Step |
Hyp |
Ref |
Expression |
1 |
|
meaiunlelem.1 |
⊢ Ⅎ 𝑛 𝜑 |
2 |
|
meaiunlelem.m |
⊢ ( 𝜑 → 𝑀 ∈ Meas ) |
3 |
|
meaiunlelem.s |
⊢ 𝑆 = dom 𝑀 |
4 |
|
meaiunlelem.z |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑁 ) |
5 |
|
meaiunlelem.e |
⊢ ( 𝜑 → 𝐸 : 𝑍 ⟶ 𝑆 ) |
6 |
|
meaiunlelem.f |
⊢ 𝐹 = ( 𝑛 ∈ 𝑍 ↦ ( ( 𝐸 ‘ 𝑛 ) ∖ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ( 𝐸 ‘ 𝑖 ) ) ) |
7 |
1 4 5 6
|
iundjiun |
⊢ ( 𝜑 → ( ( ∀ 𝑥 ∈ 𝑍 ∪ 𝑛 ∈ ( 𝑁 ... 𝑥 ) ( 𝐹 ‘ 𝑛 ) = ∪ 𝑛 ∈ ( 𝑁 ... 𝑥 ) ( 𝐸 ‘ 𝑛 ) ∧ ∪ 𝑛 ∈ 𝑍 ( 𝐹 ‘ 𝑛 ) = ∪ 𝑛 ∈ 𝑍 ( 𝐸 ‘ 𝑛 ) ) ∧ Disj 𝑛 ∈ 𝑍 ( 𝐹 ‘ 𝑛 ) ) ) |
8 |
7
|
simplrd |
⊢ ( 𝜑 → ∪ 𝑛 ∈ 𝑍 ( 𝐹 ‘ 𝑛 ) = ∪ 𝑛 ∈ 𝑍 ( 𝐸 ‘ 𝑛 ) ) |
9 |
8
|
eqcomd |
⊢ ( 𝜑 → ∪ 𝑛 ∈ 𝑍 ( 𝐸 ‘ 𝑛 ) = ∪ 𝑛 ∈ 𝑍 ( 𝐹 ‘ 𝑛 ) ) |
10 |
9
|
fveq2d |
⊢ ( 𝜑 → ( 𝑀 ‘ ∪ 𝑛 ∈ 𝑍 ( 𝐸 ‘ 𝑛 ) ) = ( 𝑀 ‘ ∪ 𝑛 ∈ 𝑍 ( 𝐹 ‘ 𝑛 ) ) ) |
11 |
2 3
|
dmmeasal |
⊢ ( 𝜑 → 𝑆 ∈ SAlg ) |
12 |
11
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → 𝑆 ∈ SAlg ) |
13 |
5
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝐸 ‘ 𝑛 ) ∈ 𝑆 ) |
14 |
|
fzofi |
⊢ ( 𝑁 ..^ 𝑛 ) ∈ Fin |
15 |
|
isfinite |
⊢ ( ( 𝑁 ..^ 𝑛 ) ∈ Fin ↔ ( 𝑁 ..^ 𝑛 ) ≺ ω ) |
16 |
15
|
biimpi |
⊢ ( ( 𝑁 ..^ 𝑛 ) ∈ Fin → ( 𝑁 ..^ 𝑛 ) ≺ ω ) |
17 |
|
sdomdom |
⊢ ( ( 𝑁 ..^ 𝑛 ) ≺ ω → ( 𝑁 ..^ 𝑛 ) ≼ ω ) |
18 |
16 17
|
syl |
⊢ ( ( 𝑁 ..^ 𝑛 ) ∈ Fin → ( 𝑁 ..^ 𝑛 ) ≼ ω ) |
19 |
14 18
|
ax-mp |
⊢ ( 𝑁 ..^ 𝑛 ) ≼ ω |
20 |
19
|
a1i |
⊢ ( 𝜑 → ( 𝑁 ..^ 𝑛 ) ≼ ω ) |
21 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ) → 𝐸 : 𝑍 ⟶ 𝑆 ) |
22 |
|
elfzouz |
⊢ ( 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) → 𝑖 ∈ ( ℤ≥ ‘ 𝑁 ) ) |
23 |
4
|
eqcomi |
⊢ ( ℤ≥ ‘ 𝑁 ) = 𝑍 |
24 |
22 23
|
eleqtrdi |
⊢ ( 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) → 𝑖 ∈ 𝑍 ) |
25 |
24
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ) → 𝑖 ∈ 𝑍 ) |
26 |
21 25
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ) → ( 𝐸 ‘ 𝑖 ) ∈ 𝑆 ) |
27 |
11 20 26
|
saliuncl |
⊢ ( 𝜑 → ∪ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ( 𝐸 ‘ 𝑖 ) ∈ 𝑆 ) |
28 |
27
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ∪ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ( 𝐸 ‘ 𝑖 ) ∈ 𝑆 ) |
29 |
|
saldifcl2 |
⊢ ( ( 𝑆 ∈ SAlg ∧ ( 𝐸 ‘ 𝑛 ) ∈ 𝑆 ∧ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ( 𝐸 ‘ 𝑖 ) ∈ 𝑆 ) → ( ( 𝐸 ‘ 𝑛 ) ∖ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ( 𝐸 ‘ 𝑖 ) ) ∈ 𝑆 ) |
30 |
12 13 28 29
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( ( 𝐸 ‘ 𝑛 ) ∖ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ( 𝐸 ‘ 𝑖 ) ) ∈ 𝑆 ) |
