Step | Hyp | Ref
| Expression |
1 | | ovolicc2.m |
. . . . . 6
⊢ 𝑀 = {𝑦 ∈ ℝ* ∣
∃𝑓 ∈ (( ≤
∩ (ℝ × ℝ)) ↑m ℕ)((𝐴[,]𝐵) ⊆ ∪ ran
((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs
∘ − ) ∘ 𝑓)), ℝ*, <
))} |
2 | 1 | elovolm 24650 |
. . . . 5
⊢ (𝑧 ∈ 𝑀 ↔ ∃𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)((𝐴[,]𝐵) ⊆ ∪ ran
((,) ∘ 𝑓) ∧ 𝑧 = sup(ran seq1( + , ((abs
∘ − ) ∘ 𝑓)), ℝ*, <
))) |
3 | | simprr 770 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran
((,) ∘ 𝑓))) →
(𝐴[,]𝐵) ⊆ ∪ ran
((,) ∘ 𝑓)) |
4 | | unieq 4856 |
. . . . . . . . . . . . . 14
⊢ (𝑢 = ran ((,) ∘ 𝑓) → ∪ 𝑢 =
∪ ran ((,) ∘ 𝑓)) |
5 | 4 | sseq2d 3958 |
. . . . . . . . . . . . 13
⊢ (𝑢 = ran ((,) ∘ 𝑓) → ((𝐴[,]𝐵) ⊆ ∪ 𝑢 ↔ (𝐴[,]𝐵) ⊆ ∪ ran
((,) ∘ 𝑓))) |
6 | | pweq 4555 |
. . . . . . . . . . . . . . 15
⊢ (𝑢 = ran ((,) ∘ 𝑓) → 𝒫 𝑢 = 𝒫 ran ((,) ∘
𝑓)) |
7 | 6 | ineq1d 4151 |
. . . . . . . . . . . . . 14
⊢ (𝑢 = ran ((,) ∘ 𝑓) → (𝒫 𝑢 ∩ Fin) = (𝒫 ran
((,) ∘ 𝑓) ∩
Fin)) |
8 | 7 | rexeqdv 3348 |
. . . . . . . . . . . . 13
⊢ (𝑢 = ran ((,) ∘ 𝑓) → (∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(𝐴[,]𝐵) ⊆ ∪ 𝑣 ↔ ∃𝑣 ∈ (𝒫 ran ((,)
∘ 𝑓) ∩ Fin)(𝐴[,]𝐵) ⊆ ∪ 𝑣)) |
9 | 5, 8 | imbi12d 345 |
. . . . . . . . . . . 12
⊢ (𝑢 = ran ((,) ∘ 𝑓) → (((𝐴[,]𝐵) ⊆ ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(𝐴[,]𝐵) ⊆ ∪ 𝑣) ↔ ((𝐴[,]𝐵) ⊆ ∪ ran
((,) ∘ 𝑓) →
∃𝑣 ∈ (𝒫
ran ((,) ∘ 𝑓) ∩
Fin)(𝐴[,]𝐵) ⊆ ∪ 𝑣))) |
10 | | ovolicc.1 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐴 ∈ ℝ) |
11 | | ovolicc.2 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐵 ∈ ℝ) |
12 | | eqid 2740 |
. . . . . . . . . . . . . . . 16
⊢
(topGen‘ran (,)) = (topGen‘ran (,)) |
13 | | eqid 2740 |
. . . . . . . . . . . . . . . 16
⊢
((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) = ((topGen‘ran (,))
↾t (𝐴[,]𝐵)) |
14 | 12, 13 | icccmp 23999 |
. . . . . . . . . . . . . . 15
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) →
((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) ∈ Comp) |
15 | 10, 11, 14 | syl2anc 584 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((topGen‘ran (,))
↾t (𝐴[,]𝐵)) ∈ Comp) |
16 | | retop 23936 |
. . . . . . . . . . . . . . 15
⊢
(topGen‘ran (,)) ∈ Top |
17 | | iccssre 13172 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) |
18 | 10, 11, 17 | syl2anc 584 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ) |
19 | | uniretop 23937 |
. . . . . . . . . . . . . . . 16
⊢ ℝ =
∪ (topGen‘ran (,)) |
20 | 19 | cmpsub 22562 |
. . . . . . . . . . . . . . 15
⊢
(((topGen‘ran (,)) ∈ Top ∧ (𝐴[,]𝐵) ⊆ ℝ) →
(((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) ∈ Comp ↔ ∀𝑢 ∈ 𝒫
(topGen‘ran (,))((𝐴[,]𝐵) ⊆ ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(𝐴[,]𝐵) ⊆ ∪ 𝑣))) |
21 | 16, 18, 20 | sylancr 587 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (((topGen‘ran (,))
↾t (𝐴[,]𝐵)) ∈ Comp ↔ ∀𝑢 ∈ 𝒫
(topGen‘ran (,))((𝐴[,]𝐵) ⊆ ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(𝐴[,]𝐵) ⊆ ∪ 𝑣))) |
