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 23050 |
. . . . 5
⊢ (𝑧 ∈ 𝑀 ↔ ∃𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑𝑚 ℕ)((𝐴[,]𝐵) ⊆ ∪ ran
((,) ∘ 𝑓) ∧ 𝑧 = sup(ran seq1( + , ((abs
∘ − ) ∘ 𝑓)), ℝ*, <
))) |
3 | | ioof 12142 |
. . . . . . . . . . . . . . . . . 18
⊢
(,):(ℝ* × ℝ*)⟶𝒫
ℝ |
4 | | ffn 5958 |
. . . . . . . . . . . . . . . . . 18
⊢
((,):(ℝ* × ℝ*)⟶𝒫
ℝ → (,) Fn (ℝ* ×
ℝ*)) |
5 | 3, 4 | ax-mp 5 |
. . . . . . . . . . . . . . . . 17
⊢ (,) Fn
(ℝ* × ℝ*) |
6 | | dffn3 5967 |
. . . . . . . . . . . . . . . . 17
⊢ ((,) Fn
(ℝ* × ℝ*) ↔
(,):(ℝ* × ℝ*)⟶ran
(,)) |
7 | 5, 6 | mpbi 219 |
. . . . . . . . . . . . . . . 16
⊢
(,):(ℝ* × ℝ*)⟶ran
(,) |
8 | | simpr 476 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑𝑚 ℕ)) → 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑𝑚 ℕ)) |
9 | | reex 9906 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ℝ
∈ V |
10 | 9, 9 | xpex 6860 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (ℝ
× ℝ) ∈ V |
11 | 10 | inex2 4728 |
. . . . . . . . . . . . . . . . . . 19
⊢ ( ≤
∩ (ℝ × ℝ)) ∈ V |
12 | | nnex 10903 |
. . . . . . . . . . . . . . . . . . 19
⊢ ℕ
∈ V |
13 | 11, 12 | elmap 7772 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑓 ∈ (( ≤ ∩ (ℝ
× ℝ)) ↑𝑚 ℕ) ↔ 𝑓:ℕ⟶( ≤ ∩ (ℝ ×
ℝ))) |
14 | 8, 13 | sylib 207 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑𝑚 ℕ)) → 𝑓:ℕ⟶( ≤ ∩ (ℝ ×
ℝ))) |
15 | | inss2 3796 |
. . . . . . . . . . . . . . . . . 18
⊢ ( ≤
∩ (ℝ × ℝ)) ⊆ (ℝ ×
ℝ) |
16 | | rexpssxrxp 9963 |
. . . . . . . . . . . . . . . . . 18
⊢ (ℝ
× ℝ) ⊆ (ℝ* ×
ℝ*) |
17 | 15, 16 | sstri 3577 |
. . . . . . . . . . . . . . . . 17
⊢ ( ≤
∩ (ℝ × ℝ)) ⊆ (ℝ* ×
ℝ*) |
18 | | fss 5969 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑓:ℕ⟶( ≤ ∩
(ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ))
⊆ (ℝ* × ℝ*)) → 𝑓:ℕ⟶(ℝ* ×
ℝ*)) |
19 | 14, 17, 18 | sylancl 693 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑𝑚 ℕ)) → 𝑓:ℕ⟶(ℝ* ×
ℝ*)) |
20 | | fco 5971 |
. . . . . . . . . . . . . . . 16
⊢
(((,):(ℝ* × ℝ*)⟶ran (,)
∧ 𝑓:ℕ⟶(ℝ* ×
ℝ*)) → ((,) ∘ 𝑓):ℕ⟶ran (,)) |
21 | 7, 19, 20 | sylancr 694 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑𝑚 ℕ)) → ((,) ∘ 𝑓):ℕ⟶ran
(,)) |
22 | 21 | adantrr 749 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑𝑚 ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran
((,) ∘ 𝑓))) →
((,) ∘ 𝑓):ℕ⟶ran (,)) |
23 | | frn 5966 |
. . . . . . . . . . . . . 14
⊢ (((,)
∘ 𝑓):ℕ⟶ran (,) → ran ((,)
∘ 𝑓) ⊆ ran
(,)) |
24 | 22, 23 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑𝑚 ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran
((,) ∘ 𝑓))) →
ran ((,) ∘ 𝑓) ⊆
ran (,)) |
25 | | retopbas 22374 |
. . . . . . . . . . . . . 14
⊢ ran (,)
∈ TopBases |
26 | | bastg 20581 |
. . . . . . . . . . . . . 14
⊢ (ran (,)
∈ TopBases → ran (,) ⊆ (topGen‘ran (,))) |
27 | 25, 26 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢ ran (,)
