Step |
Hyp |
Ref |
Expression |
1 |
|
1xr |
⊢ 1 ∈ ℝ* |
2 |
|
0le1 |
⊢ 0 ≤ 1 |
3 |
|
pnfge |
⊢ ( 1 ∈ ℝ* → 1 ≤ +∞ ) |
4 |
1 3
|
ax-mp |
⊢ 1 ≤ +∞ |
5 |
|
0xr |
⊢ 0 ∈ ℝ* |
6 |
|
pnfxr |
⊢ +∞ ∈ ℝ* |
7 |
|
elicc1 |
⊢ ( ( 0 ∈ ℝ* ∧ +∞ ∈ ℝ* ) → ( 1 ∈ ( 0 [,] +∞ ) ↔ ( 1 ∈ ℝ* ∧ 0 ≤ 1 ∧ 1 ≤ +∞ ) ) ) |
8 |
5 6 7
|
mp2an |
⊢ ( 1 ∈ ( 0 [,] +∞ ) ↔ ( 1 ∈ ℝ* ∧ 0 ≤ 1 ∧ 1 ≤ +∞ ) ) |
9 |
1 2 4 8
|
mpbir3an |
⊢ 1 ∈ ( 0 [,] +∞ ) |
10 |
|
0e0iccpnf |
⊢ 0 ∈ ( 0 [,] +∞ ) |
11 |
9 10
|
ifcli |
⊢ if ( 0 ∈ 𝑎 , 1 , 0 ) ∈ ( 0 [,] +∞ ) |
12 |
11
|
rgenw |
⊢ ∀ 𝑎 ∈ 𝒫 ℝ if ( 0 ∈ 𝑎 , 1 , 0 ) ∈ ( 0 [,] +∞ ) |
13 |
|
df-dde |
⊢ δ = ( 𝑎 ∈ 𝒫 ℝ ↦ if ( 0 ∈ 𝑎 , 1 , 0 ) ) |
14 |
13
|
fmpt |
⊢ ( ∀ 𝑎 ∈ 𝒫 ℝ if ( 0 ∈ 𝑎 , 1 , 0 ) ∈ ( 0 [,] +∞ ) ↔ δ : 𝒫 ℝ ⟶ ( 0 [,] +∞ ) ) |
15 |
12 14
|
mpbi |
⊢ δ : 𝒫 ℝ ⟶ ( 0 [,] +∞ ) |
16 |
|
0ss |
⊢ ∅ ⊆ ℝ |
17 |
|
noel |
⊢ ¬ 0 ∈ ∅ |
18 |
|
ddeval0 |
⊢ ( ( ∅ ⊆ ℝ ∧ ¬ 0 ∈ ∅ ) → ( δ ‘ ∅ ) = 0 ) |
19 |
16 17 18
|
mp2an |
⊢ ( δ ‘ ∅ ) = 0 |
20 |
|
rabxm |
⊢ 𝑥 = ( { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } ∪ { 𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎 } ) |
21 |
|
esumeq1 |
⊢ ( 𝑥 = ( { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } ∪ { 𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎 } ) → Σ* 𝑦 ∈ 𝑥 ( δ ‘ 𝑦 ) = Σ* 𝑦 ∈ ( { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } ∪ { 𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎 } ) ( δ ‘ 𝑦 ) ) |
22 |
20 21
|
ax-mp |
⊢ Σ* 𝑦 ∈ 𝑥 ( δ ‘ 𝑦 ) = Σ* 𝑦 ∈ ( { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } ∪ { 𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎 } ) ( δ ‘ 𝑦 ) |
23 |
|
nfv |
⊢ Ⅎ 𝑦 𝑥 ∈ 𝒫 𝒫 ℝ |
24 |
|
nfcv |
⊢ Ⅎ 𝑦 { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } |
25 |
|
nfcv |
⊢ Ⅎ 𝑦 { 𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎 } |
26 |
|
rabexg |
⊢ ( 𝑥 ∈ 𝒫 𝒫 ℝ → { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } ∈ V ) |
27 |
|
rabexg |
⊢ ( 𝑥 ∈ 𝒫 𝒫 ℝ → { 𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎 } ∈ V ) |
