Step | Hyp | Ref
| Expression |
1 | | 1re 9918 |
. . . . . . 7
⊢ 1 ∈
ℝ |
2 | 1 | rexri 9976 |
. . . . . 6
⊢ 1 ∈
ℝ* |
3 | | 0le1 10430 |
. . . . . 6
⊢ 0 ≤
1 |
4 | | pnfge 11840 |
. . . . . . 7
⊢ (1 ∈
ℝ* → 1 ≤ +∞) |
5 | 2, 4 | ax-mp 5 |
. . . . . 6
⊢ 1 ≤
+∞ |
6 | | 0xr 9965 |
. . . . . . 7
⊢ 0 ∈
ℝ* |
7 | | pnfxr 9971 |
. . . . . . 7
⊢ +∞
∈ ℝ* |
8 | | elicc1 12090 |
. . . . . . 7
⊢ ((0
∈ ℝ* ∧ +∞ ∈ ℝ*) → (1
∈ (0[,]+∞) ↔ (1 ∈ ℝ* ∧ 0 ≤ 1 ∧
1 ≤ +∞))) |
9 | 6, 7, 8 | mp2an 704 |
. . . . . 6
⊢ (1 ∈
(0[,]+∞) ↔ (1 ∈ ℝ* ∧ 0 ≤ 1 ∧ 1 ≤
+∞)) |
10 | 2, 3, 5, 9 | mpbir3an 1237 |
. . . . 5
⊢ 1 ∈
(0[,]+∞) |
11 | | 0e0iccpnf 12154 |
. . . . 5
⊢ 0 ∈
(0[,]+∞) |
12 | 10, 11 | keepel 4105 |
. . . 4
⊢ if(0
∈ 𝑎, 1, 0) ∈
(0[,]+∞) |
13 | 12 | rgenw 2908 |
. . 3
⊢
∀𝑎 ∈
𝒫 ℝif(0 ∈ 𝑎, 1, 0) ∈
(0[,]+∞) |
14 | | df-dde 29623 |
. . . 4
⊢ δ =
(𝑎 ∈ 𝒫 ℝ
↦ if(0 ∈ 𝑎, 1,
0)) |
15 | 14 | fmpt 6289 |
. . 3
⊢
(∀𝑎 ∈
𝒫 ℝif(0 ∈ 𝑎, 1, 0) ∈ (0[,]+∞) ↔
δ:𝒫 ℝ⟶(0[,]+∞)) |
16 | 13, 15 | mpbi 219 |
. 2
⊢
δ:𝒫 ℝ⟶(0[,]+∞) |
17 | | 0ss 3924 |
. . 3
⊢ ∅
⊆ ℝ |
18 | | noel 3878 |
. . 3
⊢ ¬ 0
∈ ∅ |
19 | | ddeval0 29625 |
. . 3
⊢ ((∅
⊆ ℝ ∧ ¬ 0 ∈ ∅) → (δ‘∅) =
0) |
20 | 17, 18, 19 | mp2an 704 |
. 2
⊢
(δ‘∅) = 0 |
21 | | rabxm 3915 |
. . . . . . . . 9
⊢ 𝑥 = ({𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎} ∪ {𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎}) |
22 | | esumeq1 29423 |
. . . . . . . . 9
⊢ (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)) |
23 | 21, 22 | ax-mp 5 |
. . . . . . . 8
⊢
Σ*𝑦
∈ 𝑥(δ‘𝑦) = Σ*𝑦 ∈ ({𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎} ∪ {𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎})(δ‘𝑦) |
24 | | nfv 1830 |
. . . . . . . . 9
⊢
Ⅎ𝑦 𝑥 ∈ 𝒫 𝒫
ℝ |
25 | | nfcv 2751 |
. . . . . . . . 9
⊢
Ⅎ𝑦{𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎} |
26 | | nfcv 2751 |
. . . . . . . . 9
⊢
Ⅎ𝑦{𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎} |
27 | | rabexg 4739 |
. . . . . . . . 9
⊢ (𝑥 ∈ 𝒫 𝒫
ℝ → {𝑎 ∈
𝑥 ∣ 0 ∈ 𝑎} ∈ V) |
28 | | rabexg 4739 |
. . . . . . . . 9
⊢ (𝑥 ∈ 𝒫 𝒫
ℝ → {𝑎 ∈
𝑥 ∣ ¬ 0 ∈
𝑎} ∈
V) |
29 | | rabnc 3916 |
. . . . . . . . . 10
⊢ ({𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎} ∩ {𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎}) = ∅ |
