| Step | Hyp | Ref
| Expression |
| 1 | | nfv 1914 |
. . . 4
⊢
Ⅎ𝑛𝜑 |
| 2 | | vonicclem2.x |
. . . . 5
⊢ (𝜑 → 𝑋 ∈ Fin) |
| 3 | 2 | vonmea 46589 |
. . . 4
⊢ (𝜑 → (voln‘𝑋) ∈ Meas) |
| 4 | | 1zzd 12648 |
. . . 4
⊢ (𝜑 → 1 ∈
ℤ) |
| 5 | | nnuz 12921 |
. . . 4
⊢ ℕ =
(ℤ≥‘1) |
| 6 | 2 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ Fin) |
| 7 | | eqid 2737 |
. . . . . 6
⊢ dom
(voln‘𝑋) = dom
(voln‘𝑋) |
| 8 | | vonicclem2.a |
. . . . . . 7
⊢ (𝜑 → 𝐴:𝑋⟶ℝ) |
| 9 | 8 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴:𝑋⟶ℝ) |
| 10 | | vonicclem2.b |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐵:𝑋⟶ℝ) |
| 11 | 10 | ffvelcdmda 7104 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ) |
| 12 | 11 | adantlr 715 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ) |
| 13 | | nnrecre 12308 |
. . . . . . . . . 10
⊢ (𝑛 ∈ ℕ → (1 /
𝑛) ∈
ℝ) |
| 14 | 13 | ad2antlr 727 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) ∈ ℝ) |
| 15 | 12, 14 | readdcld 11290 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) + (1 / 𝑛)) ∈ ℝ) |
| 16 | 15 | fmpttd 7135 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑛))):𝑋⟶ℝ) |
| 17 | | vonicclem2.c |
. . . . . . . . . 10
⊢ 𝐶 = (𝑛 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑛)))) |
| 18 | 17 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → 𝐶 = (𝑛 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑛))))) |
| 19 | 2 | mptexd 7244 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑛))) ∈ V) |
| 20 | 19 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑛))) ∈ V) |
| 21 | 18, 20 | fvmpt2d 7029 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘𝑛) = (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑛)))) |
| 22 | 21 | feq1d 6720 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐶‘𝑛):𝑋⟶ℝ ↔ (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑛))):𝑋⟶ℝ)) |
| 23 | 16, 22 | mpbird 257 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘𝑛):𝑋⟶ℝ) |
| 24 | 6, 7, 9, 23 | hoimbl 46646 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈
𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)) ∈ dom (voln‘𝑋)) |
| 25 | | vonicclem2.d |
. . . . 5
⊢ 𝐷 = (𝑛 ∈ ℕ ↦ X𝑘 ∈
𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘))) |
| 26 | 24, 25 | fmptd 7134 |
. . . 4
⊢ (𝜑 → 𝐷:ℕ⟶dom (voln‘𝑋)) |
| 27 | | nfv 1914 |
. . . . . 6
⊢
Ⅎ𝑘(𝜑 ∧ 𝑛 ∈ ℕ) |
| 28 | | ressxr 11305 |
. . . . . . . . 9
⊢ ℝ
⊆ ℝ* |
| 29 | 8 | ffvelcdmda 7104 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ) |
| 30 | 28, 29 | sselid 3981 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈
ℝ*) |
| 31 | 30 | adantlr 715 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈
ℝ*) |
| 32 | | ovexd 7466 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) + (1 / 𝑛)) ∈ V) |
| 33 | 21, 32 | fvmpt2d 7029 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑛)‘𝑘) = ((𝐵‘𝑘) + (1 / 𝑛))) |
| 34 | 33, 15 | eqeltrd 2841 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑛)‘𝑘) ∈ ℝ) |
| 35 | 34 | rexrd 11311 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑛)‘𝑘) ∈
ℝ*) |
| 36 | 9 | ffvelcdmda 7104 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ) |