28 |
|
rabnc |
⊢ ( { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } ∩ { 𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎 } ) = ∅ |
29 |
28
|
a1i |
⊢ ( 𝑥 ∈ 𝒫 𝒫 ℝ → ( { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } ∩ { 𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎 } ) = ∅ ) |
30 |
|
elrabi |
⊢ ( 𝑦 ∈ { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } → 𝑦 ∈ 𝑥 ) |
31 |
30
|
adantl |
⊢ ( ( 𝑥 ∈ 𝒫 𝒫 ℝ ∧ 𝑦 ∈ { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } ) → 𝑦 ∈ 𝑥 ) |
32 |
|
simpl |
⊢ ( ( 𝑥 ∈ 𝒫 𝒫 ℝ ∧ 𝑦 ∈ { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } ) → 𝑥 ∈ 𝒫 𝒫 ℝ ) |
33 |
|
elelpwi |
⊢ ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝒫 𝒫 ℝ ) → 𝑦 ∈ 𝒫 ℝ ) |
34 |
31 32 33
|
syl2anc |
⊢ ( ( 𝑥 ∈ 𝒫 𝒫 ℝ ∧ 𝑦 ∈ { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } ) → 𝑦 ∈ 𝒫 ℝ ) |
35 |
15
|
ffvelrni |
⊢ ( 𝑦 ∈ 𝒫 ℝ → ( δ ‘ 𝑦 ) ∈ ( 0 [,] +∞ ) ) |
36 |
34 35
|
syl |
⊢ ( ( 𝑥 ∈ 𝒫 𝒫 ℝ ∧ 𝑦 ∈ { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } ) → ( δ ‘ 𝑦 ) ∈ ( 0 [,] +∞ ) ) |
37 |
|
elrabi |
⊢ ( 𝑦 ∈ { 𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎 } → 𝑦 ∈ 𝑥 ) |
38 |
37
|
adantl |
⊢ ( ( 𝑥 ∈ 𝒫 𝒫 ℝ ∧ 𝑦 ∈ { 𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎 } ) → 𝑦 ∈ 𝑥 ) |
39 |
|
simpl |
⊢ ( ( 𝑥 ∈ 𝒫 𝒫 ℝ ∧ 𝑦 ∈ { 𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎 } ) → 𝑥 ∈ 𝒫 𝒫 ℝ ) |
40 |
38 39 33
|
syl2anc |
⊢ ( ( 𝑥 ∈ 𝒫 𝒫 ℝ ∧ 𝑦 ∈ { 𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎 } ) → 𝑦 ∈ 𝒫 ℝ ) |
41 |
40 35
|
syl |
⊢ ( ( 𝑥 ∈ 𝒫 𝒫 ℝ ∧ 𝑦 ∈ { 𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎 } ) → ( δ ‘ 𝑦 ) ∈ ( 0 [,] +∞ ) ) |
42 |
23 24 25 26 27 29 36 41
|
esumsplit |
⊢ ( 𝑥 ∈ 𝒫 𝒫 ℝ → Σ* 𝑦 ∈ ( { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } ∪ { 𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎 } ) ( δ ‘ 𝑦 ) = ( Σ* 𝑦 ∈ { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } ( δ ‘ 𝑦 ) +𝑒 Σ* 𝑦 ∈ { 𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎 } ( δ ‘ 𝑦 ) ) ) |