30 | 29 | a1i 11 |
. . . . . . . . 9
⊢ (𝑥 ∈ 𝒫 𝒫
ℝ → ({𝑎 ∈
𝑥 ∣ 0 ∈ 𝑎} ∩ {𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎}) = ∅) |
31 | | elrabi 3328 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ {𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎} → 𝑦 ∈ 𝑥) |
32 | 31 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ 𝒫 𝒫
ℝ ∧ 𝑦 ∈
{𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎}) → 𝑦 ∈ 𝑥) |
33 | | simpl 472 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ 𝒫 𝒫
ℝ ∧ 𝑦 ∈
{𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎}) → 𝑥 ∈ 𝒫 𝒫
ℝ) |
34 | | elelpwi 4119 |
. . . . . . . . . . 11
⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝒫 𝒫 ℝ) →
𝑦 ∈ 𝒫
ℝ) |
35 | 32, 33, 34 | syl2anc 691 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ 𝒫 𝒫
ℝ ∧ 𝑦 ∈
{𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎}) → 𝑦 ∈ 𝒫 ℝ) |
36 | 16 | ffvelrni 6266 |
. . . . . . . . . 10
⊢ (𝑦 ∈ 𝒫 ℝ →
(δ‘𝑦) ∈
(0[,]+∞)) |
37 | 35, 36 | syl 17 |
. . . . . . . . 9
⊢ ((𝑥 ∈ 𝒫 𝒫
ℝ ∧ 𝑦 ∈
{𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎}) → (δ‘𝑦) ∈
(0[,]+∞)) |
38 | | elrabi 3328 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ {𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎} → 𝑦 ∈ 𝑥) |
39 | 38 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ 𝒫 𝒫
ℝ ∧ 𝑦 ∈
{𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎}) → 𝑦 ∈ 𝑥) |
40 | | simpl 472 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ 𝒫 𝒫
ℝ ∧ 𝑦 ∈
{𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎}) → 𝑥 ∈ 𝒫 𝒫
ℝ) |
41 | 39, 40, 34 | syl2anc 691 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ 𝒫 𝒫
ℝ ∧ 𝑦 ∈
{𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎}) → 𝑦 ∈ 𝒫 ℝ) |
42 | 41, 36 | syl 17 |
. . . . . . . . 9
⊢ ((𝑥 ∈ 𝒫 𝒫
ℝ ∧ 𝑦 ∈
{𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎}) → (δ‘𝑦) ∈
(0[,]+∞)) |
43 | 24, 25, 26, 27, 28, 30, 37, 42 | esumsplit 29442 |
. . . . . . . 8
⊢ (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))) |
44 | 23, 43 | syl5eq 2656 |
. . . . . . 7
⊢ (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))) |
45 | 44 | adantr 480 |
. . . . . 6
