| 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 25510 |
. . . . 5
⊢ (𝑧 ∈ 𝑀 ↔ ∃𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)((𝐴[,]𝐵) ⊆ ∪ ran
((,) ∘ 𝑓) ∧ 𝑧 = sup(ran seq1( + , ((abs
∘ − ) ∘ 𝑓)), ℝ*, <
))) |
| 3 | | simprr 773 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran
((,) ∘ 𝑓))) →
(𝐴[,]𝐵) ⊆ ∪ ran
((,) ∘ 𝑓)) |
| 4 | | unieq 4918 |
. . . . . . . . . . . . . 14
⊢ (𝑢 = ran ((,) ∘ 𝑓) → ∪ 𝑢 =
∪ ran ((,) ∘ 𝑓)) |
| 5 | 4 | sseq2d 4016 |
. . . . . . . . . . . . 13
⊢ (𝑢 = ran ((,) ∘ 𝑓) → ((𝐴[,]𝐵) ⊆ ∪ 𝑢 ↔ (𝐴[,]𝐵) ⊆ ∪ ran
((,) ∘ 𝑓))) |
| 6 | | pweq 4614 |
. . . . . . . . . . . . . . 15
⊢ (𝑢 = ran ((,) ∘ 𝑓) → 𝒫 𝑢 = 𝒫 ran ((,) ∘
𝑓)) |
| 7 | 6 | ineq1d 4219 |
. . . . . . . . . . . . . 14
⊢ (𝑢 = ran ((,) ∘ 𝑓) → (𝒫 𝑢 ∩ Fin) = (𝒫 ran
((,) ∘ 𝑓) ∩
Fin)) |
| 8 | 7 | rexeqdv 3327 |
. . . . . . . . . . . . 13
⊢ (𝑢 = ran ((,) ∘ 𝑓) → (∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(𝐴[,]𝐵) ⊆ ∪ 𝑣 ↔ ∃𝑣 ∈ (𝒫 ran ((,)
∘ 𝑓) ∩ Fin)(𝐴[,]𝐵) ⊆ ∪ 𝑣)) |
| 9 | 5, 8 | imbi12d 344 |
. . . . . . . . . . . 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 2737 |
. . . . . . . . . . . . . . . 16
⊢
(topGen‘ran (,)) = (topGen‘ran (,)) |
| 13 | | eqid 2737 |
. . . . . . . . . . . . . . . 16
⊢
((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) = ((topGen‘ran (,))
↾t (𝐴[,]𝐵)) |
| 14 | 12, 13 | icccmp 24847 |
. . . . . . . . . . . . . . 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 24782 |
. . . . . . . . . . . . . . 15
⊢
(topGen‘ran (,)) ∈ Top |
| 17 | | iccssre 13469 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) |
| 18 | 10, 11, 17 | syl2anc 584 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ) |
| 19 | | uniretop 24783 |
. . . . . . . . . . . . . . . 16
⊢ ℝ =
∪ (topGen‘ran (,)) |
| 20 | 19 | cmpsub 23408 |
. . . . . . . . . . . . . . 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 232 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ∀𝑢 ∈ 𝒫 (topGen‘ran
(,))((𝐴[,]𝐵) ⊆ ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(𝐴[,]𝐵) ⊆ ∪ 𝑣)) |
| 23 | 22 | adantr 480 |
. . . . . . . . . . . 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 13487 |
. . . . . . . . . . . . . . . . . . 19
⊢
(,):(ℝ* × ℝ*)⟶𝒫
ℝ |
| 25 | | ffn 6736 |
. . . . . . . . . . . . . . . . . . 19
⊢
((,):(ℝ* × ℝ*)⟶𝒫
ℝ → (,) Fn (ℝ* ×
ℝ*)) |
| 26 | 24, 25 | ax-mp 5 |
. . . . . . . . . . . . . . . . . 18
⊢ (,) Fn
(ℝ* × ℝ*) |
| 27 | | dffn3 6748 |
. . . . . . . . . . . . . . . . . 18
⊢ ((,) Fn
(ℝ* × ℝ*) ↔
(,):(ℝ* × ℝ*)⟶ran
(,)) |
| 28 | 26, 27 | mpbi 230 |
. . . . . . . . . . . . . . . . 17
⊢
(,):(ℝ* × ℝ*)⟶ran
(,) |
| 29 | | simpr 484 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)) → 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)) |
| 30 | | elovolmlem 25509 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑓 ∈ (( ≤ ∩ (ℝ
× ℝ)) ↑m ℕ) ↔ 𝑓:ℕ⟶( ≤ ∩ (ℝ ×
ℝ))) |
| 31 | 29, 30 | sylib 218 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)) → 𝑓:ℕ⟶( ≤ ∩ (ℝ ×
ℝ))) |
| 32 | | inss2 4238 |
. . . . . . . . . . . . . . . . . . 19
⊢ ( ≤
∩ (ℝ × ℝ)) ⊆ (ℝ ×
ℝ) |
| 33 | | rexpssxrxp 11306 |
. . . . . . . . . . . . . . . . . . 19
⊢ (ℝ
× ℝ) ⊆ (ℝ* ×
ℝ*) |
| 34 | 32, 33 | sstri 3993 |
. . . . . . . . . . . . . . . . . 18