43 |
22 42
|
syl5eq |
⊢ ( 𝑥 ∈ 𝒫 𝒫 ℝ → Σ* 𝑦 ∈ 𝑥 ( δ ‘ 𝑦 ) = ( Σ* 𝑦 ∈ { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } ( δ ‘ 𝑦 ) +𝑒 Σ* 𝑦 ∈ { 𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎 } ( δ ‘ 𝑦 ) ) ) |
44 |
43
|
adantr |
⊢ ( ( 𝑥 ∈ 𝒫 𝒫 ℝ ∧ Disj 𝑦 ∈ 𝑥 𝑦 ) → Σ* 𝑦 ∈ 𝑥 ( δ ‘ 𝑦 ) = ( Σ* 𝑦 ∈ { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } ( δ ‘ 𝑦 ) +𝑒 Σ* 𝑦 ∈ { 𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎 } ( δ ‘ 𝑦 ) ) ) |
45 |
|
esumeq1 |
⊢ ( { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } = { 𝑘 } → Σ* 𝑦 ∈ { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } ( δ ‘ 𝑦 ) = Σ* 𝑦 ∈ { 𝑘 } ( δ ‘ 𝑦 ) ) |
46 |
45
|
adantl |
⊢ ( ( ( ( ( 𝑥 ∈ 𝒫 𝒫 ℝ ∧ Disj 𝑦 ∈ 𝑥 𝑦 ) ∧ ∃ 𝑦 ∈ 𝑥 0 ∈ 𝑦 ) ∧ 𝑘 ∈ 𝑥 ) ∧ { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } = { 𝑘 } ) → Σ* 𝑦 ∈ { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } ( δ ‘ 𝑦 ) = Σ* 𝑦 ∈ { 𝑘 } ( δ ‘ 𝑦 ) ) |
47 |
|
simp-4l |
⊢ ( ( ( ( ( 𝑥 ∈ 𝒫 𝒫 ℝ ∧ Disj 𝑦 ∈ 𝑥 𝑦 ) ∧ ∃ 𝑦 ∈ 𝑥 0 ∈ 𝑦 ) ∧ 𝑘 ∈ 𝑥 ) ∧ { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } = { 𝑘 } ) → 𝑥 ∈ 𝒫 𝒫 ℝ ) |
48 |
|
vex |
⊢ 𝑘 ∈ V |
49 |
48
|
rabsnel |
⊢ ( { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } = { 𝑘 } → 𝑘 ∈ 𝑥 ) |
50 |
49
|
adantl |
⊢ ( ( ( ( ( 𝑥 ∈ 𝒫 𝒫 ℝ ∧ Disj 𝑦 ∈ 𝑥 𝑦 ) ∧ ∃ 𝑦 ∈ 𝑥 0 ∈ 𝑦 ) ∧ 𝑘 ∈ 𝑥 ) ∧ { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } = { 𝑘 } ) → 𝑘 ∈ 𝑥 ) |
51 |
|
eleq2w |
⊢ ( 𝑎 = 𝑘 → ( 0 ∈ 𝑎 ↔ 0 ∈ 𝑘 ) ) |
52 |
48 51
|
rabsnt |
⊢ ( { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } = { 𝑘 } → 0 ∈ 𝑘 ) |
53 |
52
|
adantl |
⊢ ( ( ( ( ( 𝑥 ∈ 𝒫 𝒫 ℝ ∧ Disj 𝑦 ∈ 𝑥 𝑦 ) ∧ ∃ 𝑦 ∈ 𝑥 0 ∈ 𝑦 ) ∧ 𝑘 ∈ 𝑥 ) ∧ { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } = { 𝑘 } ) → 0 ∈ 𝑘 ) |
54 |
|
elelpwi |
⊢ ( ( 𝑘 ∈ 𝑥 ∧ 𝑥 ∈ 𝒫 𝒫 ℝ ) → 𝑘 ∈ 𝒫 ℝ ) |
55 |
54
|
ancoms |
⊢ ( ( 𝑥 ∈ 𝒫 𝒫 ℝ ∧ 𝑘 ∈ 𝑥 ) → 𝑘 ∈ 𝒫 ℝ ) |
56 |
55
|
adantrr |
⊢ ( ( 𝑥 ∈ 𝒫 𝒫 ℝ ∧ ( 𝑘 ∈ 𝑥 ∧ 0 ∈ 𝑘 ) ) → 𝑘 ∈ 𝒫 ℝ ) |
57 |
|
simpr |