| 37 | 36 | leidd 11829 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ≤ (𝐴‘𝑘)) |
| 38 | | 1red 11262 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ ℕ → 1 ∈
ℝ) |
| 39 | | nnre 12273 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℝ) |
| 40 | 39, 38 | readdcld 11290 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈
ℝ) |
| 41 | | peano2nn 12278 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈
ℕ) |
| 42 | | nnne0 12300 |
. . . . . . . . . . . 12
⊢ ((𝑛 + 1) ∈ ℕ →
(𝑛 + 1) ≠
0) |
| 43 | 41, 42 | syl 17 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ≠ 0) |
| 44 | 38, 40, 43 | redivcld 12095 |
. . . . . . . . . 10
⊢ (𝑛 ∈ ℕ → (1 /
(𝑛 + 1)) ∈
ℝ) |
| 45 | 44 | ad2antlr 727 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / (𝑛 + 1)) ∈ ℝ) |
| 46 | 39 | ltp1d 12198 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ ℕ → 𝑛 < (𝑛 + 1)) |
| 47 | | nnrp 13046 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℝ+) |
| 48 | 41 | nnrpd 13075 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈
ℝ+) |
| 49 | 47, 48 | ltrecd 13095 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ ℕ → (𝑛 < (𝑛 + 1) ↔ (1 / (𝑛 + 1)) < (1 / 𝑛))) |
| 50 | 46, 49 | mpbid 232 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ ℕ → (1 /
(𝑛 + 1)) < (1 / 𝑛)) |
| 51 | 44, 13, 50 | ltled 11409 |
. . . . . . . . . 10
⊢ (𝑛 ∈ ℕ → (1 /
(𝑛 + 1)) ≤ (1 / 𝑛)) |
| 52 | 51 | ad2antlr 727 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / (𝑛 + 1)) ≤ (1 / 𝑛)) |
| 53 | 45, 14, 12, 52 | leadd2dd 11878 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) + (1 / (𝑛 + 1))) ≤ ((𝐵‘𝑘) + (1 / 𝑛))) |
| 54 | | oveq2 7439 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 𝑚 → (1 / 𝑛) = (1 / 𝑚)) |
| 55 | 54 | oveq2d 7447 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = 𝑚 → ((𝐵‘𝑘) + (1 / 𝑛)) = ((𝐵‘𝑘) + (1 / 𝑚))) |
| 56 | 55 | mpteq2dv 5244 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 𝑚 → (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑛))) = (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑚)))) |
| 57 | 56 | cbvmptv 5255 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑛)))) = (𝑚 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑚)))) |
| 58 | 17, 57 | eqtri 2765 |
. . . . . . . . . . 11
⊢ 𝐶 = (𝑚 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑚)))) |
| 59 | | oveq2 7439 |
. . . . . . . . . . . . 13
⊢ (𝑚 = (𝑛 + 1) → (1 / 𝑚) = (1 / (𝑛 + 1))) |
| 60 | 59 | oveq2d 7447 |
. . . . . . . . . . . 12
⊢ (𝑚 = (𝑛 + 1) → ((𝐵‘𝑘) + (1 / 𝑚)) = ((𝐵‘𝑘) + (1 / (𝑛 + 1)))) |
| 61 | 60 | mpteq2dv 5244 |
. . . . . . . . . . 11
⊢ (𝑚 = (𝑛 + 1) → (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑚))) = (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / (𝑛 + 1))))) |
| 62 | | simpr 484 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ) |
| 63 | 62 | peano2nnd 12283 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℕ) |
| 64 | 6 | mptexd 7244 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / (𝑛 + 1)))) ∈ V) |
| 65 | 58, 61, 63, 64 | fvmptd3 7039 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘(𝑛 + 1)) = (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / (𝑛 + 1))))) |
