| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | iccssxr 13471 | . . 3
⊢
(0[,]+∞) ⊆ ℝ* | 
| 2 |  | omeiunle.o | . . . 4
⊢ (𝜑 → 𝑂 ∈ OutMeas) | 
| 3 |  | omeiunle.x | . . . 4
⊢ 𝑋 = ∪
dom 𝑂 | 
| 4 |  | omeiunle.nph | . . . . . 6
⊢
Ⅎ𝑛𝜑 | 
| 5 |  | omeiunle.e | . . . . . . . . 9
⊢ (𝜑 → 𝐸:𝑍⟶𝒫 𝑋) | 
| 6 | 5 | ffvelcdmda 7103 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ∈ 𝒫 𝑋) | 
| 7 |  | elpwi 4606 | . . . . . . . 8
⊢ ((𝐸‘𝑛) ∈ 𝒫 𝑋 → (𝐸‘𝑛) ⊆ 𝑋) | 
| 8 | 6, 7 | syl 17 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ⊆ 𝑋) | 
| 9 | 8 | ex 412 | . . . . . 6
⊢ (𝜑 → (𝑛 ∈ 𝑍 → (𝐸‘𝑛) ⊆ 𝑋)) | 
| 10 | 4, 9 | ralrimi 3256 | . . . . 5
⊢ (𝜑 → ∀𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ 𝑋) | 
| 11 |  | iunss 5044 | . . . . 5
⊢ (∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ 𝑋 ↔ ∀𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ 𝑋) | 
| 12 | 10, 11 | sylibr 234 | . . . 4
⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ 𝑋) | 
| 13 | 2, 3, 12 | omecl 46523 | . . 3
⊢ (𝜑 → (𝑂‘∪
𝑛 ∈ 𝑍 (𝐸‘𝑛)) ∈ (0[,]+∞)) | 
| 14 | 1, 13 | sselid 3980 | . 2
⊢ (𝜑 → (𝑂‘∪
𝑛 ∈ 𝑍 (𝐸‘𝑛)) ∈
ℝ*) | 
| 15 | 5 | ffnd 6736 | . . . . 5
⊢ (𝜑 → 𝐸 Fn 𝑍) | 
| 16 |  | omeiunle.z | . . . . . . 7
⊢ 𝑍 =
(ℤ≥‘𝑁) | 
| 17 | 16 | fvexi 6919 | . . . . . 6
⊢ 𝑍 ∈ V | 
| 18 | 17 | a1i 11 | . . . . 5
⊢ (𝜑 → 𝑍 ∈ V) | 
| 19 |  | fnex 7238 | . . . . 5
⊢ ((𝐸 Fn 𝑍 ∧ 𝑍 ∈ V) → 𝐸 ∈ V) | 
| 20 | 15, 18, 19 | syl2anc 584 | . . . 4
⊢ (𝜑 → 𝐸 ∈ V) | 
| 21 |  | rnexg 7925 | . . . 4
⊢ (𝐸 ∈ V → ran 𝐸 ∈ V) | 
| 22 | 20, 21 | syl 17 | . . 3
⊢ (𝜑 → ran 𝐸 ∈ V) | 
| 23 | 2, 3 | omef 46516 | . . . 4
⊢ (𝜑 → 𝑂:𝒫 𝑋⟶(0[,]+∞)) | 
| 24 | 5 | frnd 6743 | . . . 4
⊢ (𝜑 → ran 𝐸 ⊆ 𝒫 𝑋) | 
| 25 | 23, 24 | fssresd 6774 | . . 3
⊢ (𝜑 → (𝑂 ↾ ran 𝐸):ran 𝐸⟶(0[,]+∞)) | 
| 26 | 22, 25 | sge0xrcl 46405 | . 2
⊢ (𝜑 →
(Σ^‘(𝑂 ↾ ran 𝐸)) ∈
ℝ*) | 
| 27 | 2 | adantr 480 | . . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑂 ∈ OutMeas) | 
| 28 | 27, 3, 8 | omecl 46523 | . . . 4
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑂‘(𝐸‘𝑛)) ∈ (0[,]+∞)) | 
| 29 |  | eqid 2736 | . . . 4
