Theorem uniioovol 25643

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 25618.) Lemma 565Ca of [Fremlin5] p. 213. (Contributed by Mario Carneiro, 26-Mar-2015.)

Hypotheses

Ref	Expression
uniioombl.1	⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
uniioombl.2	⊢ (𝜑 → Disj 𝑥 ∈ ℕ ((,)‘(𝐹‘𝑥)))
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.

Step	Hyp	Ref	Expression
1		ioof 13453	. . . . . 6 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ
2		uniioombl.1	. . . . . . 7 ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
3		inss2 4191	. . . . . . . 8 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
4		rexpssxrxp 11229	. . . . . . . 8 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
5	3, 4	sstri 3947	. . . . . . 7 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
6		fss 6710	. . . . . . 7 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)) → 𝐹:ℕ⟶(ℝ* × ℝ*))
7	2, 5, 6	sylancl 595	. . . . . 6 ⊢ (𝜑 → 𝐹:ℕ⟶(ℝ* × ℝ*))
8		fco 6718	. . . . . 6 ⊢ (((,):(ℝ* × ℝ*)⟶𝒫 ℝ ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
9	1, 7, 8	sylancr 596	. . . . 5 ⊢ (𝜑 → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
10	9	frnd 6702	. . . 4 ⊢ (𝜑 → ran ((,) ∘ 𝐹) ⊆ 𝒫 ℝ)
11		sspwuni 5059	. . . 4 ⊢ (ran ((,) ∘ 𝐹) ⊆ 𝒫 ℝ ↔ ∪ ran ((,) ∘ 𝐹) ⊆ ℝ)
12	10, 11	sylib 220	. . 3 ⊢ (𝜑 → ∪ ran ((,) ∘ 𝐹) ⊆ ℝ)
13		ovolcl 25542	. . 3 ⊢ (∪ ran ((,) ∘ 𝐹) ⊆ ℝ → (vol*‘∪ ran ((,) ∘ 𝐹)) ∈ ℝ*)
14	12, 13	syl 17	. 2 ⊢ (𝜑 → (vol*‘∪ ran ((,) ∘ 𝐹)) ∈ ℝ*)
15		eqid 2764	. . . . . 6 ⊢ ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
16		uniioombl.3	. . . . . 6 ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
17	15, 16	ovolsf 25536	. . . . 5 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
18		frn 6701	. . . . 5 ⊢ (𝑆:ℕ⟶(0[,)+∞) → ran 𝑆 ⊆ (0[,)+∞))
19	2, 17, 18	3syl 18	. . . 4 ⊢ (𝜑 → ran 𝑆 ⊆ (0[,)+∞))
20		icossxr 13438	. . . 4 ⊢ (0[,)+∞) ⊆ ℝ*
21	19, 20	sstrdi 3950	. . 3 ⊢ (𝜑 → ran 𝑆 ⊆ ℝ*)
22		supxrcl 13320	. . 3 ⊢ (ran 𝑆 ⊆ ℝ* → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
23	21, 22	syl 17	. 2 ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
24		ssid 3960	. . 3 ⊢ ∪ ran ((,) ∘ 𝐹) ⊆ ∪ ran ((,) ∘ 𝐹)
25	16	ovollb 25543	. . 3 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ∪ ran ((,) ∘ 𝐹) ⊆ ∪ ran ((,) ∘ 𝐹)) → (vol*‘∪ ran ((,) ∘ 𝐹)) ≤ sup(ran 𝑆, ℝ*, < ))
26	2, 24, 25	sylancl 595	. 2 ⊢ (𝜑 → (vol*‘∪ ran ((,) ∘ 𝐹)) ≤ sup(ran 𝑆, ℝ*, < ))
27	16	fveq1i 6870	. . . . . . . 8 ⊢ (𝑆‘𝑛) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑛)
28	2	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
29		elfznn 13560	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (1...𝑛) → 𝑥 ∈ ℕ)
30	15	ovolfsval 25534	. . . . . . . . . . 11 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑥) = ((2nd ‘(𝐹‘𝑥)) − (1st ‘(𝐹‘𝑥))))
31	28, 29, 30	syl2an 605	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (((abs ∘ − ) ∘ 𝐹)‘𝑥) = ((2nd ‘(𝐹‘𝑥)) − (1st ‘(𝐹‘𝑥))))
32		fvco3 6969	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹‘𝑥)))
33	28, 29, 32	syl2an 605	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹‘𝑥)))
34		ffvelcdm 7064	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (𝐹‘𝑥) ∈ ( ≤ ∩ (ℝ × ℝ)))
35	28, 29, 34	syl2an 605	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (𝐹‘𝑥) ∈ ( ≤ ∩ (ℝ × ℝ)))
36	35	elin2d 4159	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (𝐹‘𝑥) ∈ (ℝ × ℝ))
37		1st2nd2 8011	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹‘𝑥) ∈ (ℝ × ℝ) → (𝐹‘𝑥) = ⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩)
