Step | Hyp | Ref
| Expression |
1 | | fveq2 6103 |
. . . . . . . . 9
⊢ (𝑛 = 𝑚 → (𝐺‘𝑛) = (𝐺‘𝑚)) |
2 | 1 | oveq2d 6565 |
. . . . . . . 8
⊢ (𝑛 = 𝑚 → (𝑠 − (𝐺‘𝑛)) = (𝑠 − (𝐺‘𝑚))) |
3 | 2 | eleq1d 2672 |
. . . . . . 7
⊢ (𝑛 = 𝑚 → ((𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹 ↔ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹)) |
4 | 3 | rabbidv 3164 |
. . . . . 6
⊢ (𝑛 = 𝑚 → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹} = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹}) |
5 | | vitali.6 |
. . . . . 6
⊢ 𝑇 = (𝑛 ∈ ℕ ↦ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹}) |
6 | | reex 9906 |
. . . . . . 7
⊢ ℝ
∈ V |
7 | 6 | rabex 4740 |
. . . . . 6
⊢ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹} ∈ V |
8 | 4, 5, 7 | fvmpt 6191 |
. . . . 5
⊢ (𝑚 ∈ ℕ → (𝑇‘𝑚) = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹}) |
9 | 8 | adantl 481 |
. . . 4
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑇‘𝑚) = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹}) |
10 | 9 | fveq2d 6107 |
. . 3
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol*‘(𝑇‘𝑚)) = (vol*‘{𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹})) |
11 | | vitali.1 |
. . . . . . . 8
⊢ ∼ =
{〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥 − 𝑦) ∈ ℚ)} |
12 | | vitali.2 |
. . . . . . . 8
⊢ 𝑆 = ((0[,]1) / ∼
) |
13 | | vitali.3 |
. . . . . . . 8
⊢ (𝜑 → 𝐹 Fn 𝑆) |
14 | | vitali.4 |
. . . . . . . 8
⊢ (𝜑 → ∀𝑧 ∈ 𝑆 (𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧)) |
15 | | vitali.5 |
. . . . . . . 8
⊢ (𝜑 → 𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1))) |
16 | | vitali.7 |
. . . . . . . 8
⊢ (𝜑 → ¬ ran 𝐹 ∈ (𝒫 ℝ
∖ dom vol)) |
17 | 11, 12, 13, 14, 15, 5, 16 | vitalilem2 23184 |
. . . . . . 7
⊢ (𝜑 → (ran 𝐹 ⊆ (0[,]1) ∧ (0[,]1) ⊆
∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ∧ ∪
𝑚 ∈ ℕ (𝑇‘𝑚) ⊆ (-1[,]2))) |
18 | 17 | simp1d 1066 |
. . . . . 6
⊢ (𝜑 → ran 𝐹 ⊆ (0[,]1)) |
19 | | unitssre 12190 |
. . . . . 6
⊢ (0[,]1)
⊆ ℝ |
20 | 18, 19 | syl6ss 3580 |
. . . . 5
⊢ (𝜑 → ran 𝐹 ⊆ ℝ) |
21 | 20 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ran 𝐹 ⊆ ℝ) |
22 | | neg1rr 11002 |
. . . . . 6
⊢ -1 ∈
ℝ |
23 | | 1re 9918 |
. . . . . 6
⊢ 1 ∈
ℝ |
24 | | iccssre 12126 |
. . . . . 6
⊢ ((-1
∈ ℝ ∧ 1 ∈ ℝ) → (-1[,]1) ⊆
ℝ) |
25 | 22, 23, 24 | mp2an 704 |
. . . . 5
⊢ (-1[,]1)
⊆ ℝ |
26 | | inss2 3796 |
. . . . . 6
⊢ (ℚ
∩ (-1[,]1)) ⊆ (-1[,]1) |
27 | | f1of 6050 |
. . . . . . . 8
⊢ (𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)) → 𝐺:ℕ⟶(ℚ ∩
(-1[,]1))) |