| 66 | | ovexd 7466 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) + (1 / (𝑛 + 1))) ∈ V) |
| 67 | 65, 66 | fvmpt2d 7029 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘(𝑛 + 1))‘𝑘) = ((𝐵‘𝑘) + (1 / (𝑛 + 1)))) |
| 68 | 67, 33 | breq12d 5156 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝐶‘(𝑛 + 1))‘𝑘) ≤ ((𝐶‘𝑛)‘𝑘) ↔ ((𝐵‘𝑘) + (1 / (𝑛 + 1))) ≤ ((𝐵‘𝑘) + (1 / 𝑛)))) |
| 69 | 53, 68 | mpbird 257 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘(𝑛 + 1))‘𝑘) ≤ ((𝐶‘𝑛)‘𝑘)) |
| 70 | | icossico 13457 |
. . . . . . 7
⊢ ((((𝐴‘𝑘) ∈ ℝ* ∧ ((𝐶‘𝑛)‘𝑘) ∈ ℝ*) ∧ ((𝐴‘𝑘) ≤ (𝐴‘𝑘) ∧ ((𝐶‘(𝑛 + 1))‘𝑘) ≤ ((𝐶‘𝑛)‘𝑘))) → ((𝐴‘𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ⊆ ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘))) |
| 71 | 31, 35, 37, 69, 70 | syl22anc 839 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ⊆ ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘))) |
| 72 | 27, 71 | ixpssixp 45097 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈
𝑋 ((𝐴‘𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ⊆ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘))) |
| 73 | | fveq2 6906 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝑚 → (𝐶‘𝑛) = (𝐶‘𝑚)) |
| 74 | 73 | fveq1d 6908 |
. . . . . . . . . . 11
⊢ (𝑛 = 𝑚 → ((𝐶‘𝑛)‘𝑘) = ((𝐶‘𝑚)‘𝑘)) |
| 75 | 74 | oveq2d 7447 |
. . . . . . . . . 10
⊢ (𝑛 = 𝑚 → ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)) = ((𝐴‘𝑘)[,)((𝐶‘𝑚)‘𝑘))) |
| 76 | 75 | ixpeq2dv 8953 |
. . . . . . . . 9
⊢ (𝑛 = 𝑚 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑚)‘𝑘))) |
| 77 | 76 | cbvmptv 5255 |
. . . . . . . 8
⊢ (𝑛 ∈ ℕ ↦ X𝑘 ∈
𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘))) = (𝑚 ∈ ℕ ↦ X𝑘 ∈
𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑚)‘𝑘))) |
| 78 | 25, 77 | eqtri 2765 |
. . . . . . 7
⊢ 𝐷 = (𝑚 ∈ ℕ ↦ X𝑘 ∈
𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑚)‘𝑘))) |
| 79 | | fveq2 6906 |
. . . . . . . . . 10
⊢ (𝑚 = (𝑛 + 1) → (𝐶‘𝑚) = (𝐶‘(𝑛 + 1))) |
| 80 | 79 | fveq1d 6908 |
. . . . . . . . 9
⊢ (𝑚 = (𝑛 + 1) → ((𝐶‘𝑚)‘𝑘) = ((𝐶‘(𝑛 + 1))‘𝑘)) |
| 81 | 80 | oveq2d 7447 |
. . . . . . . 8
⊢ (𝑚 = (𝑛 + 1) → ((𝐴‘𝑘)[,)((𝐶‘𝑚)‘𝑘)) = ((𝐴‘𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘))) |
| 82 | 81 | ixpeq2dv 8953 |
. . . . . . 7
⊢ (𝑚 = (𝑛 + 1) → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑚)‘𝑘)) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘))) |
| 83 | | ovex 7464 |
. . . . . . . . . 10
⊢ ((𝐴‘𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ∈ V |
| 84 | 83 | rgenw 3065 |
. . . . . . . . 9
⊢
∀𝑘 ∈
𝑋 ((𝐴‘𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ∈ V |
| 85 | | ixpexg 8962 |
. . . . . . . . 9
⊢
(∀𝑘 ∈
𝑋 ((𝐴‘𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ∈ V → X𝑘 ∈
𝑋 ((𝐴‘𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ∈ V) |
| 86 | 84, 85 | ax-mp 5 |
. . . . . . . 8
⊢ X𝑘 ∈
𝑋 ((𝐴‘𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ∈ V |
| 87 | 86 | a1i 11 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈
𝑋 ((𝐴‘𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ∈ V) |