38	36, 37	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (𝐹‘𝑥) = ⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩)
39	38	fveq2d 6873	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → ((,)‘(𝐹‘𝑥)) = ((,)‘⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩))
40		df-ov 7401	. . . . . . . . . . . . . . 15 ⊢ ((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥))) = ((,)‘⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩)
41	39, 40	eqtr4di 2817	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → ((,)‘(𝐹‘𝑥)) = ((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥))))
42	33, 41	eqtrd 2799	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (((,) ∘ 𝐹)‘𝑥) = ((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥))))
43		ioombl 25629	. . . . . . . . . . . . 13 ⊢ ((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥))) ∈ dom vol
44	42, 43	eqeltrdi 2872	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (((,) ∘ 𝐹)‘𝑥) ∈ dom vol)
45		mblvol 25594	. . . . . . . . . . . 12 ⊢ ((((,) ∘ 𝐹)‘𝑥) ∈ dom vol → (vol‘(((,) ∘ 𝐹)‘𝑥)) = (vol*‘(((,) ∘ 𝐹)‘𝑥)))
46	44, 45	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (vol‘(((,) ∘ 𝐹)‘𝑥)) = (vol*‘(((,) ∘ 𝐹)‘𝑥)))
47	42	fveq2d 6873	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (vol*‘(((,) ∘ 𝐹)‘𝑥)) = (vol*‘((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥)))))
48		ovolfcl 25530	. . . . . . . . . . . . 13 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((1st ‘(𝐹‘𝑥)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑥)) ∈ ℝ ∧ (1st ‘(𝐹‘𝑥)) ≤ (2nd ‘(𝐹‘𝑥))))
49	28, 29, 48	syl2an 605	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → ((1st ‘(𝐹‘𝑥)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑥)) ∈ ℝ ∧ (1st ‘(𝐹‘𝑥)) ≤ (2nd ‘(𝐹‘𝑥))))
50		ovolioo 25632	. . . . . . . . . . . 12 ⊢ (((1st ‘(𝐹‘𝑥)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑥)) ∈ ℝ ∧ (1st ‘(𝐹‘𝑥)) ≤ (2nd ‘(𝐹‘𝑥))) → (vol*‘((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥)))) = ((2nd ‘(𝐹‘𝑥)) − (1st ‘(𝐹‘𝑥))))
51	49, 50	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (vol*‘((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥)))) = ((2nd ‘(𝐹‘𝑥)) − (1st ‘(𝐹‘𝑥))))
52	46, 47, 51	3eqtrd 2803	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (vol‘(((,) ∘ 𝐹)‘𝑥)) = ((2nd ‘(𝐹‘𝑥)) − (1st ‘(𝐹‘𝑥))))
53	31, 52	eqtr4d 2802	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (((abs ∘ − ) ∘ 𝐹)‘𝑥) = (vol‘(((,) ∘ 𝐹)‘𝑥)))
54		simpr 488	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
55		nnuz 12880	. . . . . . . . . 10 ⊢ ℕ = (ℤ≥‘1)
56	54, 55	eleqtrdi 2874	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ≥‘1))
57	49	simp2d 1157	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (2nd ‘(𝐹‘𝑥)) ∈ ℝ)
58	49	simp1d 1156	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (1st ‘(𝐹‘𝑥)) ∈ ℝ)
59	57, 58	resubcld 11617	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → ((2nd ‘(𝐹‘𝑥)) − (1st ‘(𝐹‘𝑥))) ∈ ℝ)
60	52, 59	eqeltrd 2864	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (vol‘(((,) ∘ 𝐹)‘𝑥)) ∈ ℝ)
61	60	recnd 11212	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (vol‘(((,) ∘ 𝐹)‘𝑥)) ∈ ℂ)
62	53, 56, 61	fsumser 15759	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Σ𝑥 ∈ (1...𝑛)(vol‘(((,) ∘ 𝐹)‘𝑥)) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑛))
63	27, 62	eqtr4id 2818	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑆‘𝑛) = Σ𝑥 ∈ (1...𝑛)(vol‘(((,) ∘ 𝐹)‘𝑥)))
64		fzfid 13988	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1...𝑛) ∈ Fin)