22 | 15, 21 | mpbid 231 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ∀𝑢 ∈ 𝒫 (topGen‘ran
(,))((𝐴[,]𝐵) ⊆ ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(𝐴[,]𝐵) ⊆ ∪ 𝑣)) |
23 | 22 | adantr 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran
((,) ∘ 𝑓))) →
∀𝑢 ∈ 𝒫
(topGen‘ran (,))((𝐴[,]𝐵) ⊆ ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(𝐴[,]𝐵) ⊆ ∪ 𝑣)) |
24 | | ioof 13190 |
. . . . . . . . . . . . . . . . . . 19
⊢
(,):(ℝ* × ℝ*)⟶𝒫
ℝ |
25 | | ffn 6598 |
. . . . . . . . . . . . . . . . . . 19
⊢
((,):(ℝ* × ℝ*)⟶𝒫
ℝ → (,) Fn (ℝ* ×
ℝ*)) |
26 | 24, 25 | ax-mp 5 |
. . . . . . . . . . . . . . . . . 18
⊢ (,) Fn
(ℝ* × ℝ*) |
27 | | dffn3 6611 |
. . . . . . . . . . . . . . . . . 18
⊢ ((,) Fn
(ℝ* × ℝ*) ↔
(,):(ℝ* × ℝ*)⟶ran
(,)) |
28 | 26, 27 | mpbi 229 |
. . . . . . . . . . . . . . . . 17
⊢
(,):(ℝ* × ℝ*)⟶ran
(,) |
29 | | simpr 485 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)) → 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)) |
30 | | elovolmlem 24649 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑓 ∈ (( ≤ ∩ (ℝ
× ℝ)) ↑m ℕ) ↔ 𝑓:ℕ⟶( ≤ ∩ (ℝ ×
ℝ))) |
31 | 29, 30 | sylib 217 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)) → 𝑓:ℕ⟶( ≤ ∩ (ℝ ×
ℝ))) |
32 | | inss2 4169 |
. . . . . . . . . . . . . . . . . . 19
⊢ ( ≤
∩ (ℝ × ℝ)) ⊆ (ℝ ×
ℝ) |
33 | | rexpssxrxp 11031 |
. . . . . . . . . . . . . . . . . . 19
⊢ (ℝ
× ℝ) ⊆ (ℝ* ×
ℝ*) |
34 | 32, 33 | sstri 3935 |
. . . . . . . . . . . . . . . . . 18
⊢ ( ≤
∩ (ℝ × ℝ)) ⊆ (ℝ* ×
ℝ*) |
35 | | fss 6615 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑓:ℕ⟶( ≤ ∩
(ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ))
⊆ (ℝ* × ℝ*)) → 𝑓:ℕ⟶(ℝ* ×
ℝ*)) |
36 | 31, 34, 35 | sylancl 586 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)) → 𝑓:ℕ⟶(ℝ* ×
ℝ*)) |
37 | | fco 6622 |
. . . . . . . . . . . . . . . . 17
⊢
(((,):(ℝ* × ℝ*)⟶ran (,)
∧ 𝑓:ℕ⟶(ℝ* ×
ℝ*)) → ((,) ∘ 𝑓):ℕ⟶ran (,)) |
38 | 28, 36, 37 | sylancr 587 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)) → ((,) ∘ 𝑓):ℕ⟶ran (,)) |
39 | 38 | adantrr 714 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran
((,) ∘ 𝑓))) →
((,) ∘ 𝑓):ℕ⟶ran (,)) |
40 | 39 | frnd 6606 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran
((,) ∘ 𝑓))) →
ran ((,) ∘ 𝑓) ⊆
ran (,)) |
41 | | retopbas 23935 |
. . . . . . . . . . . . . . 15
⊢ ran (,)
∈ TopBases |
42 | | bastg 22127 |
. . . . . . . . . . . . . . 15
⊢ (ran (,)
∈ TopBases → ran (,) ⊆ (topGen‘ran (,))) |
43 | 41, 42 | ax-mp 5 |
. . . . . . . . . . . . . 14
⊢ ran (,)
⊆ (topGen‘ran (,)) |
44 | 40, 43 | sstrdi 3938 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran
((,) ∘ 𝑓))) →
ran ((,) ∘ 𝑓) ⊆
(topGen‘ran (,))) |
45 | | fvex 6784 |
. . . . . . . . . . . . . 14
⊢
(topGen‘ran (,)) ∈ V |
46 | 45 | elpw2 5273 |
. . . . . . . . . . . . 13
⊢ (ran ((,)
∘ 𝑓) ∈ 𝒫
(topGen‘ran (,)) ↔ ran ((,) ∘ 𝑓) ⊆ (topGen‘ran
(,))) |
47 | 44, 46 | sylibr 233 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran
