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Theorem uniioovol 23547
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 23522.) 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 3765 . . 3 ran ((,) ∘ 𝐹) ⊆ ran ((,) ∘ 𝐹)
3 uniioombl.3 . . . 4 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
43ovollb 23447 . . 3 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ran ((,) ∘ 𝐹) ⊆ ran ((,) ∘ 𝐹)) → (vol*‘ ran ((,) ∘ 𝐹)) ≤ sup(ran 𝑆, ℝ*, < ))
51, 2, 4sylancl 697 . 2 (𝜑 → (vol*‘ ran ((,) ∘ 𝐹)) ≤ sup(ran 𝑆, ℝ*, < ))
61adantr 472 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
7 elfznn 12563 . . . . . . . . . . 11 (𝑥 ∈ (1...𝑛) → 𝑥 ∈ ℕ)
8 eqid 2760 . . . . . . . . . . . 12 ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
98ovolfsval 23439 . . . . . . . . . . 11 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑥) = ((2nd ‘(𝐹𝑥)) − (1st ‘(𝐹𝑥))))
106, 7, 9syl2an 495 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (((abs ∘ − ) ∘ 𝐹)‘𝑥) = ((2nd ‘(𝐹𝑥)) − (1st ‘(𝐹𝑥))))
11 fvco3 6437 . . . . . . . . . . . . . . 15 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹𝑥)))
126, 7, 11syl2an 495 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹𝑥)))
13 inss2 3977 . . . . . . . . . . . . . . . . . 18 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
14 ffvelrn 6520 . . . . . . . . . . . . . . . . . . 19 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (𝐹𝑥) ∈ ( ≤ ∩ (ℝ × ℝ)))
156, 7, 14syl2an 495 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (𝐹𝑥) ∈ ( ≤ ∩ (ℝ × ℝ)))
1613, 15sseldi 3742 . . . . . . . . . . . . . . . . 17 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (𝐹𝑥) ∈ (ℝ × ℝ))
17 1st2nd2 7372 . . . . . . . . . . . . . . . . 17 ((𝐹𝑥) ∈ (ℝ × ℝ) → (𝐹𝑥) = ⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
1816, 17syl 17 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (𝐹𝑥) = ⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
1918fveq2d 6356 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → ((,)‘(𝐹𝑥)) = ((,)‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩))
20 df-ov 6816 . . . . . . . . . . . . . . 15 ((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))) = ((,)‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
2119, 20syl6eqr 2812 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → ((,)‘(𝐹𝑥)) = ((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))))
2212, 21eqtrd 2794 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (((,) ∘ 𝐹)‘𝑥) = ((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))))
23 ioombl 23533 . . . . . . . . . . . . 13 ((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))) ∈ dom vol
2422, 23syl6eqel 2847 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (((,) ∘ 𝐹)‘𝑥) ∈ dom vol)
25 mblvol 23498 . . . . . . . . . . . 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 6356 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (vol*‘(((,) ∘ 𝐹)‘𝑥)) = (vol*‘((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥)))))
28 ovolfcl 23435 . . . . . . . . . . . . 13 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((1st ‘(𝐹𝑥)) ∈ ℝ ∧ (2nd ‘(𝐹𝑥)) ∈ ℝ ∧ (1st ‘(𝐹𝑥)) ≤ (2nd ‘(𝐹𝑥))))
296, 7, 28syl2an 495 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → ((1st ‘(𝐹𝑥)) ∈ ℝ ∧ (2nd ‘(𝐹𝑥)) ∈ ℝ ∧ (1st ‘(𝐹𝑥)) ≤ (2nd ‘(𝐹𝑥))))
30 ovolioo 23536 . . . . . . . . . . . 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 2798 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (vol‘(((,) ∘ 𝐹)‘𝑥)) = ((2nd ‘(𝐹𝑥)) − (1st ‘(𝐹𝑥))))
3310, 32eqtr4d 2797 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (((abs ∘ − ) ∘ 𝐹)‘𝑥) = (vol‘(((,) ∘ 𝐹)‘𝑥)))
34 simpr 479 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
35 nnuz 11916 . . . . . . . . . 10 ℕ = (ℤ‘1)
3634, 35syl6eleq 2849 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ‘1))