28 | 15, 27 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝐺:ℕ⟶(ℚ ∩
(-1[,]1))) |
29 | 28 | ffvelrnda 6267 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚) ∈ (ℚ ∩
(-1[,]1))) |
30 | 26, 29 | sseldi 3566 |
. . . . 5
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚) ∈ (-1[,]1)) |
31 | 25, 30 | sseldi 3566 |
. . . 4
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚) ∈ ℝ) |
32 | | eqidd 2611 |
. . . 4
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹} = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹}) |
33 | 21, 31, 32 | ovolshft 23086 |
. . 3
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol*‘ran 𝐹) = (vol*‘{𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹})) |
34 | 10, 33 | eqtr4d 2647 |
. 2
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol*‘(𝑇‘𝑚)) = (vol*‘ran 𝐹)) |
35 | | 3re 10971 |
. . . . . . . 8
⊢ 3 ∈
ℝ |
36 | 35 | rexri 9976 |
. . . . . . 7
⊢ 3 ∈
ℝ* |
37 | 36 | a1i 11 |
. . . . . 6
⊢ ((𝜑 ∧ 0 < (vol*‘ran
𝐹)) → 3 ∈
ℝ*) |
38 | | 3nn 11063 |
. . . . . . . . . . . . . 14
⊢ 3 ∈
ℕ |
39 | | nnrp 11718 |
. . . . . . . . . . . . . 14
⊢ (3 ∈
ℕ → 3 ∈ ℝ+) |
40 | 38, 39 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢ 3 ∈
ℝ+ |
41 | | 0re 9919 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 0 ∈
ℝ |
42 | | 0le1 10430 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 0 ≤
1 |
43 | | ovolicc 23098 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((0
∈ ℝ ∧ 1 ∈ ℝ ∧ 0 ≤ 1) →
(vol*‘(0[,]1)) = (1 − 0)) |
44 | 41, 23, 42, 43 | mp3an 1416 |
. . . . . . . . . . . . . . . . . . 19
⊢
(vol*‘(0[,]1)) = (1 − 0) |
45 | | 1m0e1 11008 |
. . . . . . . . . . . . . . . . . . 19
⊢ (1
− 0) = 1 |
46 | 44, 45 | eqtri 2632 |
. . . . . . . . . . . . . . . . . 18
⊢
(vol*‘(0[,]1)) = 1 |
47 | 46, 23 | eqeltri 2684 |
. . . . . . . . . . . . . . . . 17
⊢
(vol*‘(0[,]1)) ∈ ℝ |
48 | | ovolsscl 23061 |
. . . . . . . . . . . . . . . . 17
⊢ ((ran
𝐹 ⊆ (0[,]1) ∧
(0[,]1) ⊆ ℝ ∧ (vol*‘(0[,]1)) ∈ ℝ) →
(vol*‘ran 𝐹) ∈
ℝ) |
49 | 19, 47, 48 | mp3an23 1408 |
. . . . . . . . . . . . . . . 16
⊢ (ran
𝐹 ⊆ (0[,]1) →
(vol*‘ran 𝐹) ∈
ℝ) |
50 | 18, 49 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (vol*‘ran 𝐹) ∈
ℝ) |
51 | 50 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 0 < (vol*‘ran
𝐹)) → (vol*‘ran
𝐹) ∈
ℝ) |
52 | | simpr 476 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 0 < (vol*‘ran
𝐹)) → 0 <
(vol*‘ran 𝐹)) |
53 | 51, 52 | elrpd 11745 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 0 < (vol*‘ran
𝐹)) → (vol*‘ran
𝐹) ∈
ℝ+) |
54 | | rpdivcl 11732 |
. . . . . . . . . . . . 13
⊢ ((3
