| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | vonioolem2.x | . . . . 5
⊢ (𝜑 → 𝑋 ∈ Fin) | 
| 2 | 1 | vonmea 46594 | . . . 4
⊢ (𝜑 → (voln‘𝑋) ∈ Meas) | 
| 3 |  | 1zzd 12650 | . . . 4
⊢ (𝜑 → 1 ∈
ℤ) | 
| 4 |  | nnuz 12922 | . . . 4
⊢ ℕ =
(ℤ≥‘1) | 
| 5 | 1 | adantr 480 | . . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ Fin) | 
| 6 |  | eqid 2736 | . . . . . 6
⊢ dom
(voln‘𝑋) = dom
(voln‘𝑋) | 
| 7 |  | vonioolem2.a | . . . . . . . . . . 11
⊢ (𝜑 → 𝐴:𝑋⟶ℝ) | 
| 8 | 7 | adantr 480 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴:𝑋⟶ℝ) | 
| 9 | 8 | ffvelcdmda 7103 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ) | 
| 10 |  | nnrecre 12309 | . . . . . . . . . 10
⊢ (𝑛 ∈ ℕ → (1 /
𝑛) ∈
ℝ) | 
| 11 | 10 | ad2antlr 727 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) ∈ ℝ) | 
| 12 | 9, 11 | readdcld 11291 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘) + (1 / 𝑛)) ∈ ℝ) | 
| 13 | 12 | fmpttd 7134 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))):𝑋⟶ℝ) | 
| 14 |  | vonioolem2.c | . . . . . . . . . 10
⊢ 𝐶 = (𝑛 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛)))) | 
| 15 | 14 | a1i 11 | . . . . . . . . 9
⊢ (𝜑 → 𝐶 = (𝑛 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))))) | 
| 16 | 1 | mptexd 7245 | . . . . . . . . . 10
⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))) ∈ V) | 
| 17 | 16 | adantr 480 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))) ∈ V) | 
| 18 | 15, 17 | fvmpt2d 7028 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘𝑛) = (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛)))) | 
| 19 | 18 | feq1d 6719 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐶‘𝑛):𝑋⟶ℝ ↔ (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))):𝑋⟶ℝ)) | 
| 20 | 13, 19 | mpbird 257 | . . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘𝑛):𝑋⟶ℝ) | 
| 21 |  | vonioolem2.b | . . . . . . 7
⊢ (𝜑 → 𝐵:𝑋⟶ℝ) | 
| 22 | 21 | adantr 480 | . . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐵:𝑋⟶ℝ) | 
| 23 | 5, 6, 20, 22 | hoimbl 46651 | . . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈
𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) ∈ dom (voln‘𝑋)) | 
| 24 |  | vonioolem2.d | . . . . 5
⊢ 𝐷 = (𝑛 ∈ ℕ ↦ X𝑘 ∈
𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))) | 
| 25 | 23, 24 | fmptd 7133 | . . . 4
⊢ (𝜑 → 𝐷:ℕ⟶dom (voln‘𝑋)) | 
| 26 |  | nfv 1913 | . . . . . 6
⊢
Ⅎ𝑘(𝜑 ∧ 𝑛 ∈ ℕ) | 
| 27 |  | oveq2 7440 | . . . . . . . . . . . . . . 15
⊢ (𝑛 = 𝑚 → (1 / 𝑛) = (1 / 𝑚)) | 
| 28 | 27 | oveq2d 7448 | . . . . . . . . . . . . . 14
⊢ (𝑛 = 𝑚 → ((𝐴‘𝑘) + (1 / 𝑛)) = ((𝐴‘𝑘) + (1 / 𝑚))) | 
| 29 | 28 | mpteq2dv 5243 | . . . . . . . . . . . . 13
⊢ (𝑛 = 𝑚 → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))) = (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑚)))) | 
| 30 | 29 | cbvmptv 5254 | . . . . . . . . . . . 12