⊢ ((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))) |
46 | | esumeq1 29423 |
. . . . . . . . . . . 12
⊢ ({𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎} = {𝑘} → Σ*𝑦 ∈ {𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎} (δ‘𝑦) = Σ*𝑦 ∈ {𝑘} (δ‘𝑦)) |
47 | 46 | adantl 481 |
. . . . . . . . . . 11
⊢
(((((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)) |
48 | | simp-4l 802 |
. . . . . . . . . . . 12
⊢
(((((𝑥 ∈
𝒫 𝒫 ℝ ∧ Disj 𝑦 ∈ 𝑥 𝑦) ∧ ∃𝑦 ∈ 𝑥 0 ∈ 𝑦) ∧ 𝑘 ∈ 𝑥) ∧ {𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎} = {𝑘}) → 𝑥 ∈ 𝒫 𝒫
ℝ) |
49 | | vex 3176 |
. . . . . . . . . . . . . 14
⊢ 𝑘 ∈ V |
50 | 49 | rabsnel 28726 |
. . . . . . . . . . . . 13
⊢ ({𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎} = {𝑘} → 𝑘 ∈ 𝑥) |
51 | 50 | adantl 481 |
. . . . . . . . . . . 12
⊢
(((((𝑥 ∈
𝒫 𝒫 ℝ ∧ Disj 𝑦 ∈ 𝑥 𝑦) ∧ ∃𝑦 ∈ 𝑥 0 ∈ 𝑦) ∧ 𝑘 ∈ 𝑥) ∧ {𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎} = {𝑘}) → 𝑘 ∈ 𝑥) |
52 | | eleq2 2677 |
. . . . . . . . . . . . . 14
⊢ (𝑎 = 𝑘 → (0 ∈ 𝑎 ↔ 0 ∈ 𝑘)) |
53 | 49, 52 | rabsnt 4210 |
. . . . . . . . . . . . 13
⊢ ({𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎} = {𝑘} → 0 ∈ 𝑘) |
54 | 53 | adantl 481 |
. . . . . . . . . . . 12
⊢
(((((𝑥 ∈
𝒫 𝒫 ℝ ∧ Disj 𝑦 ∈ 𝑥 𝑦) ∧ ∃𝑦 ∈ 𝑥 0 ∈ 𝑦) ∧ 𝑘 ∈ 𝑥) ∧ {𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎} = {𝑘}) → 0 ∈ 𝑘) |
55 | | elelpwi 4119 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑘 ∈ 𝑥 ∧ 𝑥 ∈ 𝒫 𝒫 ℝ) →
𝑘 ∈ 𝒫
ℝ) |
56 | 55 | ancoms 468 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ 𝒫 𝒫
ℝ ∧ 𝑘 ∈
𝑥) → 𝑘 ∈ 𝒫
ℝ) |
57 | 56 | adantrr 749 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ 𝒫 𝒫
ℝ ∧ (𝑘 ∈
𝑥 ∧ 0 ∈ 𝑘)) → 𝑘 ∈ 𝒫 ℝ) |
58 | | simpr 476 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑘 ∈ 𝒫 ℝ ∧
𝑦 = 𝑘) → 𝑦 = 𝑘) |
59 | 58 | fveq2d 6107 |
. . . . . . . . . . . . . . 15
⊢ ((𝑘 ∈ 𝒫 ℝ ∧
𝑦 = 𝑘) → (δ‘𝑦) = (δ‘𝑘)) |
60 | 49 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ 𝒫 ℝ →
𝑘 ∈
V) |
61 | 16 | ffvelrni 6266 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ 𝒫 ℝ →
(δ‘𝑘) ∈
(0[,]+∞)) |
62 | 59, 60, 61 | esumsn 29454 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ 𝒫 ℝ →
Σ*𝑦 ∈
{𝑘} (δ‘𝑦) = (δ‘𝑘)) |
