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Theorem ovolgelb 24010
Description: The outer volume is the greatest lower bound on the sum of all interval coverings of 𝐴. (Contributed by Mario Carneiro, 15-Jun-2014.)
Hypothesis
Ref Expression
ovolgelb.1 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝑔))
Assertion
Ref Expression
ovolgelb ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + 𝐵)))
Distinct variable groups:   𝐴,𝑔   𝐵,𝑔
Allowed substitution hint:   𝑆(𝑔)

Proof of Theorem ovolgelb
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp2 1129 . . . . . 6 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (vol*‘𝐴) ∈ ℝ)
2 simp3 1130 . . . . . 6 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → 𝐵 ∈ ℝ+)
31, 2ltaddrpd 12454 . . . . 5 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (vol*‘𝐴) < ((vol*‘𝐴) + 𝐵))
42rpred 12421 . . . . . . 7 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → 𝐵 ∈ ℝ)
51, 4readdcld 10659 . . . . . 6 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((vol*‘𝐴) + 𝐵) ∈ ℝ)
61, 5ltnled 10776 . . . . 5 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((vol*‘𝐴) < ((vol*‘𝐴) + 𝐵) ↔ ¬ ((vol*‘𝐴) + 𝐵) ≤ (vol*‘𝐴)))
73, 6mpbid 233 . . . 4 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ¬ ((vol*‘𝐴) + 𝐵) ≤ (vol*‘𝐴))
8 eqid 2821 . . . . . . . 8 {𝑦 ∈ ℝ* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ))} = {𝑦 ∈ ℝ* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ))}
98ovolval 24003 . . . . . . 7 (𝐴 ⊆ ℝ → (vol*‘𝐴) = inf({𝑦 ∈ ℝ* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ))}, ℝ*, < ))
1093ad2ant1 1125 . . . . . 6 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (vol*‘𝐴) = inf({𝑦 ∈ ℝ* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ))}, ℝ*, < ))
1110breq2d 5070 . . . . 5 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (((vol*‘𝐴) + 𝐵) ≤ (vol*‘𝐴) ↔ ((vol*‘𝐴) + 𝐵) ≤ inf({𝑦 ∈ ℝ* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ))}, ℝ*, < )))
12 ssrab2 4055 . . . . . . 7 {𝑦 ∈ ℝ* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ))} ⊆ ℝ*
135rexrd 10680 . . . . . . 7 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((vol*‘𝐴) + 𝐵) ∈ ℝ*)
14 infxrgelb 12718 . . . . . . 7 (({𝑦 ∈ ℝ* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ))} ⊆ ℝ* ∧ ((vol*‘𝐴) + 𝐵) ∈ ℝ*) → (((vol*‘𝐴) + 𝐵) ≤ inf({𝑦 ∈ ℝ* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ))}, ℝ*, < ) ↔ ∀𝑥 ∈ {𝑦 ∈ ℝ* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ))} ((vol*‘𝐴) + 𝐵) ≤ 𝑥))
1512, 13, 14sylancr 587 . . . . . 6 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (((vol*‘𝐴) + 𝐵) ≤ inf({𝑦 ∈ ℝ* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ))}, ℝ*, < ) ↔ ∀𝑥 ∈ {𝑦 ∈ ℝ* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ))} ((vol*‘𝐴) + 𝐵) ≤ 𝑥))
16 eqeq1 2825 . . . . . . . . . . 11 (𝑦 = 𝑥 → (𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ↔ 𝑥 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < )))
