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Theorem uniioovol 23637
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 23612.) 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 3783 . . 3 ran ((,) ∘ 𝐹) ⊆ ran ((,) ∘ 𝐹)
3 uniioombl.3 . . . 4 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
43ovollb 23537 . . 3 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ran ((,) ∘ 𝐹) ⊆ ran ((,) ∘ 𝐹)) → (vol*‘ ran ((,) ∘ 𝐹)) ≤ sup(ran 𝑆, ℝ*, < ))
51, 2, 4sylancl 580 . 2 (𝜑 → (vol*‘ ran ((,) ∘ 𝐹)) ≤ sup(ran 𝑆, ℝ*, < ))
61adantr 472 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
7 elfznn 12577 . . . . . . . . . . 11 (𝑥 ∈ (1...𝑛) → 𝑥 ∈ ℕ)
8 eqid 2765 . . . . . . . . . . . 12 ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
98ovolfsval 23528 . . . . . . . . . . 11 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑥) = ((2nd ‘(𝐹𝑥)) − (1st ‘(𝐹𝑥))))
106, 7, 9syl2an 589 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (((abs ∘ − ) ∘ 𝐹)‘𝑥) = ((2nd ‘(𝐹𝑥)) − (1st ‘(𝐹𝑥))))
11 fvco3 6464 . . . . . . . . . . . . . . 15 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹𝑥)))
126, 7, 11syl2an 589 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹𝑥)))
13 inss2 3993 . . . . . . . . . . . . . . . . . 18 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
14 ffvelrn 6547 . . . . . . . . . . . . . . . . . . 19 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (𝐹𝑥) ∈ ( ≤ ∩ (ℝ × ℝ)))
156, 7, 14syl2an 589 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (𝐹𝑥) ∈ ( ≤ ∩ (ℝ × ℝ)))
1613, 15sseldi 3759 . . . . . . . . . . . . . . . . 17 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (𝐹𝑥) ∈ (ℝ × ℝ))
17 1st2nd2 7405 . . . . . . . . . . . . . . . . 17 ((𝐹𝑥) ∈ (ℝ × ℝ) → (𝐹𝑥) = ⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
1816, 17syl 17 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (𝐹𝑥) = ⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
1918fveq2d 6379 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → ((,)‘(𝐹𝑥)) = ((,)‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩))
20 df-ov 6845 . . . . . . . . . . . . . . 15 ((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))) = ((,)‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
2119, 20syl6eqr 2817 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → ((,)‘(𝐹𝑥)) = ((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))))
2212, 21eqtrd 2799 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (((,) ∘ 𝐹)‘𝑥) = ((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))))
23 ioombl 23623 . . . . . . . . . . . . 13 ((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))) ∈ dom vol
2422, 23syl6eqel 2852 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (((,) ∘ 𝐹)‘𝑥) ∈ dom vol)
25 mblvol 23588 . . . . . . . . . . . 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 6379 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (vol*‘(((,) ∘ 𝐹)‘𝑥)) = (vol*‘((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥)))))