⊆ (topGen‘ran (,)) |
28 | 24, 27 | syl6ss 3580 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑𝑚 ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran
((,) ∘ 𝑓))) →
ran ((,) ∘ 𝑓) ⊆
(topGen‘ran (,))) |
29 | | fvex 6113 |
. . . . . . . . . . . . 13
⊢
(topGen‘ran (,)) ∈ V |
30 | 29 | elpw2 4755 |
. . . . . . . . . . . 12
⊢ (ran ((,)
∘ 𝑓) ∈ 𝒫
(topGen‘ran (,)) ↔ ran ((,) ∘ 𝑓) ⊆ (topGen‘ran
(,))) |
31 | 28, 30 | sylibr 223 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑𝑚 ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran
((,) ∘ 𝑓))) →
ran ((,) ∘ 𝑓) ∈
𝒫 (topGen‘ran (,))) |
32 | | ovolicc.1 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐴 ∈ ℝ) |
33 | | ovolicc.2 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐵 ∈ ℝ) |
34 | | eqid 2610 |
. . . . . . . . . . . . . . 15
⊢
(topGen‘ran (,)) = (topGen‘ran (,)) |
35 | | eqid 2610 |
. . . . . . . . . . . . . . 15
⊢
((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) = ((topGen‘ran (,))
↾t (𝐴[,]𝐵)) |
36 | 34, 35 | icccmp 22436 |
. . . . . . . . . . . . . 14
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) →
((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) ∈ Comp) |
37 | 32, 33, 36 | syl2anc 691 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((topGen‘ran (,))
↾t (𝐴[,]𝐵)) ∈ Comp) |
38 | | retop 22375 |
. . . . . . . . . . . . . 14
⊢
(topGen‘ran (,)) ∈ Top |
39 | | iccssre 12126 |
. . . . . . . . . . . . . . 15
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) |
40 | 32, 33, 39 | syl2anc 691 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ) |
41 | | uniretop 22376 |
. . . . . . . . . . . . . . 15
⊢ ℝ =
∪ (topGen‘ran (,)) |
42 | 41 | cmpsub 21013 |
. . . . . . . . . . . . . 14
⊢
(((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))) |
43 | 38, 40, 42 | sylancr 694 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (((topGen‘ran (,))
↾t (𝐴[,]𝐵)) ∈ Comp ↔ ∀𝑢 ∈ 𝒫
(topGen‘ran (,))((𝐴[,]𝐵) ⊆ ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(𝐴[,]𝐵) ⊆ ∪ 𝑣))) |
44 | 37, 43 | mpbid 221 |
. . . . . . . . . . . 12
⊢ (𝜑 → ∀𝑢 ∈ 𝒫 (topGen‘ran
(,))((𝐴[,]𝐵) ⊆ ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(𝐴[,]𝐵) ⊆ ∪ 𝑣)) |
45 | 44 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑𝑚 ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran
((,) ∘ 𝑓))) →
∀𝑢 ∈ 𝒫
(topGen‘ran (,))((𝐴[,]𝐵) ⊆ ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(𝐴[,]𝐵) ⊆ ∪ 𝑣)) |
46 | | simprr 792 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑𝑚 ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran
((,) ∘ 𝑓))) →
(𝐴[,]𝐵) ⊆ ∪ ran
((,) ∘ 𝑓)) |
47 | | unieq 4380 |
. . . . . . . . . . . . . 14
⊢ (𝑢 = ran ((,) ∘ 𝑓) → ∪ 𝑢 =
∪ ran ((,) ∘ 𝑓)) |
48 | 47 | sseq2d 3596 |
. . . . . . . . . . . . 13
⊢ (𝑢 = ran ((,) ∘ 𝑓) → ((𝐴[,]𝐵) ⊆ ∪ 𝑢 ↔ (𝐴[,]𝐵) ⊆ ∪ ran
((,) ∘ 𝑓))) |
49 | | pweq 4111 |
. . . . . . . . . . . . . . 15