∈ ℝ+ ∧ (vol*‘ran 𝐹) ∈ ℝ+) → (3 /
(vol*‘ran 𝐹)) ∈
ℝ+) |
55 | 40, 53, 54 | sylancr 694 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 0 < (vol*‘ran
𝐹)) → (3 /
(vol*‘ran 𝐹)) ∈
ℝ+) |
56 | 55 | rpred 11748 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 0 < (vol*‘ran
𝐹)) → (3 /
(vol*‘ran 𝐹)) ∈
ℝ) |
57 | 55 | rpge0d 11752 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 0 < (vol*‘ran
𝐹)) → 0 ≤ (3 /
(vol*‘ran 𝐹))) |
58 | | flge0nn0 12483 |
. . . . . . . . . . 11
⊢ (((3 /
(vol*‘ran 𝐹)) ∈
ℝ ∧ 0 ≤ (3 / (vol*‘ran 𝐹))) → (⌊‘(3 /
(vol*‘ran 𝐹))) ∈
ℕ0) |
59 | 56, 57, 58 | syl2anc 691 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 0 < (vol*‘ran
𝐹)) →
(⌊‘(3 / (vol*‘ran 𝐹))) ∈
ℕ0) |
60 | | nn0p1nn 11209 |
. . . . . . . . . 10
⊢
((⌊‘(3 / (vol*‘ran 𝐹))) ∈ ℕ0 →
((⌊‘(3 / (vol*‘ran 𝐹))) + 1) ∈ ℕ) |
61 | 59, 60 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 0 < (vol*‘ran
𝐹)) →
((⌊‘(3 / (vol*‘ran 𝐹))) + 1) ∈ ℕ) |
62 | 61 | nnred 10912 |
. . . . . . . 8
⊢ ((𝜑 ∧ 0 < (vol*‘ran
𝐹)) →
((⌊‘(3 / (vol*‘ran 𝐹))) + 1) ∈ ℝ) |
63 | 62, 51 | remulcld 9949 |
. . . . . . 7
⊢ ((𝜑 ∧ 0 < (vol*‘ran
𝐹)) →
(((⌊‘(3 / (vol*‘ran 𝐹))) + 1) · (vol*‘ran 𝐹)) ∈
ℝ) |
64 | 63 | rexrd 9968 |
. . . . . 6
⊢ ((𝜑 ∧ 0 < (vol*‘ran
𝐹)) →
(((⌊‘(3 / (vol*‘ran 𝐹))) + 1) · (vol*‘ran 𝐹)) ∈
ℝ*) |
65 | 6 | elpw2 4755 |
. . . . . . . . . . . . . . . . . . 19
⊢ (ran
𝐹 ∈ 𝒫 ℝ
↔ ran 𝐹 ⊆
ℝ) |
66 | 20, 65 | sylibr 223 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ran 𝐹 ∈ 𝒫 ℝ) |
67 | 66 | anim1i 590 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ ¬ ran 𝐹 ∈ dom vol) → (ran 𝐹 ∈ 𝒫 ℝ ∧
¬ ran 𝐹 ∈ dom
vol)) |
68 | | eldif 3550 |
. . . . . . . . . . . . . . . . 17
⊢ (ran
𝐹 ∈ (𝒫 ℝ
∖ dom vol) ↔ (ran 𝐹 ∈ 𝒫 ℝ ∧ ¬ ran
𝐹 ∈ dom
vol)) |
69 | 67, 68 | sylibr 223 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ ¬ ran 𝐹 ∈ dom vol) → ran 𝐹 ∈ (𝒫 ℝ
∖ dom vol)) |
70 | 69 | ex 449 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (¬ ran 𝐹 ∈ dom vol → ran 𝐹 ∈ (𝒫 ℝ
∖ dom vol))) |
71 | 16, 70 | mt3d 139 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ran 𝐹 ∈ dom vol) |
72 | 71 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ran 𝐹 ∈ dom vol) |
73 | | inss1 3795 |
. . . . . . . . . . . . . . . 16
⊢ (ℚ
∩ (-1[,]1)) ⊆ ℚ |
74 | | qssre 11674 |
. . . . . . . . . . . . . . . 16
⊢ ℚ
⊆ ℝ |
75 | 73, 74 | sstri 3577 |
. . . . . . . . . . . . . . 15
⊢ (ℚ
∩ (-1[,]1)) ⊆ ℝ |
76 | | fss 5969 |
. . . . . . . . . . . . . . 15
⊢ ((𝐺:ℕ⟶(ℚ ∩
(-1[,]1)) ∧ (ℚ ∩ (-1[,]1)) ⊆ ℝ) → 𝐺:ℕ⟶ℝ) |
77 | 28, 75, 76 | sylancl 693 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐺:ℕ⟶ℝ) |