⊢ ( ( 𝑘 ∈ 𝒫 ℝ ∧ 𝑦 = 𝑘 ) → 𝑦 = 𝑘 ) |
58 |
57
|
fveq2d |
⊢ ( ( 𝑘 ∈ 𝒫 ℝ ∧ 𝑦 = 𝑘 ) → ( δ ‘ 𝑦 ) = ( δ ‘ 𝑘 ) ) |
59 |
48
|
a1i |
⊢ ( 𝑘 ∈ 𝒫 ℝ → 𝑘 ∈ V ) |
60 |
15
|
ffvelrni |
⊢ ( 𝑘 ∈ 𝒫 ℝ → ( δ ‘ 𝑘 ) ∈ ( 0 [,] +∞ ) ) |
61 |
58 59 60
|
esumsn |
⊢ ( 𝑘 ∈ 𝒫 ℝ → Σ* 𝑦 ∈ { 𝑘 } ( δ ‘ 𝑦 ) = ( δ ‘ 𝑘 ) ) |
62 |
56 61
|
syl |
⊢ ( ( 𝑥 ∈ 𝒫 𝒫 ℝ ∧ ( 𝑘 ∈ 𝑥 ∧ 0 ∈ 𝑘 ) ) → Σ* 𝑦 ∈ { 𝑘 } ( δ ‘ 𝑦 ) = ( δ ‘ 𝑘 ) ) |
63 |
56
|
elpwid |
⊢ ( ( 𝑥 ∈ 𝒫 𝒫 ℝ ∧ ( 𝑘 ∈ 𝑥 ∧ 0 ∈ 𝑘 ) ) → 𝑘 ⊆ ℝ ) |
64 |
|
simprr |
⊢ ( ( 𝑥 ∈ 𝒫 𝒫 ℝ ∧ ( 𝑘 ∈ 𝑥 ∧ 0 ∈ 𝑘 ) ) → 0 ∈ 𝑘 ) |
65 |
|
ddeval1 |
⊢ ( ( 𝑘 ⊆ ℝ ∧ 0 ∈ 𝑘 ) → ( δ ‘ 𝑘 ) = 1 ) |
66 |
63 64 65
|
syl2anc |
⊢ ( ( 𝑥 ∈ 𝒫 𝒫 ℝ ∧ ( 𝑘 ∈ 𝑥 ∧ 0 ∈ 𝑘 ) ) → ( δ ‘ 𝑘 ) = 1 ) |
67 |
62 66
|
eqtrd |
⊢ ( ( 𝑥 ∈ 𝒫 𝒫 ℝ ∧ ( 𝑘 ∈ 𝑥 ∧ 0 ∈ 𝑘 ) ) → Σ* 𝑦 ∈ { 𝑘 } ( δ ‘ 𝑦 ) = 1 ) |
68 |
47 50 53 67
|
syl12anc |
⊢ ( ( ( ( ( 𝑥 ∈ 𝒫 𝒫 ℝ ∧ Disj 𝑦 ∈ 𝑥 𝑦 ) ∧ ∃ 𝑦 ∈ 𝑥 0 ∈ 𝑦 ) ∧ 𝑘 ∈ 𝑥 ) ∧ { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } = { 𝑘 } ) → Σ* 𝑦 ∈ { 𝑘 } ( δ ‘ 𝑦 ) = 1 ) |
69 |
46 68
|
eqtrd |
⊢ ( ( ( ( ( 𝑥 ∈ 𝒫 𝒫 ℝ ∧ Disj 𝑦 ∈ 𝑥 𝑦 ) ∧ ∃ 𝑦 ∈ 𝑥 0 ∈ 𝑦 ) ∧ 𝑘 ∈ 𝑥 ) ∧ { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } = { 𝑘 } ) → Σ* 𝑦 ∈ { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } ( δ ‘ 𝑦 ) = 1 ) |
70 |
|
df-disj |
⊢ ( Disj 𝑦 ∈ 𝑥 𝑦 ↔ ∀ 𝑘 ∃* 𝑦 ∈ 𝑥 𝑘 ∈ 𝑦 ) |
71 |
|
c0ex |
⊢ 0 ∈ V |
72 |
|
eleq1 |
⊢ ( 𝑘 = 0 → ( 𝑘 ∈ 𝑦 ↔ 0 ∈ 𝑦 ) ) |
73 |
72
|
rmobidv |
⊢ ( 𝑘 = 0 → ( ∃* 𝑦 ∈ 𝑥 𝑘 ∈ 𝑦 ↔ ∃* 𝑦 ∈ 𝑥 0 ∈ 𝑦 ) ) |
74 |
71 73
|
spcv |
⊢ ( ∀ 𝑘 ∃* 𝑦 ∈ 𝑥 𝑘 ∈ 𝑦 → ∃* 𝑦 ∈ 𝑥 0 ∈ 𝑦 ) |
75 |
70 74
|
sylbi |
⊢ ( Disj 𝑦 ∈ 𝑥 𝑦 → ∃* 𝑦 ∈ 𝑥 0 ∈ 𝑦 ) |
76 |
|
rmo5 |
⊢ ( ∃* 𝑦 ∈ 𝑥 0 ∈ 𝑦 ↔ ( ∃ 𝑦 ∈ 𝑥 0 ∈ 𝑦 → ∃! 𝑦 ∈ 𝑥 0 ∈ 𝑦 ) ) |
77 |
76
|
biimpi |
⊢ ( ∃* 𝑦 ∈ 𝑥 0 ∈ 𝑦 → ( ∃ 𝑦 ∈ 𝑥 0 ∈ 𝑦 → ∃! 𝑦 ∈ 𝑥 0 ∈ 𝑦 ) ) |
78 |
77
|
imp |
⊢ ( ( ∃* 𝑦 ∈ 𝑥 0 ∈ 𝑦 ∧ ∃ 𝑦 ∈ 𝑥 0 ∈ 𝑦 ) → ∃! 𝑦 ∈ 𝑥 0 ∈ 𝑦 ) |
79 |
75 78
|
sylan |
⊢ ( ( Disj 𝑦 ∈ 𝑥 𝑦 ∧ ∃ 𝑦 ∈ 𝑥 0 ∈ 𝑦 ) → ∃! 𝑦 ∈ 𝑥 0 ∈ 𝑦 ) |
80 |
|
reusn |
⊢ ( ∃! 𝑦 ∈ 𝑥 0 ∈ 𝑦 ↔ ∃ 𝑘 { 𝑦 ∈ 𝑥 ∣ 0 ∈ 𝑦 } = { 𝑘 } ) |
81 |
79 80
|
sylib |