17 ovolgelb.1 . . . . . . . . . . . . . 14 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝑔))
1817rneqi 5801 . . . . . . . . . . . . 13 ran 𝑆 = ran seq1( + , ((abs ∘ − ) ∘ 𝑔))
1918supeq1i 8900 . . . . . . . . . . . 12 sup(ran 𝑆, ℝ*, < ) = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < )
2019eqeq2i 2834 . . . . . . . . . . 11 (𝑥 = sup(ran 𝑆, ℝ*, < ) ↔ 𝑥 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ))
2116, 20syl6bbr 290 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ↔ 𝑥 = sup(ran 𝑆, ℝ*, < )))
2221anbi2d 628 . . . . . . . . 9 (𝑦 = 𝑥 → ((𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < )) ↔ (𝐴 ran ((,) ∘ 𝑔) ∧ 𝑥 = sup(ran 𝑆, ℝ*, < ))))
2322rexbidv 3297 . . . . . . . 8 (𝑦 = 𝑥 → (∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < )) ↔ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ 𝑥 = sup(ran 𝑆, ℝ*, < ))))
2423ralrab 3684 . . . . . . 7 (∀𝑥 ∈ {𝑦 ∈ ℝ* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ))} ((vol*‘𝐴) + 𝐵) ≤ 𝑥 ↔ ∀𝑥 ∈ ℝ* (∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ 𝑥 = sup(ran 𝑆, ℝ*, < )) → ((vol*‘𝐴) + 𝐵) ≤ 𝑥))
25 ralcom 3354 . . . . . . . 8 (∀𝑥 ∈ ℝ*𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)((𝐴 ran ((,) ∘ 𝑔) ∧ 𝑥 = sup(ran 𝑆, ℝ*, < )) → ((vol*‘𝐴) + 𝐵) ≤ 𝑥) ↔ ∀𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)∀𝑥 ∈ ℝ* ((𝐴 ran ((,) ∘ 𝑔) ∧ 𝑥 = sup(ran 𝑆, ℝ*, < )) → ((vol*‘𝐴) + 𝐵) ≤ 𝑥))
26 r19.23v 3279 . . . . . . . . 9 (∀𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)((𝐴 ran ((,) ∘ 𝑔) ∧ 𝑥 = sup(ran 𝑆, ℝ*, < )) → ((vol*‘𝐴) + 𝐵) ≤ 𝑥) ↔ (∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ 𝑥 = sup(ran 𝑆, ℝ*, < )) → ((vol*‘𝐴) + 𝐵) ≤ 𝑥))
2726ralbii 3165 . . . . . . . 8 (∀𝑥 ∈ ℝ*𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)((𝐴 ran ((,) ∘ 𝑔) ∧ 𝑥 = sup(ran 𝑆, ℝ*, < )) → ((vol*‘𝐴) + 𝐵) ≤ 𝑥) ↔ ∀𝑥 ∈ ℝ* (∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ 𝑥 = sup(ran 𝑆, ℝ*, < )) → ((vol*‘𝐴) + 𝐵) ≤ 𝑥))
28 ancomst 465 . . . . . . . . . . . 12 (((𝐴 ran ((,) ∘ 𝑔) ∧ 𝑥 = sup(ran 𝑆, ℝ*, < )) → ((vol*‘𝐴) + 𝐵) ≤ 𝑥) ↔ ((𝑥 = sup(ran 𝑆, ℝ*, < ) ∧ 𝐴 ran ((,) ∘ 𝑔)) → ((vol*‘𝐴) + 𝐵) ≤ 𝑥))
29 impexp 451 . . . . . . . . . . . 12 (((𝑥 = sup(ran 𝑆, ℝ*, < ) ∧ 𝐴 ran ((,) ∘ 𝑔)) → ((vol*‘𝐴) + 𝐵) ≤ 𝑥) ↔ (𝑥 = sup(ran 𝑆, ℝ*, < ) → (𝐴 ran ((,) ∘ 𝑔) → ((vol*‘𝐴) + 𝐵) ≤ 𝑥)))
3028, 29bitri 276 . . . . . . . . . . 11 (((𝐴 ran ((,) ∘ 𝑔) ∧ 𝑥 = sup(ran 𝑆, ℝ*, < )) → ((vol*‘𝐴) + 𝐵) ≤ 𝑥) ↔ (𝑥 = sup(ran 𝑆, ℝ*, < ) → (𝐴 ran ((,) ∘ 𝑔) → ((vol*‘𝐴) + 𝐵) ≤ 𝑥)))
3130ralbii 3165 . . . . . . . . . 10 (∀𝑥 ∈ ℝ* ((𝐴 ran ((,) ∘ 𝑔) ∧ 𝑥 = sup(ran 𝑆, ℝ*, < )) → ((vol*‘𝐴) + 𝐵) ≤ 𝑥) ↔ ∀𝑥 ∈ ℝ* (𝑥 = sup(ran 𝑆, ℝ*, < ) → (𝐴 ran ((,) ∘ 𝑔) → ((vol*‘𝐴) + 𝐵) ≤ 𝑥)))
32 elovolmlem 24004 . . . . . . . . . . . . . . 15 (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ↔ 𝑔:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
33 eqid 2821 . . . . . . . . . . . . . . . 16 ((abs ∘ − ) ∘ 𝑔) = ((abs ∘ − ) ∘ 𝑔)
