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Theorem uniioovol 24943
Description: A disjoint union of open intervals has volume equal to the sum of the volume of the intervals. (This proof does not use countable choice, unlike voliun 24918.) Lemma 565Ca of [Fremlin5] p. 213. (Contributed by Mario Carneiro, 26-Mar-2015.)
Hypotheses
Ref Expression
uniioombl.1 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
uniioombl.2 (𝜑Disj 𝑥 ∈ ℕ ((,)‘(𝐹𝑥)))
uniioombl.3 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
Assertion
Ref Expression
uniioovol (𝜑 → (vol*‘ ran ((,) ∘ 𝐹)) = sup(ran 𝑆, ℝ*, < ))
Distinct variable groups:   𝑥,𝐹   𝜑,𝑥
Allowed substitution hint:   𝑆(𝑥)

Proof of Theorem uniioovol
Dummy variables 𝑛 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ioof 13364 . . . . . 6 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
2 uniioombl.1 . . . . . . 7 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
3 inss2 4189 . . . . . . . 8 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
4 rexpssxrxp 11200 . . . . . . . 8 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
53, 4sstri 3953 . . . . . . 7 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
6 fss 6685 . . . . . . 7 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)) → 𝐹:ℕ⟶(ℝ* × ℝ*))
72, 5, 6sylancl 586 . . . . . 6 (𝜑𝐹:ℕ⟶(ℝ* × ℝ*))
8 fco 6692 . . . . . 6 (((,):(ℝ* × ℝ*)⟶𝒫 ℝ ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
91, 7, 8sylancr 587 . . . . 5 (𝜑 → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
109frnd 6676 . . . 4 (𝜑 → ran ((,) ∘ 𝐹) ⊆ 𝒫 ℝ)
11 sspwuni 5060 . . . 4 (ran ((,) ∘ 𝐹) ⊆ 𝒫 ℝ ↔ ran ((,) ∘ 𝐹) ⊆ ℝ)
1210, 11sylib 217 . . 3 (𝜑 ran ((,) ∘ 𝐹) ⊆ ℝ)
13 ovolcl 24842 . . 3 ( ran ((,) ∘ 𝐹) ⊆ ℝ → (vol*‘ ran ((,) ∘ 𝐹)) ∈ ℝ*)
1412, 13syl 17 . 2 (𝜑 → (vol*‘ ran ((,) ∘ 𝐹)) ∈ ℝ*)
15 eqid 2736 . . . . . 6 ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
16 uniioombl.3 . . . . . 6 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
1715, 16ovolsf 24836 . . . . 5 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
18 frn 6675 . . . . 5 (𝑆:ℕ⟶(0[,)+∞) → ran 𝑆 ⊆ (0[,)+∞))
192, 17, 183syl 18 . . . 4 (𝜑 → ran 𝑆 ⊆ (0[,)+∞))
20 icossxr 13349 . . . 4 (0[,)+∞) ⊆ ℝ*
2119, 20sstrdi 3956 . . 3 (𝜑 → ran 𝑆 ⊆ ℝ*)
22 supxrcl 13234 . . 3 (ran 𝑆 ⊆ ℝ* → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
2321, 22syl 17 . 2 (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
24 ssid 3966 . . 3 ran ((,) ∘ 𝐹) ⊆ ran ((,) ∘ 𝐹)
2516ovollb 24843 . . 3 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ran ((,) ∘ 𝐹) ⊆ ran ((,) ∘ 𝐹)) → (vol*‘ ran ((,) ∘ 𝐹)) ≤ sup(ran 𝑆, ℝ*, < ))
262, 24, 25sylancl 586 . 2 (𝜑 → (vol*‘ ran ((,) ∘ 𝐹)) ≤ sup(ran 𝑆, ℝ*, < ))
2716fveq1i 6843 . . . . . . . 8 (𝑆𝑛) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑛)
282adantr 481 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
29 elfznn 13470 . . . . . . . . . . 11 (𝑥 ∈ (1...𝑛) → 𝑥 ∈ ℕ)
3015ovolfsval 24834 . . . . . . . . . . 11 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑥) = ((2nd ‘(𝐹𝑥)) − (1st ‘(𝐹𝑥))))
3128, 29, 30syl2an 596 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (((abs ∘ − ) ∘ 𝐹)‘𝑥) = ((2nd ‘(𝐹𝑥)) − (1st ‘(𝐹𝑥))))
32 fvco3 6940 . . . . . . . . . . . . . . 15 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹𝑥)))
3328, 29, 32syl2an 596 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹𝑥)))