⊢ (𝑢 = ran ((,) ∘ 𝑓) → 𝒫 𝑢 = 𝒫 ran ((,) ∘
𝑓)) |
50 | 49 | ineq1d 3775 |
. . . . . . . . . . . . . 14
⊢ (𝑢 = ran ((,) ∘ 𝑓) → (𝒫 𝑢 ∩ Fin) = (𝒫 ran
((,) ∘ 𝑓) ∩
Fin)) |
51 | 50 | rexeqdv 3122 |
. . . . . . . . . . . . 13
⊢ (𝑢 = ran ((,) ∘ 𝑓) → (∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(𝐴[,]𝐵) ⊆ ∪ 𝑣 ↔ ∃𝑣 ∈ (𝒫 ran ((,)
∘ 𝑓) ∩ Fin)(𝐴[,]𝐵) ⊆ ∪ 𝑣)) |
52 | 48, 51 | imbi12d 333 |
. . . . . . . . . . . 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))) |
53 | 52 | rspcv 3278 |
. . . . . . . . . . 11
⊢ (ran ((,)
∘ 𝑓) ∈ 𝒫
(topGen‘ran (,)) → (∀𝑢 ∈ 𝒫 (topGen‘ran
(,))((𝐴[,]𝐵) ⊆ ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(𝐴[,]𝐵) ⊆ ∪ 𝑣) → ((𝐴[,]𝐵) ⊆ ∪ ran
((,) ∘ 𝑓) →
∃𝑣 ∈ (𝒫
ran ((,) ∘ 𝑓) ∩
Fin)(𝐴[,]𝐵) ⊆ ∪ 𝑣))) |
54 | 31, 45, 46, 53 | syl3c 64 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑𝑚 ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran
((,) ∘ 𝑓))) →
∃𝑣 ∈ (𝒫
ran ((,) ∘ 𝑓) ∩
Fin)(𝐴[,]𝐵) ⊆ ∪ 𝑣) |
55 | | simprl 790 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑𝑚 ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → 𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin)) |
56 | | elin 3758 |
. . . . . . . . . . . . . . . 16
⊢ (𝑣 ∈ (𝒫 ran ((,)
∘ 𝑓) ∩ Fin)
↔ (𝑣 ∈ 𝒫
ran ((,) ∘ 𝑓) ∧
𝑣 ∈
Fin)) |
57 | 55, 56 | sylib 207 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑𝑚 ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → (𝑣 ∈ 𝒫 ran ((,) ∘ 𝑓) ∧ 𝑣 ∈ Fin)) |
58 | 57 | simprd 478 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑𝑚 ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → 𝑣 ∈ Fin) |
59 | 57 | simpld 474 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑𝑚 ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → 𝑣 ∈ 𝒫 ran ((,) ∘ 𝑓)) |
60 | 59 | elpwid 4118 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑𝑚 ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → 𝑣 ⊆ ran ((,) ∘ 𝑓)) |
61 | 60 | sseld 3567 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑𝑚 ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → (𝑡 ∈ 𝑣 → 𝑡 ∈ ran ((,) ∘ 𝑓))) |
62 | | ffn 5958 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((,)
∘ 𝑓):ℕ⟶ran (,) → ((,) ∘
𝑓) Fn
ℕ) |
63 | 21, 62 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑𝑚 ℕ)) → ((,) ∘ 𝑓) Fn ℕ) |
64 | 63 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑𝑚 ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → ((,) ∘ 𝑓) Fn ℕ) |
65 | | fvelrnb 6153 |
. . . . . . . . . . . . . . . . 17
⊢ (((,)
∘ 𝑓) Fn ℕ
→ (𝑡 ∈ ran ((,)
∘ 𝑓) ↔
∃𝑘 ∈ ℕ
(((,) ∘ 𝑓)‘𝑘) = 𝑡)) |
66 | 64, 65 | 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)) |
67 | 61, 66 | sylibd 228 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑𝑚 ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → (𝑡 ∈ 𝑣 → ∃𝑘 ∈ ℕ (((,) ∘ 𝑓)‘𝑘) = 𝑡)) |
68 | 67 | ralrimiv 2948 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑𝑚 ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → ∀𝑡 ∈ 𝑣 ∃𝑘 ∈ ℕ (((,) ∘ 𝑓)‘𝑘) = 𝑡) |
