝐴 ‘ 𝑛 ) , ( 𝐹 ‘ 𝑥 ) , 0 ) ≠ 0 ↔ 𝑘 ≠ 0 ) ) |
214 |
212 213
|
syl5ibrcom |
⊢ ( ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ∧ 𝑥 ∈ ℝ ) → ( if ( 𝑥 ∈ ( 𝐴 ‘ 𝑛 ) , ( 𝐹 ‘ 𝑥 ) , 0 ) = 𝑘 → if ( 𝑥 ∈ ( 𝐴 ‘ 𝑛 ) , ( 𝐹 ‘ 𝑥 ) , 0 ) ≠ 0 ) ) |
215 |
|
iffalse |
⊢ ( ¬ 𝑥 ∈ ( 𝐴 ‘ 𝑛 ) → if ( 𝑥 ∈ ( 𝐴 ‘ 𝑛 ) , ( 𝐹 ‘ 𝑥 ) , 0 ) = 0 ) |
216 |
215
|
necon1ai |
⊢ ( if ( 𝑥 ∈ ( 𝐴 ‘ 𝑛 ) , ( 𝐹 ‘ 𝑥 ) , 0 ) ≠ 0 → 𝑥 ∈ ( 𝐴 ‘ 𝑛 ) ) |
217 |
214 216
|
syl6 |
⊢ ( ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ∧ 𝑥 ∈ ℝ ) → ( if ( 𝑥 ∈ ( 𝐴 ‘ 𝑛 ) , ( 𝐹 ‘ 𝑥 ) , 0 ) = 𝑘 → 𝑥 ∈ ( 𝐴 ‘ 𝑛 ) ) ) |
218 |
217
|
pm4.71rd |
⊢ ( ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ∧ 𝑥 ∈ ℝ ) → ( if ( 𝑥 ∈ ( 𝐴 ‘ 𝑛 ) , ( 𝐹 ‘ 𝑥 ) , 0 ) = 𝑘 ↔ ( 𝑥 ∈ ( 𝐴 ‘ 𝑛 ) ∧ if ( 𝑥 ∈ ( 𝐴 ‘ 𝑛 ) , ( 𝐹 ‘ 𝑥 ) , 0 ) = 𝑘 ) ) ) |
219 |
210 218
|
bitrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ∧ 𝑥 ∈ ℝ ) → ( ( 𝐺 ‘ 𝑥 ) = 𝑘 ↔ ( 𝑥 ∈ ( 𝐴 ‘ 𝑛 ) ∧ if ( 𝑥 ∈ ( 𝐴 ‘ 𝑛 ) , ( 𝐹 ‘ 𝑥 ) , 0 ) = 𝑘 ) ) ) |
220 |
|
iftrue |
⊢ ( 𝑥 ∈ ( 𝐴 ‘ 𝑛 ) → if ( 𝑥 ∈ ( 𝐴 ‘ 𝑛 ) , ( 𝐹 ‘ 𝑥 ) , 0 ) = ( 𝐹 ‘ 𝑥 ) ) |
221 |
220
|
eqeq1d |
⊢ ( 𝑥 ∈ ( 𝐴 ‘ 𝑛 ) → ( if ( 𝑥 ∈ ( 𝐴 ‘ 𝑛 ) , ( 𝐹 ‘ 𝑥 ) , 0 ) = 𝑘 ↔ ( 𝐹 ‘ 𝑥 ) = 𝑘 ) ) |
222 |
221
|
pm5.32i |
⊢ ( ( 𝑥 ∈ ( 𝐴 ‘ 𝑛 ) ∧ if ( 𝑥 ∈ ( 𝐴 ‘ 𝑛 ) , ( 𝐹 ‘ 𝑥 ) , 0 ) = 𝑘 ) ↔ ( 𝑥 ∈ ( 𝐴 ‘ 𝑛 ) ∧ ( 𝐹 ‘ 𝑥 ) = 𝑘 ) ) |
223 |
222
|
biancomi |
⊢ ( ( 𝑥 ∈ ( 𝐴 ‘ 𝑛 ) ∧ if ( 𝑥 ∈ ( 𝐴 ‘ 𝑛 ) , ( 𝐹 ‘ 𝑥 ) , 0 ) = 𝑘 ) ↔ ( ( 𝐹 ‘ 𝑥 ) = 𝑘 ∧ 𝑥 ∈ ( 𝐴 ‘ 𝑛 ) ) ) |
224 |
219 223
|
bitrdi |
⊢ ( ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ∧ 𝑥 ∈ ℝ ) → ( ( 𝐺 ‘ 𝑥 ) = 𝑘 ↔ ( ( 𝐹 ‘ 𝑥 ) = 𝑘 ∧ 𝑥 ∈ ( 𝐴 ‘ 𝑛 ) ) ) ) |
225 |
224
|
pm5.32da |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( ( 𝑥 ∈ ℝ ∧ ( 𝐺 ‘ 𝑥 ) = 𝑘 ) ↔ ( 𝑥 ∈ ℝ ∧ ( ( 𝐹 ‘ 𝑥 ) = 𝑘 ∧ 𝑥 ∈ ( 𝐴 ‘ 𝑛 ) ) ) ) ) |
226 |
|
anass |
⊢ ( ( ( 𝑥 ∈ ℝ ∧ ( 𝐹 ‘ 𝑥 ) = 𝑘 ) ∧ 𝑥 ∈ ( 𝐴 ‘ 𝑛 ) ) ↔ ( 𝑥 ∈ ℝ ∧ ( ( 𝐹 ‘ 𝑥 ) = 𝑘 ∧ 𝑥 ∈ ( 𝐴 ‘ 𝑛 ) ) ) ) |
227 |
225 226
|
bitr4di |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( ( 𝑥 ∈ ℝ ∧ ( 𝐺 ‘ 𝑥 ) = 𝑘 ) ↔ ( ( 𝑥 ∈ ℝ ∧ ( 𝐹 ‘ 𝑥 ) = 𝑘 ) ∧ 𝑥 ∈ ( 𝐴 ‘ 𝑛 ) ) ) ) |
228 |
|
i1ff |
⊢ ( 𝐺 ∈ dom ∫1 → 𝐺 : ℝ ⟶ ℝ ) |
229 |
|
ffn |
⊢ ( 𝐺 : ℝ ⟶ ℝ → 𝐺 Fn ℝ ) |
230 |
185 228 229
|
3syl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → 𝐺 Fn ℝ ) |
231 |
230
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → 𝐺 Fn ℝ ) |
232 |
|
fniniseg |
⊢ ( 𝐺 Fn ℝ → ( 𝑥 ∈ ( ◡ 𝐺 “ { 𝑘 } ) ↔ ( 𝑥 ∈ ℝ ∧ ( 𝐺 ‘ 𝑥 ) = 𝑘 ) ) ) |
233 |
231 232
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( 𝑥 ∈ ( ◡ 𝐺 “ { 𝑘 } ) ↔ ( 𝑥 ∈ ℝ ∧ ( 𝐺 ‘ 𝑥 ) = 𝑘 ) ) ) |
234 |
|
elin |
⊢ ( 𝑥 ∈ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ↔ ( 𝑥 ∈ ( ◡ 𝐹 “ { 𝑘 } ) ∧ 𝑥 ∈ ( 𝐴 ‘ 𝑛 ) ) ) |
235 |
189
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → 𝐹 Fn ℝ ) |