3729simp2d 1138 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (2nd ‘(𝐹𝑥)) ∈ ℝ)
3829simp1d 1137 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (1st ‘(𝐹𝑥)) ∈ ℝ)
3937, 38resubcld 10650 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → ((2nd ‘(𝐹𝑥)) − (1st ‘(𝐹𝑥))) ∈ ℝ)
4032, 39eqeltrd 2839 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (vol‘(((,) ∘ 𝐹)‘𝑥)) ∈ ℝ)
4140recnd 10260 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (vol‘(((,) ∘ 𝐹)‘𝑥)) ∈ ℂ)
4233, 36, 41fsumser 14660 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → Σ𝑥 ∈ (1...𝑛)(vol‘(((,) ∘ 𝐹)‘𝑥)) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑛))
433fveq1i 6353 . . . . . . . 8 (𝑆𝑛) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑛)
4442, 43syl6reqr 2813 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (𝑆𝑛) = Σ𝑥 ∈ (1...𝑛)(vol‘(((,) ∘ 𝐹)‘𝑥)))
45 fzfid 12966 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (1...𝑛) ∈ Fin)
4624, 40jca 555 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → ((((,) ∘ 𝐹)‘𝑥) ∈ dom vol ∧ (vol‘(((,) ∘ 𝐹)‘𝑥)) ∈ ℝ))
4746ralrimiva 3104 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ∀𝑥 ∈ (1...𝑛)((((,) ∘ 𝐹)‘𝑥) ∈ dom vol ∧ (vol‘(((,) ∘ 𝐹)‘𝑥)) ∈ ℝ))
487ssriv 3748 . . . . . . . . 9 (1...𝑛) ⊆ ℕ
49 uniioombl.2 . . . . . . . . . . 11 (𝜑Disj 𝑥 ∈ ℕ ((,)‘(𝐹𝑥)))
501, 11sylan 489 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹𝑥)))
5150disjeq2dv 4777 . . . . . . . . . . 11 (𝜑 → (Disj 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) ↔ Disj 𝑥 ∈ ℕ ((,)‘(𝐹𝑥))))
5249, 51mpbird 247 . . . . . . . . . 10 (𝜑Disj 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥))
5352adantr 472 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → Disj 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥))
54 disjss1 4778 . . . . . . . . 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 23515 . . . . . . . 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 1477 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (vol‘ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥)) = Σ𝑥 ∈ (1...𝑛)(vol‘(((,) ∘ 𝐹)‘𝑥)))
5824ralrimiva 3104 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ∀𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ∈ dom vol)
59 finiunmbl 23512 . . . . . . . . 9 (((1...𝑛) ∈ Fin ∧ ∀𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ∈ dom vol) → 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ∈ dom vol)
6045, 58, 59syl2anc 696 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ∈ dom vol)
61 mblvol 23498 . . . . . . . 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 2800 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝑆𝑛) = (vol*‘ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥)))
64 iunss1 4684 . . . . . . . . 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 12464 . . . . . . . . . . 11 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
67 rexpssxrxp 10276 . . . . . . . . . . . . 13 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
6813, 67sstri 3753 . . . . . . . . . . . 12 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
69 fss 6217 . . . . . . . . . . . 12 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)) → 𝐹:ℕ⟶(ℝ* × ℝ*))
701, 68, 69sylancl 697 . . . . . . . . . . 11 (𝜑𝐹:ℕ⟶(ℝ* × ℝ*))
71 fco 6219 . . . . . . . . . . 11 (((,):(ℝ* × ℝ*)⟶𝒫 ℝ ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
7266, 70, 71sylancr 698 . . . . . . . . . 10 (𝜑 → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
7372adantr 472 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
74 ffn 6206 . . . . . . . . 9 (((,) ∘ 𝐹):ℕ⟶𝒫 ℝ → ((,) ∘ 𝐹) Fn ℕ)
75 fniunfv 6668 . . . . . . . . 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 3782 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ⊆ ran ((,) ∘ 𝐹))
78 frn 6214 . . . . . . . . . 10 (((,) ∘ 𝐹):ℕ⟶𝒫 ℝ → ran ((,) ∘ 𝐹) ⊆ 𝒫 ℝ)