⊢ ( ( Disj 𝑦 ∈ 𝑥 𝑦 ∧ ∃ 𝑦 ∈ 𝑥 0 ∈ 𝑦 ) → ∃ 𝑘 { 𝑦 ∈ 𝑥 ∣ 0 ∈ 𝑦 } = { 𝑘 } ) |
82 |
|
eleq2w |
⊢ ( 𝑎 = 𝑦 → ( 0 ∈ 𝑎 ↔ 0 ∈ 𝑦 ) ) |
83 |
82
|
cbvrabv |
⊢ { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } = { 𝑦 ∈ 𝑥 ∣ 0 ∈ 𝑦 } |
84 |
83
|
eqeq1i |
⊢ ( { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } = { 𝑘 } ↔ { 𝑦 ∈ 𝑥 ∣ 0 ∈ 𝑦 } = { 𝑘 } ) |
85 |
49
|
ancri |
⊢ ( { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } = { 𝑘 } → ( 𝑘 ∈ 𝑥 ∧ { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } = { 𝑘 } ) ) |
86 |
84 85
|
sylbir |
⊢ ( { 𝑦 ∈ 𝑥 ∣ 0 ∈ 𝑦 } = { 𝑘 } → ( 𝑘 ∈ 𝑥 ∧ { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } = { 𝑘 } ) ) |
87 |
86
|
eximi |
⊢ ( ∃ 𝑘 { 𝑦 ∈ 𝑥 ∣ 0 ∈ 𝑦 } = { 𝑘 } → ∃ 𝑘 ( 𝑘 ∈ 𝑥 ∧ { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } = { 𝑘 } ) ) |
88 |
|
df-rex |
⊢ ( ∃ 𝑘 ∈ 𝑥 { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } = { 𝑘 } ↔ ∃ 𝑘 ( 𝑘 ∈ 𝑥 ∧ { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } = { 𝑘 } ) ) |
89 |
87 88
|
sylibr |
⊢ ( ∃ 𝑘 { 𝑦 ∈ 𝑥 ∣ 0 ∈ 𝑦 } = { 𝑘 } → ∃ 𝑘 ∈ 𝑥 { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } = { 𝑘 } ) |
90 |
81 89
|
syl |
⊢ ( ( Disj 𝑦 ∈ 𝑥 𝑦 ∧ ∃ 𝑦 ∈ 𝑥 0 ∈ 𝑦 ) → ∃ 𝑘 ∈ 𝑥 { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } = { 𝑘 } ) |
91 |
90
|
adantll |
⊢ ( ( ( 𝑥 ∈ 𝒫 𝒫 ℝ ∧ Disj 𝑦 ∈ 𝑥 𝑦 ) ∧ ∃ 𝑦 ∈ 𝑥 0 ∈ 𝑦 ) → ∃ 𝑘 ∈ 𝑥 { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } = { 𝑘 } ) |
92 |
69 91
|
r19.29a |
⊢ ( ( ( 𝑥 ∈ 𝒫 𝒫 ℝ ∧ Disj 𝑦 ∈ 𝑥 𝑦 ) ∧ ∃ 𝑦 ∈ 𝑥 0 ∈ 𝑦 ) → Σ* 𝑦 ∈ { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } ( δ ‘ 𝑦 ) = 1 ) |
93 |
|
elpwi |
⊢ ( 𝑥 ∈ 𝒫 𝒫 ℝ → 𝑥 ⊆ 𝒫 ℝ ) |
94 |
|
sspwuni |
⊢ ( 𝑥 ⊆ 𝒫 ℝ ↔ ∪ 𝑥 ⊆ ℝ ) |
95 |
93 94
|
sylib |
⊢ ( 𝑥 ∈ 𝒫 𝒫 ℝ → ∪ 𝑥 ⊆ ℝ ) |
96 |
|
eluni2 |
⊢ ( 0 ∈ ∪ 𝑥 ↔ ∃ 𝑦 ∈ 𝑥 0 ∈ 𝑦 ) |
97 |
96
|
biimpri |
⊢ ( ∃ 𝑦 ∈ 𝑥 0 ∈ 𝑦 → 0 ∈ ∪ 𝑥 ) |
98 |
|
ddeval1 |
⊢ ( ( ∪ 𝑥 ⊆ ℝ ∧ 0 ∈ ∪ 𝑥 ) → ( δ ‘ ∪ 𝑥 ) = 1 ) |
99 |
95 97 98
|
syl2an |
⊢ ( ( 𝑥 ∈ 𝒫 𝒫 ℝ ∧ ∃ 𝑦 ∈ 𝑥 0 ∈ 𝑦 ) → ( δ ‘ ∪ 𝑥 ) = 1 ) |
100 |
99
|
adantlr |