⊢ (𝑛 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛)))) = (𝑚 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑚)))) | 
| 31 | 14, 30 | eqtri 2764 | . . . . . . . . . . 11
⊢ 𝐶 = (𝑚 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑚)))) | 
| 32 |  | oveq2 7440 | . . . . . . . . . . . . 13
⊢ (𝑚 = (𝑛 + 1) → (1 / 𝑚) = (1 / (𝑛 + 1))) | 
| 33 | 32 | oveq2d 7448 | . . . . . . . . . . . 12
⊢ (𝑚 = (𝑛 + 1) → ((𝐴‘𝑘) + (1 / 𝑚)) = ((𝐴‘𝑘) + (1 / (𝑛 + 1)))) | 
| 34 | 33 | mpteq2dv 5243 | . . . . . . . . . . 11
⊢ (𝑚 = (𝑛 + 1) → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑚))) = (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / (𝑛 + 1))))) | 
| 35 |  | simpr 484 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ) | 
| 36 | 35 | peano2nnd 12284 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℕ) | 
| 37 | 5 | mptexd 7245 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / (𝑛 + 1)))) ∈ V) | 
| 38 | 31, 34, 36, 37 | fvmptd3 7038 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘(𝑛 + 1)) = (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / (𝑛 + 1))))) | 
| 39 |  | ovexd 7467 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘) + (1 / (𝑛 + 1))) ∈ V) | 
| 40 | 38, 39 | fvmpt2d 7028 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘(𝑛 + 1))‘𝑘) = ((𝐴‘𝑘) + (1 / (𝑛 + 1)))) | 
| 41 |  | 1red 11263 | . . . . . . . . . . . 12
⊢ (𝑛 ∈ ℕ → 1 ∈
ℝ) | 
| 42 |  | nnre 12274 | . . . . . . . . . . . . 13
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℝ) | 
| 43 | 42, 41 | readdcld 11291 | . . . . . . . . . . . 12
⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈
ℝ) | 
| 44 |  | peano2nn 12279 | . . . . . . . . . . . . 13
⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈
ℕ) | 
| 45 |  | nnne0 12301 | . . . . . . . . . . . . 13
⊢ ((𝑛 + 1) ∈ ℕ →
(𝑛 + 1) ≠
0) | 
| 46 | 44, 45 | syl 17 | . . . . . . . . . . . 12
⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ≠ 0) | 
| 47 | 41, 43, 46 | redivcld 12096 | . . . . . . . . . . 11
⊢ (𝑛 ∈ ℕ → (1 /
(𝑛 + 1)) ∈
ℝ) | 
| 48 | 47 | ad2antlr 727 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / (𝑛 + 1)) ∈ ℝ) | 
| 49 | 9, 48 | readdcld 11291 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘) + (1 / (𝑛 + 1))) ∈ ℝ) | 
| 50 | 40, 49 | eqeltrd 2840 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘(𝑛 + 1))‘𝑘) ∈ ℝ) | 
| 51 | 50 | rexrd 11312 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘(𝑛 + 1))‘𝑘) ∈
ℝ*) | 
| 52 |  | ressxr 11306 | . . . . . . . . 9
⊢ ℝ
⊆ ℝ* | 
| 53 | 21 | ffvelcdmda 7103 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ) | 
| 54 | 52, 53 | sselid 3980 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈
ℝ*) | 
| 55 | 54 | adantlr 715 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈
ℝ*) | 
| 56 | 42 | ltp1d 12199 | . . . . . . . . . . . 12
⊢ (𝑛 ∈ ℕ → 𝑛 < (𝑛 + 1)) | 
| 57 |  | nnrp 13047 | . . . . . . . . . . . . 13