((,) ∘ 𝑓))) →
ran ((,) ∘ 𝑓) ∈
𝒫 (topGen‘ran (,))) |
48 | 9, 23, 47 | rspcdva 3563 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran
((,) ∘ 𝑓))) →
((𝐴[,]𝐵) ⊆ ∪ ran
((,) ∘ 𝑓) →
∃𝑣 ∈ (𝒫
ran ((,) ∘ 𝑓) ∩
Fin)(𝐴[,]𝐵) ⊆ ∪ 𝑣)) |
49 | 3, 48 | mpd 15 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran
((,) ∘ 𝑓))) →
∃𝑣 ∈ (𝒫
ran ((,) ∘ 𝑓) ∩
Fin)(𝐴[,]𝐵) ⊆ ∪ 𝑣) |
50 | | simprl 768 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → 𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin)) |
51 | | elin 3908 |
. . . . . . . . . . . . . . . 16
⊢ (𝑣 ∈ (𝒫 ran ((,)
∘ 𝑓) ∩ Fin)
↔ (𝑣 ∈ 𝒫
ran ((,) ∘ 𝑓) ∧
𝑣 ∈
Fin)) |
52 | 50, 51 | sylib 217 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → (𝑣 ∈ 𝒫 ran ((,) ∘ 𝑓) ∧ 𝑣 ∈ Fin)) |
53 | 52 | simprd 496 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → 𝑣 ∈ Fin) |
54 | 52 | simpld 495 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → 𝑣 ∈ 𝒫 ran ((,) ∘ 𝑓)) |
55 | 54 | elpwid 4550 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → 𝑣 ⊆ ran ((,) ∘ 𝑓)) |
56 | 55 | sseld 3925 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → (𝑡 ∈ 𝑣 → 𝑡 ∈ ran ((,) ∘ 𝑓))) |
57 | 38 | ffnd 6599 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)) → ((,) ∘ 𝑓) Fn ℕ) |
58 | 57 | adantr 481 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → ((,) ∘ 𝑓) Fn ℕ) |
59 | | fvelrnb 6827 |
. . . . . . . . . . . . . . . . 17
⊢ (((,)
∘ 𝑓) Fn ℕ
→ (𝑡 ∈ ran ((,)
∘ 𝑓) ↔
∃𝑘 ∈ ℕ
(((,) ∘ 𝑓)‘𝑘) = 𝑡)) |
60 | 58, 59 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → (𝑡 ∈ ran ((,) ∘ 𝑓) ↔ ∃𝑘 ∈ ℕ (((,) ∘ 𝑓)‘𝑘) = 𝑡)) |
61 | 56, 60 | sylibd 238 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → (𝑡 ∈ 𝑣 → ∃𝑘 ∈ ℕ (((,) ∘ 𝑓)‘𝑘) = 𝑡)) |
62 | 61 | ralrimiv 3109 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → ∀𝑡 ∈ 𝑣 ∃𝑘 ∈ ℕ (((,) ∘ 𝑓)‘𝑘) = 𝑡) |
63 | | fveqeq2 6780 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = (𝑔‘𝑡) → ((((,) ∘ 𝑓)‘𝑘) = 𝑡 ↔ (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡)) |
64 | 63 | ac6sfi 9046 |
. . . . . . . . . . . . . 14
⊢ ((𝑣 ∈ Fin ∧ ∀𝑡 ∈ 𝑣 ∃𝑘 ∈ ℕ (((,) ∘ 𝑓)‘𝑘) = 𝑡) → ∃𝑔(𝑔:𝑣⟶ℕ ∧ ∀𝑡 ∈ 𝑣 (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡)) |
65 | 53, 62, 64 | syl2anc 584 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → ∃𝑔(𝑔:𝑣⟶ℕ ∧ ∀𝑡 ∈ 𝑣 (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡)) |
66 | 10 | ad2antrr 723 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)) ∧ ((𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣) ∧ (𝑔:𝑣⟶ℕ ∧ ∀𝑡 ∈ 𝑣 (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡))) → 𝐴 ∈ ℝ) |
67 | 11 | ad2antrr 723 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)) ∧ ((𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣) ∧ (𝑔:𝑣⟶ℕ ∧ ∀𝑡 ∈ 𝑣 (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡))) → 𝐵 ∈ ℝ) |
68 | | ovolicc.3 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐴 ≤ 𝐵) |