⊢ ( ( ( 𝑥 ∈ 𝒫 𝒫 ℝ ∧ Disj 𝑦 ∈ 𝑥 𝑦 ) ∧ ∃ 𝑦 ∈ 𝑥 0 ∈ 𝑦 ) → ( δ ‘ ∪ 𝑥 ) = 1 ) |
101 |
92 100
|
eqtr4d |
⊢ ( ( ( 𝑥 ∈ 𝒫 𝒫 ℝ ∧ Disj 𝑦 ∈ 𝑥 𝑦 ) ∧ ∃ 𝑦 ∈ 𝑥 0 ∈ 𝑦 ) → Σ* 𝑦 ∈ { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } ( δ ‘ 𝑦 ) = ( δ ‘ ∪ 𝑥 ) ) |
102 |
|
nfre1 |
⊢ Ⅎ 𝑦 ∃ 𝑦 ∈ 𝑥 0 ∈ 𝑦 |
103 |
102
|
nfn |
⊢ Ⅎ 𝑦 ¬ ∃ 𝑦 ∈ 𝑥 0 ∈ 𝑦 |
104 |
82
|
elrab |
⊢ ( 𝑦 ∈ { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } ↔ ( 𝑦 ∈ 𝑥 ∧ 0 ∈ 𝑦 ) ) |
105 |
104
|
exbii |
⊢ ( ∃ 𝑦 𝑦 ∈ { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } ↔ ∃ 𝑦 ( 𝑦 ∈ 𝑥 ∧ 0 ∈ 𝑦 ) ) |
106 |
|
neq0 |
⊢ ( ¬ { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } = ∅ ↔ ∃ 𝑦 𝑦 ∈ { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } ) |
107 |
|
df-rex |
⊢ ( ∃ 𝑦 ∈ 𝑥 0 ∈ 𝑦 ↔ ∃ 𝑦 ( 𝑦 ∈ 𝑥 ∧ 0 ∈ 𝑦 ) ) |
108 |
105 106 107
|
3bitr4i |
⊢ ( ¬ { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } = ∅ ↔ ∃ 𝑦 ∈ 𝑥 0 ∈ 𝑦 ) |
109 |
108
|
biimpi |
⊢ ( ¬ { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } = ∅ → ∃ 𝑦 ∈ 𝑥 0 ∈ 𝑦 ) |
110 |
109
|
con1i |
⊢ ( ¬ ∃ 𝑦 ∈ 𝑥 0 ∈ 𝑦 → { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } = ∅ ) |
111 |
103 110
|
esumeq1d |
⊢ ( ¬ ∃ 𝑦 ∈ 𝑥 0 ∈ 𝑦 → Σ* 𝑦 ∈ { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } ( δ ‘ 𝑦 ) = Σ* 𝑦 ∈ ∅ ( δ ‘ 𝑦 ) ) |
112 |
|
esumnul |
⊢ Σ* 𝑦 ∈ ∅ ( δ ‘ 𝑦 ) = 0 |
113 |
111 112
|
eqtrdi |
⊢ ( ¬ ∃ 𝑦 ∈ 𝑥 0 ∈ 𝑦 → Σ* 𝑦 ∈ { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } ( δ ‘ 𝑦 ) = 0 ) |
114 |
113
|
adantl |
⊢ ( ( ( 𝑥 ∈ 𝒫 𝒫 ℝ ∧ Disj 𝑦 ∈ 𝑥 𝑦 ) ∧ ¬ ∃ 𝑦 ∈ 𝑥 0 ∈ 𝑦 ) → Σ* 𝑦 ∈ { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } ( δ ‘ 𝑦 ) = 0 ) |
115 |
96
|
biimpi |
⊢ ( 0 ∈ ∪ 𝑥 → ∃ 𝑦 ∈ 𝑥 0 ∈ 𝑦 ) |
116 |
115
|
con3i |
⊢ ( ¬ ∃ 𝑦 ∈ 𝑥 0 ∈ 𝑦 → ¬ 0 ∈ ∪ 𝑥 ) |
117 |
|
ddeval0 |
⊢ ( ( ∪ 𝑥 ⊆ ℝ ∧ ¬ 0 ∈ ∪ 𝑥 ) → ( δ ‘ ∪ 𝑥 ) = 0 ) |
118 |
95 116 117
|
syl2an |
⊢ ( ( 𝑥 ∈ 𝒫 𝒫 ℝ ∧ ¬ ∃ 𝑦 ∈ 𝑥 0 ∈ 𝑦 ) → ( δ ‘ ∪ 𝑥 ) = 0 ) |
119 |
118
|
adantlr |
⊢ ( ( ( 𝑥 ∈ 𝒫 𝒫 ℝ ∧ Disj 𝑦 ∈ 𝑥 𝑦 ) ∧ ¬ ∃ 𝑦 ∈ 𝑥 0 ∈ 𝑦 ) → ( δ ‘ ∪ 𝑥 ) = 0 ) |
120 |
114 119
|
eqtr4d |
