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Theorem uniioovol 23253
 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 23229.) 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 uniioombl.1 . . 3 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
2 ssid 3603 . . 3 ran ((,) ∘ 𝐹) ⊆ ran ((,) ∘ 𝐹)
3 uniioombl.3 . . . 4 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
43ovollb 23154 . . 3 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ran ((,) ∘ 𝐹) ⊆ ran ((,) ∘ 𝐹)) → (vol*‘ ran ((,) ∘ 𝐹)) ≤ sup(ran 𝑆, ℝ*, < ))
51, 2, 4sylancl 693 . 2 (𝜑 → (vol*‘ ran ((,) ∘ 𝐹)) ≤ sup(ran 𝑆, ℝ*, < ))
61adantr 481 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
7 elfznn 12312 . . . . . . . . . . 11 (𝑥 ∈ (1...𝑛) → 𝑥 ∈ ℕ)
8 eqid 2621 . . . . . . . . . . . 12 ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
98ovolfsval 23146 . . . . . . . . . . 11 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑥) = ((2nd ‘(𝐹𝑥)) − (1st ‘(𝐹𝑥))))
106, 7, 9syl2an 494 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (((abs ∘ − ) ∘ 𝐹)‘𝑥) = ((2nd ‘(𝐹𝑥)) − (1st ‘(𝐹𝑥))))
11 fvco3 6232 . . . . . . . . . . . . . . 15 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹𝑥)))
126, 7, 11syl2an 494 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹𝑥)))
13 inss2 3812 . . . . . . . . . . . . . . . . . 18 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
14 ffvelrn 6313 . . . . . . . . . . . . . . . . . . 19 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (𝐹𝑥) ∈ ( ≤ ∩ (ℝ × ℝ)))
156, 7, 14syl2an 494 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (𝐹𝑥) ∈ ( ≤ ∩ (ℝ × ℝ)))
1613, 15sseldi 3581 . . . . . . . . . . . . . . . . 17 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (𝐹𝑥) ∈ (ℝ × ℝ))
17 1st2nd2 7150 . . . . . . . . . . . . . . . . 17 ((𝐹𝑥) ∈ (ℝ × ℝ) → (𝐹𝑥) = ⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
1816, 17syl 17 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (𝐹𝑥) = ⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
1918fveq2d 6152 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → ((,)‘(𝐹𝑥)) = ((,)‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩))
20 df-ov 6607 . . . . . . . . . . . . . . 15 ((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))) = ((,)‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
2119, 20syl6eqr 2673 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → ((,)‘(𝐹𝑥)) = ((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))))
2212, 21eqtrd 2655 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (((,) ∘ 𝐹)‘𝑥) = ((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))))
23 ioombl 23240 . . . . . . . . . . . . 13 ((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))) ∈ dom vol
2422, 23syl6eqel 2706 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (((,) ∘ 𝐹)‘𝑥) ∈ dom vol)
25 mblvol 23205 . . . . . . . . . . . 12 ((((,) ∘ 𝐹)‘𝑥) ∈ dom vol → (vol‘(((,) ∘ 𝐹)‘𝑥)) = (vol*‘(((,) ∘ 𝐹)‘𝑥)))
2624, 25syl 17 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (vol‘(((,) ∘ 𝐹)‘𝑥)) = (vol*‘(((,) ∘ 𝐹)‘𝑥)))
2722fveq2d 6152 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (vol*‘(((,) ∘ 𝐹)‘𝑥)) = (vol*‘((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥)))))
28 ovolfcl 23142 . . . . . . . . . . . . 13 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((1st ‘(𝐹𝑥)) ∈ ℝ ∧ (2nd ‘(𝐹𝑥)) ∈ ℝ ∧ (1st ‘(𝐹𝑥)) ≤ (2nd ‘(𝐹𝑥))))