78 | 77 | ffvelrnda 6267 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) ∈ ℝ) |
79 | | shftmbl 23113 |
. . . . . . . . . . . . 13
⊢ ((ran
𝐹 ∈ dom vol ∧
(𝐺‘𝑛) ∈ ℝ) → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹} ∈ dom vol) |
80 | 72, 78, 79 | syl2anc 691 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹} ∈ dom vol) |
81 | 80, 5 | fmptd 6292 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑇:ℕ⟶dom vol) |
82 | 81 | ffvelrnda 6267 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑇‘𝑚) ∈ dom vol) |
83 | 82 | ralrimiva 2949 |
. . . . . . . . 9
⊢ (𝜑 → ∀𝑚 ∈ ℕ (𝑇‘𝑚) ∈ dom vol) |
84 | | iunmbl 23128 |
. . . . . . . . 9
⊢
(∀𝑚 ∈
ℕ (𝑇‘𝑚) ∈ dom vol → ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ∈ dom vol) |
85 | 83, 84 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ∈ dom vol) |
86 | | mblss 23106 |
. . . . . . . 8
⊢ (∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ∈ dom vol → ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ⊆ ℝ) |
87 | | ovolcl 23053 |
. . . . . . . 8
⊢ (∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ⊆ ℝ → (vol*‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) ∈
ℝ*) |
88 | 85, 86, 87 | 3syl 18 |
. . . . . . 7
⊢ (𝜑 → (vol*‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) ∈
ℝ*) |
89 | 88 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 0 < (vol*‘ran
𝐹)) →
(vol*‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) ∈
ℝ*) |
90 | | flltp1 12463 |
. . . . . . . 8
⊢ ((3 /
(vol*‘ran 𝐹)) ∈
ℝ → (3 / (vol*‘ran 𝐹)) < ((⌊‘(3 / (vol*‘ran
𝐹))) + 1)) |
91 | 56, 90 | syl 17 |
. . . . . . 7
⊢ ((𝜑 ∧ 0 < (vol*‘ran
𝐹)) → (3 /
(vol*‘ran 𝐹)) <
((⌊‘(3 / (vol*‘ran 𝐹))) + 1)) |
92 | 35 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ 0 < (vol*‘ran
𝐹)) → 3 ∈
ℝ) |
93 | 92, 62, 53 | ltdivmul2d 11800 |
. . . . . . 7
⊢ ((𝜑 ∧ 0 < (vol*‘ran
𝐹)) → ((3 /
(vol*‘ran 𝐹)) <
((⌊‘(3 / (vol*‘ran 𝐹))) + 1) ↔ 3 < (((⌊‘(3 /
(vol*‘ran 𝐹))) + 1)
· (vol*‘ran 𝐹)))) |
94 | 91, 93 | mpbid 221 |
. . . . . 6
⊢ ((𝜑 ∧ 0 < (vol*‘ran
𝐹)) → 3 <
(((⌊‘(3 / (vol*‘ran 𝐹))) + 1) · (vol*‘ran 𝐹))) |
95 | | nnuz 11599 |
. . . . . . . . . . 11
⊢ ℕ =
(ℤ≥‘1) |
96 | | 1zzd 11285 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 0 < (vol*‘ran
𝐹)) → 1 ∈
ℤ) |
97 | | mblvol 23105 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑇‘𝑚) ∈ dom vol → (vol‘(𝑇‘𝑚)) = (vol*‘(𝑇‘𝑚))) |
98 | 82, 97 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol‘(𝑇‘𝑚)) = (vol*‘(𝑇‘𝑚))) |
99 | 98, 34 | eqtrd 2644 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol‘(𝑇‘𝑚)) = (vol*‘ran 𝐹)) |