⊢ (𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))) | 
| 30 | 4, 28, 29 | fmptdf 7136 | . . 3
⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))):𝑍⟶(0[,]+∞)) | 
| 31 | 18, 30 | sge0xrcl 46405 | . 2
⊢ (𝜑 →
(Σ^‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈
ℝ*) | 
| 32 |  | fvex 6918 | . . . . . . . 8
⊢ (𝐸‘𝑛) ∈ V | 
| 33 | 32 | rgenw 3064 | . . . . . . 7
⊢
∀𝑛 ∈
𝑍 (𝐸‘𝑛) ∈ V | 
| 34 |  | dfiun3g 5977 | . . . . . . 7
⊢
(∀𝑛 ∈
𝑍 (𝐸‘𝑛) ∈ V → ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) = ∪ ran (𝑛 ∈ 𝑍 ↦ (𝐸‘𝑛))) | 
| 35 | 33, 34 | ax-mp 5 | . . . . . 6
⊢ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) = ∪ ran (𝑛 ∈ 𝑍 ↦ (𝐸‘𝑛)) | 
| 36 | 35 | a1i 11 | . . . . 5
⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) = ∪ ran (𝑛 ∈ 𝑍 ↦ (𝐸‘𝑛))) | 
| 37 | 5 | feqmptd 6976 | . . . . . . . 8
⊢ (𝜑 → 𝐸 = (𝑚 ∈ 𝑍 ↦ (𝐸‘𝑚))) | 
| 38 |  | omeiunle.ne | . . . . . . . . . . 11
⊢
Ⅎ𝑛𝐸 | 
| 39 |  | nfcv 2904 | . . . . . . . . . . 11
⊢
Ⅎ𝑛𝑚 | 
| 40 | 38, 39 | nffv 6915 | . . . . . . . . . 10
⊢
Ⅎ𝑛(𝐸‘𝑚) | 
| 41 |  | nfcv 2904 | . . . . . . . . . 10
⊢
Ⅎ𝑚(𝐸‘𝑛) | 
| 42 |  | fveq2 6905 | . . . . . . . . . 10
⊢ (𝑚 = 𝑛 → (𝐸‘𝑚) = (𝐸‘𝑛)) | 
| 43 | 40, 41, 42 | cbvmpt 5252 | . . . . . . . . 9
⊢ (𝑚 ∈ 𝑍 ↦ (𝐸‘𝑚)) = (𝑛 ∈ 𝑍 ↦ (𝐸‘𝑛)) | 
| 44 | 43 | a1i 11 | . . . . . . . 8
⊢ (𝜑 → (𝑚 ∈ 𝑍 ↦ (𝐸‘𝑚)) = (𝑛 ∈ 𝑍 ↦ (𝐸‘𝑛))) | 
| 45 | 37, 44 | eqtrd 2776 | . . . . . . 7
⊢ (𝜑 → 𝐸 = (𝑛 ∈ 𝑍 ↦ (𝐸‘𝑛))) | 
| 46 | 45 | rneqd 5948 | . . . . . 6
⊢ (𝜑 → ran 𝐸 = ran (𝑛 ∈ 𝑍 ↦ (𝐸‘𝑛))) | 
| 47 | 46 | unieqd 4919 | . . . . 5
⊢ (𝜑 → ∪ ran 𝐸 = ∪ ran (𝑛 ∈ 𝑍 ↦ (𝐸‘𝑛))) | 
| 48 | 36, 47 | eqtr4d 2779 | . . . 4
⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) = ∪ ran 𝐸) | 
| 49 | 48 | fveq2d 6909 | . . 3
⊢ (𝜑 → (𝑂‘∪
𝑛 ∈ 𝑍 (𝐸‘𝑛)) = (𝑂‘∪ ran
𝐸)) | 
| 50 |  | fnrndomg 10577 | . . . . . 6
⊢ (𝑍 ∈ V → (𝐸 Fn 𝑍 → ran 𝐸 ≼ 𝑍)) | 
| 51 | 18, 15, 50 | sylc 65 | . . . . 5
⊢ (𝜑 → ran 𝐸 ≼ 𝑍) | 
| 52 | 16 | uzct 45073 | . . . . . 6
⊢ 𝑍 ≼
ω | 
| 53 | 52 | a1i 11 | . . . . 5
⊢ (𝜑 → 𝑍 ≼ ω) | 
| 54 |  | domtr 9048 | . . . . 5
⊢ ((ran
𝐸 ≼ 𝑍 ∧ 𝑍 ≼ ω) → ran 𝐸 ≼
