Step | Hyp | Ref
| Expression |
1 | | simp2 1133 |
. . . 4
⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ⊆ ∪ 𝑄)
→ 𝑅:𝑄⟶(0[,]+∞)) |
2 | | simp1 1132 |
. . . 4
⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ⊆ ∪ 𝑄)
→ 𝑄 ∈ 𝑉) |
3 | | fex 6991 |
. . . 4
⊢ ((𝑅:𝑄⟶(0[,]+∞) ∧ 𝑄 ∈ 𝑉) → 𝑅 ∈ V) |
4 | 1, 2, 3 | syl2anc 586 |
. . 3
⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ⊆ ∪ 𝑄)
→ 𝑅 ∈
V) |
5 | | omsval 31553 |
. . 3
⊢ (𝑅 ∈ V →
(toOMeas‘𝑅) = (𝑎 ∈ 𝒫 ∪ dom 𝑅 ↦ inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦
Σ*𝑦 ∈
𝑥(𝑅‘𝑦)), (0[,]+∞), < ))) |
6 | 4, 5 | syl 17 |
. 2
⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ⊆ ∪ 𝑄)
→ (toOMeas‘𝑅) =
(𝑎 ∈ 𝒫 ∪ dom 𝑅 ↦ inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦
Σ*𝑦 ∈
𝑥(𝑅‘𝑦)), (0[,]+∞), < ))) |
7 | | simpr 487 |
. . . . . . . 8
⊢ (((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ⊆ ∪ 𝑄)
∧ 𝑎 = 𝐴) → 𝑎 = 𝐴) |
8 | 7 | sseq1d 4000 |
. . . . . . 7
⊢ (((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ⊆ ∪ 𝑄)
∧ 𝑎 = 𝐴) → (𝑎 ⊆ ∪ 𝑧 ↔ 𝐴 ⊆ ∪ 𝑧)) |
9 | 8 | anbi1d 631 |
. . . . . 6
⊢ (((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ⊆ ∪ 𝑄)
∧ 𝑎 = 𝐴) → ((𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω) ↔ (𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω))) |
10 | 9 | rabbidv 3482 |
. . . . 5
⊢ (((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ⊆ ∪ 𝑄)
∧ 𝑎 = 𝐴) → {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} = {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)}) |
11 | 10 | mpteq1d 5157 |
. . . 4
⊢ (((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ⊆ ∪ 𝑄)
∧ 𝑎 = 𝐴) → (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦
Σ*𝑦 ∈
𝑥(𝑅‘𝑦)) = (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦
Σ*𝑦 ∈
𝑥(𝑅‘𝑦))) |
12 | 11 | rneqd 5810 |
. . 3
⊢ (((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ⊆ ∪ 𝑄)
∧ 𝑎 = 𝐴) → ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦
Σ*𝑦 ∈
𝑥(𝑅‘𝑦)) = ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦
Σ*𝑦 ∈
𝑥(𝑅‘𝑦))) |
13 | 12 | infeq1d 8943 |
. 2
⊢ (((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ⊆ ∪ 𝑄)
∧ 𝑎 = 𝐴) → inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦
Σ*𝑦 ∈
𝑥(𝑅‘𝑦)), (0[,]+∞), < ) = inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦
Σ*𝑦 ∈
𝑥(𝑅‘𝑦)), (0[,]+∞), < )) |
14 | | simp3 1134 |
. . . 4
⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ⊆ ∪ 𝑄)
→ 𝐴 ⊆ ∪ 𝑄) |
15 | | fdm 6524 |
. . . . . 6
⊢ (𝑅:𝑄⟶(0[,]+∞) → dom 𝑅 = 𝑄) |
16 | 15 | 3ad2ant2 1130 |
. . . . 5
⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ⊆ ∪ 𝑄)
→ dom 𝑅 = 𝑄) |
17 | 16 | unieqd 4854 |
. . . 4
⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ⊆ ∪ 𝑄)
→ ∪ dom 𝑅 = ∪ 𝑄) |
18 | 14, 17 | sseqtrrd 4010 |
. . 3
⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ⊆ ∪ 𝑄)
→ 𝐴 ⊆ ∪ dom 𝑅) |
19 | 2 | uniexd 7470 |
. . . . 5
⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ⊆ ∪ 𝑄)
→ ∪ 𝑄 ∈ V) |
20 | | ssexg 5229 |
. . . . 5
⊢ ((𝐴 ⊆ ∪ 𝑄
∧ ∪ 𝑄 ∈ V) → 𝐴 ∈ V) |
21 | 14, 19, 20 | syl2anc 586 |
. . . 4
⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ⊆ ∪ 𝑄)
→ 𝐴 ∈
V) |
22 | | elpwg 4544 |
. . . 4
⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 ∪ dom 𝑅 ↔ 𝐴 ⊆ ∪ dom
𝑅)) |
23 | 21, 22 | syl 17 |
. . 3
⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ⊆ ∪ 𝑄)
→ (𝐴 ∈ 𝒫
∪ dom 𝑅 ↔ 𝐴 ⊆ ∪ dom
𝑅)) |
24 | 18, 23 | mpbird 259 |
. 2
⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ⊆ ∪ 𝑄)
→ 𝐴 ∈ 𝒫
∪ dom 𝑅) |
25 | | xrltso 12537 |
. . . 4
⊢ < Or
ℝ* |
26 | | iccssxr 12822 |
. . . . 5
⊢
(0[,]+∞) ⊆ ℝ* |
27 | | soss 5495 |
. . . . 5
⊢
((0[,]+∞) ⊆ ℝ* → ( < Or
ℝ* → < Or (0[,]+∞))) |
28 | 26, 27 | ax-mp 5 |
. . . 4
⊢ ( < Or
ℝ* → < Or (0[,]+∞)) |
29 | 25, 28 | mp1i 13 |
. . 3
⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ⊆ ∪ 𝑄)
→ < Or (0[,]+∞)) |
30 | 29 | infexd 8949 |
. 2
⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ⊆ ∪ 𝑄)
→ inf(ran (𝑥 ∈
{𝑧 ∈ 𝒫 dom
𝑅 ∣ (𝐴 ⊆ ∪ 𝑧
∧ 𝑧 ≼ ω)}
↦ Σ*𝑦 ∈ 𝑥(𝑅‘𝑦)), (0[,]+∞), < ) ∈
V) |
31 | 6, 13, 24, 30 | fvmptd 6777 |
1
⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ⊆ ∪ 𝑄)
→ ((toOMeas‘𝑅)‘𝐴) = inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦
Σ*𝑦 ∈
𝑥(𝑅‘𝑦)), (0[,]+∞), < )) |