34 ffvelcdm 7032 . . . . . . . . . . . . . . . . . . 19 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (𝐹𝑥) ∈ ( ≤ ∩ (ℝ × ℝ)))
3528, 29, 34syl2an 596 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (𝐹𝑥) ∈ ( ≤ ∩ (ℝ × ℝ)))
3635elin2d 4159 . . . . . . . . . . . . . . . . 17 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (𝐹𝑥) ∈ (ℝ × ℝ))
37 1st2nd2 7960 . . . . . . . . . . . . . . . . 17 ((𝐹𝑥) ∈ (ℝ × ℝ) → (𝐹𝑥) = ⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
3836, 37syl 17 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (𝐹𝑥) = ⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
3938fveq2d 6846 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → ((,)‘(𝐹𝑥)) = ((,)‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩))
40 df-ov 7360 . . . . . . . . . . . . . . 15 ((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))) = ((,)‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
4139, 40eqtr4di 2794 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → ((,)‘(𝐹𝑥)) = ((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))))
4233, 41eqtrd 2776 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (((,) ∘ 𝐹)‘𝑥) = ((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))))
43 ioombl 24929 . . . . . . . . . . . . 13 ((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))) ∈ dom vol
4442, 43eqeltrdi 2846 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (((,) ∘ 𝐹)‘𝑥) ∈ dom vol)
45 mblvol 24894 . . . . . . . . . . . 12 ((((,) ∘ 𝐹)‘𝑥) ∈ dom vol → (vol‘(((,) ∘ 𝐹)‘𝑥)) = (vol*‘(((,) ∘ 𝐹)‘𝑥)))
4644, 45syl 17 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (vol‘(((,) ∘ 𝐹)‘𝑥)) = (vol*‘(((,) ∘ 𝐹)‘𝑥)))
4742fveq2d 6846 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (vol*‘(((,) ∘ 𝐹)‘𝑥)) = (vol*‘((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥)))))
48 ovolfcl 24830 . . . . . . . . . . . . 13 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((1st ‘(𝐹𝑥)) ∈ ℝ ∧ (2nd ‘(𝐹𝑥)) ∈ ℝ ∧ (1st ‘(𝐹𝑥)) ≤ (2nd ‘(𝐹𝑥))))
4928, 29, 48syl2an 596 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → ((1st ‘(𝐹𝑥)) ∈ ℝ ∧ (2nd ‘(𝐹𝑥)) ∈ ℝ ∧ (1st ‘(𝐹𝑥)) ≤ (2nd ‘(𝐹𝑥))))
50 ovolioo 24932 . . . . . . . . . . . 12 (((1st ‘(𝐹𝑥)) ∈ ℝ ∧ (2nd ‘(𝐹𝑥)) ∈ ℝ ∧ (1st ‘(𝐹𝑥)) ≤ (2nd ‘(𝐹𝑥))) → (vol*‘((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥)))) = ((2nd ‘(𝐹𝑥)) − (1st ‘(𝐹𝑥))))
5149, 50syl 17 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (vol*‘((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥)))) = ((2nd ‘(𝐹𝑥)) − (1st ‘(𝐹𝑥))))
5246, 47, 513eqtrd 2780 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (vol‘(((,) ∘ 𝐹)‘𝑥)) = ((2nd ‘(𝐹𝑥)) − (1st ‘(𝐹𝑥))))
5331, 52eqtr4d 2779 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (((abs ∘ − ) ∘ 𝐹)‘𝑥) = (vol‘(((,) ∘ 𝐹)‘𝑥)))
54 simpr 485 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
55 nnuz 12806 . . . . . . . . . 10 ℕ = (ℤ‘1)
5654, 55eleqtrdi 2848 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ‘1))
5749simp2d 1143 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (2nd ‘(𝐹𝑥)) ∈ ℝ)
5849simp1d 1142 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (1st ‘(𝐹𝑥)) ∈ ℝ)
5957, 58resubcld 11583 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → ((2nd ‘(𝐹𝑥)) − (1st ‘(𝐹𝑥))) ∈ ℝ)
6052, 59eqeltrd 2838 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (vol‘(((,) ∘ 𝐹)‘𝑥)) ∈ ℝ)