69 | 68 | ad2antrr 723 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)) ∧ ((𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣) ∧ (𝑔:𝑣⟶ℕ ∧ ∀𝑡 ∈ 𝑣 (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡))) → 𝐴 ≤ 𝐵) |
70 | | eqid 2740 |
. . . . . . . . . . . . . . . 16
⊢ seq1( + ,
((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − )
∘ 𝑓)) |
71 | 31 | adantr 481 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)) ∧ ((𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣) ∧ (𝑔:𝑣⟶ℕ ∧ ∀𝑡 ∈ 𝑣 (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡))) → 𝑓:ℕ⟶( ≤ ∩ (ℝ ×
ℝ))) |
72 | | simprll 776 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)) ∧ ((𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣) ∧ (𝑔:𝑣⟶ℕ ∧ ∀𝑡 ∈ 𝑣 (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡))) → 𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin)) |
73 | | simprlr 777 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)) ∧ ((𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣) ∧ (𝑔:𝑣⟶ℕ ∧ ∀𝑡 ∈ 𝑣 (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡))) → (𝐴[,]𝐵) ⊆ ∪ 𝑣) |
74 | | simprrl 778 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)) ∧ ((𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣) ∧ (𝑔:𝑣⟶ℕ ∧ ∀𝑡 ∈ 𝑣 (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡))) → 𝑔:𝑣⟶ℕ) |
75 | | simprrr 779 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)) ∧ ((𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣) ∧ (𝑔:𝑣⟶ℕ ∧ ∀𝑡 ∈ 𝑣 (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡))) → ∀𝑡 ∈ 𝑣 (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡) |
76 | | 2fveq3 6776 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑥 → (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = (((,) ∘ 𝑓)‘(𝑔‘𝑥))) |
77 | | id 22 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑥 → 𝑡 = 𝑥) |
78 | 76, 77 | eqeq12d 2756 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑥 → ((((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡 ↔ (((,) ∘ 𝑓)‘(𝑔‘𝑥)) = 𝑥)) |
79 | 78 | rspccva 3560 |
. . . . . . . . . . . . . . . . 17
⊢
((∀𝑡 ∈
𝑣 (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡 ∧ 𝑥 ∈ 𝑣) → (((,) ∘ 𝑓)‘(𝑔‘𝑥)) = 𝑥) |
80 | 75, 79 | sylan 580 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)) ∧ ((𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣) ∧ (𝑔:𝑣⟶ℕ ∧ ∀𝑡 ∈ 𝑣 (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡))) ∧ 𝑥 ∈ 𝑣) → (((,) ∘ 𝑓)‘(𝑔‘𝑥)) = 𝑥) |
81 | | eqid 2740 |
. . . . . . . . . . . . . . . 16
⊢ {𝑢 ∈ 𝑣 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅} = {𝑢 ∈ 𝑣 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅} |
82 | 66, 67, 69, 70, 71, 72, 73, 74, 80, 81 | ovolicc2lem5 24696 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)) ∧ ((𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣) ∧ (𝑔:𝑣⟶ℕ ∧ ∀𝑡 ∈ 𝑣 (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡))) → (𝐵 − 𝐴) ≤ sup(ran seq1( + , ((abs ∘
− ) ∘ 𝑓)),
ℝ*, < )) |
