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Theorem ovolgelb 25229
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 1135 . . . . . 6 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (vol*‘𝐴) ∈ ℝ)
2 simp3 1136 . . . . . 6 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → 𝐵 ∈ ℝ+)
31, 2ltaddrpd 13053 . . . . 5 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (vol*‘𝐴) < ((vol*‘𝐴) + 𝐵))
42rpred 13020 . . . . . . 7 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → 𝐵 ∈ ℝ)
51, 4readdcld 11247 . . . . . 6 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((vol*‘𝐴) + 𝐵) ∈ ℝ)
61, 5ltnled 11365 . . . . 5 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((vol*‘𝐴) < ((vol*‘𝐴) + 𝐵) ↔ ¬ ((vol*‘𝐴) + 𝐵) ≤ (vol*‘𝐴)))
73, 6mpbid 231 . . . 4 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ¬ ((vol*‘𝐴) + 𝐵) ≤ (vol*‘𝐴))
8 eqid 2730 . . . . . . . 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 25222 . . . . . . 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 1131 . . . . . 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 5159 . . . . 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 4076 . . . . . . 7 {𝑦 ∈ ℝ* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ))} ⊆ ℝ*
135rexrd 11268 . . . . . . 7 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((vol*‘𝐴) + 𝐵) ∈ ℝ*)
14 infxrgelb 13318 . . . . . . 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 585 . . . . . 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 2734 . . . . . . . . . . 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 5935 . . . . . . . . . . . . 13 ran 𝑆 = ran seq1( + , ((abs ∘ − ) ∘ 𝑔))
1918supeq1i 9444 . . . . . . . . . . . 12 sup(ran 𝑆, ℝ*, < ) = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < )
2019eqeq2i 2743 . . . . . . . . . . 11 (𝑥 = sup(ran 𝑆, ℝ*, < ) ↔ 𝑥 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ))
2116, 20bitr4di 288 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ↔ 𝑥 = sup(ran 𝑆, ℝ*, < )))
2221anbi2d 627 . . . . . . . . 9 (𝑦 = 𝑥 → ((𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < )) ↔ (𝐴 ran ((,) ∘ 𝑔) ∧ 𝑥 = sup(ran 𝑆, ℝ*, < ))))
2322rexbidv 3176 . . . . . . . 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 3688 . . . . . . 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 3284 . . . . . . . 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 3180 . . . . . . . . 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 3091 . . . . . . . 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 463 . . . . . . . . . . . 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 449 . . . . . . . . . . . 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 274 . . . . . . . . . . 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 3091 . . . . . . . . . 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 25223 . . . . . . . . . . . . . . 15 (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ↔ 𝑔:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
33 eqid 2730 . . . . . . . . . . . . . . . 16 ((abs ∘ − ) ∘ 𝑔) = ((abs ∘ − ) ∘ 𝑔)
3433, 17ovolsf 25221 . . . . . . . . . . . . . . 15 (𝑔:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
3532, 34sylbi 216 . . . . . . . . . . . . . 14 (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) → 𝑆:ℕ⟶(0[,)+∞))
3635frnd 6724 . . . . . . . . . . . . 13 (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) → ran 𝑆 ⊆ (0[,)+∞))
37 icossxr 13413 . . . . . . . . . . . . 13 (0[,)+∞) ⊆ ℝ*
3836, 37sstrdi 3993 . . . . . . . . . . . 12 (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) → ran 𝑆 ⊆ ℝ*)
39 supxrcl 13298 . . . . . . . . . . . 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 5151 . . . . . . . . . . . . 13 (𝑥 = sup(ran 𝑆, ℝ*, < ) → (((vol*‘𝐴) + 𝐵) ≤ 𝑥 ↔ ((vol*‘𝐴) + 𝐵) ≤ sup(ran 𝑆, ℝ*, < )))
4241imbi2d 339 . . . . . . . . . . . 12 (𝑥 = sup(ran 𝑆, ℝ*, < ) → ((𝐴 ran ((,) ∘ 𝑔) → ((vol*‘𝐴) + 𝐵) ≤ 𝑥) ↔ (𝐴 ran ((,) ∘ 𝑔) → ((vol*‘𝐴) + 𝐵) ≤ sup(ran 𝑆, ℝ*, < ))))
4342ceqsralv 3512 . . . . . . . . . . 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, 44bitrid 282 . . . . . . . . 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 3089 . . . . . . . 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 300 . . . . . . 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 274 . . . . . 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, 48bitr2di 287 . . . . 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 281 . . . 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 323 . . 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 3100 . . 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 233 . 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 11285 . . . . . 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 13132 . . . . . 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 259 . . . . 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 595 . . . 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 610 . . 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 3166 . 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 205  wa 394  w3a 1085   = wceq 1539  wcel 2104  wral 3059  wrex 3068  {crab 3430  cin 3946  wss 3947   cuni 4907   class class class wbr 5147   × cxp 5673  ran crn 5676  ccom 5679  wf 6538  cfv 6542  (class class class)co 7411  m cmap 8822  supcsup 9437  infcinf 9438  cr 11111  0cc0 11112  1c1 11113   + caddc 11115  +∞cpnf 11249  *cxr 11251   < clt 11252  cle 11253  cmin 11448  cn 12216  +crp 12978  (,)cioo 13328  [,)cico 13330  seqcseq 13970  abscabs 15185  vol*covol 25211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-sup 9439  df-inf 9440  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-n0 12477  df-z 12563  df-uz 12827  df-rp 12979  df-ico 13334  df-fz 13489  df-seq 13971  df-exp 14032  df-cj 15050  df-re 15051  df-im 15052  df-sqrt 15186  df-abs 15187  df-ovol 25213
This theorem is referenced by:  ovolunlem2  25247  ovoliunlem3  25253  ovolscalem2  25263  ioombl1  25311  uniioombl  25338  mblfinlem3  36830  mblfinlem4  36831
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