63 | 57, 62 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ 𝒫 𝒫
ℝ ∧ (𝑘 ∈
𝑥 ∧ 0 ∈ 𝑘)) →
Σ*𝑦 ∈
{𝑘} (δ‘𝑦) = (δ‘𝑘)) |
64 | 57 | elpwid 4118 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ 𝒫 𝒫
ℝ ∧ (𝑘 ∈
𝑥 ∧ 0 ∈ 𝑘)) → 𝑘 ⊆ ℝ) |
65 | | simprr 792 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ 𝒫 𝒫
ℝ ∧ (𝑘 ∈
𝑥 ∧ 0 ∈ 𝑘)) → 0 ∈ 𝑘) |
66 | | ddeval1 29624 |
. . . . . . . . . . . . . 14
⊢ ((𝑘 ⊆ ℝ ∧ 0 ∈
𝑘) →
(δ‘𝑘) =
1) |
67 | 64, 65, 66 | syl2anc 691 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ 𝒫 𝒫
ℝ ∧ (𝑘 ∈
𝑥 ∧ 0 ∈ 𝑘)) → (δ‘𝑘) = 1) |
68 | 63, 67 | eqtrd 2644 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ 𝒫 𝒫
ℝ ∧ (𝑘 ∈
𝑥 ∧ 0 ∈ 𝑘)) →
Σ*𝑦 ∈
{𝑘} (δ‘𝑦) = 1) |
69 | 48, 51, 54, 68 | syl12anc 1316 |
. . . . . . . . . . 11
⊢
(((((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) |
70 | 47, 69 | eqtrd 2644 |
. . . . . . . . . 10
⊢
(((((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) |
71 | | df-disj 4554 |
. . . . . . . . . . . . . . 15
⊢
(Disj 𝑦
∈ 𝑥 𝑦 ↔ ∀𝑘∃*𝑦 ∈ 𝑥 𝑘 ∈ 𝑦) |
72 | | c0ex 9913 |
. . . . . . . . . . . . . . . 16
⊢ 0 ∈
V |
73 | | eleq1 2676 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 0 → (𝑘 ∈ 𝑦 ↔ 0 ∈ 𝑦)) |
74 | 73 | rmobidv 3108 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 0 → (∃*𝑦 ∈ 𝑥 𝑘 ∈ 𝑦 ↔ ∃*𝑦 ∈ 𝑥 0 ∈ 𝑦)) |
75 | 72, 74 | spcv 3272 |
. . . . . . . . . . . . . . 15
⊢
(∀𝑘∃*𝑦 ∈ 𝑥 𝑘 ∈ 𝑦 → ∃*𝑦 ∈ 𝑥 0 ∈ 𝑦) |
76 | 71, 75 | sylbi 206 |
. . . . . . . . . . . . . 14
⊢
(Disj 𝑦
∈ 𝑥 𝑦 → ∃*𝑦 ∈ 𝑥 0 ∈ 𝑦) |
77 | | rmo5 3139 |
. . . . . . . . . . . . . . . 16
⊢
(∃*𝑦 ∈
𝑥 0 ∈ 𝑦 ↔ (∃𝑦 ∈ 𝑥 0 ∈ 𝑦 → ∃!𝑦 ∈ 𝑥 0 ∈ 𝑦)) |
78 | 77 | biimpi 205 |
. . . . . . . . . . . . . . 15
⊢
(∃*𝑦 ∈
𝑥 0 ∈ 𝑦 → (∃𝑦 ∈ 𝑥 0 ∈ 𝑦 → ∃!𝑦 ∈ 𝑥 0 ∈ 𝑦)) |
79 | 78 | imp 444 |
. . . . . . . . . . . . . 14
⊢
((∃*𝑦 ∈
𝑥 0 ∈ 𝑦 ∧ ∃𝑦 ∈ 𝑥 0 ∈ 𝑦) → ∃!𝑦 ∈ 𝑥 0 ∈ 𝑦) |
80 | 76, 79 | sylan 487 |
. . . . . . . . . . . . 13
⊢
((Disj 𝑦
∈ 𝑥 𝑦 ∧ ∃𝑦 ∈ 𝑥 0 ∈ 𝑦) → ∃!𝑦 ∈ 𝑥 0 ∈ 𝑦) |
81 | | reusn 4206 |
. . . . . . . . . . . . 13
⊢
(∃!𝑦 ∈
𝑥 0 ∈ 𝑦 ↔ ∃𝑘{𝑦 ∈ 𝑥 ∣ 0 ∈ 𝑦} = {𝑘}) |
82 | 80, 81 | sylib 207 |
. . . . . . . . . . . 12
⊢
((Disj 𝑦
∈ 𝑥 𝑦 ∧ ∃𝑦 ∈ 𝑥 0 ∈ 𝑦) → ∃𝑘{𝑦 ∈ 𝑥 ∣ 0 ∈ 𝑦} = {𝑘}) |
83 | | eleq2 2677 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑎 = 𝑦 → (0 ∈ 𝑎 ↔ 0 ∈ 𝑦)) |