65	44, 60	jca 519	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → ((((,) ∘ 𝐹)‘𝑥) ∈ dom vol ∧ (vol‘(((,) ∘ 𝐹)‘𝑥)) ∈ ℝ))
66	65	ralrimiva 3156	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑥 ∈ (1...𝑛)((((,) ∘ 𝐹)‘𝑥) ∈ dom vol ∧ (vol‘(((,) ∘ 𝐹)‘𝑥)) ∈ ℝ))
67		fz1ssnn 13562	. . . . . . . . 9 ⊢ (1...𝑛) ⊆ ℕ
68		uniioombl.2	. . . . . . . . . . 11 ⊢ (𝜑 → Disj 𝑥 ∈ ℕ ((,)‘(𝐹‘𝑥)))
69	2, 32	sylan 589	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹‘𝑥)))
70	69	disjeq2dv 5074	. . . . . . . . . . 11 ⊢ (𝜑 → (Disj 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) ↔ Disj 𝑥 ∈ ℕ ((,)‘(𝐹‘𝑥))))
71	68, 70	mpbird 259	. . . . . . . . . 10 ⊢ (𝜑 → Disj 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥))
72	71	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Disj 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥))
73		disjss1 5075	. . . . . . . . 9 ⊢ ((1...𝑛) ⊆ ℕ → (Disj 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) → Disj 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥)))
74	67, 72, 73	mpsyl 68	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Disj 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥))
75		volfiniun 25611	. . . . . . . 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)))
76	64, 66, 74, 75	syl3anc 1392	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol‘∪ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥)) = Σ𝑥 ∈ (1...𝑛)(vol‘(((,) ∘ 𝐹)‘𝑥)))
77	44	ralrimiva 3156	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ∈ dom vol)
78		finiunmbl 25608	. . . . . . . . 9 ⊢ (((1...𝑛) ∈ Fin ∧ ∀𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ∈ dom vol) → ∪ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ∈ dom vol)
79	64, 77, 78	syl2anc 593	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∪ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ∈ dom vol)
80		mblvol 25594	. . . . . . . 8 ⊢ (∪ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ∈ dom vol → (vol‘∪ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥)) = (vol*‘∪ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥)))
81	79, 80	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol‘∪ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥)) = (vol*‘∪ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥)))
82	63, 76, 81	3eqtr2d 2805	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑆‘𝑛) = (vol*‘∪ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥)))
83		iunss1 4966	. . . . . . . . 9 ⊢ ((1...𝑛) ⊆ ℕ → ∪ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ⊆ ∪ 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥))
84	67, 83	mp1i 13	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∪ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ⊆ ∪ 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥))
85	9	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
86		ffn 6693	. . . . . . . . 9 ⊢ (((,) ∘ 𝐹):ℕ⟶𝒫 ℝ → ((,) ∘ 𝐹) Fn ℕ)
87		fniunfv 7233	. . . . . . . . 9 ⊢ (((,) ∘ 𝐹) Fn ℕ → ∪ 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) = ∪ ran ((,) ∘ 𝐹))
88	85, 86, 87	3syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∪ 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) = ∪ ran ((,) ∘ 𝐹))
89	84, 88	sseqtrd 3974	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∪ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ⊆ ∪ ran ((,) ∘ 𝐹))
90	12	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∪ ran ((,) ∘ 𝐹) ⊆ ℝ)
91		ovolss 25549	. . . . . . 7 ⊢ ((∪ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ⊆ ∪ ran ((,) ∘ 𝐹) ∧ ∪ ran ((,) ∘ 𝐹) ⊆ ℝ) → (vol*‘∪ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥)) ≤ (vol*‘∪ ran ((,) ∘ 𝐹)))