296, 7, 28syl2an 494 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → ((1st ‘(𝐹𝑥)) ∈ ℝ ∧ (2nd ‘(𝐹𝑥)) ∈ ℝ ∧ (1st ‘(𝐹𝑥)) ≤ (2nd ‘(𝐹𝑥))))
30 ovolioo 23243 . . . . . . . . . . . 12 (((1st ‘(𝐹𝑥)) ∈ ℝ ∧ (2nd ‘(𝐹𝑥)) ∈ ℝ ∧ (1st ‘(𝐹𝑥)) ≤ (2nd ‘(𝐹𝑥))) → (vol*‘((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥)))) = ((2nd ‘(𝐹𝑥)) − (1st ‘(𝐹𝑥))))
3129, 30syl 17 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (vol*‘((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥)))) = ((2nd ‘(𝐹𝑥)) − (1st ‘(𝐹𝑥))))
3226, 27, 313eqtrd 2659 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (vol‘(((,) ∘ 𝐹)‘𝑥)) = ((2nd ‘(𝐹𝑥)) − (1st ‘(𝐹𝑥))))
3310, 32eqtr4d 2658 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (((abs ∘ − ) ∘ 𝐹)‘𝑥) = (vol‘(((,) ∘ 𝐹)‘𝑥)))
34 simpr 477 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
35 nnuz 11667 . . . . . . . . . 10 ℕ = (ℤ‘1)
3634, 35syl6eleq 2708 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ‘1))
3729simp2d 1072 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (2nd ‘(𝐹𝑥)) ∈ ℝ)
3829simp1d 1071 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (1st ‘(𝐹𝑥)) ∈ ℝ)
3937, 38resubcld 10402 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → ((2nd ‘(𝐹𝑥)) − (1st ‘(𝐹𝑥))) ∈ ℝ)
4032, 39eqeltrd 2698 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (vol‘(((,) ∘ 𝐹)‘𝑥)) ∈ ℝ)
4140recnd 10012 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (vol‘(((,) ∘ 𝐹)‘𝑥)) ∈ ℂ)
4233, 36, 41fsumser 14394 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → Σ𝑥 ∈ (1...𝑛)(vol‘(((,) ∘ 𝐹)‘𝑥)) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑛))
433fveq1i 6149 . . . . . . . 8 (𝑆𝑛) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑛)
4442, 43syl6reqr 2674 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (𝑆𝑛) = Σ𝑥 ∈ (1...𝑛)(vol‘(((,) ∘ 𝐹)‘𝑥)))
45 fzfid 12712 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (1...𝑛) ∈ Fin)
4624, 40jca 554 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → ((((,) ∘ 𝐹)‘𝑥) ∈ dom vol ∧ (vol‘(((,) ∘ 𝐹)‘𝑥)) ∈ ℝ))
4746ralrimiva 2960 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ∀𝑥 ∈ (1...𝑛)((((,) ∘ 𝐹)‘𝑥) ∈ dom vol ∧ (vol‘(((,) ∘ 𝐹)‘𝑥)) ∈ ℝ))
487ssriv 3587 . . . . . . . . 9 (1...𝑛) ⊆ ℕ
49 uniioombl.2 . . . . . . . . . . 11 (𝜑Disj 𝑥 ∈ ℕ ((,)‘(𝐹𝑥)))
501, 11sylan 488 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹𝑥)))
5150disjeq2dv 4588 . . . . . . . . . . 11 (𝜑 → (Disj 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) ↔ Disj 𝑥 ∈ ℕ ((,)‘(𝐹𝑥))))
5249, 51mpbird 247 . . . . . . . . . 10 (𝜑Disj 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥))
5352adantr 481 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → Disj 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥))
54 disjss1 4589 . . . . . . . . 9 ((1...𝑛) ⊆ ℕ → (Disj 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) → Disj 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥)))
5548, 53, 54mpsyl 68 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → Disj 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥))
56 volfiniun 23222 . . . . . . . 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)))
5745, 47, 55, 56syl3anc 1323 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (vol‘ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥)) = Σ𝑥 ∈ (1...𝑛)(vol‘(((,) ∘ 𝐹)‘𝑥)))
5824ralrimiva 2960 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ∀𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ∈ dom vol)