3433, 17ovolsf 24002 . . . . . . . . . . . . . . 15 (𝑔:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
3532, 34sylbi 218 . . . . . . . . . . . . . 14 (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) → 𝑆:ℕ⟶(0[,)+∞))
3635frnd 6515 . . . . . . . . . . . . 13 (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) → ran 𝑆 ⊆ (0[,)+∞))
37 icossxr 12811 . . . . . . . . . . . . 13 (0[,)+∞) ⊆ ℝ*
3836, 37sstrdi 3978 . . . . . . . . . . . 12 (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) → ran 𝑆 ⊆ ℝ*)
39 supxrcl 12698 . . . . . . . . . . . 12 (ran 𝑆 ⊆ ℝ* → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
4038, 39syl 17 . . . . . . . . . . 11 (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
41 breq2 5062 . . . . . . . . . . . . 13 (𝑥 = sup(ran 𝑆, ℝ*, < ) → (((vol*‘𝐴) + 𝐵) ≤ 𝑥 ↔ ((vol*‘𝐴) + 𝐵) ≤ sup(ran 𝑆, ℝ*, < )))
4241imbi2d 342 . . . . . . . . . . . 12 (𝑥 = sup(ran 𝑆, ℝ*, < ) → ((𝐴 ran ((,) ∘ 𝑔) → ((vol*‘𝐴) + 𝐵) ≤ 𝑥) ↔ (𝐴 ran ((,) ∘ 𝑔) → ((vol*‘𝐴) + 𝐵) ≤ sup(ran 𝑆, ℝ*, < ))))
4342ceqsralv 3534 . . . . . . . . . . 11 (sup(ran 𝑆, ℝ*, < ) ∈ ℝ* → (∀𝑥 ∈ ℝ* (𝑥 = sup(ran 𝑆, ℝ*, < ) → (𝐴 ran ((,) ∘ 𝑔) → ((vol*‘𝐴) + 𝐵) ≤ 𝑥)) ↔ (𝐴 ran ((,) ∘ 𝑔) → ((vol*‘𝐴) + 𝐵) ≤ sup(ran 𝑆, ℝ*, < ))))
4440, 43syl 17 . . . . . . . . . 10 (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) → (∀𝑥 ∈ ℝ* (𝑥 = sup(ran 𝑆, ℝ*, < ) → (𝐴 ran ((,) ∘ 𝑔) → ((vol*‘𝐴) + 𝐵) ≤ 𝑥)) ↔ (𝐴 ran ((,) ∘ 𝑔) → ((vol*‘𝐴) + 𝐵) ≤ sup(ran 𝑆, ℝ*, < ))))
4531, 44syl5bb 284 . . . . . . . . 9 (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) → (∀𝑥 ∈ ℝ* ((𝐴 ran ((,) ∘ 𝑔) ∧ 𝑥 = sup(ran 𝑆, ℝ*, < )) → ((vol*‘𝐴) + 𝐵) ≤ 𝑥) ↔ (𝐴 ran ((,) ∘ 𝑔) → ((vol*‘𝐴) + 𝐵) ≤ sup(ran 𝑆, ℝ*, < ))))
4645ralbiia 3164 . . . . . . . 8 (∀𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)∀𝑥 ∈ ℝ* ((𝐴 ran ((,) ∘ 𝑔) ∧ 𝑥 = sup(ran 𝑆, ℝ*, < )) → ((vol*‘𝐴) + 𝐵) ≤ 𝑥) ↔ ∀𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑔) → ((vol*‘𝐴) + 𝐵) ≤ sup(ran 𝑆, ℝ*, < )))
4725, 27, 463bitr3i 302 . . . . . . 7 (∀𝑥 ∈ ℝ* (∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ 𝑥 = sup(ran 𝑆, ℝ*, < )) → ((vol*‘𝐴) + 𝐵) ≤ 𝑥) ↔ ∀𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑔) → ((vol*‘𝐴) + 𝐵) ≤ sup(ran 𝑆, ℝ*, < )))
4824, 47bitri 276 . . . . . 6 (∀𝑥 ∈ {𝑦 ∈ ℝ* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ))} ((vol*‘𝐴) + 𝐵) ≤ 𝑥 ↔ ∀𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑔) → ((vol*‘𝐴) + 𝐵) ≤ sup(ran 𝑆, ℝ*, < )))
4915, 48syl6rbb 289 . . . . 5 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (∀𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑔) → ((vol*‘𝐴) + 𝐵) ≤ sup(ran 𝑆, ℝ*, < )) ↔ ((vol*‘𝐴) + 𝐵) ≤ inf({𝑦 ∈ ℝ* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ))}, ℝ*, < )))
5011, 49bitr4d 283 . . . 4 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (((vol*‘𝐴) + 𝐵) ≤ (vol*‘𝐴) ↔ ∀𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑔) → ((vol*‘𝐴) + 𝐵) ≤ sup(ran 𝑆, ℝ*, < ))))