⊢ ( ( ( 𝑥 ∈ 𝒫 𝒫 ℝ ∧ Disj 𝑦 ∈ 𝑥 𝑦 ) ∧ ¬ ∃ 𝑦 ∈ 𝑥 0 ∈ 𝑦 ) → Σ* 𝑦 ∈ { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } ( δ ‘ 𝑦 ) = ( δ ‘ ∪ 𝑥 ) ) |
121 |
101 120
|
pm2.61dan |
⊢ ( ( 𝑥 ∈ 𝒫 𝒫 ℝ ∧ Disj 𝑦 ∈ 𝑥 𝑦 ) → Σ* 𝑦 ∈ { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } ( δ ‘ 𝑦 ) = ( δ ‘ ∪ 𝑥 ) ) |
122 |
40
|
elpwid |
⊢ ( ( 𝑥 ∈ 𝒫 𝒫 ℝ ∧ 𝑦 ∈ { 𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎 } ) → 𝑦 ⊆ ℝ ) |
123 |
82
|
notbid |
⊢ ( 𝑎 = 𝑦 → ( ¬ 0 ∈ 𝑎 ↔ ¬ 0 ∈ 𝑦 ) ) |
124 |
123
|
elrab |
⊢ ( 𝑦 ∈ { 𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎 } ↔ ( 𝑦 ∈ 𝑥 ∧ ¬ 0 ∈ 𝑦 ) ) |
125 |
124
|
simprbi |
⊢ ( 𝑦 ∈ { 𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎 } → ¬ 0 ∈ 𝑦 ) |
126 |
125
|
adantl |
⊢ ( ( 𝑥 ∈ 𝒫 𝒫 ℝ ∧ 𝑦 ∈ { 𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎 } ) → ¬ 0 ∈ 𝑦 ) |
127 |
|
ddeval0 |
⊢ ( ( 𝑦 ⊆ ℝ ∧ ¬ 0 ∈ 𝑦 ) → ( δ ‘ 𝑦 ) = 0 ) |
128 |
122 126 127
|
syl2anc |
⊢ ( ( 𝑥 ∈ 𝒫 𝒫 ℝ ∧ 𝑦 ∈ { 𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎 } ) → ( δ ‘ 𝑦 ) = 0 ) |
129 |
128
|
esumeq2dv |
⊢ ( 𝑥 ∈ 𝒫 𝒫 ℝ → Σ* 𝑦 ∈ { 𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎 } ( δ ‘ 𝑦 ) = Σ* 𝑦 ∈ { 𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎 } 0 ) |
130 |
|
vex |
⊢ 𝑥 ∈ V |
131 |
130
|
rabex |
⊢ { 𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎 } ∈ V |
132 |
25
|
esum0 |
⊢ ( { 𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎 } ∈ V → Σ* 𝑦 ∈ { 𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎 } 0 = 0 ) |
133 |
131 132
|
ax-mp |
⊢ Σ* 𝑦 ∈ { 𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎 } 0 = 0 |
134 |
129 133
|
eqtrdi |
⊢ ( 𝑥 ∈ 𝒫 𝒫 ℝ → Σ* 𝑦 ∈ { 𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎 } ( δ ‘ 𝑦 ) = 0 ) |
135 |
134
|
adantr |
⊢ ( ( 𝑥 ∈ 𝒫 𝒫 ℝ ∧ Disj 𝑦 ∈ 𝑥 𝑦 ) → Σ* 𝑦 ∈ { 𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎 } ( δ ‘ 𝑦 ) = 0 ) |
136 |
121 135
|
oveq12d |
