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Theorem uniioovol 24187
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 24162.) 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 12829 . . . . . 6 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
2 uniioombl.1 . . . . . . 7 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
3 inss2 4159 . . . . . . . 8 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
4 rexpssxrxp 10679 . . . . . . . 8 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
53, 4sstri 3927 . . . . . . 7 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
6 fss 6505 . . . . . . 7 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)) → 𝐹:ℕ⟶(ℝ* × ℝ*))
72, 5, 6sylancl 589 . . . . . 6 (𝜑𝐹:ℕ⟶(ℝ* × ℝ*))
8 fco 6509 . . . . . 6 (((,):(ℝ* × ℝ*)⟶𝒫 ℝ ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
91, 7, 8sylancr 590 . . . . 5 (𝜑 → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
109frnd 6498 . . . 4 (𝜑 → ran ((,) ∘ 𝐹) ⊆ 𝒫 ℝ)
11 sspwuni 4988 . . . 4 (ran ((,) ∘ 𝐹) ⊆ 𝒫 ℝ ↔ ran ((,) ∘ 𝐹) ⊆ ℝ)
1210, 11sylib 221 . . 3 (𝜑 ran ((,) ∘ 𝐹) ⊆ ℝ)
13 ovolcl 24086 . . 3 ( ran ((,) ∘ 𝐹) ⊆ ℝ → (vol*‘ ran ((,) ∘ 𝐹)) ∈ ℝ*)
1412, 13syl 17 . 2 (𝜑 → (vol*‘ ran ((,) ∘ 𝐹)) ∈ ℝ*)
15 eqid 2801 . . . . . 6 ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
16 uniioombl.3 . . . . . 6 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
1715, 16ovolsf 24080 . . . . 5 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
18 frn 6497 . . . . 5 (𝑆:ℕ⟶(0[,)+∞) → ran 𝑆 ⊆ (0[,)+∞))
192, 17, 183syl 18 . . . 4 (𝜑 → ran 𝑆 ⊆ (0[,)+∞))
20 icossxr 12814 . . . 4 (0[,)+∞) ⊆ ℝ*
2119, 20sstrdi 3930 . . 3 (𝜑 → ran 𝑆 ⊆ ℝ*)
22 supxrcl 12700 . . 3 (ran 𝑆 ⊆ ℝ* → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
2321, 22syl 17 . 2 (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
24 ssid 3940 . . 3 ran ((,) ∘ 𝐹) ⊆ ran ((,) ∘ 𝐹)
2516ovollb 24087 . . 3 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ran ((,) ∘ 𝐹) ⊆ ran ((,) ∘ 𝐹)) → (vol*‘ ran ((,) ∘ 𝐹)) ≤ sup(ran 𝑆, ℝ*, < ))
262, 24, 25sylancl 589 . 2 (𝜑 → (vol*‘ ran ((,) ∘ 𝐹)) ≤ sup(ran 𝑆, ℝ*, < ))
2716fveq1i 6650 . . . . . . . 8 (𝑆𝑛) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑛)
282adantr 484 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
29 elfznn 12935 . . . . . . . . . . 11 (𝑥 ∈ (1...𝑛) → 𝑥 ∈ ℕ)
3015ovolfsval 24078 . . . . . . . . . . 11 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑥) = ((2nd ‘(𝐹𝑥)) − (1st ‘(𝐹𝑥))))
3128, 29, 30syl2an 598 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (((abs ∘ − ) ∘ 𝐹)‘𝑥) = ((2nd ‘(𝐹𝑥)) − (1st ‘(𝐹𝑥))))
32 fvco3 6741 . . . . . . . . . . . . . . 15 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹𝑥)))
3328, 29, 32syl2an 598 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹𝑥)))
34 ffvelrn 6830 . . . . . . . . . . . . . . . . . . 19 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (𝐹𝑥) ∈ ( ≤ ∩ (ℝ × ℝ)))
3528, 29, 34syl2an 598 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (𝐹𝑥) ∈ ( ≤ ∩ (ℝ × ℝ)))
3635elin2d 4129 . . . . . . . . . . . . . . . . 17 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (𝐹𝑥) ∈ (ℝ × ℝ))
37 1st2nd2 7714 . . . . . . . . . . . . . . . . 17 ((𝐹𝑥) ∈ (ℝ × ℝ) → (𝐹𝑥) = ⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
3836, 37syl 17 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (𝐹𝑥) = ⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
3938fveq2d 6653 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → ((,)‘(𝐹𝑥)) = ((,)‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩))
40 df-ov 7142 . . . . . . . . . . . . . . 15 ((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))) = ((,)‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
4139, 40eqtr4di 2854 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → ((,)‘(𝐹𝑥)) = ((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))))