59 finiunmbl 23219 . . . . . . . . 9 (((1...𝑛) ∈ Fin ∧ ∀𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ∈ dom vol) → 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ∈ dom vol)
6045, 58, 59syl2anc 692 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ∈ dom vol)
61 mblvol 23205 . . . . . . . 8 ( 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ∈ dom vol → (vol‘ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥)) = (vol*‘ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥)))
6260, 61syl 17 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (vol‘ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥)) = (vol*‘ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥)))
6344, 57, 623eqtr2d 2661 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝑆𝑛) = (vol*‘ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥)))
64 iunss1 4498 . . . . . . . . 9 ((1...𝑛) ⊆ ℕ → 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ⊆ 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥))
6548, 64mp1i 13 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ⊆ 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥))
66 ioof 12213 . . . . . . . . . . 11 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
67 rexpssxrxp 10028 . . . . . . . . . . . . 13 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
6813, 67sstri 3592 . . . . . . . . . . . 12 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
69 fss 6013 . . . . . . . . . . . 12 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)) → 𝐹:ℕ⟶(ℝ* × ℝ*))
701, 68, 69sylancl 693 . . . . . . . . . . 11 (𝜑𝐹:ℕ⟶(ℝ* × ℝ*))
71 fco 6015 . . . . . . . . . . 11 (((,):(ℝ* × ℝ*)⟶𝒫 ℝ ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
7266, 70, 71sylancr 694 . . . . . . . . . 10 (𝜑 → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
7372adantr 481 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
74 ffn 6002 . . . . . . . . 9 (((,) ∘ 𝐹):ℕ⟶𝒫 ℝ → ((,) ∘ 𝐹) Fn ℕ)
75 fniunfv 6459 . . . . . . . . 9 (((,) ∘ 𝐹) Fn ℕ → 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) = ran ((,) ∘ 𝐹))
7673, 74, 753syl 18 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) = ran ((,) ∘ 𝐹))
7765, 76sseqtrd 3620 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ⊆ ran ((,) ∘ 𝐹))
78 frn 6010 . . . . . . . . . 10 (((,) ∘ 𝐹):ℕ⟶𝒫 ℝ → ran ((,) ∘ 𝐹) ⊆ 𝒫 ℝ)
7972, 78syl 17 . . . . . . . . 9 (𝜑 → ran ((,) ∘ 𝐹) ⊆ 𝒫 ℝ)
80 sspwuni 4577 . . . . . . . . 9 (ran ((,) ∘ 𝐹) ⊆ 𝒫 ℝ ↔ ran ((,) ∘ 𝐹) ⊆ ℝ)
8179, 80sylib 208 . . . . . . . 8 (𝜑 ran ((,) ∘ 𝐹) ⊆ ℝ)
8281adantr 481 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → ran ((,) ∘ 𝐹) ⊆ ℝ)
83 ovolss 23160 . . . . . . 7 (( 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ⊆ ran ((,) ∘ 𝐹) ∧ ran ((,) ∘ 𝐹) ⊆ ℝ) → (vol*‘ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥)) ≤ (vol*‘ ran ((,) ∘ 𝐹)))
8477, 82, 83syl2anc 692 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (vol*‘ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥)) ≤ (vol*‘ ran ((,) ∘ 𝐹)))
8563, 84eqbrtrd 4635 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (𝑆𝑛) ≤ (vol*‘ ran ((,) ∘ 𝐹)))
8685ralrimiva 2960 . . . 4 (𝜑 → ∀𝑛 ∈ ℕ (𝑆𝑛) ≤ (vol*‘ ran ((,) ∘ 𝐹)))
878, 3ovolsf 23148 . . . . . 6 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
881, 87syl 17 . . . . 5 (𝜑𝑆:ℕ⟶(0[,)+∞))