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℝ+) | 
| 58 | 44 | nnrpd 13076 | . . . . . . . . . . . . 13
⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈
ℝ+) | 
| 59 | 57, 58 | ltrecd 13096 | . . . . . . . . . . . 12
⊢ (𝑛 ∈ ℕ → (𝑛 < (𝑛 + 1) ↔ (1 / (𝑛 + 1)) < (1 / 𝑛))) | 
| 60 | 56, 59 | mpbid 232 | . . . . . . . . . . 11
⊢ (𝑛 ∈ ℕ → (1 /
(𝑛 + 1)) < (1 / 𝑛)) | 
| 61 | 47, 10, 60 | ltled 11410 | . . . . . . . . . 10
⊢ (𝑛 ∈ ℕ → (1 /
(𝑛 + 1)) ≤ (1 / 𝑛)) | 
| 62 | 61 | ad2antlr 727 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / (𝑛 + 1)) ≤ (1 / 𝑛)) | 
| 63 | 48, 11, 9, 62 | leadd2dd 11879 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘) + (1 / (𝑛 + 1))) ≤ ((𝐴‘𝑘) + (1 / 𝑛))) | 
| 64 |  | ovexd 7467 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘) + (1 / 𝑛)) ∈ V) | 
| 65 | 18, 64 | fvmpt2d 7028 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑛)‘𝑘) = ((𝐴‘𝑘) + (1 / 𝑛))) | 
| 66 | 40, 65 | breq12d 5155 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝐶‘(𝑛 + 1))‘𝑘) ≤ ((𝐶‘𝑛)‘𝑘) ↔ ((𝐴‘𝑘) + (1 / (𝑛 + 1))) ≤ ((𝐴‘𝑘) + (1 / 𝑛)))) | 
| 67 | 63, 66 | mpbird 257 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘(𝑛 + 1))‘𝑘) ≤ ((𝐶‘𝑛)‘𝑘)) | 
| 68 | 53 | adantlr 715 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ) | 
| 69 |  | eqidd 2737 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) = (𝐵‘𝑘)) | 
| 70 | 68, 69 | eqled 11365 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ≤ (𝐵‘𝑘)) | 
| 71 |  | icossico 13458 | . . . . . . 7
⊢
(((((𝐶‘(𝑛 + 1))‘𝑘) ∈ ℝ* ∧ (𝐵‘𝑘) ∈ ℝ*) ∧ (((𝐶‘(𝑛 + 1))‘𝑘) ≤ ((𝐶‘𝑛)‘𝑘) ∧ (𝐵‘𝑘) ≤ (𝐵‘𝑘))) → (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) ⊆ (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘))) | 
| 72 | 51, 55, 67, 70, 71 | syl22anc 838 | . . . . . 6
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) ⊆ (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘))) | 
| 73 | 26, 72 | ixpssixp 45102 | . . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈
𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) ⊆ X𝑘 ∈ 𝑋 (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘))) | 
| 74 | 24 | a1i 11 | . . . . . . 7
⊢ (𝜑 → 𝐷 = (𝑛 ∈ ℕ ↦ X𝑘 ∈
𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)))) | 
| 75 | 23 | elexd 3503 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈
𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) ∈ V) | 
| 76 | 74, 75 | fvmpt2d 7028 | . . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) = X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))) | 
| 77 |  | fveq2 6905 | . . . . . . . . . . . 12
⊢ (𝑛 = 𝑚 → (𝐶‘𝑛) = (𝐶‘𝑚)) | 
| 78 | 77 | fveq1d 6907 | . . . . . . . . . . 11
⊢ (𝑛 = 𝑚 → ((𝐶‘𝑛)‘𝑘) = ((𝐶‘𝑚)‘𝑘)) | 
| 79 | 78 | oveq1d 7447 | . . . . . . . . . 10
⊢ (𝑛 = 𝑚 → (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) = (((𝐶‘𝑚)‘𝑘)[,)(𝐵‘𝑘))) | 
| 80 | 79 | ixpeq2dv 8954 | . . . . . . . . 9