| 88 | 78, 82, 63, 87 | fvmptd3 7039 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘(𝑛 + 1)) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘))) |
| 89 | 25 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → 𝐷 = (𝑛 ∈ ℕ ↦ X𝑘 ∈
𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)))) |
| 90 | 24 | elexd 3504 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈
𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)) ∈ V) |
| 91 | 89, 90 | fvmpt2d 7029 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘))) |
| 92 | 88, 91 | sseq12d 4017 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐷‘(𝑛 + 1)) ⊆ (𝐷‘𝑛) ↔ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ⊆ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)))) |
| 93 | 72, 92 | mpbird 257 |
. . . 4
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘(𝑛 + 1)) ⊆ (𝐷‘𝑛)) |
| 94 | | 1nn 12277 |
. . . . . 6
⊢ 1 ∈
ℕ |
| 95 | 94, 5 | eleqtri 2839 |
. . . . 5
⊢ 1 ∈
(ℤ≥‘1) |
| 96 | 95 | a1i 11 |
. . . 4
⊢ (𝜑 → 1 ∈
(ℤ≥‘1)) |
| 97 | | fveq2 6906 |
. . . . . . . . . 10
⊢ (𝑛 = 1 → (𝐶‘𝑛) = (𝐶‘1)) |
| 98 | 97 | fveq1d 6908 |
. . . . . . . . 9
⊢ (𝑛 = 1 → ((𝐶‘𝑛)‘𝑘) = ((𝐶‘1)‘𝑘)) |
| 99 | 98 | oveq2d 7447 |
. . . . . . . 8
⊢ (𝑛 = 1 → ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)) = ((𝐴‘𝑘)[,)((𝐶‘1)‘𝑘))) |
| 100 | 99 | ixpeq2dv 8953 |
. . . . . . 7
⊢ (𝑛 = 1 → X𝑘 ∈
𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘1)‘𝑘))) |
| 101 | 94 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → 1 ∈
ℕ) |
| 102 | | ovex 7464 |
. . . . . . . . . 10
⊢ ((𝐴‘𝑘)[,)((𝐶‘1)‘𝑘)) ∈ V |
| 103 | 102 | rgenw 3065 |
. . . . . . . . 9
⊢
∀𝑘 ∈
𝑋 ((𝐴‘𝑘)[,)((𝐶‘1)‘𝑘)) ∈ V |
| 104 | | ixpexg 8962 |
. . . . . . . . 9
⊢
(∀𝑘 ∈
𝑋 ((𝐴‘𝑘)[,)((𝐶‘1)‘𝑘)) ∈ V → X𝑘 ∈
𝑋 ((𝐴‘𝑘)[,)((𝐶‘1)‘𝑘)) ∈ V) |
| 105 | 103, 104 | ax-mp 5 |
. . . . . . . 8
⊢ X𝑘 ∈
𝑋 ((𝐴‘𝑘)[,)((𝐶‘1)‘𝑘)) ∈ V |
| 106 | 105 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → X𝑘 ∈
𝑋 ((𝐴‘𝑘)[,)((𝐶‘1)‘𝑘)) ∈ V) |
| 107 | 25, 100, 101, 106 | fvmptd3 7039 |
. . . . . 6
⊢ (𝜑 → (𝐷‘1) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘1)‘𝑘))) |
| 108 | 107 | fveq2d 6910 |
. . . . 5
⊢ (𝜑 → ((voln‘𝑋)‘(𝐷‘1)) = ((voln‘𝑋)‘X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘1)‘𝑘)))) |
| 109 | | nfv 1914 |
. . . . . 6
⊢
Ⅎ𝑘𝜑 |
| 110 | | simpl 482 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝜑) |
| 111 | 94 | a1i 11 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 1 ∈ ℕ) |
| 112 | | simpr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋) |
| 113 | 94 | elexi 3503 |
. . . . . . . 8
⊢ 1 ∈
V |
| 114 | | eleq1 2829 |
. . . . . . . . . . 11
⊢ (𝑛 = 1 → (𝑛 ∈ ℕ ↔ 1 ∈
ℕ)) |
| 115 | 114 | anbi2d 630 |
. . . . . . . . . 10
⊢ (𝑛 = 1 → ((𝜑 ∧ 𝑛 ∈ ℕ) ↔ (𝜑 ∧ 1 ∈ ℕ))) |
| 116 | 115 | anbi1d 631 |
. . . . . . . . 9
⊢ (𝑛 = 1 → (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) ↔ ((𝜑 ∧ 1 ∈ ℕ) ∧ 𝑘 ∈ 𝑋))) |
| 117 | 98 | eleq1d 2826 |
. . . . . . . . 9
⊢ (𝑛 = 1 → (((𝐶‘𝑛)‘𝑘) ∈ ℝ ↔ ((𝐶‘1)‘𝑘) ∈ ℝ)) |
| 118 | 116, 117 | imbi12d 344 |
. . . . . . . 8
⊢ (𝑛 = 1 → ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑛)‘𝑘) ∈ ℝ) ↔ (((𝜑 ∧ 1 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘1)‘𝑘) ∈ ℝ))) |