⊢ ( ≤
∩ (ℝ × ℝ)) ⊆ (ℝ* ×
ℝ*) |
| 35 | | fss 6752 |
. . . . . . . . . . . . . . . . . 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 6760 |
. . . . . . . . . . . . . . . . 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 717 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran
((,) ∘ 𝑓))) →
((,) ∘ 𝑓):ℕ⟶ran (,)) |
| 40 | 39 | frnd 6744 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran
((,) ∘ 𝑓))) →
ran ((,) ∘ 𝑓) ⊆
ran (,)) |
| 41 | | retopbas 24781 |
. . . . . . . . . . . . . . 15
⊢ ran (,)
∈ TopBases |
| 42 | | bastg 22973 |
. . . . . . . . . . . . . . 15
⊢ (ran (,)
∈ TopBases → ran (,) ⊆ (topGen‘ran (,))) |
| 43 | 41, 42 | ax-mp 5 |
. . . . . . . . . . . . . 14
⊢ ran (,)
⊆ (topGen‘ran (,)) |
| 44 | 40, 43 | sstrdi 3996 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran
((,) ∘ 𝑓))) →
ran ((,) ∘ 𝑓) ⊆
(topGen‘ran (,))) |
| 45 | | fvex 6919 |
. . . . . . . . . . . . . 14
⊢
(topGen‘ran (,)) ∈ V |
| 46 | 45 | elpw2 5334 |
. . . . . . . . . . . . 13
⊢ (ran ((,)
∘ 𝑓) ∈ 𝒫
(topGen‘ran (,)) ↔ ran ((,) ∘ 𝑓) ⊆ (topGen‘ran
(,))) |
| 47 | 44, 46 | sylibr 234 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran
((,) ∘ 𝑓))) →
ran ((,) ∘ 𝑓) ∈
𝒫 (topGen‘ran (,))) |
| 48 | 9, 23, 47 | rspcdva 3623 |
. . . . . . . . . . 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 771 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → 𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin)) |
| 51 | | elin 3967 |
. . . . . . . . . . . . . . . 16
⊢ (𝑣 ∈ (𝒫 ran ((,)
∘ 𝑓) ∩ Fin)
↔ (𝑣 ∈ 𝒫
ran ((,) ∘ 𝑓) ∧
𝑣 ∈
Fin)) |
| 52 | 50, 51 | sylib 218 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → (𝑣 ∈ 𝒫 ran ((,) ∘ 𝑓) ∧ 𝑣 ∈ Fin)) |
| 53 | 52 | simprd 495 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → 𝑣 ∈ Fin) |
| 54 | 52 | simpld 494 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → 𝑣 ∈ 𝒫 ran ((,) ∘ 𝑓)) |
| 55 | 54 | elpwid 4609 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → 𝑣 ⊆ ran ((,) ∘ 𝑓)) |
| 56 | 55 | sseld 3982 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → (𝑡 ∈ 𝑣 → 𝑡 ∈ ran ((,) ∘ 𝑓))) |
| 57 | 38 | ffnd 6737 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)) → ((,) ∘ 𝑓) Fn ℕ) |
| 58 | 57 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → ((,) ∘ 𝑓) Fn ℕ) |
| 59 | | fvelrnb 6969 |
. . . . . . . . . . . . . . . . 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 239 |
. . . . . . . . . . . . . . 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 3145 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → ∀𝑡 ∈ 𝑣 ∃𝑘 ∈ ℕ (((,) ∘ 𝑓)‘𝑘) = 𝑡) |
| 63 | | fveqeq2 6915 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = (𝑔‘𝑡) → ((((,) ∘ 𝑓)‘𝑘) = 𝑡 ↔ (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡)) |
| 64 | 63 | ac6sfi 9320 |
. . . . . . . . . . . . . 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 726 |
. . . . . . . . . . . . . . . 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 726 |
. . . . . . . . . . . . . . . 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 726 |
. . . . . . . . . . . . . . . 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 2737 |
. . . . . . . . . . . . . . . 16
⊢ seq1( + ,
((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − )
∘ 𝑓)) |
| 71 | 31 | 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))) |
| 72 | | simprll 779 |
. . . . . . . . . . . . . . . 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 780 |
. . . . . . . . . . . . . . . 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 781 |
. . . . . . . . . . . . . . . 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 782 |
. . . . . . . . . . . . . . . . 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 6911 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑥 → (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = (((,) ∘ 𝑓)‘(𝑔‘𝑥))) |
| 77 | | id 22 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑥 → 𝑡 = 𝑥) |
| 78 | 76, 77 | eqeq12d 2753 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑥 → ((((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡 ↔ (((,) ∘ 𝑓)‘(𝑔‘𝑥)) = 𝑥)) |
| 79 | 78 | rspccva 3621 |
. . . . . . . . . . . . . . . . 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 2737 |
. . . . . . . . . . . . . . . 16
⊢ {𝑢 ∈ 𝑣 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅} = {𝑢 ∈ 𝑣 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅} |
| 82 | 66, 67, 69, 70, 71, 72, 73, 74, 80, 81 | ovolicc2lem5 25556 |
. . . . . . . . . . . . . . 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 456 |
. . . . . . . . . . . . . 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 1933 |
. . . . . . . . . . . . 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 3156 |
. . . . . . . . . . 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 717 |
. . . . . . . . . 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 5147 |
. . . . . . . . 9
⊢ (𝑧 = sup(ran seq1( + , ((abs
∘ − ) ∘ 𝑓)), ℝ*, < ) →
((𝐵 − 𝐴) ≤ 𝑧 ↔ (𝐵 − 𝐴) ≤ sup(ran seq1( + , ((abs ∘
− ) ∘ 𝑓)),
ℝ*, < ))) |
| 90 | 88, 89 | syl5ibrcom 247 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran
((,) ∘ 𝑓))) →
(𝑧 = sup(ran seq1( + ,
((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) → (𝐵 − 𝐴) ≤ 𝑧)) |
| 91 | 90 | expr 456 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)) → ((𝐴[,]𝐵) ⊆ ∪ ran
((,) ∘ 𝑓) →
(𝑧 = sup(ran seq1( + ,
((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) → (𝐵 − 𝐴) ≤ 𝑧))) |
| 92 | 91 | impd 410 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)) → (((𝐴[,]𝐵) ⊆ ∪ ran
((,) ∘ 𝑓) ∧ 𝑧 = sup(ran seq1( + , ((abs
∘ − ) ∘ 𝑓)), ℝ*, < )) →
(𝐵 − 𝐴) ≤ 𝑧)) |
| 93 | 92 | rexlimdva 3155 |
. . . . 5
⊢ (𝜑 → (∃𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)((𝐴[,]𝐵) ⊆ ∪ ran
((,) ∘ 𝑓) ∧ 𝑧 = sup(ran seq1( + , ((abs
∘ − ) ∘ 𝑓)), ℝ*, < )) →
(𝐵 − 𝐴) ≤ 𝑧)) |
| 94 | 2, 93 | biimtrid 242 |
. . . 4
⊢ (𝜑 → (𝑧 ∈ 𝑀 → (𝐵 − 𝐴) ≤ 𝑧)) |
| 95 | 94 | ralrimiv 3145 |
. . 3
⊢ (𝜑 → ∀𝑧 ∈ 𝑀 (𝐵 − 𝐴) ≤ 𝑧) |
| 96 | 1 | ssrab3 4082 |
. . . 4
⊢ 𝑀 ⊆
ℝ* |
| 97 | 11, 10 | resubcld 11691 |
. . . . 5
⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ) |
| 98 | 97 | rexrd 11311 |
. . . 4
⊢ (𝜑 → (𝐵 − 𝐴) ∈
ℝ*) |
| 99 | | infxrgelb 13377 |
. . . 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 257 |
. 2
⊢ (𝜑 → (𝐵 − 𝐴) ≤ inf(𝑀, ℝ*, <
)) |
| 102 | 1 | ovolval 25508 |
. . 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 5171 |
1
⊢ (𝜑 → (𝐵 − 𝐴) ≤ (vol*‘(𝐴[,]𝐵))) |