100 | 50 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol*‘ran 𝐹) ∈
ℝ) |
101 | 99, 100 | eqeltrd 2688 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol‘(𝑇‘𝑚)) ∈ ℝ) |
102 | 101 | adantlr 747 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 0 < (vol*‘ran
𝐹)) ∧ 𝑚 ∈ ℕ) →
(vol‘(𝑇‘𝑚)) ∈
ℝ) |
103 | | eqid 2610 |
. . . . . . . . . . . . 13
⊢ (𝑚 ∈ ℕ ↦
(vol‘(𝑇‘𝑚))) = (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚))) |
104 | 102, 103 | fmptd 6292 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 0 < (vol*‘ran
𝐹)) → (𝑚 ∈ ℕ ↦
(vol‘(𝑇‘𝑚))):ℕ⟶ℝ) |
105 | 104 | ffvelrnda 6267 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 0 < (vol*‘ran
𝐹)) ∧ 𝑘 ∈ ℕ) → ((𝑚 ∈ ℕ ↦
(vol‘(𝑇‘𝑚)))‘𝑘) ∈ ℝ) |
106 | 95, 96, 105 | serfre 12692 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 0 < (vol*‘ran
𝐹)) → seq1( + , (𝑚 ∈ ℕ ↦
(vol‘(𝑇‘𝑚)))):ℕ⟶ℝ) |
107 | | frn 5966 |
. . . . . . . . . 10
⊢ (seq1( +
, (𝑚 ∈ ℕ ↦
(vol‘(𝑇‘𝑚)))):ℕ⟶ℝ
→ ran seq1( + , (𝑚
∈ ℕ ↦ (vol‘(𝑇‘𝑚)))) ⊆ ℝ) |
108 | 106, 107 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 0 < (vol*‘ran
𝐹)) → ran seq1( + ,
(𝑚 ∈ ℕ ↦
(vol‘(𝑇‘𝑚)))) ⊆
ℝ) |
109 | | ressxr 9962 |
. . . . . . . . 9
⊢ ℝ
⊆ ℝ* |
110 | 108, 109 | syl6ss 3580 |
. . . . . . . 8
⊢ ((𝜑 ∧ 0 < (vol*‘ran
𝐹)) → ran seq1( + ,
(𝑚 ∈ ℕ ↦
(vol‘(𝑇‘𝑚)))) ⊆
ℝ*) |
111 | 99 | adantlr 747 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 0 < (vol*‘ran
𝐹)) ∧ 𝑚 ∈ ℕ) →
(vol‘(𝑇‘𝑚)) = (vol*‘ran 𝐹)) |
112 | 111 | mpteq2dva 4672 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 0 < (vol*‘ran
𝐹)) → (𝑚 ∈ ℕ ↦
(vol‘(𝑇‘𝑚))) = (𝑚 ∈ ℕ ↦ (vol*‘ran 𝐹))) |
113 | | fconstmpt 5085 |
. . . . . . . . . . . . 13
⊢ (ℕ
× {(vol*‘ran 𝐹)}) = (𝑚 ∈ ℕ ↦ (vol*‘ran 𝐹)) |
114 | 112, 113 | syl6eqr 2662 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 0 < (vol*‘ran
𝐹)) → (𝑚 ∈ ℕ ↦
(vol‘(𝑇‘𝑚))) = (ℕ ×
{(vol*‘ran 𝐹)})) |
115 | 114 | seqeq3d 12671 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 0 < (vol*‘ran
𝐹)) → seq1( + , (𝑚 ∈ ℕ ↦
(vol‘(𝑇‘𝑚)))) = seq1( + , (ℕ
× {(vol*‘ran 𝐹)}))) |
116 | 115 | fveq1d 6105 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 0 < (vol*‘ran
𝐹)) → (seq1( + ,
(𝑚 ∈ ℕ ↦
(vol‘(𝑇‘𝑚))))‘((⌊‘(3 /
(vol*‘ran 𝐹))) + 1))
= (seq1( + , (ℕ × {(vol*‘ran 𝐹)}))‘((⌊‘(3 /
(vol*‘ran 𝐹))) +
1))) |
117 | 51 | recnd 9947 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 0 < (vol*‘ran
𝐹)) → (vol*‘ran
𝐹) ∈
ℂ) |
118 | | ser1const 12719 |
. . . . . . . . . . 11
⊢
(((vol*‘ran 𝐹)
∈ ℂ ∧ ((⌊‘(3 / (vol*‘ran 𝐹))) + 1) ∈ ℕ) → (seq1( + ,
(ℕ × {(vol*‘ran 𝐹)}))‘((⌊‘(3 /