31 |
1 30 6
|
fmptdf |
⊢ ( 𝜑 → 𝐹 : 𝑍 ⟶ 𝑆 ) |
32 |
31
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑛 ) ∈ 𝑆 ) |
33 |
|
eqid |
⊢ ( ℤ≥ ‘ 𝑁 ) = ( ℤ≥ ‘ 𝑁 ) |
34 |
33
|
uzct |
⊢ ( ℤ≥ ‘ 𝑁 ) ≼ ω |
35 |
4 34
|
eqbrtri |
⊢ 𝑍 ≼ ω |
36 |
35
|
a1i |
⊢ ( 𝜑 → 𝑍 ≼ ω ) |
37 |
7
|
simprd |
⊢ ( 𝜑 → Disj 𝑛 ∈ 𝑍 ( 𝐹 ‘ 𝑛 ) ) |
38 |
1 2 3 32 36 37
|
meadjiun |
⊢ ( 𝜑 → ( 𝑀 ‘ ∪ 𝑛 ∈ 𝑍 ( 𝐹 ‘ 𝑛 ) ) = ( Σ^ ‘ ( 𝑛 ∈ 𝑍 ↦ ( 𝑀 ‘ ( 𝐹 ‘ 𝑛 ) ) ) ) ) |
39 |
|
eqidd |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑛 ∈ 𝑍 ↦ ( 𝑀 ‘ ( 𝐹 ‘ 𝑛 ) ) ) ) = ( Σ^ ‘ ( 𝑛 ∈ 𝑍 ↦ ( 𝑀 ‘ ( 𝐹 ‘ 𝑛 ) ) ) ) ) |
40 |
10 38 39
|
3eqtrd |
⊢ ( 𝜑 → ( 𝑀 ‘ ∪ 𝑛 ∈ 𝑍 ( 𝐸 ‘ 𝑛 ) ) = ( Σ^ ‘ ( 𝑛 ∈ 𝑍 ↦ ( 𝑀 ‘ ( 𝐹 ‘ 𝑛 ) ) ) ) ) |
41 |
4
|
fvexi |
⊢ 𝑍 ∈ V |
42 |
41
|
a1i |
⊢ ( 𝜑 → 𝑍 ∈ V ) |
43 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → 𝑀 ∈ Meas ) |
44 |
43 3 32
|
meacl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑀 ‘ ( 𝐹 ‘ 𝑛 ) ) ∈ ( 0 [,] +∞ ) ) |
45 |
43 3 13
|
meacl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ∈ ( 0 [,] +∞ ) ) |
46 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → 𝑛 ∈ 𝑍 ) |
47 |
13
|
difexd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( ( 𝐸 ‘ 𝑛 ) ∖ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ( 𝐸 ‘ 𝑖 ) ) ∈ V ) |
48 |
6
|
fvmpt2 |
⊢ ( ( 𝑛 ∈ 𝑍 ∧ ( ( 𝐸 ‘ 𝑛 ) ∖ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ( 𝐸 ‘ 𝑖 ) ) ∈ V ) → ( 𝐹 ‘ 𝑛 ) = ( ( 𝐸 ‘ 𝑛 ) ∖ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ( 𝐸 ‘ 𝑖 ) ) ) |
49 |
46 47 48
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑛 ) = ( ( 𝐸 ‘ 𝑛 ) ∖ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ( 𝐸 ‘ 𝑖 ) ) ) |
50 |
|
difssd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( ( 𝐸 ‘ 𝑛 ) ∖ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ( 𝐸 ‘ 𝑖 ) ) ⊆ ( 𝐸 ‘ 𝑛 ) ) |
51 |
49 50
|
eqsstrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑛 ) ⊆ ( 𝐸 ‘ 𝑛 ) ) |
52 |
43 3 32 13 51
|
meassle |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑀 ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ) |
53 |
1 42 44 45 52
|
sge0lempt |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑛 ∈ 𝑍 ↦ ( 𝑀 ‘ ( 𝐹 ‘ 𝑛 ) ) ) ) ≤ ( Σ^ ‘ ( 𝑛 ∈ 𝑍 ↦ ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ) ) ) |
54 |
40 53
|
eqbrtrd |
⊢ ( 𝜑 → ( 𝑀 ‘ ∪ 𝑛 ∈ 𝑍 ( 𝐸 ‘ 𝑛 ) ) ≤ ( Σ^ ‘ ( 𝑛 ∈ 𝑍 ↦ ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ) ) ) |