28 ovolfcl 23524 . . . . . . . . . . . . 13 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((1st ‘(𝐹𝑥)) ∈ ℝ ∧ (2nd ‘(𝐹𝑥)) ∈ ℝ ∧ (1st ‘(𝐹𝑥)) ≤ (2nd ‘(𝐹𝑥))))
296, 7, 28syl2an 589 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → ((1st ‘(𝐹𝑥)) ∈ ℝ ∧ (2nd ‘(𝐹𝑥)) ∈ ℝ ∧ (1st ‘(𝐹𝑥)) ≤ (2nd ‘(𝐹𝑥))))
30 ovolioo 23626 . . . . . . . . . . . 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 2803 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (vol‘(((,) ∘ 𝐹)‘𝑥)) = ((2nd ‘(𝐹𝑥)) − (1st ‘(𝐹𝑥))))
3310, 32eqtr4d 2802 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (((abs ∘ − ) ∘ 𝐹)‘𝑥) = (vol‘(((,) ∘ 𝐹)‘𝑥)))
34 simpr 477 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
35 nnuz 11923 . . . . . . . . . 10 ℕ = (ℤ‘1)
3634, 35syl6eleq 2854 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ‘1))
3729simp2d 1173 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (2nd ‘(𝐹𝑥)) ∈ ℝ)
3829simp1d 1172 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (1st ‘(𝐹𝑥)) ∈ ℝ)
3937, 38resubcld 10712 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → ((2nd ‘(𝐹𝑥)) − (1st ‘(𝐹𝑥))) ∈ ℝ)
4032, 39eqeltrd 2844 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (vol‘(((,) ∘ 𝐹)‘𝑥)) ∈ ℝ)
4140recnd 10322 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (vol‘(((,) ∘ 𝐹)‘𝑥)) ∈ ℂ)
4233, 36, 41fsumser 14746 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → Σ𝑥 ∈ (1...𝑛)(vol‘(((,) ∘ 𝐹)‘𝑥)) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑛))
433fveq1i 6376 . . . . . . . 8 (𝑆𝑛) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑛)
4442, 43syl6reqr 2818 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (𝑆𝑛) = Σ𝑥 ∈ (1...𝑛)(vol‘(((,) ∘ 𝐹)‘𝑥)))
45 fzfid 12980 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (1...𝑛) ∈ Fin)
4624, 40jca 507 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → ((((,) ∘ 𝐹)‘𝑥) ∈ dom vol ∧ (vol‘(((,) ∘ 𝐹)‘𝑥)) ∈ ℝ))
4746ralrimiva 3113 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ∀𝑥 ∈ (1...𝑛)((((,) ∘ 𝐹)‘𝑥) ∈ dom vol ∧ (vol‘(((,) ∘ 𝐹)‘𝑥)) ∈ ℝ))
487ssriv 3765 . . . . . . . . 9 (1...𝑛) ⊆ ℕ
49 uniioombl.2 . . . . . . . . . . 11 (𝜑Disj 𝑥 ∈ ℕ ((,)‘(𝐹𝑥)))
501, 11sylan 575 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹𝑥)))
5150disjeq2dv 4782 . . . . . . . . . . 11 (𝜑 → (Disj 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) ↔ Disj 𝑥 ∈ ℕ ((,)‘(𝐹𝑥))))
5249, 51mpbird 248 . . . . . . . . . 10 (𝜑Disj 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥))
5352adantr 472 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → Disj 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥))
54 disjss1 4783 . . . . . . . . 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 23605 . . . . . . . 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 1490 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (vol‘ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥)) = Σ𝑥 ∈ (1...𝑛)(vol‘(((,) ∘ 𝐹)‘𝑥)))
5824ralrimiva 3113 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ∀𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ∈ dom vol)