(vol*‘ran 𝐹))) + 1))
= (((⌊‘(3 / (vol*‘ran 𝐹))) + 1) · (vol*‘ran 𝐹))) |
119 | 117, 61, 118 | syl2anc 691 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 0 < (vol*‘ran
𝐹)) → (seq1( + ,
(ℕ × {(vol*‘ran 𝐹)}))‘((⌊‘(3 /
(vol*‘ran 𝐹))) + 1))
= (((⌊‘(3 / (vol*‘ran 𝐹))) + 1) · (vol*‘ran 𝐹))) |
120 | 116, 119 | eqtrd 2644 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 0 < (vol*‘ran
𝐹)) → (seq1( + ,
(𝑚 ∈ ℕ ↦
(vol‘(𝑇‘𝑚))))‘((⌊‘(3 /
(vol*‘ran 𝐹))) + 1))
= (((⌊‘(3 / (vol*‘ran 𝐹))) + 1) · (vol*‘ran 𝐹))) |
121 | | ffn 5958 |
. . . . . . . . . . 11
⊢ (seq1( +
, (𝑚 ∈ ℕ ↦
(vol‘(𝑇‘𝑚)))):ℕ⟶ℝ
→ seq1( + , (𝑚 ∈
ℕ ↦ (vol‘(𝑇‘𝑚)))) Fn ℕ) |
122 | 106, 121 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 0 < (vol*‘ran
𝐹)) → seq1( + , (𝑚 ∈ ℕ ↦
(vol‘(𝑇‘𝑚)))) Fn
ℕ) |
123 | | fnfvelrn 6264 |
. . . . . . . . . 10
⊢ ((seq1( +
, (𝑚 ∈ ℕ ↦
(vol‘(𝑇‘𝑚)))) Fn ℕ ∧
((⌊‘(3 / (vol*‘ran 𝐹))) + 1) ∈ ℕ) → (seq1( + ,
(𝑚 ∈ ℕ ↦
(vol‘(𝑇‘𝑚))))‘((⌊‘(3 /
(vol*‘ran 𝐹))) + 1))
∈ ran seq1( + , (𝑚
∈ ℕ ↦ (vol‘(𝑇‘𝑚))))) |
124 | 122, 61, 123 | syl2anc 691 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 0 < (vol*‘ran
𝐹)) → (seq1( + ,
(𝑚 ∈ ℕ ↦
(vol‘(𝑇‘𝑚))))‘((⌊‘(3 /
(vol*‘ran 𝐹))) + 1))
∈ ran seq1( + , (𝑚
∈ ℕ ↦ (vol‘(𝑇‘𝑚))))) |
125 | 120, 124 | eqeltrrd 2689 |
. . . . . . . 8
⊢ ((𝜑 ∧ 0 < (vol*‘ran
𝐹)) →
(((⌊‘(3 / (vol*‘ran 𝐹))) + 1) · (vol*‘ran 𝐹)) ∈ ran seq1( + , (𝑚 ∈ ℕ ↦
(vol‘(𝑇‘𝑚))))) |
126 | | supxrub 12026 |
. . . . . . . 8
⊢ ((ran
seq1( + , (𝑚 ∈ ℕ
↦ (vol‘(𝑇‘𝑚)))) ⊆ ℝ* ∧
(((⌊‘(3 / (vol*‘ran 𝐹))) + 1) · (vol*‘ran 𝐹)) ∈ ran seq1( + , (𝑚 ∈ ℕ ↦
(vol‘(𝑇‘𝑚))))) → (((⌊‘(3
/ (vol*‘ran 𝐹))) + 1)
· (vol*‘ran 𝐹)) ≤ sup(ran seq1( + , (𝑚 ∈ ℕ ↦
(vol‘(𝑇‘𝑚)))), ℝ*, <
)) |
127 | 110, 125,
126 | syl2anc 691 |
. . . . . . 7
⊢ ((𝜑 ∧ 0 < (vol*‘ran
𝐹)) →
(((⌊‘(3 / (vol*‘ran 𝐹))) + 1) · (vol*‘ran 𝐹)) ≤ sup(ran seq1( + , (𝑚 ∈ ℕ ↦
(vol‘(𝑇‘𝑚)))), ℝ*, <
)) |
128 | 85 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 0 < (vol*‘ran
𝐹)) → ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ∈ dom vol) |
129 | | mblvol 23105 |
. . . . . . . . 9
⊢ (∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ∈ dom vol → (vol‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) = (vol*‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚))) |
130 | 128, 129 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 0 < (vol*‘ran
𝐹)) →
(vol‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) = (vol*‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚))) |
131 | 82, 101 | jca 553 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑇‘𝑚) ∈ dom vol ∧ (vol‘(𝑇‘𝑚)) ∈ ℝ)) |