4233, 41eqtrd 2836 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (((,) ∘ 𝐹)‘𝑥) = ((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))))
43 ioombl 24173 . . . . . . . . . . . . 13 ((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))) ∈ dom vol
4442, 43eqeltrdi 2901 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (((,) ∘ 𝐹)‘𝑥) ∈ dom vol)
45 mblvol 24138 . . . . . . . . . . . 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 6653 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (vol*‘(((,) ∘ 𝐹)‘𝑥)) = (vol*‘((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥)))))
48 ovolfcl 24074 . . . . . . . . . . . . 13 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((1st ‘(𝐹𝑥)) ∈ ℝ ∧ (2nd ‘(𝐹𝑥)) ∈ ℝ ∧ (1st ‘(𝐹𝑥)) ≤ (2nd ‘(𝐹𝑥))))
4928, 29, 48syl2an 598 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → ((1st ‘(𝐹𝑥)) ∈ ℝ ∧ (2nd ‘(𝐹𝑥)) ∈ ℝ ∧ (1st ‘(𝐹𝑥)) ≤ (2nd ‘(𝐹𝑥))))
50 ovolioo 24176 . . . . . . . . . . . 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 2840 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (vol‘(((,) ∘ 𝐹)‘𝑥)) = ((2nd ‘(𝐹𝑥)) − (1st ‘(𝐹𝑥))))
5331, 52eqtr4d 2839 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (((abs ∘ − ) ∘ 𝐹)‘𝑥) = (vol‘(((,) ∘ 𝐹)‘𝑥)))
54 simpr 488 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
55 nnuz 12273 . . . . . . . . . 10 ℕ = (ℤ‘1)
5654, 55eleqtrdi 2903 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ‘1))
5749simp2d 1140 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (2nd ‘(𝐹𝑥)) ∈ ℝ)
5849simp1d 1139 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (1st ‘(𝐹𝑥)) ∈ ℝ)
5957, 58resubcld 11061 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → ((2nd ‘(𝐹𝑥)) − (1st ‘(𝐹𝑥))) ∈ ℝ)
6052, 59eqeltrd 2893 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (vol‘(((,) ∘ 𝐹)‘𝑥)) ∈ ℝ)
6160recnd 10662 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (vol‘(((,) ∘ 𝐹)‘𝑥)) ∈ ℂ)
6253, 56, 61fsumser 15083 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → Σ𝑥 ∈ (1...𝑛)(vol‘(((,) ∘ 𝐹)‘𝑥)) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑛))
6327, 62eqtr4id 2855 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (𝑆𝑛) = Σ𝑥 ∈ (1...𝑛)(vol‘(((,) ∘ 𝐹)‘𝑥)))
64 fzfid 13340 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (1...𝑛) ∈ Fin)
6544, 60jca 515 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → ((((,) ∘ 𝐹)‘𝑥) ∈ dom vol ∧ (vol‘(((,) ∘ 𝐹)‘𝑥)) ∈ ℝ))
6665ralrimiva 3152 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ∀𝑥 ∈ (1...𝑛)((((,) ∘ 𝐹)‘𝑥) ∈ dom vol ∧ (vol‘(((,) ∘ 𝐹)‘𝑥)) ∈ ℝ))
67 fz1ssnn 12937 . . . . . . . . 9 (1...𝑛) ⊆ ℕ
68 uniioombl.2 . . . . . . . . . . 11 (𝜑Disj 𝑥 ∈ ℕ ((,)‘(𝐹𝑥)))
692, 32sylan 583 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹𝑥)))
7069disjeq2dv 5003 . . . . . . . . . . 11 (𝜑 → (Disj 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) ↔ Disj 𝑥 ∈ ℕ ((,)‘(𝐹𝑥))))
7168, 70mpbird 260 . . . . . . . . . 10 (𝜑Disj 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥))
7271adantr 484 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → Disj 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥))
73 disjss1 5004 . . . . . . . . 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 24155 . . . . . . . 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 1368 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (vol‘ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥)) = Σ𝑥 ∈ (1...𝑛)(vol‘(((,) ∘ 𝐹)‘𝑥)))
7744ralrimiva 3152 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ∀𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ∈ dom vol)
78 finiunmbl 24152 . . . . . . . . 9 (((1...𝑛) ∈ Fin ∧ ∀𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ∈ dom vol) → 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ∈ dom vol)