236 |
|
fniniseg |
⊢ ( 𝐹 Fn ℝ → ( 𝑥 ∈ ( ◡ 𝐹 “ { 𝑘 } ) ↔ ( 𝑥 ∈ ℝ ∧ ( 𝐹 ‘ 𝑥 ) = 𝑘 ) ) ) |
237 |
235 236
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( 𝑥 ∈ ( ◡ 𝐹 “ { 𝑘 } ) ↔ ( 𝑥 ∈ ℝ ∧ ( 𝐹 ‘ 𝑥 ) = 𝑘 ) ) ) |
238 |
237
|
anbi1d |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( ( 𝑥 ∈ ( ◡ 𝐹 “ { 𝑘 } ) ∧ 𝑥 ∈ ( 𝐴 ‘ 𝑛 ) ) ↔ ( ( 𝑥 ∈ ℝ ∧ ( 𝐹 ‘ 𝑥 ) = 𝑘 ) ∧ 𝑥 ∈ ( 𝐴 ‘ 𝑛 ) ) ) ) |
239 |
234 238
|
syl5bb |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( 𝑥 ∈ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ↔ ( ( 𝑥 ∈ ℝ ∧ ( 𝐹 ‘ 𝑥 ) = 𝑘 ) ∧ 𝑥 ∈ ( 𝐴 ‘ 𝑛 ) ) ) ) |
240 |
227 233 239
|
3bitr4d |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( 𝑥 ∈ ( ◡ 𝐺 “ { 𝑘 } ) ↔ 𝑥 ∈ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) |
241 |
240
|
alrimiv |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ∀ 𝑥 ( 𝑥 ∈ ( ◡ 𝐺 “ { 𝑘 } ) ↔ 𝑥 ∈ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) |
242 |
|
nfmpt1 |
⊢ Ⅎ 𝑥 ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ ( 𝐴 ‘ 𝑛 ) , ( 𝐹 ‘ 𝑥 ) , 0 ) ) |
243 |
5 242
|
nfcxfr |
⊢ Ⅎ 𝑥 𝐺 |
244 |
243
|
nfcnv |
⊢ Ⅎ 𝑥 ◡ 𝐺 |
245 |
|
nfcv |
⊢ Ⅎ 𝑥 { 𝑘 } |
246 |
244 245
|
nfima |
⊢ Ⅎ 𝑥 ( ◡ 𝐺 “ { 𝑘 } ) |
247 |
|
nfcv |
⊢ Ⅎ 𝑥 ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) |
248 |
246 247
|
cleqf |
⊢ ( ( ◡ 𝐺 “ { 𝑘 } ) = ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ↔ ∀ 𝑥 ( 𝑥 ∈ ( ◡ 𝐺 “ { 𝑘 } ) ↔ 𝑥 ∈ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) |
249 |
241 248
|
sylibr |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( ◡ 𝐺 “ { 𝑘 } ) = ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) |
250 |
249
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( vol ‘ ( ◡ 𝐺 “ { 𝑘 } ) ) = ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) |
251 |
250
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( 𝑘 · ( vol ‘ ( ◡ 𝐺 “ { 𝑘 } ) ) ) = ( 𝑘 · ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ) |
252 |
251
|
sumeq2dv |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → Σ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ( 𝑘 · ( vol ‘ ( ◡ 𝐺 “ { 𝑘 } ) ) ) = Σ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ( 𝑘 · ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ) |
253 |
203 252
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ∫1 ‘ 𝐺 ) = Σ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ( 𝑘 · ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ) |
254 |
253
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑛 ∈ ℕ ↦ ( ∫1 ‘ 𝐺 ) ) = ( 𝑛 ∈ ℕ ↦ Σ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ( 𝑘 · ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ) ) |
255 |
254
|
fveq1d |