⊢ (𝑛 = 𝑚 → X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) = X𝑘 ∈ 𝑋 (((𝐶‘𝑚)‘𝑘)[,)(𝐵‘𝑘))) | 
| 81 | 80 | cbvmptv 5254 | . . . . . . . 8
⊢ (𝑛 ∈ ℕ ↦ X𝑘 ∈
𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))) = (𝑚 ∈ ℕ ↦ X𝑘 ∈
𝑋 (((𝐶‘𝑚)‘𝑘)[,)(𝐵‘𝑘))) | 
| 82 | 24, 81 | eqtri 2764 | . . . . . . 7
⊢ 𝐷 = (𝑚 ∈ ℕ ↦ X𝑘 ∈
𝑋 (((𝐶‘𝑚)‘𝑘)[,)(𝐵‘𝑘))) | 
| 83 |  | fveq2 6905 | . . . . . . . . . 10
⊢ (𝑚 = (𝑛 + 1) → (𝐶‘𝑚) = (𝐶‘(𝑛 + 1))) | 
| 84 | 83 | fveq1d 6907 | . . . . . . . . 9
⊢ (𝑚 = (𝑛 + 1) → ((𝐶‘𝑚)‘𝑘) = ((𝐶‘(𝑛 + 1))‘𝑘)) | 
| 85 | 84 | oveq1d 7447 | . . . . . . . 8
⊢ (𝑚 = (𝑛 + 1) → (((𝐶‘𝑚)‘𝑘)[,)(𝐵‘𝑘)) = (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘))) | 
| 86 | 85 | ixpeq2dv 8954 | . . . . . . 7
⊢ (𝑚 = (𝑛 + 1) → X𝑘 ∈ 𝑋 (((𝐶‘𝑚)‘𝑘)[,)(𝐵‘𝑘)) = X𝑘 ∈ 𝑋 (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘))) | 
| 87 |  | ovex 7465 | . . . . . . . . . 10
⊢ (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘)) ∈ V | 
| 88 | 87 | rgenw 3064 | . . . . . . . . 9
⊢
∀𝑘 ∈
𝑋 (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘)) ∈ V | 
| 89 |  | ixpexg 8963 | . . . . . . . . 9
⊢
(∀𝑘 ∈
𝑋 (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘)) ∈ V → X𝑘 ∈
𝑋 (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘)) ∈ V) | 
| 90 | 88, 89 | ax-mp 5 | . . . . . . . 8
⊢ X𝑘 ∈
𝑋 (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘)) ∈ V | 
| 91 | 90 | a1i 11 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈
𝑋 (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘)) ∈ V) | 
| 92 | 82, 86, 36, 91 | fvmptd3 7038 | . . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘(𝑛 + 1)) = X𝑘 ∈ 𝑋 (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘))) | 
| 93 | 76, 92 | sseq12d 4016 | . . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐷‘𝑛) ⊆ (𝐷‘(𝑛 + 1)) ↔ X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) ⊆ X𝑘 ∈ 𝑋 (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘)))) | 
| 94 | 73, 93 | mpbird 257 | . . . 4
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) ⊆ (𝐷‘(𝑛 + 1))) | 
| 95 | 1, 6, 7, 21 | hoimbl 46651 | . . . . 5
⊢ (𝜑 → X𝑘 ∈
𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∈ dom (voln‘𝑋)) | 
| 96 |  | nfv 1913 | . . . . . 6
⊢
Ⅎ𝑘𝜑 | 
| 97 | 7 | ffvelcdmda 7103 | . . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ) | 
| 98 | 96, 1, 97, 53 | vonhoire 46692 | . . . . 5
⊢ (𝜑 → ((voln‘𝑋)‘X𝑘 ∈
𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ) | 
| 99 |  | vonioolem2.i | . . . . . . 7
⊢ 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)) | 
| 100 | 99 | a1i 11 | . . . . . 6
⊢ (𝜑 → 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘))) | 
| 101 |  | nftru 1803 | . . . . . . . . 9
⊢
Ⅎ𝑘⊤ | 
| 102 |  | ioossico 13479 | . . . . . . . . . 10
⊢ ((𝐴‘𝑘)(,)(𝐵‘𝑘)) ⊆ ((𝐴‘𝑘)[,)(𝐵‘𝑘)) | 
| 103 | 102 | a1i 11 | . . . . . . . . 9
⊢
((⊤ ∧ 𝑘
∈ 𝑋) → ((𝐴‘𝑘)(,)(𝐵‘𝑘)) ⊆ ((𝐴‘𝑘)[,)(𝐵‘𝑘))) | 