7964, 77, 78syl2anc 587 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ∈ dom vol)
80 mblvol 24138 . . . . . . . 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 2842 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝑆𝑛) = (vol*‘ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥)))
83 iunss1 4898 . . . . . . . . 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 484 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
86 ffn 6491 . . . . . . . . 9 (((,) ∘ 𝐹):ℕ⟶𝒫 ℝ → ((,) ∘ 𝐹) Fn ℕ)
87 fniunfv 6988 . . . . . . . . 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 3958 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ⊆ ran ((,) ∘ 𝐹))
9012adantr 484 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → ran ((,) ∘ 𝐹) ⊆ ℝ)
91 ovolss 24093 . . . . . . 7 (( 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ⊆ ran ((,) ∘ 𝐹) ∧ ran ((,) ∘ 𝐹) ⊆ ℝ) → (vol*‘ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥)) ≤ (vol*‘ ran ((,) ∘ 𝐹)))
9289, 90, 91syl2anc 587 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (vol*‘ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥)) ≤ (vol*‘ ran ((,) ∘ 𝐹)))
9382, 92eqbrtrd 5055 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (𝑆𝑛) ≤ (vol*‘ ran ((,) ∘ 𝐹)))
9493ralrimiva 3152 . . . 4 (𝜑 → ∀𝑛 ∈ ℕ (𝑆𝑛) ≤ (vol*‘ ran ((,) ∘ 𝐹)))
952, 17syl 17 . . . . 5 (𝜑𝑆:ℕ⟶(0[,)+∞))
96 ffn 6491 . . . . 5 (𝑆:ℕ⟶(0[,)+∞) → 𝑆 Fn ℕ)
97 breq1 5036 . . . . . 6 (𝑦 = (𝑆𝑛) → (𝑦 ≤ (vol*‘ ran ((,) ∘ 𝐹)) ↔ (𝑆𝑛) ≤ (vol*‘ ran ((,) ∘ 𝐹))))
9897ralrn 6835 . . . . 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 260 . . 3 (𝜑 → ∀𝑦 ∈ ran 𝑆 𝑦 ≤ (vol*‘ ran ((,) ∘ 𝐹)))
101 supxrleub 12711 . . . 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 587 . . 3 (𝜑 → (sup(ran 𝑆, ℝ*, < ) ≤ (vol*‘ ran ((,) ∘ 𝐹)) ↔ ∀𝑦 ∈ ran 𝑆 𝑦 ≤ (vol*‘ ran ((,) ∘ 𝐹))))
103100, 102mpbird 260 . 2 (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ (vol*‘ ran ((,) ∘ 𝐹)))
10414, 23, 26, 103xrletrid 12540 1 (𝜑 → (vol*‘ ran ((,) ∘ 𝐹)) = sup(ran 𝑆, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  wral 3109  cin 3883  wss 3884  𝒫 cpw 4500  cop 4534   cuni 4803   ciun 4884  Disj wdisj 4998   class class class wbr 5033   × cxp 5521  dom cdm 5523  ran crn 5524  ccom 5527   Fn wfn 6323  wf 6324  cfv 6328  (class class class)co 7139  1st c1st 7673  2nd c2nd 7674  Fincfn 8496  supcsup 8892  cr 10529  0cc0 10530  1c1 10531   + caddc 10533  +∞cpnf 10665  *cxr 10667   < clt 10668  cle 10669  cmin 10863  cn 11629  cuz 12235  (,)cioo 12730  [,)cico 12732  ...cfz 12889  seqcseq 13368  abscabs 14589  Σcsu 15038  vol*covol 24070  volcvol 24071
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-inf2 9092  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607  ax-pre-sup 10608
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-disj 4999  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-of 7393  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-er 8276  df-map 8395  df-pm 8396  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-fi 8863  df-sup 8894  df-inf 8895  df-oi 8962  df-dju 9318  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-div 11291  df-nn 11630  df-2 11692  df-3 11693  df-n0 11890  df-z 11974  df-uz 12236  df-q 12341  df-rp 12382  df-xneg 12499  df-xadd 12500  df-xmul 12501  df-ioo 12734  df-ico 12736  df-icc 12737  df-fz 12890  df-fzo 13033  df-fl 13161  df-seq 13369  df-exp 13430  df-hash 13691  df-cj 14454  df-re 14455  df-im 14456  df-sqrt 14590  df-abs 14591  df-clim 14841  df-rlim 14842  df-sum 15039  df-rest 16692  df-topgen 16713  df-psmet 20087  df-xmet 20088  df-met 20089  df-bl 20090  df-mopn 20091  df-top 21503  df-topon 21520  df-bases 21555  df-cmp 21996  df-ovol 24072  df-vol 24073
This theorem is referenced by:  uniiccvol  24188  uniioombllem2  24191
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