84 | 83 | cbvrabv 3172 |
. . . . . . . . . . . . . . . 16
⊢ {𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎} = {𝑦 ∈ 𝑥 ∣ 0 ∈ 𝑦} |
85 | 84 | eqeq1i 2615 |
. . . . . . . . . . . . . . 15
⊢ ({𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎} = {𝑘} ↔ {𝑦 ∈ 𝑥 ∣ 0 ∈ 𝑦} = {𝑘}) |
86 | 50 | ancri 573 |
. . . . . . . . . . . . . . 15
⊢ ({𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎} = {𝑘} → (𝑘 ∈ 𝑥 ∧ {𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎} = {𝑘})) |
87 | 85, 86 | sylbir 224 |
. . . . . . . . . . . . . 14
⊢ ({𝑦 ∈ 𝑥 ∣ 0 ∈ 𝑦} = {𝑘} → (𝑘 ∈ 𝑥 ∧ {𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎} = {𝑘})) |
88 | 87 | eximi 1752 |
. . . . . . . . . . . . 13
⊢
(∃𝑘{𝑦 ∈ 𝑥 ∣ 0 ∈ 𝑦} = {𝑘} → ∃𝑘(𝑘 ∈ 𝑥 ∧ {𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎} = {𝑘})) |
89 | | df-rex 2902 |
. . . . . . . . . . . . 13
⊢
(∃𝑘 ∈
𝑥 {𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎} = {𝑘} ↔ ∃𝑘(𝑘 ∈ 𝑥 ∧ {𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎} = {𝑘})) |
90 | 88, 89 | sylibr 223 |
. . . . . . . . . . . 12
⊢
(∃𝑘{𝑦 ∈ 𝑥 ∣ 0 ∈ 𝑦} = {𝑘} → ∃𝑘 ∈ 𝑥 {𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎} = {𝑘}) |
91 | 82, 90 | syl 17 |
. . . . . . . . . . 11
⊢
((Disj 𝑦
∈ 𝑥 𝑦 ∧ ∃𝑦 ∈ 𝑥 0 ∈ 𝑦) → ∃𝑘 ∈ 𝑥 {𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎} = {𝑘}) |
92 | 91 | adantll 746 |
. . . . . . . . . 10
⊢ (((𝑥 ∈ 𝒫 𝒫
ℝ ∧ Disj 𝑦
∈ 𝑥 𝑦) ∧ ∃𝑦 ∈ 𝑥 0 ∈ 𝑦) → ∃𝑘 ∈ 𝑥 {𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎} = {𝑘}) |
93 | 70, 92 | r19.29a 3060 |
. . . . . . . . 9
⊢ (((𝑥 ∈ 𝒫 𝒫
ℝ ∧ Disj 𝑦
∈ 𝑥 𝑦) ∧ ∃𝑦 ∈ 𝑥 0 ∈ 𝑦) → Σ*𝑦 ∈ {𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎} (δ‘𝑦) = 1) |
94 | | elpwi 4117 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ 𝒫 𝒫
ℝ → 𝑥 ⊆
𝒫 ℝ) |
95 | | sspwuni 4547 |
. . . . . . . . . . . 12
⊢ (𝑥 ⊆ 𝒫 ℝ
↔ ∪ 𝑥 ⊆ ℝ) |
96 | 94, 95 | sylib 207 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ 𝒫 𝒫
ℝ → ∪ 𝑥 ⊆ ℝ) |
97 | | eluni2 4376 |
. . . . . . . . . . . 12
⊢ (0 ∈
∪ 𝑥 ↔ ∃𝑦 ∈ 𝑥 0 ∈ 𝑦) |
98 | 97 | biimpri 217 |
. . . . . . . . . . 11
⊢
(∃𝑦 ∈
𝑥 0 ∈ 𝑦 → 0 ∈ ∪ 𝑥) |
99 | | ddeval1 29624 |
. . . . . . . . . . 11
⊢ ((∪ 𝑥
⊆ ℝ ∧ 0 ∈ ∪ 𝑥) → (δ‘∪ 𝑥) =
1) |