| 104 | 101, 103 | ixpssixp 45102 | . . . . . . . 8
⊢ (⊤
→ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)) ⊆ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) | 
| 105 | 104 | mptru 1546 | . . . . . . 7
⊢ X𝑘 ∈
𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)) ⊆ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) | 
| 106 | 105 | a1i 11 | . . . . . 6
⊢ (𝜑 → X𝑘 ∈
𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)) ⊆ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) | 
| 107 | 100, 106 | eqsstrd 4017 | . . . . 5
⊢ (𝜑 → 𝐼 ⊆ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) | 
| 108 | 52 | a1i 11 | . . . . . . . 8
⊢ (𝜑 → ℝ ⊆
ℝ*) | 
| 109 | 7, 108 | fssd 6752 | . . . . . . 7
⊢ (𝜑 → 𝐴:𝑋⟶ℝ*) | 
| 110 | 21, 108 | fssd 6752 | . . . . . . 7
⊢ (𝜑 → 𝐵:𝑋⟶ℝ*) | 
| 111 | 1, 6, 109, 110 | ioovonmbl 46697 | . . . . . 6
⊢ (𝜑 → X𝑘 ∈
𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)) ∈ dom (voln‘𝑋)) | 
| 112 | 99, 111 | eqeltrid 2844 | . . . . 5
⊢ (𝜑 → 𝐼 ∈ dom (voln‘𝑋)) | 
| 113 | 2, 95, 98, 107, 112 | meassre 46497 | . . . 4
⊢ (𝜑 → ((voln‘𝑋)‘𝐼) ∈ ℝ) | 
| 114 | 2 | adantr 480 | . . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (voln‘𝑋) ∈ Meas) | 
| 115 | 76, 23 | eqeltrd 2840 | . . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) ∈ dom (voln‘𝑋)) | 
| 116 | 112 | adantr 480 | . . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐼 ∈ dom (voln‘𝑋)) | 
| 117 | 52, 97 | sselid 3980 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈
ℝ*) | 
| 118 | 117 | adantlr 715 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈
ℝ*) | 
| 119 | 57 | rpreccld 13088 | . . . . . . . . . 10
⊢ (𝑛 ∈ ℕ → (1 /
𝑛) ∈
ℝ+) | 
| 120 | 119 | ad2antlr 727 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) ∈
ℝ+) | 
| 121 | 9, 120 | ltaddrpd 13111 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) < ((𝐴‘𝑘) + (1 / 𝑛))) | 
| 122 |  | icossioo 13481 | . . . . . . . 8
⊢ ((((𝐴‘𝑘) ∈ ℝ* ∧ (𝐵‘𝑘) ∈ ℝ*) ∧ ((𝐴‘𝑘) < ((𝐴‘𝑘) + (1 / 𝑛)) ∧ (𝐵‘𝑘) ≤ (𝐵‘𝑘))) → (((𝐴‘𝑘) + (1 / 𝑛))[,)(𝐵‘𝑘)) ⊆ ((𝐴‘𝑘)(,)(𝐵‘𝑘))) | 
| 123 | 118, 55, 121, 70, 122 | syl22anc 838 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝐴‘𝑘) + (1 / 𝑛))[,)(𝐵‘𝑘)) ⊆ ((𝐴‘𝑘)(,)(𝐵‘𝑘))) | 
| 124 | 26, 123 | ixpssixp 45102 | . . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈
𝑋 (((𝐴‘𝑘) + (1 / 𝑛))[,)(𝐵‘𝑘)) ⊆ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘))) | 
| 125 | 65 | oveq1d 7447 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) = (((𝐴‘𝑘) + (1 / 𝑛))[,)(𝐵‘𝑘))) | 
| 126 | 125 | ixpeq2dva 8953 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈
𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) = X𝑘 ∈ 𝑋 (((𝐴‘𝑘) + (1 / 𝑛))[,)(𝐵‘𝑘))) | 
| 127 | 76, 126 | eqtrd 2776 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) = X𝑘 ∈ 𝑋 (((𝐴‘𝑘) + (1 / 𝑛))[,)(𝐵‘𝑘))) | 
| 128 | 99 | a1i 11 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘))) | 