6160recnd 11183 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (vol‘(((,) ∘ 𝐹)‘𝑥)) ∈ ℂ)
6253, 56, 61fsumser 15615 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → Σ𝑥 ∈ (1...𝑛)(vol‘(((,) ∘ 𝐹)‘𝑥)) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑛))
6327, 62eqtr4id 2795 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (𝑆𝑛) = Σ𝑥 ∈ (1...𝑛)(vol‘(((,) ∘ 𝐹)‘𝑥)))
64 fzfid 13878 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (1...𝑛) ∈ Fin)
6544, 60jca 512 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → ((((,) ∘ 𝐹)‘𝑥) ∈ dom vol ∧ (vol‘(((,) ∘ 𝐹)‘𝑥)) ∈ ℝ))
6665ralrimiva 3143 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ∀𝑥 ∈ (1...𝑛)((((,) ∘ 𝐹)‘𝑥) ∈ dom vol ∧ (vol‘(((,) ∘ 𝐹)‘𝑥)) ∈ ℝ))
67 fz1ssnn 13472 . . . . . . . . 9 (1...𝑛) ⊆ ℕ
68 uniioombl.2 . . . . . . . . . . 11 (𝜑Disj 𝑥 ∈ ℕ ((,)‘(𝐹𝑥)))
692, 32sylan 580 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹𝑥)))
7069disjeq2dv 5075 . . . . . . . . . . 11 (𝜑 → (Disj 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) ↔ Disj 𝑥 ∈ ℕ ((,)‘(𝐹𝑥))))
7168, 70mpbird 256 . . . . . . . . . 10 (𝜑Disj 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥))
7271adantr 481 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → Disj 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥))
73 disjss1 5076 . . . . . . . . 9 ((1...𝑛) ⊆ ℕ → (Disj 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) → Disj 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥)))
7467, 72, 73mpsyl 68 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → Disj 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥))
75 volfiniun 24911 . . . . . . . 8 (((1...𝑛) ∈ Fin ∧ ∀𝑥 ∈ (1...𝑛)((((,) ∘ 𝐹)‘𝑥) ∈ dom vol ∧ (vol‘(((,) ∘ 𝐹)‘𝑥)) ∈ ℝ) ∧ Disj 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥)) → (vol‘ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥)) = Σ𝑥 ∈ (1...𝑛)(vol‘(((,) ∘ 𝐹)‘𝑥)))
7664, 66, 74, 75syl3anc 1371 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (vol‘ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥)) = Σ𝑥 ∈ (1...𝑛)(vol‘(((,) ∘ 𝐹)‘𝑥)))
7744ralrimiva 3143 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ∀𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ∈ dom vol)
78 finiunmbl 24908 . . . . . . . . 9 (((1...𝑛) ∈ Fin ∧ ∀𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ∈ dom vol) → 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ∈ dom vol)
7964, 77, 78syl2anc 584 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ∈ dom vol)
80 mblvol 24894 . . . . . . . 8 ( 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ∈ dom vol → (vol‘ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥)) = (vol*‘ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥)))
8179, 80syl 17 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (vol‘ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥)) = (vol*‘ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥)))
8263, 76, 813eqtr2d 2782 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝑆𝑛) = (vol*‘ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥)))
83 iunss1 4968 . . . . . . . . 9 ((1...𝑛) ⊆ ℕ → 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ⊆ 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥))
8467, 83mp1i 13 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ⊆ 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥))