ω) | 
| 55 | 51, 53, 54 | syl2anc 584 | . . . 4
⊢ (𝜑 → ran 𝐸 ≼ ω) | 
| 56 | 2, 3, 24, 55 | omeunile 46525 | . . 3
⊢ (𝜑 → (𝑂‘∪ ran
𝐸) ≤
(Σ^‘(𝑂 ↾ ran 𝐸))) | 
| 57 | 49, 56 | eqbrtrd 5164 | . 2
⊢ (𝜑 → (𝑂‘∪
𝑛 ∈ 𝑍 (𝐸‘𝑛)) ≤
(Σ^‘(𝑂 ↾ ran 𝐸))) | 
| 58 |  | ltweuz 14003 | . . . . . 6
⊢  < We
(ℤ≥‘𝑁) | 
| 59 |  | weeq2 5672 | . . . . . . 7
⊢ (𝑍 =
(ℤ≥‘𝑁) → ( < We 𝑍 ↔ < We
(ℤ≥‘𝑁))) | 
| 60 | 16, 59 | ax-mp 5 | . . . . . 6
⊢ ( < We
𝑍 ↔ < We
(ℤ≥‘𝑁)) | 
| 61 | 58, 60 | mpbir 231 | . . . . 5
⊢  < We
𝑍 | 
| 62 | 61 | a1i 11 | . . . 4
⊢ (𝜑 → < We 𝑍) | 
| 63 | 18, 23, 5, 62 | sge0resrn 46424 | . . 3
⊢ (𝜑 →
(Σ^‘(𝑂 ↾ ran 𝐸)) ≤
(Σ^‘(𝑂 ∘ 𝐸))) | 
| 64 |  | fcompt 7152 | . . . . . 6
⊢ ((𝑂:𝒫 𝑋⟶(0[,]+∞) ∧ 𝐸:𝑍⟶𝒫 𝑋) → (𝑂 ∘ 𝐸) = (𝑚 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑚)))) | 
| 65 |  | nfcv 2904 | . . . . . . . . 9
⊢
Ⅎ𝑛𝑂 | 
| 66 | 65, 40 | nffv 6915 | . . . . . . . 8
⊢
Ⅎ𝑛(𝑂‘(𝐸‘𝑚)) | 
| 67 |  | nfcv 2904 | . . . . . . . 8
⊢
Ⅎ𝑚(𝑂‘(𝐸‘𝑛)) | 
| 68 |  | 2fveq3 6910 | . . . . . . . 8
⊢ (𝑚 = 𝑛 → (𝑂‘(𝐸‘𝑚)) = (𝑂‘(𝐸‘𝑛))) | 
| 69 | 66, 67, 68 | cbvmpt 5252 | . . . . . . 7
⊢ (𝑚 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑚))) = (𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))) | 
| 70 | 69 | a1i 11 | . . . . . 6
⊢ ((𝑂:𝒫 𝑋⟶(0[,]+∞) ∧ 𝐸:𝑍⟶𝒫 𝑋) → (𝑚 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑚))) = (𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) | 
| 71 | 64, 70 | eqtrd 2776 | . . . . 5
⊢ ((𝑂:𝒫 𝑋⟶(0[,]+∞) ∧ 𝐸:𝑍⟶𝒫 𝑋) → (𝑂 ∘ 𝐸) = (𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) | 
| 72 | 23, 5, 71 | syl2anc 584 | . . . 4
⊢ (𝜑 → (𝑂 ∘ 𝐸) = (𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) | 
| 73 | 72 | fveq2d 6909 | . . 3
⊢ (𝜑 →
(Σ^‘(𝑂 ∘ 𝐸)) =
(Σ^‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))))) | 
| 74 | 63, 73 | breqtrd 5168 | . 2
⊢ (𝜑 →
(Σ^‘(𝑂 ↾ ran 𝐸)) ≤
(Σ^‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))))) | 
| 75 | 14, 26, 31, 57, 74 | xrletrd 13205 | 1
⊢ (𝜑 → (𝑂‘∪
𝑛 ∈ 𝑍 (𝐸‘𝑛)) ≤
(Σ^‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))))) |