83 | 82 | expr 457 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → ((𝑔:𝑣⟶ℕ ∧ ∀𝑡 ∈ 𝑣 (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡) → (𝐵 − 𝐴) ≤ sup(ran seq1( + , ((abs ∘
− ) ∘ 𝑓)),
ℝ*, < ))) |
84 | 83 | exlimdv 1940 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → (∃𝑔(𝑔:𝑣⟶ℕ ∧ ∀𝑡 ∈ 𝑣 (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡) → (𝐵 − 𝐴) ≤ sup(ran seq1( + , ((abs ∘
− ) ∘ 𝑓)),
ℝ*, < ))) |
85 | 65, 84 | mpd 15 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → (𝐵 − 𝐴) ≤ sup(ran seq1( + , ((abs ∘
− ) ∘ 𝑓)),
ℝ*, < )) |
86 | 85 | rexlimdvaa 3216 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)) → (∃𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin)(𝐴[,]𝐵) ⊆ ∪ 𝑣 → (𝐵 − 𝐴) ≤ sup(ran seq1( + , ((abs ∘
− ) ∘ 𝑓)),
ℝ*, < ))) |
87 | 86 | adantrr 714 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran
((,) ∘ 𝑓))) →
(∃𝑣 ∈ (𝒫
ran ((,) ∘ 𝑓) ∩
Fin)(𝐴[,]𝐵) ⊆ ∪ 𝑣 → (𝐵 − 𝐴) ≤ sup(ran seq1( + , ((abs ∘
− ) ∘ 𝑓)),
ℝ*, < ))) |
88 | 49, 87 | mpd 15 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran
((,) ∘ 𝑓))) →
(𝐵 − 𝐴) ≤ sup(ran seq1( + , ((abs
∘ − ) ∘ 𝑓)), ℝ*, <
)) |
89 | | breq2 5083 |
. . . . . . . . 9
⊢ (𝑧 = sup(ran seq1( + , ((abs
∘ − ) ∘ 𝑓)), ℝ*, < ) →
((𝐵 − 𝐴) ≤ 𝑧 ↔ (𝐵 − 𝐴) ≤ sup(ran seq1( + , ((abs ∘
− ) ∘ 𝑓)),
ℝ*, < ))) |
90 | 88, 89 | syl5ibrcom 246 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran
((,) ∘ 𝑓))) →
(𝑧 = sup(ran seq1( + ,
((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) → (𝐵 − 𝐴) ≤ 𝑧)) |
91 | 90 | expr 457 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)) → ((𝐴[,]𝐵) ⊆ ∪ ran
((,) ∘ 𝑓) →
(𝑧 = sup(ran seq1( + ,
((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) → (𝐵 − 𝐴) ≤ 𝑧))) |
92 | 91 | impd 411 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)) → (((𝐴[,]𝐵) ⊆ ∪ ran
((,) ∘ 𝑓) ∧ 𝑧 = sup(ran seq1( + , ((abs
∘ − ) ∘ 𝑓)), ℝ*, < )) →
(𝐵 − 𝐴) ≤ 𝑧)) |
93 | 92 | rexlimdva 3215 |
. . . . 5
⊢ (𝜑 → (∃𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)((𝐴[,]𝐵) ⊆ ∪ ran
((,) ∘ 𝑓) ∧ 𝑧 = sup(ran seq1( + , ((abs
∘ − ) ∘ 𝑓)), ℝ*, < )) →
(𝐵 − 𝐴) ≤ 𝑧)) |
94 | 2, 93 | syl5bi 241 |
. . . 4
⊢ (𝜑 → (𝑧 ∈ 𝑀 → (𝐵 − 𝐴) ≤ 𝑧)) |
95 | 94 | ralrimiv 3109 |
. . 3
⊢ (𝜑 → ∀𝑧 ∈ 𝑀 (𝐵 − 𝐴) ≤ 𝑧) |
96 | 1 | ssrab3 4020 |
. . . 4
⊢ 𝑀 ⊆
ℝ* |
97 | 11, 10 | resubcld 11414 |
. . . . 5
⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ) |
98 | 97 | rexrd 11036 |
. . . 4
⊢ (𝜑 → (𝐵 − 𝐴) ∈
ℝ*) |
99 | | infxrgelb 13080 |
. . . 4
⊢ ((𝑀 ⊆ ℝ*
∧ (𝐵 − 𝐴) ∈ ℝ*)
→ ((𝐵 − 𝐴) ≤ inf(𝑀, ℝ*, < ) ↔
∀𝑧 ∈ 𝑀 (𝐵 − 𝐴) ≤ 𝑧)) |
100 | 96, 98, 99 | sylancr 587 |
. . 3
⊢ (𝜑 → ((𝐵 − 𝐴) ≤ inf(𝑀, ℝ*, < ) ↔
∀𝑧 ∈ 𝑀 (𝐵 − 𝐴) ≤ 𝑧)) |
101 | 95, 100 | mpbird 256 |
. 2
⊢ (𝜑 → (𝐵 − 𝐴) ≤ inf(𝑀, ℝ*, <
)) |
102 | 1 | ovolval 24648 |
. . 3
⊢ ((𝐴[,]𝐵) ⊆ ℝ → (vol*‘(𝐴[,]𝐵)) = inf(𝑀, ℝ*, <
)) |
103 | 18, 102 | syl 17 |
. 2
⊢ (𝜑 → (vol*‘(𝐴[,]𝐵)) = inf(𝑀, ℝ*, <
)) |
104 | 101, 103 | breqtrrd 5107 |
1
⊢ (𝜑 → (𝐵 − 𝐴) ≤ (vol*‘(𝐴[,]𝐵))) |