⊢ ( ( 𝑥 ∈ 𝒫 𝒫 ℝ ∧ Disj 𝑦 ∈ 𝑥 𝑦 ) → ( Σ* 𝑦 ∈ { 𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎 } ( δ ‘ 𝑦 ) +𝑒 Σ* 𝑦 ∈ { 𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎 } ( δ ‘ 𝑦 ) ) = ( ( δ ‘ ∪ 𝑥 ) +𝑒 0 ) ) |
137 |
|
vuniex |
⊢ ∪ 𝑥 ∈ V |
138 |
137
|
elpw |
⊢ ( ∪ 𝑥 ∈ 𝒫 ℝ ↔ ∪ 𝑥 ⊆ ℝ ) |
139 |
138
|
biimpri |
⊢ ( ∪ 𝑥 ⊆ ℝ → ∪ 𝑥 ∈ 𝒫 ℝ ) |
140 |
|
iccssxr |
⊢ ( 0 [,] +∞ ) ⊆ ℝ* |
141 |
15
|
ffvelrni |
⊢ ( ∪ 𝑥 ∈ 𝒫 ℝ → ( δ ‘ ∪ 𝑥 ) ∈ ( 0 [,] +∞ ) ) |
142 |
140 141
|
sselid |
⊢ ( ∪ 𝑥 ∈ 𝒫 ℝ → ( δ ‘ ∪ 𝑥 ) ∈ ℝ* ) |
143 |
|
xaddid1 |
⊢ ( ( δ ‘ ∪ 𝑥 ) ∈ ℝ* → ( ( δ ‘ ∪ 𝑥 ) +𝑒 0 ) = ( δ ‘ ∪ 𝑥 ) ) |
144 |
95 139 142 143
|
4syl |
⊢ ( 𝑥 ∈ 𝒫 𝒫 ℝ → ( ( δ ‘ ∪ 𝑥 ) +𝑒 0 ) = ( δ ‘ ∪ 𝑥 ) ) |
145 |
144
|
adantr |
⊢ ( ( 𝑥 ∈ 𝒫 𝒫 ℝ ∧ Disj 𝑦 ∈ 𝑥 𝑦 ) → ( ( δ ‘ ∪ 𝑥 ) +𝑒 0 ) = ( δ ‘ ∪ 𝑥 ) ) |
146 |
44 136 145
|
3eqtrrd |
⊢ ( ( 𝑥 ∈ 𝒫 𝒫 ℝ ∧ Disj 𝑦 ∈ 𝑥 𝑦 ) → ( δ ‘ ∪ 𝑥 ) = Σ* 𝑦 ∈ 𝑥 ( δ ‘ 𝑦 ) ) |
147 |
146
|
adantrl |
⊢ ( ( 𝑥 ∈ 𝒫 𝒫 ℝ ∧ ( 𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦 ) ) → ( δ ‘ ∪ 𝑥 ) = Σ* 𝑦 ∈ 𝑥 ( δ ‘ 𝑦 ) ) |
148 |
147
|
ex |
⊢ ( 𝑥 ∈ 𝒫 𝒫 ℝ → ( ( 𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦 ) → ( δ ‘ ∪ 𝑥 ) = Σ* 𝑦 ∈ 𝑥 ( δ ‘ 𝑦 ) ) ) |
149 |
148
|
rgen |
⊢ ∀ 𝑥 ∈ 𝒫 𝒫 ℝ ( ( 𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦 ) → ( δ ‘ ∪ 𝑥 ) = Σ* 𝑦 ∈ 𝑥 ( δ ‘ 𝑦 ) ) |
150 |
|
reex |
⊢ ℝ ∈ V |
151 |
|
pwsiga |
⊢ ( ℝ ∈ V → 𝒫 ℝ ∈ ( sigAlgebra ‘ ℝ ) ) |
152 |
150 151
|
ax-mp |
⊢ 𝒫 ℝ ∈ ( sigAlgebra ‘ ℝ ) |
153 |
|
elrnsiga |
⊢ ( 𝒫 ℝ ∈ ( sigAlgebra ‘ ℝ ) → 𝒫 ℝ ∈ ∪ ran sigAlgebra ) |
154 |
|
ismeas |
⊢ ( 𝒫 ℝ ∈ ∪ ran sigAlgebra → ( δ ∈ ( measures ‘ 𝒫 ℝ ) ↔ ( δ : 𝒫 ℝ ⟶ ( 0 [,] +∞ ) ∧ ( δ ‘ ∅ ) = 0 ∧ ∀ 𝑥 ∈ 𝒫 𝒫 ℝ ( ( 𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦 ) → ( δ ‘ ∪ 𝑥 ) = Σ* 𝑦 ∈ 𝑥 ( δ ‘ 𝑦 ) ) ) ) ) |
155 |
152 153 154
|
mp2b |
⊢ ( δ ∈ ( measures ‘ 𝒫 ℝ ) ↔ ( δ : 𝒫 ℝ ⟶ ( 0 [,] +∞ ) ∧ ( δ ‘ ∅ ) = 0 ∧ ∀ 𝑥 ∈ 𝒫 𝒫 ℝ ( ( 𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦 ) → ( δ ‘ ∪ 𝑥 ) = Σ* 𝑦 ∈ 𝑥 ( δ ‘ 𝑦 ) ) ) ) |
156 |
15 19 149 155
|
mpbir3an |
⊢ δ ∈ ( measures ‘ 𝒫 ℝ ) |