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Theorem ovolgelb 23055
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 1055 . . . . . 6 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (vol*‘𝐴) ∈ ℝ)
2 simp3 1056 . . . . . 6 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → 𝐵 ∈ ℝ+)
31, 2ltaddrpd 11781 . . . . 5 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (vol*‘𝐴) < ((vol*‘𝐴) + 𝐵))
42rpred 11748 . . . . . . 7 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → 𝐵 ∈ ℝ)
51, 4readdcld 9948 . . . . . 6 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((vol*‘𝐴) + 𝐵) ∈ ℝ)
61, 5ltnled 10063 . . . . 5 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((vol*‘𝐴) < ((vol*‘𝐴) + 𝐵) ↔ ¬ ((vol*‘𝐴) + 𝐵) ≤ (vol*‘𝐴)))
73, 6mpbid 221 . . . 4 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ¬ ((vol*‘𝐴) + 𝐵) ≤ (vol*‘𝐴))
8 eqid 2610 . . . . . . . 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 23049 . . . . . . 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 1075 . . . . . 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 4595 . . . . 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 3650 . . . . . . 7 {𝑦 ∈ ℝ* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ))} ⊆ ℝ*
135rexrd 9968 . . . . . . 7 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((vol*‘𝐴) + 𝐵) ∈ ℝ*)
14 infxrgelb 12037 . . . . . . 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 694 . . . . . 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 2614 . . . . . . . . . . 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 5273 . . . . . . . . . . . . 13 ran 𝑆 = ran seq1( + , ((abs ∘ − ) ∘ 𝑔))
1918supeq1i 8236 . . . . . . . . . . . 12 sup(ran 𝑆, ℝ*, < ) = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < )
2019eqeq2i 2622 . . . . . . . . . . 11 (𝑥 = sup(ran 𝑆, ℝ*, < ) ↔ 𝑥 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ))
2116, 20syl6bbr 277 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ↔ 𝑥 = sup(ran 𝑆, ℝ*, < )))
2221anbi2d 736 . . . . . . . . 9 (𝑦 = 𝑥 → ((𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < )) ↔ (𝐴 ran ((,) ∘ 𝑔) ∧ 𝑥 = sup(ran 𝑆, ℝ*, < ))))
2322rexbidv 3034 . . . . . . . 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 3335 . . . . . . 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 3079 . . . . . . . 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 3005 . . . . . . . . 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 2963 . . . . . . . 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 467 . . . . . . . . . . . 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 461 . . . . . . . . . . . 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 263 . . . . . . . . . . 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 2963 . . . . . . . . . 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 reex 9906 . . . . . . . . . . . . . . . . . 18 ℝ ∈ V
3332, 32xpex 6860 . . . . . . . . . . . . . . . . 17 (ℝ × ℝ) ∈ V
3433inex2 4728 . . . . . . . . . . . . . . . 16 ( ≤ ∩ (ℝ × ℝ)) ∈ V
35 nnex 10903 . . . . . . . . . . . . . . . 16 ℕ ∈ V
3634, 35elmap 7772 . . . . . . . . . . . . . . 15 (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ↔ 𝑔:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
37 eqid 2610 . . . . . . . . . . . . . . . 16 ((abs ∘ − ) ∘ 𝑔) = ((abs ∘ − ) ∘ 𝑔)
3837, 17ovolsf 23048 . . . . . . . . . . . . . . 15 (𝑔:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
3936, 38sylbi 206 . . . . . . . . . . . . . 14 (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → 𝑆:ℕ⟶(0[,)+∞))
40 frn 5966 . . . . . . . . . . . . . 14 (𝑆:ℕ⟶(0[,)+∞) → ran 𝑆 ⊆ (0[,)+∞))