132 | 131 | ralrimiva 2949 |
. . . . . . . . . 10
⊢ (𝜑 → ∀𝑚 ∈ ℕ ((𝑇‘𝑚) ∈ dom vol ∧ (vol‘(𝑇‘𝑚)) ∈ ℝ)) |
133 | 132 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 0 < (vol*‘ran
𝐹)) → ∀𝑚 ∈ ℕ ((𝑇‘𝑚) ∈ dom vol ∧ (vol‘(𝑇‘𝑚)) ∈ ℝ)) |
134 | 11, 12, 13, 14, 15, 5, 16 | vitalilem3 23185 |
. . . . . . . . . 10
⊢ (𝜑 → Disj 𝑚 ∈ ℕ (𝑇‘𝑚)) |
135 | 134 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 0 < (vol*‘ran
𝐹)) → Disj
𝑚 ∈ ℕ (𝑇‘𝑚)) |
136 | | eqid 2610 |
. . . . . . . . . 10
⊢ seq1( + ,
(𝑚 ∈ ℕ ↦
(vol‘(𝑇‘𝑚)))) = seq1( + , (𝑚 ∈ ℕ ↦
(vol‘(𝑇‘𝑚)))) |
137 | 136, 103 | voliun 23129 |
. . . . . . . . 9
⊢
((∀𝑚 ∈
ℕ ((𝑇‘𝑚) ∈ dom vol ∧
(vol‘(𝑇‘𝑚)) ∈ ℝ) ∧
Disj 𝑚 ∈
ℕ (𝑇‘𝑚)) → (vol‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) = sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))), ℝ*, <
)) |
138 | 133, 135,
137 | syl2anc 691 |
. . . . . . . 8
⊢ ((𝜑 ∧ 0 < (vol*‘ran
𝐹)) →
(vol‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) = sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))), ℝ*, <
)) |
139 | 130, 138 | eqtr3d 2646 |
. . . . . . 7
⊢ ((𝜑 ∧ 0 < (vol*‘ran
𝐹)) →
(vol*‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) = sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))), ℝ*, <
)) |
140 | 127, 139 | breqtrrd 4611 |
. . . . . 6
⊢ ((𝜑 ∧ 0 < (vol*‘ran
𝐹)) →
(((⌊‘(3 / (vol*‘ran 𝐹))) + 1) · (vol*‘ran 𝐹)) ≤ (vol*‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚))) |
141 | 37, 64, 89, 94, 140 | xrltletrd 11868 |
. . . . 5
⊢ ((𝜑 ∧ 0 < (vol*‘ran
𝐹)) → 3 <
(vol*‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚))) |
142 | 17 | simp3d 1068 |
. . . . . . . . 9
⊢ (𝜑 → ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ⊆ (-1[,]2)) |
143 | 142 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 0 < (vol*‘ran
𝐹)) → ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ⊆ (-1[,]2)) |
144 | | 2re 10967 |
. . . . . . . . 9
⊢ 2 ∈
ℝ |
145 | | iccssre 12126 |
. . . . . . . . 9
⊢ ((-1
∈ ℝ ∧ 2 ∈ ℝ) → (-1[,]2) ⊆
ℝ) |
146 | 22, 144, 145 | mp2an 704 |
. . . . . . . 8
⊢ (-1[,]2)
⊆ ℝ |
147 | | ovolss 23060 |
. . . . . . . 8
⊢
((∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ⊆ (-1[,]2) ∧ (-1[,]2) ⊆
ℝ) → (vol*‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) ≤
(vol*‘(-1[,]2))) |
148 | 143, 146,
147 | sylancl 693 |
. . . . . . 7
⊢ ((𝜑 ∧ 0 < (vol*‘ran
𝐹)) →
(vol*‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) ≤
(vol*‘(-1[,]2))) |
149 | | 2cn 10968 |
. . . . . . . . 9
⊢ 2 ∈
ℂ |
150 | | ax-1cn 9873 |
. . . . . . . . 9
⊢ 1 ∈
ℂ |