89 ffn 6002 . . . . 5 (𝑆:ℕ⟶(0[,)+∞) → 𝑆 Fn ℕ)
90 breq1 4616 . . . . . 6 (𝑦 = (𝑆𝑛) → (𝑦 ≤ (vol*‘ ran ((,) ∘ 𝐹)) ↔ (𝑆𝑛) ≤ (vol*‘ ran ((,) ∘ 𝐹))))
9190ralrn 6318 . . . . 5 (𝑆 Fn ℕ → (∀𝑦 ∈ ran 𝑆 𝑦 ≤ (vol*‘ ran ((,) ∘ 𝐹)) ↔ ∀𝑛 ∈ ℕ (𝑆𝑛) ≤ (vol*‘ ran ((,) ∘ 𝐹))))
9288, 89, 913syl 18 . . . 4 (𝜑 → (∀𝑦 ∈ ran 𝑆 𝑦 ≤ (vol*‘ ran ((,) ∘ 𝐹)) ↔ ∀𝑛 ∈ ℕ (𝑆𝑛) ≤ (vol*‘ ran ((,) ∘ 𝐹))))
9386, 92mpbird 247 . . 3 (𝜑 → ∀𝑦 ∈ ran 𝑆 𝑦 ≤ (vol*‘ ran ((,) ∘ 𝐹)))
94 frn 6010 . . . . . 6 (𝑆:ℕ⟶(0[,)+∞) → ran 𝑆 ⊆ (0[,)+∞))
951, 87, 943syl 18 . . . . 5 (𝜑 → ran 𝑆 ⊆ (0[,)+∞))
96 icossxr 12200 . . . . 5 (0[,)+∞) ⊆ ℝ*
9795, 96syl6ss 3595 . . . 4 (𝜑 → ran 𝑆 ⊆ ℝ*)
98 ovolcl 23153 . . . . 5 ( ran ((,) ∘ 𝐹) ⊆ ℝ → (vol*‘ ran ((,) ∘ 𝐹)) ∈ ℝ*)
9981, 98syl 17 . . . 4 (𝜑 → (vol*‘ ran ((,) ∘ 𝐹)) ∈ ℝ*)
100 supxrleub 12099 . . . 4 ((ran 𝑆 ⊆ ℝ* ∧ (vol*‘ ran ((,) ∘ 𝐹)) ∈ ℝ*) → (sup(ran 𝑆, ℝ*, < ) ≤ (vol*‘ ran ((,) ∘ 𝐹)) ↔ ∀𝑦 ∈ ran 𝑆 𝑦 ≤ (vol*‘ ran ((,) ∘ 𝐹))))
10197, 99, 100syl2anc 692 . . 3 (𝜑 → (sup(ran 𝑆, ℝ*, < ) ≤ (vol*‘ ran ((,) ∘ 𝐹)) ↔ ∀𝑦 ∈ ran 𝑆 𝑦 ≤ (vol*‘ ran ((,) ∘ 𝐹))))
10293, 101mpbird 247 . 2 (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ (vol*‘ ran ((,) ∘ 𝐹)))
103 supxrcl 12088 . . . 4 (ran 𝑆 ⊆ ℝ* → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
10497, 103syl 17 . . 3 (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
105 xrletri3 11929 . . 3 (((vol*‘ ran ((,) ∘ 𝐹)) ∈ ℝ* ∧ sup(ran 𝑆, ℝ*, < ) ∈ ℝ*) → ((vol*‘ ran ((,) ∘ 𝐹)) = sup(ran 𝑆, ℝ*, < ) ↔ ((vol*‘ ran ((,) ∘ 𝐹)) ≤ sup(ran 𝑆, ℝ*, < ) ∧ sup(ran 𝑆, ℝ*, < ) ≤ (vol*‘ ran ((,) ∘ 𝐹)))))
10699, 104, 105syl2anc 692 . 2 (𝜑 → ((vol*‘ ran ((,) ∘ 𝐹)) = sup(ran 𝑆, ℝ*, < ) ↔ ((vol*‘ ran ((,) ∘ 𝐹)) ≤ sup(ran 𝑆, ℝ*, < ) ∧ sup(ran 𝑆, ℝ*, < ) ≤ (vol*‘ ran ((,) ∘ 𝐹)))))
1075, 102, 106mpbir2and 956 1 (𝜑 → (vol*‘ ran ((,) ∘ 𝐹)) = sup(ran 𝑆, ℝ*, < ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987  ∀wral 2907   ∩ cin 3554   ⊆ wss 3555  𝒫 cpw 4130  ⟨cop 4154  ∪ cuni 4402  ∪ ciun 4485  Disj wdisj 4583   class class class wbr 4613   × cxp 5072  dom cdm 5074  ran crn 5075   ∘ ccom 5078   Fn wfn 5842  ⟶wf 5843  ‘cfv 5847  (class class class)co 6604  1st c1st 7111  2nd c2nd 7112  Fincfn 7899  supcsup 8290  ℝcr 9879  0cc0 9880  1c1 9881   + caddc 9883  +∞cpnf 10015  ℝ*cxr 10017   < clt 10018   ≤ cle 10019   − cmin 10210  ℕcn 10964  ℤ≥cuz 11631  (,)cioo 12117  [,)cico 12119  ...cfz 12268  seqcseq 12741  abscabs 13908  Σcsu 14350  vol*covol 23138  volcvol 23139 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-disj 4584  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-of 6850  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fi 8261  df-sup 8292  df-inf 8293  df-oi 8359  df-card 8709  df-cda 8934  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-n0 11237  df-z 11322  df-uz 11632  df-q 11733  df-rp 11777  df-xneg 11890  df-xadd 11891  df-xmul 11892  df-ioo 12121  df-ico 12123  df-icc 12124  df-fz 12269  df-fzo 12407  df-fl 12533  df-seq 12742  df-exp 12801  df-hash 13058  df-cj 13773  df-re 13774  df-im 13775  df-sqrt 13909  df-abs 13910  df-clim 14153  df-rlim 14154  df-sum 14351  df-rest 16004  df-topgen 16025  df-psmet 19657  df-xmet 19658  df-met 19659  df-bl 19660  df-mopn 19661  df-top 20621  df-bases 20622  df-topon 20623  df-cmp 21100  df-ovol 23140  df-vol 23141 This theorem is referenced by:  uniiccvol  23254  uniioombllem2  23257
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