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Theorem ovolgelb 25515
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 1138 . . . . . 6 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (vol*‘𝐴) ∈ ℝ)
2 simp3 1139 . . . . . 6 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → 𝐵 ∈ ℝ+)
31, 2ltaddrpd 13110 . . . . 5 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (vol*‘𝐴) < ((vol*‘𝐴) + 𝐵))
42rpred 13077 . . . . . . 7 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → 𝐵 ∈ ℝ)
51, 4readdcld 11290 . . . . . 6 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((vol*‘𝐴) + 𝐵) ∈ ℝ)
61, 5ltnled 11408 . . . . 5 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((vol*‘𝐴) < ((vol*‘𝐴) + 𝐵) ↔ ¬ ((vol*‘𝐴) + 𝐵) ≤ (vol*‘𝐴)))
73, 6mpbid 232 . . . 4 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ¬ ((vol*‘𝐴) + 𝐵) ≤ (vol*‘𝐴))
8 eqid 2737 . . . . . . . 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 25508 . . . . . . 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 1134 . . . . . 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 5155 . . . . 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 4080 . . . . . . 7 {𝑦 ∈ ℝ* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ))} ⊆ ℝ*
135rexrd 11311 . . . . . . 7 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((vol*‘𝐴) + 𝐵) ∈ ℝ*)
14 infxrgelb 13377 . . . . . . 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 2741 . . . . . . . . . . 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 5948 . . . . . . . . . . . . 13 ran 𝑆 = ran seq1( + , ((abs ∘ − ) ∘ 𝑔))
1918supeq1i 9487 . . . . . . . . . . . 12 sup(ran 𝑆, ℝ*, < ) = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < )
2019eqeq2i 2750 . . . . . . . . . . 11 (𝑥 = sup(ran 𝑆, ℝ*, < ) ↔ 𝑥 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ))
2116, 20bitr4di 289 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ↔ 𝑥 = sup(ran 𝑆, ℝ*, < )))
2221anbi2d 630 . . . . . . . . 9 (𝑦 = 𝑥 → ((𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < )) ↔ (𝐴 ran ((,) ∘ 𝑔) ∧ 𝑥 = sup(ran 𝑆, ℝ*, < ))))
2322rexbidv 3179 . . . . . . . 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 3699 . . . . . . 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 3289 . . . . . . . 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 3183 . . . . . . . . 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 3093 . . . . . . . 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 464 . . . . . . . . . . . 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 450 . . . . . . . . . . . 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 275 . . . . . . . . . . 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 3093 . . . . . . . . . 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 25509 . . . . . . . . . . . . . . 15 (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ↔ 𝑔:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
33 eqid 2737 . . . . . . . . . . . . . . . 16 ((abs ∘ − ) ∘ 𝑔) = ((abs ∘ − ) ∘ 𝑔)
3433, 17ovolsf 25507 . . . . . . . . . . . . . . 15 (𝑔:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
3532, 34sylbi 217 . . . . . . . . . . . . . 14 (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) → 𝑆:ℕ⟶(0[,)+∞))
3635frnd 6744 . . . . . . . . . . . . 13 (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) → ran 𝑆 ⊆ (0[,)+∞))
37 icossxr 13472 . . . . . . . . . . . . 13 (0[,)+∞) ⊆ ℝ*
3836, 37sstrdi 3996 . . . . . . . . . . . 12 (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) → ran 𝑆 ⊆ ℝ*)
39 supxrcl 13357 . . . . . . . . . . . 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 5147 . . . . . . . . . . . . 13 (𝑥 = sup(ran 𝑆, ℝ*, < ) → (((vol*‘𝐴) + 𝐵) ≤ 𝑥 ↔ ((vol*‘𝐴) + 𝐵) ≤ sup(ran 𝑆, ℝ*, < )))
4241imbi2d 340 . . . . . . . . . . . 12 (𝑥 = sup(ran 𝑆, ℝ*, < ) → ((𝐴 ran ((,) ∘ 𝑔) → ((vol*‘𝐴) + 𝐵) ≤ 𝑥) ↔ (𝐴 ran ((,) ∘ 𝑔) → ((vol*‘𝐴) + 𝐵) ≤ sup(ran 𝑆, ℝ*, < ))))
4342ceqsralv 3522 . . . . . . . . . . 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 283 . . . . . . . . 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 3091 . . . . . . . 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 301 . . . . . . 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 275 . . . . . 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 288 . . . . 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 282 . . . 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 324 . . 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 3102 . . 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 234 . 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 11328 . . . . . 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 13191 . . . . . 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 260 . . . . 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 597 . . . 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 612 . . 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 3168 . 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 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070  {crab 3436  cin 3950  wss 3951   cuni 4907   class class class wbr 5143   × cxp 5683  ran crn 5686  ccom 5689  wf 6557  cfv 6561  (class class class)co 7431  m cmap 8866  supcsup 9480  infcinf 9481  cr 11154  0cc0 11155  1c1 11156   + caddc 11158  +∞cpnf 11292  *cxr 11294   < clt 11295  cle 11296  cmin 11492  cn 12266  +crp 13034  (,)cioo 13387  [,)cico 13389  seqcseq 14042  abscabs 15273  vol*covol 25497
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-sup 9482  df-inf 9483  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-rp 13035  df-ico 13393  df-fz 13548  df-seq 14043  df-exp 14103  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-ovol 25499
This theorem is referenced by:  ovolunlem2  25533  ovoliunlem3  25539  ovolscalem2  25549  ioombl1  25597  uniioombl  25624  mblfinlem3  37666  mblfinlem4  37667
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