7972, 78syl 17 . . . . . . . . 9 (𝜑 → ran ((,) ∘ 𝐹) ⊆ 𝒫 ℝ)
80 sspwuni 4763 . . . . . . . . 9 (ran ((,) ∘ 𝐹) ⊆ 𝒫 ℝ ↔ ran ((,) ∘ 𝐹) ⊆ ℝ)
8179, 80sylib 208 . . . . . . . 8 (𝜑 ran ((,) ∘ 𝐹) ⊆ ℝ)
8281adantr 472 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → ran ((,) ∘ 𝐹) ⊆ ℝ)
83 ovolss 23453 . . . . . . 7 (( 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ⊆ ran ((,) ∘ 𝐹) ∧ ran ((,) ∘ 𝐹) ⊆ ℝ) → (vol*‘ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥)) ≤ (vol*‘ ran ((,) ∘ 𝐹)))
8477, 82, 83syl2anc 696 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (vol*‘ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥)) ≤ (vol*‘ ran ((,) ∘ 𝐹)))
8563, 84eqbrtrd 4826 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (𝑆𝑛) ≤ (vol*‘ ran ((,) ∘ 𝐹)))
8685ralrimiva 3104 . . . 4 (𝜑 → ∀𝑛 ∈ ℕ (𝑆𝑛) ≤ (vol*‘ ran ((,) ∘ 𝐹)))
878, 3ovolsf 23441 . . . . . 6 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
881, 87syl 17 . . . . 5 (𝜑𝑆:ℕ⟶(0[,)+∞))
89 ffn 6206 . . . . 5 (𝑆:ℕ⟶(0[,)+∞) → 𝑆 Fn ℕ)
90 breq1 4807 . . . . . 6 (𝑦 = (𝑆𝑛) → (𝑦 ≤ (vol*‘ ran ((,) ∘ 𝐹)) ↔ (𝑆𝑛) ≤ (vol*‘ ran ((,) ∘ 𝐹))))
9190ralrn 6525 . . . . 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 6214 . . . . . 6 (𝑆:ℕ⟶(0[,)+∞) → ran 𝑆 ⊆ (0[,)+∞))
951, 87, 943syl 18 . . . . 5 (𝜑 → ran 𝑆 ⊆ (0[,)+∞))
96 icossxr 12451 . . . . 5 (0[,)+∞) ⊆ ℝ*
9795, 96syl6ss 3756 . . . 4 (𝜑 → ran 𝑆 ⊆ ℝ*)
98 ovolcl 23446 . . . . 5 ( ran ((,) ∘ 𝐹) ⊆ ℝ → (vol*‘ ran ((,) ∘ 𝐹)) ∈ ℝ*)
9981, 98syl 17 . . . 4 (𝜑 → (vol*‘ ran ((,) ∘ 𝐹)) ∈ ℝ*)
100 supxrleub 12349 . . . 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 696 . . 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 12338 . . . 4 (ran 𝑆 ⊆ ℝ* → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
10497, 103syl 17 . . 3 (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
105 xrletri3 12178 . . 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 696 . 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 995 1 (𝜑 → (vol*‘ ran ((,) ∘ 𝐹)) = sup(ran 𝑆, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  cin 3714  wss 3715  𝒫 cpw 4302  cop 4327   cuni 4588   ciun 4672  Disj wdisj 4772   class class class wbr 4804   × cxp 5264  dom cdm 5266  ran crn 5267  ccom 5270   Fn wfn 6044  wf 6045  cfv 6049  (class class class)co 6813  1st c1st 7331  2nd c2nd 7332  Fincfn 8121  supcsup 8511  cr 10127  0cc0 10128  1c1 10129   + caddc 10131  +∞cpnf 10263  *cxr 10265   < clt 10266  cle 10267  cmin 10458  cn 11212  cuz 11879  (,)cioo 12368  [,)cico 12370  ...cfz 12519  seqcseq 12995  abscabs 14173  Σcsu 14615  vol*covol 23431  volcvol 23432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205  ax-pre-sup 10206
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-disj 4773  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-of 7062  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-2o 7730  df-oadd 7733  df-er 7911  df-map 8025  df-pm 8026  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-fi 8482  df-sup 8513  df-inf 8514  df-oi 8580  df-card 8955  df-cda 9182  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-div 10877  df-nn 11213  df-2 11271  df-3 11272  df-n0 11485  df-z 11570  df-uz 11880  df-q 11982  df-rp 12026  df-xneg 12139  df-xadd 12140  df-xmul 12141  df-ioo 12372  df-ico 12374  df-icc 12375  df-fz 12520  df-fzo 12660  df-fl 12787  df-seq 12996  df-exp 13055  df-hash 13312  df-cj 14038  df-re 14039  df-im 14040  df-sqrt 14174  df-abs 14175  df-clim 14418  df-rlim 14419  df-sum 14616  df-rest 16285  df-topgen 16306  df-psmet 19940  df-xmet 19941  df-met 19942  df-bl 19943  df-mopn 19944  df-top 20901  df-topon 20918  df-bases 20952  df-cmp 21392  df-ovol 23433  df-vol 23434
This theorem is referenced by:  uniiccvol  23548  uniioombllem2  23551
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