⊢ ( 𝜑 → ( ( 𝑛 ∈ ℕ ↦ ( ∫1 ‘ 𝐺 ) ) ‘ 𝑗 ) = ( ( 𝑛 ∈ ℕ ↦ Σ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ( 𝑘 · ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ) ‘ 𝑗 ) ) |
256 |
167
|
sumeq2sdv |
⊢ ( 𝑛 = 𝑗 → Σ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ( 𝑘 · ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) = Σ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ( 𝑘 · ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑗 ) ) ) ) ) |
257 |
|
eqid |
⊢ ( 𝑛 ∈ ℕ ↦ Σ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ( 𝑘 · ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ) = ( 𝑛 ∈ ℕ ↦ Σ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ( 𝑘 · ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ) |
258 |
|
sumex |
⊢ Σ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ( 𝑘 · ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑗 ) ) ) ) ∈ V |
259 |
256 257 258
|
fvmpt |
⊢ ( 𝑗 ∈ ℕ → ( ( 𝑛 ∈ ℕ ↦ Σ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ( 𝑘 · ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ) ‘ 𝑗 ) = Σ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ( 𝑘 · ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑗 ) ) ) ) ) |
260 |
170
|
sumeq2sdv |
⊢ ( 𝑗 ∈ ℕ → Σ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ( ( 𝑛 ∈ ℕ ↦ ( 𝑘 · ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ) ‘ 𝑗 ) = Σ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ( 𝑘 · ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑗 ) ) ) ) ) |
261 |
259 260
|
eqtr4d |
⊢ ( 𝑗 ∈ ℕ → ( ( 𝑛 ∈ ℕ ↦ Σ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ( 𝑘 · ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ) ‘ 𝑗 ) = Σ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ( ( 𝑛 ∈ ℕ ↦ ( 𝑘 · ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ) ‘ 𝑗 ) ) |
262 |
255 261
|
sylan9eq |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝑛 ∈ ℕ ↦ ( ∫1 ‘ 𝐺 ) ) ‘ 𝑗 ) = Σ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ( ( 𝑛 ∈ ℕ ↦ ( 𝑘 · ( vol ‘ ( ( ◡ 𝐹 “ { 𝑘 } ) ∩ ( 𝐴 ‘ 𝑛 ) ) ) ) ) ‘ 𝑗 ) ) |
263 |
6 7 12 174 176 182 262
|
climfsum |
⊢ ( 𝜑 → ( 𝑛 ∈ ℕ ↦ ( ∫1 ‘ 𝐺 ) ) ⇝ Σ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ( 𝑘 · ( vol ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) ) ) |
264 |
|
itg1val |
⊢ ( 𝐹 ∈ dom ∫1 → ( ∫1 ‘ 𝐹 ) = Σ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ( 𝑘 · ( vol ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) ) ) |
265 |
4 264
|
syl |
⊢ ( 𝜑 → ( ∫1 ‘ 𝐹 ) = Σ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ( 𝑘 · ( vol ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) ) ) |
266 |
263 265
|
breqtrrd |
⊢ ( 𝜑 → ( 𝑛 ∈ ℕ ↦ ( ∫1 ‘ 𝐺 ) ) ⇝ ( ∫1 ‘ 𝐹 ) ) |