859adantr 481 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
86 ffn 6668 . . . . . . . . 9 (((,) ∘ 𝐹):ℕ⟶𝒫 ℝ → ((,) ∘ 𝐹) Fn ℕ)
87 fniunfv 7194 . . . . . . . . 9 (((,) ∘ 𝐹) Fn ℕ → 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) = ran ((,) ∘ 𝐹))
8885, 86, 873syl 18 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) = ran ((,) ∘ 𝐹))
8984, 88sseqtrd 3984 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ⊆ ran ((,) ∘ 𝐹))
9012adantr 481 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → ran ((,) ∘ 𝐹) ⊆ ℝ)
91 ovolss 24849 . . . . . . 7 (( 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ⊆ ran ((,) ∘ 𝐹) ∧ ran ((,) ∘ 𝐹) ⊆ ℝ) → (vol*‘ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥)) ≤ (vol*‘ ran ((,) ∘ 𝐹)))
9289, 90, 91syl2anc 584 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (vol*‘ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥)) ≤ (vol*‘ ran ((,) ∘ 𝐹)))
9382, 92eqbrtrd 5127 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (𝑆𝑛) ≤ (vol*‘ ran ((,) ∘ 𝐹)))
9493ralrimiva 3143 . . . 4 (𝜑 → ∀𝑛 ∈ ℕ (𝑆𝑛) ≤ (vol*‘ ran ((,) ∘ 𝐹)))
952, 17syl 17 . . . . 5 (𝜑𝑆:ℕ⟶(0[,)+∞))
96 ffn 6668 . . . . 5 (𝑆:ℕ⟶(0[,)+∞) → 𝑆 Fn ℕ)
97 breq1 5108 . . . . . 6 (𝑦 = (𝑆𝑛) → (𝑦 ≤ (vol*‘ ran ((,) ∘ 𝐹)) ↔ (𝑆𝑛) ≤ (vol*‘ ran ((,) ∘ 𝐹))))
9897ralrn 7038 . . . . 5 (𝑆 Fn ℕ → (∀𝑦 ∈ ran 𝑆 𝑦 ≤ (vol*‘ ran ((,) ∘ 𝐹)) ↔ ∀𝑛 ∈ ℕ (𝑆𝑛) ≤ (vol*‘ ran ((,) ∘ 𝐹))))
9995, 96, 983syl 18 . . . 4 (𝜑 → (∀𝑦 ∈ ran 𝑆 𝑦 ≤ (vol*‘ ran ((,) ∘ 𝐹)) ↔ ∀𝑛 ∈ ℕ (𝑆𝑛) ≤ (vol*‘ ran ((,) ∘ 𝐹))))
10094, 99mpbird 256 . . 3 (𝜑 → ∀𝑦 ∈ ran 𝑆 𝑦 ≤ (vol*‘ ran ((,) ∘ 𝐹)))
101 supxrleub 13245 . . . 4 ((ran 𝑆 ⊆ ℝ* ∧ (vol*‘ ran ((,) ∘ 𝐹)) ∈ ℝ*) → (sup(ran 𝑆, ℝ*, < ) ≤ (vol*‘ ran ((,) ∘ 𝐹)) ↔ ∀𝑦 ∈ ran 𝑆 𝑦 ≤ (vol*‘ ran ((,) ∘ 𝐹))))
10221, 14, 101syl2anc 584 . . 3 (𝜑 → (sup(ran 𝑆, ℝ*, < ) ≤ (vol*‘ ran ((,) ∘ 𝐹)) ↔ ∀𝑦 ∈ ran 𝑆 𝑦 ≤ (vol*‘ ran ((,) ∘ 𝐹))))
103100, 102mpbird 256 . 2 (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ (vol*‘ ran ((,) ∘ 𝐹)))
10414, 23, 26, 103xrletrid 13074 1 (𝜑 → (vol*‘ ran ((,) ∘ 𝐹)) = sup(ran 𝑆, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  cin 3909  wss 3910  𝒫 cpw 4560  cop 4592   cuni 4865   ciun 4954  Disj wdisj 5070   class class class wbr 5105   × cxp 5631  dom cdm 5633  ran crn 5634  ccom 5637   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  1st c1st 7919  2nd c2nd 7920  Fincfn 8883  supcsup 9376  cr 11050  0cc0 11051  1c1 11052   + caddc 11054  +∞cpnf 11186  *cxr 11188   < clt 11189  cle 11190  cmin 11385  cn 12153  cuz 12763  (,)cioo 13264  [,)cico 13266  ...cfz 13424  seqcseq 13906  abscabs 15119  Σcsu 15570  vol*covol 24826  volcvol 24827
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-disj 5071  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fi 9347  df-sup 9378  df-inf 9379  df-oi 9446  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ioo 13268  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-fl 13697  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-rlim 15371  df-sum 15571  df-rest 17304  df-topgen 17325  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-top 22243  df-topon 22260  df-bases 22296  df-cmp 22738  df-ovol 24828  df-vol 24829
This theorem is referenced by:  uniiccvol  24944  uniioombllem2  24947
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