100 | 96, 98, 99 | syl2an 493 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ 𝒫 𝒫
ℝ ∧ ∃𝑦
∈ 𝑥 0 ∈ 𝑦) → (δ‘∪ 𝑥) =
1) |
101 | 100 | adantlr 747 |
. . . . . . . . 9
⊢ (((𝑥 ∈ 𝒫 𝒫
ℝ ∧ Disj 𝑦
∈ 𝑥 𝑦) ∧ ∃𝑦 ∈ 𝑥 0 ∈ 𝑦) → (δ‘∪ 𝑥) =
1) |
102 | 93, 101 | eqtr4d 2647 |
. . . . . . . 8
⊢ (((𝑥 ∈ 𝒫 𝒫
ℝ ∧ Disj 𝑦
∈ 𝑥 𝑦) ∧ ∃𝑦 ∈ 𝑥 0 ∈ 𝑦) → Σ*𝑦 ∈ {𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎} (δ‘𝑦) = (δ‘∪ 𝑥)) |
103 | | nfre1 2988 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑦∃𝑦 ∈ 𝑥 0 ∈ 𝑦 |
104 | 103 | nfn 1768 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑦 ¬
∃𝑦 ∈ 𝑥 0 ∈ 𝑦 |
105 | 83 | elrab 3331 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 ∈ {𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎} ↔ (𝑦 ∈ 𝑥 ∧ 0 ∈ 𝑦)) |
106 | 105 | exbii 1764 |
. . . . . . . . . . . . . . 15
⊢
(∃𝑦 𝑦 ∈ {𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎} ↔ ∃𝑦(𝑦 ∈ 𝑥 ∧ 0 ∈ 𝑦)) |
107 | | neq0 3889 |
. . . . . . . . . . . . . . 15
⊢ (¬
{𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎} = ∅ ↔ ∃𝑦 𝑦 ∈ {𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎}) |
108 | | df-rex 2902 |
. . . . . . . . . . . . . . 15
⊢
(∃𝑦 ∈
𝑥 0 ∈ 𝑦 ↔ ∃𝑦(𝑦 ∈ 𝑥 ∧ 0 ∈ 𝑦)) |
109 | 106, 107,
108 | 3bitr4i 291 |
. . . . . . . . . . . . . 14
⊢ (¬
{𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎} = ∅ ↔ ∃𝑦 ∈ 𝑥 0 ∈ 𝑦) |
110 | 109 | biimpi 205 |
. . . . . . . . . . . . 13
⊢ (¬
{𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎} = ∅ → ∃𝑦 ∈ 𝑥 0 ∈ 𝑦) |
111 | 110 | con1i 143 |
. . . . . . . . . . . 12
⊢ (¬
∃𝑦 ∈ 𝑥 0 ∈ 𝑦 → {𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎} = ∅) |
112 | 104, 111 | esumeq1d 29424 |
. . . . . . . . . . 11
⊢ (¬
∃𝑦 ∈ 𝑥 0 ∈ 𝑦 → Σ*𝑦 ∈ {𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎} (δ‘𝑦) = Σ*𝑦 ∈ ∅(δ‘𝑦)) |
113 | | esumnul 29437 |
. . . . . . . . . . 11
⊢
Σ*𝑦
∈ ∅(δ‘𝑦) = 0 |
114 | 112, 113 | syl6eq 2660 |
. . . . . . . . . 10
⊢ (¬
∃𝑦 ∈ 𝑥 0 ∈ 𝑦 → Σ*𝑦 ∈ {𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎} (δ‘𝑦) = 0) |
115 | 114 | adantl 481 |
. . . . . . . . 9
⊢ (((𝑥 ∈ 𝒫 𝒫
ℝ ∧ Disj 𝑦
∈ 𝑥 𝑦) ∧ ¬ ∃𝑦 ∈ 𝑥 0 ∈ 𝑦) → Σ*𝑦 ∈ {𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎} (δ‘𝑦) = 0) |
116 | 97 | biimpi 205 |
. . . . . . . . . . . 12