517, 50mtbid 325 . . 3 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ¬ ∀𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑔) → ((vol*‘𝐴) + 𝐵) ≤ sup(ran 𝑆, ℝ*, < )))
52 rexanali 3265 . . 3 (∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ ¬ ((vol*‘𝐴) + 𝐵) ≤ sup(ran 𝑆, ℝ*, < )) ↔ ¬ ∀𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑔) → ((vol*‘𝐴) + 𝐵) ≤ sup(ran 𝑆, ℝ*, < )))
5351, 52sylibr 235 . 2 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ ¬ ((vol*‘𝐴) + 𝐵) ≤ sup(ran 𝑆, ℝ*, < )))
54 xrltnle 10697 . . . . . 6 ((sup(ran 𝑆, ℝ*, < ) ∈ ℝ* ∧ ((vol*‘𝐴) + 𝐵) ∈ ℝ*) → (sup(ran 𝑆, ℝ*, < ) < ((vol*‘𝐴) + 𝐵) ↔ ¬ ((vol*‘𝐴) + 𝐵) ≤ sup(ran 𝑆, ℝ*, < )))
55 xrltle 12532 . . . . . 6 ((sup(ran 𝑆, ℝ*, < ) ∈ ℝ* ∧ ((vol*‘𝐴) + 𝐵) ∈ ℝ*) → (sup(ran 𝑆, ℝ*, < ) < ((vol*‘𝐴) + 𝐵) → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + 𝐵)))
5654, 55sylbird 261 . . . . 5 ((sup(ran 𝑆, ℝ*, < ) ∈ ℝ* ∧ ((vol*‘𝐴) + 𝐵) ∈ ℝ*) → (¬ ((vol*‘𝐴) + 𝐵) ≤ sup(ran 𝑆, ℝ*, < ) → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + 𝐵)))
5740, 13, 56syl2anr 596 . . . 4 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) ∧ 𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)) → (¬ ((vol*‘𝐴) + 𝐵) ≤ sup(ran 𝑆, ℝ*, < ) → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + 𝐵)))
5857anim2d 611 . . 3 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) ∧ 𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)) → ((𝐴 ran ((,) ∘ 𝑔) ∧ ¬ ((vol*‘𝐴) + 𝐵) ≤ sup(ran 𝑆, ℝ*, < )) → (𝐴 ran ((,) ∘ 𝑔) ∧ sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + 𝐵))))
5958reximdva 3274 . 2 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ ¬ ((vol*‘𝐴) + 𝐵) ≤ sup(ran 𝑆, ℝ*, < )) → ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + 𝐵))))
6053, 59mpd 15 1 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3138  wrex 3139  {crab 3142  cin 3934  wss 3935   cuni 4832   class class class wbr 5058   × cxp 5547  ran crn 5550  ccom 5553  wf 6345  cfv 6349  (class class class)co 7145  m cmap 8396  supcsup 8893  infcinf 8894  cr 10525  0cc0 10526  1c1 10527   + caddc 10529  +∞cpnf 10661  *cxr 10663   < clt 10664  cle 10665  cmin 10859  cn 11627  +crp 12379  (,)cioo 12728  [,)cico 12730  seqcseq 13359  abscabs 14583  vol*covol 23992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4833  df-iun 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-er 8279  df-map 8398  df-en 8499  df-dom 8500  df-sdom 8501  df-sup 8895  df-inf 8896  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11628  df-2 11689  df-3 11690  df-n0 11887  df-z 11971  df-uz 12233  df-rp 12380  df-ico 12734  df-fz 12883  df-seq 13360  df-exp 13420  df-cj 14448  df-re 14449  df-im 14450  df-sqrt 14584  df-abs 14585  df-ovol 23994
This theorem is referenced by:  ovolunlem2  24028  ovoliunlem3  24034  ovolscalem2  24044  ioombl1  24092  uniioombl  24119  mblfinlem3  34813  mblfinlem4  34814
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