| 129 | 127, 128 | sseq12d 4016 | . . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐷‘𝑛) ⊆ 𝐼 ↔ X𝑘 ∈ 𝑋 (((𝐴‘𝑘) + (1 / 𝑛))[,)(𝐵‘𝑘)) ⊆ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)))) | 
| 130 | 124, 129 | mpbird 257 | . . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) ⊆ 𝐼) | 
| 131 | 114, 6, 115, 116, 130 | meassle 46483 | . . . 4
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((voln‘𝑋)‘(𝐷‘𝑛)) ≤ ((voln‘𝑋)‘𝐼)) | 
| 132 |  | eqid 2736 | . . . 4
⊢ (𝑛 ∈ ℕ ↦
((voln‘𝑋)‘(𝐷‘𝑛))) = (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) | 
| 133 | 2, 3, 4, 25, 94, 113, 131, 132 | meaiuninc2 46502 | . . 3
⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) ⇝ ((voln‘𝑋)‘∪
𝑛 ∈ ℕ (𝐷‘𝑛))) | 
| 134 | 96, 1, 97, 54 | iunhoiioo 46696 | . . . . . . 7
⊢ (𝜑 → ∪ 𝑛 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐴‘𝑘) + (1 / 𝑛))[,)(𝐵‘𝑘)) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘))) | 
| 135 | 127 | iuneq2dv 5015 | . . . . . . 7
⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐷‘𝑛) = ∪ 𝑛 ∈ ℕ X𝑘 ∈
𝑋 (((𝐴‘𝑘) + (1 / 𝑛))[,)(𝐵‘𝑘))) | 
| 136 | 134, 135,
100 | 3eqtr4d 2786 | . . . . . 6
⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐷‘𝑛) = 𝐼) | 
| 137 | 136 | eqcomd 2742 | . . . . 5
⊢ (𝜑 → 𝐼 = ∪ 𝑛 ∈ ℕ (𝐷‘𝑛)) | 
| 138 | 137 | fveq2d 6909 | . . . 4
⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = ((voln‘𝑋)‘∪
𝑛 ∈ ℕ (𝐷‘𝑛))) | 
| 139 | 138 | eqcomd 2742 | . . 3
⊢ (𝜑 → ((voln‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐷‘𝑛)) = ((voln‘𝑋)‘𝐼)) | 
| 140 | 133, 139 | breqtrd 5168 | . 2
⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) ⇝ ((voln‘𝑋)‘𝐼)) | 
| 141 |  | 2fveq3 6910 | . . . . 5
⊢ (𝑛 = 𝑚 → ((voln‘𝑋)‘(𝐷‘𝑛)) = ((voln‘𝑋)‘(𝐷‘𝑚))) | 
| 142 | 141 | cbvmptv 5254 | . . . 4
⊢ (𝑛 ∈ ℕ ↦
((voln‘𝑋)‘(𝐷‘𝑛))) = (𝑚 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑚))) | 
| 143 | 142 | a1i 11 | . . 3
⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) = (𝑚 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑚)))) | 
| 144 |  | vonioolem2.n | . . . 4
⊢ (𝜑 → 𝑋 ≠ ∅) | 
| 145 |  | vonioolem2.t | . . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) < (𝐵‘𝑘)) | 
| 146 | 142 | eqcomi 2745 | . . . 4
⊢ (𝑚 ∈ ℕ ↦
((voln‘𝑋)‘(𝐷‘𝑚))) = (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) | 
| 147 |  | eqcom 2743 | . . . . . . . . . 10
⊢ (𝑛 = 𝑚 ↔ 𝑚 = 𝑛) | 
| 148 | 147 | imbi1i 349 | . . . . . . . . 9
⊢ ((𝑛 = 𝑚 → ((𝐶‘𝑛)‘𝑘) = ((𝐶‘𝑚)‘𝑘)) ↔ (𝑚 = 𝑛 → ((𝐶‘𝑛)‘𝑘) = ((𝐶‘𝑚)‘𝑘))) | 
| 149 |  | eqcom 2743 | . . . . . . . . . 10
⊢ (((𝐶‘𝑛)‘𝑘) = ((𝐶‘𝑚)‘𝑘) ↔ ((𝐶‘𝑚)‘𝑘) = ((𝐶‘𝑛)‘𝑘)) | 
| 150 | 149 | imbi2i 336 | . . . . . . . . 9
⊢ ((𝑚 = 𝑛 → ((𝐶‘𝑛)‘𝑘) = ((𝐶‘𝑚)‘𝑘)) ↔ (𝑚 = 𝑛 → ((𝐶‘𝑚)‘𝑘) = ((𝐶‘𝑛)‘𝑘))) | 
| 151 | 148, 150 | bitri 275 | . . . . . . . 8
⊢ ((𝑛 = 𝑚 → ((𝐶‘𝑛)‘𝑘) = ((𝐶‘𝑚)‘𝑘)) ↔ (𝑚 = 𝑛 → ((𝐶‘𝑚)‘𝑘) = ((𝐶‘𝑛)‘𝑘))) | 
| 152 | 78, 151 | mpbi 230 | . . . . . . 7
⊢ (𝑚 = 𝑛 → ((𝐶‘𝑚)‘𝑘) = ((𝐶‘𝑛)‘𝑘)) | 