92	89, 90, 91	syl2anc 593	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol*‘∪ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥)) ≤ (vol*‘∪ ran ((,) ∘ 𝐹)))
93	82, 92	eqbrtrd 5124	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑆‘𝑛) ≤ (vol*‘∪ ran ((,) ∘ 𝐹)))
94	93	ralrimiva 3156	. . . 4 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝑆‘𝑛) ≤ (vol*‘∪ ran ((,) ∘ 𝐹)))
95	2, 17	syl 17	. . . . 5 ⊢ (𝜑 → 𝑆:ℕ⟶(0[,)+∞))
96		ffn 6693	. . . . 5 ⊢ (𝑆:ℕ⟶(0[,)+∞) → 𝑆 Fn ℕ)
97		breq1 5105	. . . . . 6 ⊢ (𝑦 = (𝑆‘𝑛) → (𝑦 ≤ (vol*‘∪ ran ((,) ∘ 𝐹)) ↔ (𝑆‘𝑛) ≤ (vol*‘∪ ran ((,) ∘ 𝐹))))
98	97	ralrn 7071	. . . . 5 ⊢ (𝑆 Fn ℕ → (∀𝑦 ∈ ran 𝑆 𝑦 ≤ (vol*‘∪ ran ((,) ∘ 𝐹)) ↔ ∀𝑛 ∈ ℕ (𝑆‘𝑛) ≤ (vol*‘∪ ran ((,) ∘ 𝐹))))
99	95, 96, 98	3syl 18	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ ran 𝑆 𝑦 ≤ (vol*‘∪ ran ((,) ∘ 𝐹)) ↔ ∀𝑛 ∈ ℕ (𝑆‘𝑛) ≤ (vol*‘∪ ran ((,) ∘ 𝐹))))
100	94, 99	mpbird 259	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ ran 𝑆 𝑦 ≤ (vol*‘∪ ran ((,) ∘ 𝐹)))
101		supxrleub 13331	. . . 4 ⊢ ((ran 𝑆 ⊆ ℝ* ∧ (vol*‘∪ ran ((,) ∘ 𝐹)) ∈ ℝ*) → (sup(ran 𝑆, ℝ*, < ) ≤ (vol*‘∪ ran ((,) ∘ 𝐹)) ↔ ∀𝑦 ∈ ran 𝑆 𝑦 ≤ (vol*‘∪ ran ((,) ∘ 𝐹))))
102	21, 14, 101	syl2anc 593	. . 3 ⊢ (𝜑 → (sup(ran 𝑆, ℝ*, < ) ≤ (vol*‘∪ ran ((,) ∘ 𝐹)) ↔ ∀𝑦 ∈ ran 𝑆 𝑦 ≤ (vol*‘∪ ran ((,) ∘ 𝐹))))
103	100, 102	mpbird 259	. 2 ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ (vol*‘∪ ran ((,) ∘ 𝐹)))
104	14, 23, 26, 103	xrletrid 13159	1 ⊢ (𝜑 → (vol*‘∪ ran ((,) ∘ 𝐹)) = sup(ran 𝑆, ℝ*, < ))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144  ∀wral 3078   ∩ cin 3905   ⊆ wss 3906  𝒫 cpw 4557  ⟨cop 4590  ∪ cuni 4867  ∪ ciun 4951  Disj wdisj 5069   class class class wbr 5102   × cxp 5647  dom cdm 5649  ran crn 5650   ∘ ccom 5653   Fn wfn 6518  ⟶wf 6519  ‘cfv 6523  (class class class)co 7398  1^st c1st 7970  2^nd c2nd 7971  Fincfn 8929  supcsup 9388  ℝcr 11074  0cc0 11075  1c1 11076   + caddc 11078  +∞cpnf 11215  ℝ^*cxr 11217   < clt 11218   ≤ cle 11219   − cmin 11416  ℕcn 12212  ℤ_≥cuz 12841  (,)cioo 13351  [,)cico 13353  ...cfz 13514  seqcseq 14016  abscabs 15263  Σcsu 15715  vol*covol 25526  volcvol 25527

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720  ax-inf2 9598  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-disj 5070  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-se 5603  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-isom 6532  df-riota 7355  df-ov 7401  df-oprab 7402  df-mpo 7403  df-of 7662  df-om 7849  df-1st 7972  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-1o 8439  df-2o 8440  df-er 8680  df-map 8812  df-pm 8813  df-en 8930  df-dom 8931  df-sdom 8932  df-fin 8933  df-fi 9359  df-sup 9390  df-inf 9391  df-oi 9460  df-dju 9861  df-card 9899  df-pnf 11220  df-mnf 11221  df-xr 11222  df-ltxr 11223  df-le 11224  df-sub 11418  df-neg 11419  df-div 11847  df-nn 12213  df-2 12282  df-3 12283  df-n0 12484  df-z 12571  df-uz 12842  df-q 12952  df-rp 12996  df-xneg 13116  df-xadd 13117  df-xmul 13118  df-ioo 13355  df-ico 13357  df-icc 13358  df-fz 13515  df-fzo 13662  df-fl 13804  df-seq 14017  df-exp 14077  df-hash 14346  df-cj 15128  df-re 15129  df-im 15130  df-sqrt 15264  df-abs 15265  df-clim 15517  df-rlim 15518  df-sum 15716  df-rest 17453  df-topgen 17474  df-psmet 21418  df-xmet 21419  df-met 21420  df-bl 21421  df-mopn 21422  df-top 22956  df-topon 22973  df-bases 23008  df-cmp 23449  df-ovol 25528  df-vol 25529

This theorem is referenced by:  uniiccvol  25644  uniioombllem2  25647