⊢ (0 ∈
∪ 𝑥 → ∃𝑦 ∈ 𝑥 0 ∈ 𝑦) |
117 | 116 | con3i 149 |
. . . . . . . . . . 11
⊢ (¬
∃𝑦 ∈ 𝑥 0 ∈ 𝑦 → ¬ 0 ∈ ∪ 𝑥) |
118 | | ddeval0 29625 |
. . . . . . . . . . 11
⊢ ((∪ 𝑥
⊆ ℝ ∧ ¬ 0 ∈ ∪ 𝑥) → (δ‘∪ 𝑥) =
0) |
119 | 96, 117, 118 | syl2an 493 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ 𝒫 𝒫
ℝ ∧ ¬ ∃𝑦 ∈ 𝑥 0 ∈ 𝑦) → (δ‘∪ 𝑥) =
0) |
120 | 119 | adantlr 747 |
. . . . . . . . 9
⊢ (((𝑥 ∈ 𝒫 𝒫
ℝ ∧ Disj 𝑦
∈ 𝑥 𝑦) ∧ ¬ ∃𝑦 ∈ 𝑥 0 ∈ 𝑦) → (δ‘∪ 𝑥) =
0) |
121 | 115, 120 | eqtr4d 2647 |
. . . . . . . 8
⊢ (((𝑥 ∈ 𝒫 𝒫
ℝ ∧ Disj 𝑦
∈ 𝑥 𝑦) ∧ ¬ ∃𝑦 ∈ 𝑥 0 ∈ 𝑦) → Σ*𝑦 ∈ {𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎} (δ‘𝑦) = (δ‘∪ 𝑥)) |
122 | 102, 121 | pm2.61dan 828 |
. . . . . . 7
⊢ ((𝑥 ∈ 𝒫 𝒫
ℝ ∧ Disj 𝑦
∈ 𝑥 𝑦) → Σ*𝑦 ∈ {𝑎 ∈ 𝑥 ∣ 0 ∈ 𝑎} (δ‘𝑦) = (δ‘∪ 𝑥)) |
123 | 41 | elpwid 4118 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ 𝒫 𝒫
ℝ ∧ 𝑦 ∈
{𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎}) → 𝑦 ⊆ ℝ) |
124 | 83 | notbid 307 |
. . . . . . . . . . . . . 14
⊢ (𝑎 = 𝑦 → (¬ 0 ∈ 𝑎 ↔ ¬ 0 ∈ 𝑦)) |
125 | 124 | elrab 3331 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ {𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎} ↔ (𝑦 ∈ 𝑥 ∧ ¬ 0 ∈ 𝑦)) |
126 | 125 | simprbi 479 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ {𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎} → ¬ 0 ∈ 𝑦) |
127 | 126 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ 𝒫 𝒫
ℝ ∧ 𝑦 ∈
{𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎}) → ¬ 0 ∈ 𝑦) |
128 | | ddeval0 29625 |
. . . . . . . . . . 11
⊢ ((𝑦 ⊆ ℝ ∧ ¬ 0
∈ 𝑦) →
(δ‘𝑦) =
0) |
129 | 123, 127,
128 | syl2anc 691 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ 𝒫 𝒫
ℝ ∧ 𝑦 ∈
{𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎}) → (δ‘𝑦) = 0) |
130 | 129 | esumeq2dv 29427 |
. . . . . . . . 9
⊢ (𝑥 ∈ 𝒫 𝒫
ℝ → Σ*𝑦 ∈ {𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎} (δ‘𝑦) = Σ*𝑦 ∈ {𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎}0) |
131 | | vex 3176 |
. . . . . . . . . . 11
⊢ 𝑥 ∈ V |
132 | 131 | rabex 4740 |
. . . . . . . . . 10
⊢ {𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎} ∈ V |
133 | 26 | esum0 29438 |
. . . . . . . . . 10