151 | 149, 150 | subnegi 10239 |
. . . . . . . 8
⊢ (2
− -1) = (2 + 1) |
152 | | neg1lt0 11004 |
. . . . . . . . . . 11
⊢ -1 <
0 |
153 | | 2pos 10989 |
. . . . . . . . . . 11
⊢ 0 <
2 |
154 | 22, 41, 144 | lttri 10042 |
. . . . . . . . . . 11
⊢ ((-1 <
0 ∧ 0 < 2) → -1 < 2) |
155 | 152, 153,
154 | mp2an 704 |
. . . . . . . . . 10
⊢ -1 <
2 |
156 | 22, 144, 155 | ltleii 10039 |
. . . . . . . . 9
⊢ -1 ≤
2 |
157 | | ovolicc 23098 |
. . . . . . . . 9
⊢ ((-1
∈ ℝ ∧ 2 ∈ ℝ ∧ -1 ≤ 2) →
(vol*‘(-1[,]2)) = (2 − -1)) |
158 | 22, 144, 156, 157 | mp3an 1416 |
. . . . . . . 8
⊢
(vol*‘(-1[,]2)) = (2 − -1) |
159 | | df-3 10957 |
. . . . . . . 8
⊢ 3 = (2 +
1) |
160 | 151, 158,
159 | 3eqtr4i 2642 |
. . . . . . 7
⊢
(vol*‘(-1[,]2)) = 3 |
161 | 148, 160 | syl6breq 4624 |
. . . . . 6
⊢ ((𝜑 ∧ 0 < (vol*‘ran
𝐹)) →
(vol*‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) ≤ 3) |
162 | | xrlenlt 9982 |
. . . . . . 7
⊢
(((vol*‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) ∈ ℝ* ∧ 3 ∈
ℝ*) → ((vol*‘∪
𝑚 ∈ ℕ (𝑇‘𝑚)) ≤ 3 ↔ ¬ 3 <
(vol*‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)))) |
163 | 89, 36, 162 | sylancl 693 |
. . . . . 6
⊢ ((𝜑 ∧ 0 < (vol*‘ran
𝐹)) →
((vol*‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) ≤ 3 ↔ ¬ 3 <
(vol*‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)))) |
164 | 161, 163 | mpbid 221 |
. . . . 5
⊢ ((𝜑 ∧ 0 < (vol*‘ran
𝐹)) → ¬ 3 <
(vol*‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚))) |
165 | 141, 164 | pm2.65da 598 |
. . . 4
⊢ (𝜑 → ¬ 0 <
(vol*‘ran 𝐹)) |
166 | | ovolge0 23056 |
. . . . . . 7
⊢ (ran
𝐹 ⊆ ℝ → 0
≤ (vol*‘ran 𝐹)) |
167 | 20, 166 | syl 17 |
. . . . . 6
⊢ (𝜑 → 0 ≤ (vol*‘ran
𝐹)) |
168 | | 0xr 9965 |
. . . . . . 7
⊢ 0 ∈
ℝ* |
169 | | ovolcl 23053 |
. . . . . . . 8
⊢ (ran
𝐹 ⊆ ℝ →
(vol*‘ran 𝐹) ∈
ℝ*) |
170 | 20, 169 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (vol*‘ran 𝐹) ∈
ℝ*) |
171 | | xrleloe 11853 |
. . . . . . 7
⊢ ((0
∈ ℝ* ∧ (vol*‘ran 𝐹) ∈ ℝ*) → (0 ≤
(vol*‘ran 𝐹) ↔
(0 < (vol*‘ran 𝐹)
∨ 0 = (vol*‘ran 𝐹)))) |
172 | 168, 170,
171 | sylancr 694 |
. . . . . 6
⊢ (𝜑 → (0 ≤ (vol*‘ran
𝐹) ↔ (0 <
(vol*‘ran 𝐹) ∨ 0 =
(vol*‘ran 𝐹)))) |
173 | 167, 172 | mpbid 221 |
. . . . 5
⊢ (𝜑 → (0 < (vol*‘ran
𝐹) ∨ 0 = (vol*‘ran
𝐹))) |
174 | 173 | ord 391 |
. . . 4
⊢ (𝜑 → (¬ 0 <
(vol*‘ran 𝐹) → 0
= (vol*‘ran 𝐹))) |
175 | 165, 174 | mpd 15 |
. . 3
⊢ (𝜑 → 0 = (vol*‘ran 𝐹)) |
176 | 175 | adantr 480 |
. 2
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 0 = (vol*‘ran
𝐹)) |
177 | 34, 176 | eqtr4d 2647 |
1
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol*‘(𝑇‘𝑚)) = 0) |