69 | | fveq2 6103 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = (𝑔‘𝑡) → (((,) ∘ 𝑓)‘𝑘) = (((,) ∘ 𝑓)‘(𝑔‘𝑡))) |
70 | 69 | eqeq1d 2612 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = (𝑔‘𝑡) → ((((,) ∘ 𝑓)‘𝑘) = 𝑡 ↔ (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡)) |
71 | 70 | ac6sfi 8089 |
. . . . . . . . . . . . . 14
⊢ ((𝑣 ∈ Fin ∧ ∀𝑡 ∈ 𝑣 ∃𝑘 ∈ ℕ (((,) ∘ 𝑓)‘𝑘) = 𝑡) → ∃𝑔(𝑔:𝑣⟶ℕ ∧ ∀𝑡 ∈ 𝑣 (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡)) |
72 | 58, 68, 71 | syl2anc 691 |
. . . . . . . . . . . . 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)) |
73 | 32 | ad2antrr 758 |
. . . . . . . . . . . . . . . 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) |
74 | 33 | ad2antrr 758 |
. . . . . . . . . . . . . . . 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) |
75 | | ovolicc.3 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐴 ≤ 𝐵) |
76 | 75 | ad2antrr 758 |
. . . . . . . . . . . . . . . 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) |
77 | | eqid 2610 |
. . . . . . . . . . . . . . . 16
⊢ seq1( + ,
((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − )
∘ 𝑓)) |
78 | 14 | adantr 480 |
. . . . . . . . . . . . . . . 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))) |
79 | | simprll 798 |
. . . . . . . . . . . . . . . 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)) |
80 | | simprlr 799 |
. . . . . . . . . . . . . . . 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) |
81 | | simprrl 800 |
. . . . . . . . . . . . . . . 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) |
82 | | simprrr 801 |
. . . . . . . . . . . . . . . . 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) |
83 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑥 → (𝑔‘𝑡) = (𝑔‘𝑥)) |
84 | 83 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑥 → (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = (((,) ∘ 𝑓)‘(𝑔‘𝑥))) |
85 | | id 22 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑥 → 𝑡 = 𝑥) |
86 | 84, 85 | eqeq12d 2625 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑥 → ((((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡 ↔ (((,) ∘ 𝑓)‘(𝑔‘𝑥)) = 𝑥)) |
87 | 86 | rspccva 3281 |
. . . . . . . . . . . . . . . . 17
⊢
((∀𝑡 ∈
𝑣 (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡 ∧ 𝑥 ∈ 𝑣) → (((,) ∘ 𝑓)‘(𝑔‘𝑥)) = 𝑥) |
88 | 82, 87 | sylan 487 |
. . . . . . . . . . . . . . . 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) |
89 | | eqid 2610 |
. . . . . . . . . . . . . . . 16
⊢ {𝑢 ∈ 𝑣 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅} = {𝑢 ∈ 𝑣 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅} |
90 | 73, 74, 76, 77, 78, 79, 80, 81, 88, 89 | ovolicc2lem5 23096 |
. . . . . . . . . . . . . . 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*, < )) |