| 119 | 113, 118,
34 | vtocl 3558 |
. . . . . . 7
⊢ (((𝜑 ∧ 1 ∈ ℕ) ∧
𝑘 ∈ 𝑋) → ((𝐶‘1)‘𝑘) ∈ ℝ) |
| 120 | 110, 111,
112, 119 | syl21anc 838 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐶‘1)‘𝑘) ∈ ℝ) |
| 121 | 109, 2, 29, 120 | vonhoire 46687 |
. . . . 5
⊢ (𝜑 → ((voln‘𝑋)‘X𝑘 ∈
𝑋 ((𝐴‘𝑘)[,)((𝐶‘1)‘𝑘))) ∈ ℝ) |
| 122 | 108, 121 | eqeltrd 2841 |
. . . 4
⊢ (𝜑 → ((voln‘𝑋)‘(𝐷‘1)) ∈ ℝ) |
| 123 | | eqid 2737 |
. . . 4
⊢ (𝑛 ∈ ℕ ↦
((voln‘𝑋)‘(𝐷‘𝑛))) = (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) |
| 124 | 1, 3, 4, 5, 26, 93, 96, 122, 123 | meaiininc 46502 |
. . 3
⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) ⇝ ((voln‘𝑋)‘∩
𝑛 ∈ ℕ (𝐷‘𝑛))) |
| 125 | 109, 29, 11 | iinhoiicc 46689 |
. . . . . . 7
⊢ (𝜑 → ∩ 𝑛 ∈ ℕ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐵‘𝑘) + (1 / 𝑛))) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐵‘𝑘))) |
| 126 | 33 | oveq2d 7447 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)) = ((𝐴‘𝑘)[,)((𝐵‘𝑘) + (1 / 𝑛)))) |
| 127 | 126 | ixpeq2dva 8952 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈
𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐵‘𝑘) + (1 / 𝑛)))) |
| 128 | 91, 127 | eqtrd 2777 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐵‘𝑘) + (1 / 𝑛)))) |
| 129 | 128 | iineq2dv 5017 |
. . . . . . 7
⊢ (𝜑 → ∩ 𝑛 ∈ ℕ (𝐷‘𝑛) = ∩ 𝑛 ∈ ℕ X𝑘 ∈
𝑋 ((𝐴‘𝑘)[,)((𝐵‘𝑘) + (1 / 𝑛)))) |
| 130 | | vonicclem2.i |
. . . . . . . 8
⊢ 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐵‘𝑘)) |
| 131 | 130 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐵‘𝑘))) |
| 132 | 125, 129,
131 | 3eqtr4d 2787 |
. . . . . 6
⊢ (𝜑 → ∩ 𝑛 ∈ ℕ (𝐷‘𝑛) = 𝐼) |
| 133 | 132 | eqcomd 2743 |
. . . . 5
⊢ (𝜑 → 𝐼 = ∩ 𝑛 ∈ ℕ (𝐷‘𝑛)) |
| 134 | 133 | fveq2d 6910 |
. . . 4
⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = ((voln‘𝑋)‘∩
𝑛 ∈ ℕ (𝐷‘𝑛))) |
| 135 | 134 | eqcomd 2743 |
. . 3
⊢ (𝜑 → ((voln‘𝑋)‘∩ 𝑛 ∈ ℕ (𝐷‘𝑛)) = ((voln‘𝑋)‘𝐼)) |
| 136 | 124, 135 | breqtrd 5169 |
. 2
⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) ⇝ ((voln‘𝑋)‘𝐼)) |
| 137 | | 2fveq3 6911 |
. . . . 5
⊢ (𝑛 = 𝑚 → ((voln‘𝑋)‘(𝐷‘𝑛)) = ((voln‘𝑋)‘(𝐷‘𝑚))) |
| 138 | 137 | cbvmptv 5255 |
. . . 4
⊢ (𝑛 ∈ ℕ ↦
((voln‘𝑋)‘(𝐷‘𝑛))) = (𝑚 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑚))) |
| 139 | 138 | a1i 11 |
. . 3
⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) = (𝑚 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑚)))) |
| 140 | | vonicclem2.n |
. . . 4
⊢ (𝜑 → 𝑋 ≠ ∅) |
| 141 | | vonicclem2.t |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ≤ (𝐵‘𝑘)) |
| 142 | 138 | eqcomi 2746 |
. . . 4
⊢ (𝑚 ∈ ℕ ↦
((voln‘𝑋)‘(𝐷‘𝑚))) = (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) |
| 143 | 2, 8, 10, 140, 141, 17, 25, 142 | vonicclem1 46698 |
. . 3
⊢ (𝜑 → (𝑚 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑚))) ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘))) |
| 144 | 139, 143 | eqbrtrd 5165 |
. 2
⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘))) |
| 145 | | climuni 15588 |
. 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))) |
| 146 | 136, 144,
145 | syl2anc 584 |
1
⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘))) |