4139, 40syl 17 . . . . . . . . . . . . 13 (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → ran 𝑆 ⊆ (0[,)+∞))
42 icossxr 12129 . . . . . . . . . . . . 13 (0[,)+∞) ⊆ ℝ*
4341, 42syl6ss 3580 . . . . . . . . . . . 12 (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → ran 𝑆 ⊆ ℝ*)
44 supxrcl 12017 . . . . . . . . . . . 12 (ran 𝑆 ⊆ ℝ* → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
4543, 44syl 17 . . . . . . . . . . 11 (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
46 breq2 4587 . . . . . . . . . . . . 13 (𝑥 = sup(ran 𝑆, ℝ*, < ) → (((vol*‘𝐴) + 𝐵) ≤ 𝑥 ↔ ((vol*‘𝐴) + 𝐵) ≤ sup(ran 𝑆, ℝ*, < )))
4746imbi2d 329 . . . . . . . . . . . 12 (𝑥 = sup(ran 𝑆, ℝ*, < ) → ((𝐴 ran ((,) ∘ 𝑔) → ((vol*‘𝐴) + 𝐵) ≤ 𝑥) ↔ (𝐴 ran ((,) ∘ 𝑔) → ((vol*‘𝐴) + 𝐵) ≤ sup(ran 𝑆, ℝ*, < ))))
4847ceqsralv 3207 . . . . . . . . . . 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*, < ))))
4945, 48syl 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*, < ))))
5031, 49syl5bb 271 . . . . . . . . 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*, < ))))
5150ralbiia 2962 . . . . . . . 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*, < )))
5225, 27, 513bitr3i 289 . . . . . . 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*, < )))
5324, 52bitri 263 . . . . . 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*, < )))
5415, 53syl6rbb 276 . . . . 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*, < )))
5511, 54bitr4d 270 . . . 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*, < ))))
567, 55mtbid 313 . . 3 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ¬ ∀𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑔) → ((vol*‘𝐴) + 𝐵) ≤ sup(ran 𝑆, ℝ*, < )))
57 rexanali 2981 . . 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*, < )))
5856, 57sylibr 223 . 2 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ ¬ ((vol*‘𝐴) + 𝐵) ≤ sup(ran 𝑆, ℝ*, < )))
59 xrltnle 9984 . . . . . 6 ((sup(ran 𝑆, ℝ*, < ) ∈ ℝ* ∧ ((vol*‘𝐴) + 𝐵) ∈ ℝ*) → (sup(ran 𝑆, ℝ*, < ) < ((vol*‘𝐴) + 𝐵) ↔ ¬ ((vol*‘𝐴) + 𝐵) ≤ sup(ran 𝑆, ℝ*, < )))
60 xrltle 11858 . . . . . 6 ((sup(ran 𝑆, ℝ*, < ) ∈ ℝ* ∧ ((vol*‘𝐴) + 𝐵) ∈ ℝ*) → (sup(ran 𝑆, ℝ*, < ) < ((vol*‘𝐴) + 𝐵) → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + 𝐵)))
6159, 60sylbird 249 . . . . 5 ((sup(ran 𝑆, ℝ*, < ) ∈ ℝ* ∧ ((vol*‘𝐴) + 𝐵) ∈ ℝ*) → (¬ ((vol*‘𝐴) + 𝐵) ≤ sup(ran 𝑆, ℝ*, < ) → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + 𝐵)))
6245, 13, 61syl2anr 494 . . . 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)))
6362anim2d 587 . . 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))))
6463reximdva 3000 . 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))))
6558, 64mpd 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 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wral 2896  wrex 2897  {crab 2900  cin 3539  wss 3540   cuni 4372   class class class wbr 4583   × cxp 5036  ran crn 5039  ccom 5042  wf 5800  cfv 5804  (class class class)co 6549  𝑚 cmap 7744  supcsup 8229  infcinf 8230  cr 9814  0cc0 9815  1c1 9816   + caddc 9818  +∞cpnf 9950  *cxr 9952   < clt 9953  cle 9954  cmin 10145  cn 10897  +crp 11708  (,)cioo 12046  [,)cico 12048  seqcseq 12663  abscabs 13822  vol*covol 23038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-sup 8231  df-inf 8232  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-ico 12052  df-fz 12198  df-seq 12664  df-exp 12723  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-ovol 23040
This theorem is referenced by:  ovolunlem2  23073  ovoliunlem3  23079  ovolscalem2  23089  ioombl1  23137  uniioombl  23163  mblfinlem3  32618  mblfinlem4  32619
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