| 153 | 152 | oveq2d 7448 | . . . . . 6
⊢ (𝑚 = 𝑛 → ((𝐵‘𝑘) − ((𝐶‘𝑚)‘𝑘)) = ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘))) | 
| 154 | 153 | prodeq2ad 45612 | . . . . 5
⊢ (𝑚 = 𝑛 → ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑚)‘𝑘)) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘))) | 
| 155 | 154 | cbvmptv 5254 | . . . 4
⊢ (𝑚 ∈ ℕ ↦
∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑚)‘𝑘))) = (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘))) | 
| 156 |  | eqid 2736 | . . . 4
⊢ inf(ran
(𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < ) = inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < ) | 
| 157 |  | eqid 2736 | . . . 4
⊢
((⌊‘(1 / inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < ))) + 1) =
((⌊‘(1 / inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < ))) + 1) | 
| 158 |  | fveq2 6905 | . . . . . . . . . . . 12
⊢ (𝑗 = 𝑘 → (𝐵‘𝑗) = (𝐵‘𝑘)) | 
| 159 |  | fveq2 6905 | . . . . . . . . . . . 12
⊢ (𝑗 = 𝑘 → (𝐴‘𝑗) = (𝐴‘𝑘)) | 
| 160 | 158, 159 | oveq12d 7450 | . . . . . . . . . . 11
⊢ (𝑗 = 𝑘 → ((𝐵‘𝑗) − (𝐴‘𝑗)) = ((𝐵‘𝑘) − (𝐴‘𝑘))) | 
| 161 | 160 | cbvmptv 5254 | . . . . . . . . . 10
⊢ (𝑗 ∈ 𝑋 ↦ ((𝐵‘𝑗) − (𝐴‘𝑗))) = (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) | 
| 162 | 161 | rneqi 5947 | . . . . . . . . 9
⊢ ran
(𝑗 ∈ 𝑋 ↦ ((𝐵‘𝑗) − (𝐴‘𝑗))) = ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) | 
| 163 | 162 | infeq1i 9519 | . . . . . . . 8
⊢ inf(ran
(𝑗 ∈ 𝑋 ↦ ((𝐵‘𝑗) − (𝐴‘𝑗))), ℝ, < ) = inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < ) | 
| 164 | 163 | oveq2i 7443 | . . . . . . 7
⊢ (1 /
inf(ran (𝑗 ∈ 𝑋 ↦ ((𝐵‘𝑗) − (𝐴‘𝑗))), ℝ, < )) = (1 / inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < )) | 
| 165 | 164 | fveq2i 6908 | . . . . . 6
⊢
(⌊‘(1 / inf(ran (𝑗 ∈ 𝑋 ↦ ((𝐵‘𝑗) − (𝐴‘𝑗))), ℝ, < ))) = (⌊‘(1 /
inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < ))) | 
| 166 | 165 | oveq1i 7442 | . . . . 5
⊢
((⌊‘(1 / inf(ran (𝑗 ∈ 𝑋 ↦ ((𝐵‘𝑗) − (𝐴‘𝑗))), ℝ, < ))) + 1) =
((⌊‘(1 / inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < ))) + 1) | 
| 167 | 166 | fveq2i 6908 | . . . 4
⊢
(ℤ≥‘((⌊‘(1 / inf(ran (𝑗 ∈ 𝑋 ↦ ((𝐵‘𝑗) − (𝐴‘𝑗))), ℝ, < ))) + 1)) =
(ℤ≥‘((⌊‘(1 / inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < ))) + 1)) | 
| 168 | 1, 7, 21, 144, 145, 14, 24, 146, 155, 156, 157, 167 | vonioolem1 46700 | . . 3
⊢ (𝜑 → (𝑚 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑚))) ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘))) | 
| 169 | 143, 168 | eqbrtrd 5164 | . 2
⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘))) | 
| 170 |  | climuni 15589 | . 2
⊢ (((𝑛 ∈ ℕ ↦
((voln‘𝑋)‘(𝐷‘𝑛))) ⇝ ((voln‘𝑋)‘𝐼) ∧ (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘))) → ((voln‘𝑋)‘𝐼) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘))) | 
| 171 | 140, 169,
170 | syl2anc 584 | 1
⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘))) |