59 finiunmbl 23602 . . . . . . . . 9 (((1...𝑛) ∈ Fin ∧ ∀𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ∈ dom vol) → 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ∈ dom vol)
6045, 58, 59syl2anc 579 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ∈ dom vol)
61 mblvol 23588 . . . . . . . 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 2805 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝑆𝑛) = (vol*‘ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥)))
64 iunss1 4688 . . . . . . . . 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 12474 . . . . . . . . . . 11 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
67 rexpssxrxp 10338 . . . . . . . . . . . . 13 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
6813, 67sstri 3770 . . . . . . . . . . . 12 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
69 fss 6236 . . . . . . . . . . . 12 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)) → 𝐹:ℕ⟶(ℝ* × ℝ*))
701, 68, 69sylancl 580 . . . . . . . . . . 11 (𝜑𝐹:ℕ⟶(ℝ* × ℝ*))
71 fco 6240 . . . . . . . . . . 11 (((,):(ℝ* × ℝ*)⟶𝒫 ℝ ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
7266, 70, 71sylancr 581 . . . . . . . . . 10 (𝜑 → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
7372adantr 472 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
74 ffn 6223 . . . . . . . . 9 (((,) ∘ 𝐹):ℕ⟶𝒫 ℝ → ((,) ∘ 𝐹) Fn ℕ)
75 fniunfv 6697 . . . . . . . . 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 3801 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ⊆ ran ((,) ∘ 𝐹))
7872frnd 6230 . . . . . . . . 9 (𝜑 → ran ((,) ∘ 𝐹) ⊆ 𝒫 ℝ)
79 sspwuni 4768 . . . . . . . . 9 (ran ((,) ∘ 𝐹) ⊆ 𝒫 ℝ ↔ ran ((,) ∘ 𝐹) ⊆ ℝ)
8078, 79sylib 209 . . . . . . . 8 (𝜑 ran ((,) ∘ 𝐹) ⊆ ℝ)
8180adantr 472 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → ran ((,) ∘ 𝐹) ⊆ ℝ)
82 ovolss 23543 . . . . . . 7 (( 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ⊆ ran ((,) ∘ 𝐹) ∧ ran ((,) ∘ 𝐹) ⊆ ℝ) → (vol*‘ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥)) ≤ (vol*‘ ran ((,) ∘ 𝐹)))
8377, 81, 82syl2anc 579 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (vol*‘ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥)) ≤ (vol*‘ ran ((,) ∘ 𝐹)))
8463, 83eqbrtrd 4831 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (𝑆𝑛) ≤ (vol*‘ ran ((,) ∘ 𝐹)))
8584ralrimiva 3113 . . . 4 (𝜑 → ∀𝑛 ∈ ℕ (𝑆𝑛) ≤ (vol*‘ ran ((,) ∘ 𝐹)))
868, 3ovolsf 23530 . . . . . 6 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
871, 86syl 17 . . . . 5 (𝜑𝑆:ℕ⟶(0[,)+∞))
88 ffn 6223 . . . . 5 (𝑆:ℕ⟶(0[,)+∞) → 𝑆 Fn ℕ)
89 breq1 4812 . . . . . 6 (𝑦 = (𝑆𝑛) → (𝑦 ≤ (vol*‘ ran ((,) ∘ 𝐹)) ↔ (𝑆𝑛) ≤ (vol*‘ ran ((,) ∘ 𝐹))))
9089ralrn 6552 . . . . 5 (𝑆 Fn ℕ → (∀𝑦 ∈ ran 𝑆 𝑦 ≤ (vol*‘ ran ((,) ∘ 𝐹)) ↔ ∀𝑛 ∈ ℕ (𝑆𝑛) ≤ (vol*‘ ran ((,) ∘ 𝐹))))
9187, 88, 903syl 18 . . . 4 (𝜑 → (∀𝑦 ∈ ran 𝑆 𝑦 ≤ (vol*‘ ran ((,) ∘ 𝐹)) ↔ ∀𝑛 ∈ ℕ (𝑆𝑛) ≤ (vol*‘ ran ((,) ∘ 𝐹))))
9285, 91mpbird 248 . . 3 (𝜑 → ∀𝑦 ∈ ran 𝑆 𝑦 ≤ (vol*‘ ran ((,) ∘ 𝐹)))