⊢ ({𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎} ∈ V → Σ*𝑦 ∈ {𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎}0 = 0) |
134 | 132, 133 | ax-mp 5 |
. . . . . . . . 9
⊢
Σ*𝑦
∈ {𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎}0 = 0 |
135 | 130, 134 | syl6eq 2660 |
. . . . . . . 8
⊢ (𝑥 ∈ 𝒫 𝒫
ℝ → Σ*𝑦 ∈ {𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎} (δ‘𝑦) = 0) |
136 | 135 | adantr 480 |
. . . . . . 7
⊢ ((𝑥 ∈ 𝒫 𝒫
ℝ ∧ Disj 𝑦
∈ 𝑥 𝑦) → Σ*𝑦 ∈ {𝑎 ∈ 𝑥 ∣ ¬ 0 ∈ 𝑎} (δ‘𝑦) = 0) |
137 | 122, 136 | oveq12d 6567 |
. . . . . 6
⊢ ((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)) |
138 | | vuniex 6852 |
. . . . . . . . . 10
⊢ ∪ 𝑥
∈ V |
139 | 138 | elpw 4114 |
. . . . . . . . 9
⊢ (∪ 𝑥
∈ 𝒫 ℝ ↔ ∪ 𝑥 ⊆ ℝ) |
140 | 139 | biimpri 217 |
. . . . . . . 8
⊢ (∪ 𝑥
⊆ ℝ → ∪ 𝑥 ∈ 𝒫 ℝ) |
141 | | iccssxr 12127 |
. . . . . . . . 9
⊢
(0[,]+∞) ⊆ ℝ* |
142 | 16 | ffvelrni 6266 |
. . . . . . . . 9
⊢ (∪ 𝑥
∈ 𝒫 ℝ → (δ‘∪
𝑥) ∈
(0[,]+∞)) |
143 | 141, 142 | sseldi 3566 |
. . . . . . . 8
⊢ (∪ 𝑥
∈ 𝒫 ℝ → (δ‘∪
𝑥) ∈
ℝ*) |
144 | | xaddid1 11946 |
. . . . . . . 8
⊢
((δ‘∪ 𝑥) ∈ ℝ* →
((δ‘∪ 𝑥) +𝑒 0) =
(δ‘∪ 𝑥)) |
145 | 96, 140, 143, 144 | 4syl 19 |
. . . . . . 7
⊢ (𝑥 ∈ 𝒫 𝒫
ℝ → ((δ‘∪ 𝑥) +𝑒 0) =
(δ‘∪ 𝑥)) |
146 | 145 | adantr 480 |
. . . . . 6
⊢ ((𝑥 ∈ 𝒫 𝒫
ℝ ∧ Disj 𝑦
∈ 𝑥 𝑦) → ((δ‘∪ 𝑥)
+𝑒 0) = (δ‘∪ 𝑥)) |
147 | 45, 137, 146 | 3eqtrrd 2649 |
. . . . 5
⊢ ((𝑥 ∈ 𝒫 𝒫
ℝ ∧ Disj 𝑦
∈ 𝑥 𝑦) → (δ‘∪ 𝑥) =
Σ*𝑦 ∈
𝑥(δ‘𝑦)) |
148 | 147 | adantrl 748 |
. . . 4
⊢ ((𝑥 ∈ 𝒫 𝒫
ℝ ∧ (𝑥 ≼
ω ∧ Disj 𝑦
∈ 𝑥 𝑦)) → (δ‘∪ 𝑥) =
Σ*𝑦 ∈
𝑥(δ‘𝑦)) |
149 | 148 | ex 449 |
. . 3
⊢ (𝑥 ∈ 𝒫 𝒫
ℝ → ((𝑥 ≼
ω ∧ Disj 𝑦
∈ 𝑥 𝑦) → (δ‘∪ 𝑥) =
Σ*𝑦 ∈
𝑥(δ‘𝑦))) |
150 | 149 | rgen 2906 |
. 2
⊢
∀𝑥 ∈
𝒫 𝒫 ℝ((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (δ‘∪ 𝑥) =
Σ*𝑦 ∈
𝑥(δ‘𝑦)) |
151 | | reex 9906 |
. . . 4
⊢ ℝ
∈ V |
152 | | pwsiga 29520 |
. . . 4
⊢ (ℝ
∈ V → 𝒫 ℝ ∈
(sigAlgebra‘ℝ)) |
153 | 151, 152 | ax-mp 5 |
. . 3
⊢ 𝒫
ℝ ∈ (sigAlgebra‘ℝ) |
154 | | elrnsiga 29516 |
. . 3
⊢
(𝒫 ℝ ∈ (sigAlgebra‘ℝ) → 𝒫
ℝ ∈ ∪ ran sigAlgebra) |
155 | | ismeas 29589 |
. . 3
⊢
(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))))) |
156 | 153, 154,
155 | mp2b 10 |
. 2
⊢ (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)))) |
157 | 16, 20, 150, 156 | mpbir3an 1237 |
1
⊢ δ
∈ (measures‘𝒫 ℝ) |