91 | 90 | expr 641 |
. . . . . . . . . . . . . 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*, < ))) |
92 | 91 | exlimdv 1848 |
. . . . . . . . . . . . 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*, < ))) |
93 | 72, 92 | 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*, < )) |
94 | 93 | rexlimdvaa 3014 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑𝑚 ℕ)) → (∃𝑣 ∈ (𝒫 ran ((,)
∘ 𝑓) ∩ Fin)(𝐴[,]𝐵) ⊆ ∪ 𝑣 → (𝐵 − 𝐴) ≤ sup(ran seq1( + , ((abs ∘
− ) ∘ 𝑓)),
ℝ*, < ))) |
95 | 94 | adantrr 749 |
. . . . . . . . . 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*, < ))) |
96 | 54, 95 | mpd 15 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑𝑚 ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran
((,) ∘ 𝑓))) →
(𝐵 − 𝐴) ≤ sup(ran seq1( + , ((abs
∘ − ) ∘ 𝑓)), ℝ*, <
)) |
97 | | breq2 4587 |
. . . . . . . . 9
⊢ (𝑧 = sup(ran seq1( + , ((abs
∘ − ) ∘ 𝑓)), ℝ*, < ) →
((𝐵 − 𝐴) ≤ 𝑧 ↔ (𝐵 − 𝐴) ≤ sup(ran seq1( + , ((abs ∘
− ) ∘ 𝑓)),
ℝ*, < ))) |
98 | 96, 97 | syl5ibrcom 236 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑𝑚 ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran
((,) ∘ 𝑓))) →
(𝑧 = sup(ran seq1( + ,
((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) → (𝐵 − 𝐴) ≤ 𝑧)) |
99 | 98 | expr 641 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑𝑚 ℕ)) → ((𝐴[,]𝐵) ⊆ ∪ ran
((,) ∘ 𝑓) →
(𝑧 = sup(ran seq1( + ,
((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) → (𝐵 − 𝐴) ≤ 𝑧))) |
100 | 99 | impd 446 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑𝑚 ℕ)) → (((𝐴[,]𝐵) ⊆ ∪ ran
((,) ∘ 𝑓) ∧ 𝑧 = sup(ran seq1( + , ((abs
∘ − ) ∘ 𝑓)), ℝ*, < )) →
(𝐵 − 𝐴) ≤ 𝑧)) |
101 | 100 | rexlimdva 3013 |
. . . . 5
⊢ (𝜑 → (∃𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑𝑚 ℕ)((𝐴[,]𝐵) ⊆ ∪ ran
((,) ∘ 𝑓) ∧ 𝑧 = sup(ran seq1( + , ((abs
∘ − ) ∘ 𝑓)), ℝ*, < )) →
(𝐵 − 𝐴) ≤ 𝑧)) |
102 | 2, 101 | syl5bi 231 |
. . . 4
⊢ (𝜑 → (𝑧 ∈ 𝑀 → (𝐵 − 𝐴) ≤ 𝑧)) |
103 | 102 | ralrimiv 2948 |
. . 3
⊢ (𝜑 → ∀𝑧 ∈ 𝑀 (𝐵 − 𝐴) ≤ 𝑧) |
104 | | ssrab2 3650 |
. . . . 5
⊢ {𝑦 ∈ ℝ*
∣ ∃𝑓 ∈ ((
≤ ∩ (ℝ × ℝ)) ↑𝑚
ℕ)((𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − )
∘ 𝑓)),
ℝ*, < ))} ⊆ ℝ* |
105 | 1, 104 | eqsstri 3598 |
. . . 4
⊢ 𝑀 ⊆
ℝ* |
106 | 33, 32 | resubcld 10337 |
. . . . 5
⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ) |
107 | 106 | rexrd 9968 |
. . . 4
⊢ (𝜑 → (𝐵 − 𝐴) ∈
ℝ*) |
108 | | infxrgelb 12037 |
. . . 4
⊢ ((𝑀 ⊆ ℝ*
∧ (𝐵 − 𝐴) ∈ ℝ*)
→ ((𝐵 − 𝐴) ≤ inf(𝑀, ℝ*, < ) ↔
∀𝑧 ∈ 𝑀 (𝐵 − 𝐴) ≤ 𝑧)) |
109 | 105, 107,
108 | sylancr 694 |
. . 3
⊢ (𝜑 → ((𝐵 − 𝐴) ≤ inf(𝑀, ℝ*, < ) ↔
∀𝑧 ∈ 𝑀 (𝐵 − 𝐴) ≤ 𝑧)) |
110 | 103, 109 | mpbird 246 |
. 2
⊢ (𝜑 → (𝐵 − 𝐴) ≤ inf(𝑀, ℝ*, <
)) |
111 | 1 | ovolval 23049 |
. . 3
⊢ ((𝐴[,]𝐵) ⊆ ℝ → (vol*‘(𝐴[,]𝐵)) = inf(𝑀, ℝ*, <
)) |
112 | 40, 111 | syl 17 |
. 2
⊢ (𝜑 → (vol*‘(𝐴[,]𝐵)) = inf(𝑀, ℝ*, <
)) |
113 | 110, 112 | breqtrrd 4611 |
1
⊢ (𝜑 → (𝐵 − 𝐴) ≤ (vol*‘(𝐴[,]𝐵))) |