93 frn 6229 . . . . . 6 (𝑆:ℕ⟶(0[,)+∞) → ran 𝑆 ⊆ (0[,)+∞))
941, 86, 933syl 18 . . . . 5 (𝜑 → ran 𝑆 ⊆ (0[,)+∞))
95 icossxr 12460 . . . . 5 (0[,)+∞) ⊆ ℝ*
9694, 95syl6ss 3773 . . . 4 (𝜑 → ran 𝑆 ⊆ ℝ*)
97 ovolcl 23536 . . . . 5 ( ran ((,) ∘ 𝐹) ⊆ ℝ → (vol*‘ ran ((,) ∘ 𝐹)) ∈ ℝ*)
9880, 97syl 17 . . . 4 (𝜑 → (vol*‘ ran ((,) ∘ 𝐹)) ∈ ℝ*)
99 supxrleub 12358 . . . 4 ((ran 𝑆 ⊆ ℝ* ∧ (vol*‘ ran ((,) ∘ 𝐹)) ∈ ℝ*) → (sup(ran 𝑆, ℝ*, < ) ≤ (vol*‘ ran ((,) ∘ 𝐹)) ↔ ∀𝑦 ∈ ran 𝑆 𝑦 ≤ (vol*‘ ran ((,) ∘ 𝐹))))
10096, 98, 99syl2anc 579 . . 3 (𝜑 → (sup(ran 𝑆, ℝ*, < ) ≤ (vol*‘ ran ((,) ∘ 𝐹)) ↔ ∀𝑦 ∈ ran 𝑆 𝑦 ≤ (vol*‘ ran ((,) ∘ 𝐹))))
10192, 100mpbird 248 . 2 (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ (vol*‘ ran ((,) ∘ 𝐹)))
102 supxrcl 12347 . . . 4 (ran 𝑆 ⊆ ℝ* → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
10396, 102syl 17 . . 3 (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
104 xrletri3 12187 . . 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)))))
10598, 103, 104syl2anc 579 . 2 (𝜑 → ((vol*‘ ran ((,) ∘ 𝐹)) = sup(ran 𝑆, ℝ*, < ) ↔ ((vol*‘ ran ((,) ∘ 𝐹)) ≤ sup(ran 𝑆, ℝ*, < ) ∧ sup(ran 𝑆, ℝ*, < ) ≤ (vol*‘ ran ((,) ∘ 𝐹)))))
1065, 101, 105mpbir2and 704 1 (𝜑 → (vol*‘ ran ((,) ∘ 𝐹)) = sup(ran 𝑆, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wral 3055  cin 3731  wss 3732  𝒫 cpw 4315  cop 4340   cuni 4594   ciun 4676  Disj wdisj 4777   class class class wbr 4809   × cxp 5275  dom cdm 5277  ran crn 5278  ccom 5281   Fn wfn 6063  wf 6064  cfv 6068  (class class class)co 6842  1st c1st 7364  2nd c2nd 7365  Fincfn 8160  supcsup 8553  cr 10188  0cc0 10189  1c1 10190   + caddc 10192  +∞cpnf 10325  *cxr 10327   < clt 10328  cle 10329  cmin 10520  cn 11274  cuz 11886  (,)cioo 12377  [,)cico 12379  ...cfz 12533  seqcseq 13008  abscabs 14259  Σcsu 14701  vol*covol 23520  volcvol 23521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266  ax-pre-sup 10267
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-disj 4778  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-of 7095  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-2o 7765  df-oadd 7768  df-er 7947  df-map 8062  df-pm 8063  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-fi 8524  df-sup 8555  df-inf 8556  df-oi 8622  df-card 9016  df-cda 9243  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-div 10939  df-nn 11275  df-2 11335  df-3 11336  df-n0 11539  df-z 11625  df-uz 11887  df-q 11990  df-rp 12029  df-xneg 12146  df-xadd 12147  df-xmul 12148  df-ioo 12381  df-ico 12383  df-icc 12384  df-fz 12534  df-fzo 12674  df-fl 12801  df-seq 13009  df-exp 13068  df-hash 13322  df-cj 14124  df-re 14125  df-im 14126  df-sqrt 14260  df-abs 14261  df-clim 14504  df-rlim 14505  df-sum 14702  df-rest 16349  df-topgen 16370  df-psmet 20011  df-xmet 20012  df-met 20013  df-bl 20014  df-mopn 20015  df-top 20978  df-topon 20995  df-bases 21030  df-cmp 21470  df-ovol 23522  df-vol 23523
This theorem is referenced by:  uniiccvol  23638  uniioombllem2  23641
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