Theorem uniioovol 25530

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 25505.) 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 13462	. . . . . 6 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ
2		uniioombl.1	. . . . . . 7 ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
3		inss2 4213	. . . . . . . 8 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
4		rexpssxrxp 11278	. . . . . . . 8 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
5	3, 4	sstri 3968	. . . . . . 7 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
6		fss 6721	. . . . . . 7 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)) → 𝐹:ℕ⟶(ℝ* × ℝ*))
7	2, 5, 6	sylancl 586	. . . . . 6 ⊢ (𝜑 → 𝐹:ℕ⟶(ℝ* × ℝ*))
8		fco 6729	. . . . . 6 ⊢ (((,):(ℝ* × ℝ*)⟶𝒫 ℝ ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
9	1, 7, 8	sylancr 587	. . . . 5 ⊢ (𝜑 → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
10	9	frnd 6713	. . . 4 ⊢ (𝜑 → ran ((,) ∘ 𝐹) ⊆ 𝒫 ℝ)
11		sspwuni 5076	. . . 4 ⊢ (ran ((,) ∘ 𝐹) ⊆ 𝒫 ℝ ↔ ∪ ran ((,) ∘ 𝐹) ⊆ ℝ)
12	10, 11	sylib 218	. . 3 ⊢ (𝜑 → ∪ ran ((,) ∘ 𝐹) ⊆ ℝ)
13		ovolcl 25429	. . 3 ⊢ (∪ ran ((,) ∘ 𝐹) ⊆ ℝ → (vol*‘∪ ran ((,) ∘ 𝐹)) ∈ ℝ*)
14	12, 13	syl 17	. 2 ⊢ (𝜑 → (vol*‘∪ ran ((,) ∘ 𝐹)) ∈ ℝ*)
15		eqid 2735	. . . . . 6 ⊢ ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
16		uniioombl.3	. . . . . 6 ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
17	15, 16	ovolsf 25423	. . . . 5 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
18		frn 6712	. . . . 5 ⊢ (𝑆:ℕ⟶(0[,)+∞) → ran 𝑆 ⊆ (0[,)+∞))
19	2, 17, 18	3syl 18	. . . 4 ⊢ (𝜑 → ran 𝑆 ⊆ (0[,)+∞))
20		icossxr 13447	. . . 4 ⊢ (0[,)+∞) ⊆ ℝ*
21	19, 20	sstrdi 3971	. . 3 ⊢ (𝜑 → ran 𝑆 ⊆ ℝ*)
22		supxrcl 13329	. . 3 ⊢ (ran 𝑆 ⊆ ℝ* → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
23	21, 22	syl 17	. 2 ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
24		ssid 3981	. . 3 ⊢ ∪ ran ((,) ∘ 𝐹) ⊆ ∪ ran ((,) ∘ 𝐹)
25	16	ovollb 25430	. . 3 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ∪ ran ((,) ∘ 𝐹) ⊆ ∪ ran ((,) ∘ 𝐹)) → (vol*‘∪ ran ((,) ∘ 𝐹)) ≤ sup(ran 𝑆, ℝ*, < ))
26	2, 24, 25	sylancl 586	. 2 ⊢ (𝜑 → (vol*‘∪ ran ((,) ∘ 𝐹)) ≤ sup(ran 𝑆, ℝ*, < ))
27	16	fveq1i 6876	. . . . . . . 8 ⊢ (𝑆‘𝑛) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑛)
28	2	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
29		elfznn 13568	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (1...𝑛) → 𝑥 ∈ ℕ)
30	15	ovolfsval 25421	. . . . . . . . . . 11 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑥) = ((2nd ‘(𝐹‘𝑥)) − (1st ‘(𝐹‘𝑥))))
31	28, 29, 30	syl2an 596	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (((abs ∘ − ) ∘ 𝐹)‘𝑥) = ((2nd ‘(𝐹‘𝑥)) − (1st ‘(𝐹‘𝑥))))
32		fvco3 6977	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹‘𝑥)))
33	28, 29, 32	syl2an 596	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹‘𝑥)))
34		ffvelcdm 7070	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (𝐹‘𝑥) ∈ ( ≤ ∩ (ℝ × ℝ)))
35	28, 29, 34	syl2an 596	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (𝐹‘𝑥) ∈ ( ≤ ∩ (ℝ × ℝ)))
36	35	elin2d 4180	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (𝐹‘𝑥) ∈ (ℝ × ℝ))
37		1st2nd2 8025	. . . . . . . . . . . . . . . . 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 6879	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → ((,)‘(𝐹‘𝑥)) = ((,)‘⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩))
40		df-ov 7406	. . . . . . . . . . . . . . 15 ⊢ ((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥))) = ((,)‘⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩)
41	39, 40	eqtr4di 2788	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → ((,)‘(𝐹‘𝑥)) = ((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥))))
42	33, 41	eqtrd 2770	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (((,) ∘ 𝐹)‘𝑥) = ((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥))))
43		ioombl 25516	. . . . . . . . . . . . 13 ⊢ ((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥))) ∈ dom vol
44	42, 43	eqeltrdi 2842	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (((,) ∘ 𝐹)‘𝑥) ∈ dom vol)
45		mblvol 25481	. . . . . . . . . . . 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 6879	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (vol*‘(((,) ∘ 𝐹)‘𝑥)) = (vol*‘((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥)))))
48		ovolfcl 25417	. . . . . . . . . . . . 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 596	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → ((1st ‘(𝐹‘𝑥)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑥)) ∈ ℝ ∧ (1st ‘(𝐹‘𝑥)) ≤ (2nd ‘(𝐹‘𝑥))))
50		ovolioo 25519	. . . . . . . . . . . 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 2774	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (vol‘(((,) ∘ 𝐹)‘𝑥)) = ((2nd ‘(𝐹‘𝑥)) − (1st ‘(𝐹‘𝑥))))
53	31, 52	eqtr4d 2773	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (((abs ∘ − ) ∘ 𝐹)‘𝑥) = (vol‘(((,) ∘ 𝐹)‘𝑥)))
54		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
55		nnuz 12893	. . . . . . . . . 10 ⊢ ℕ = (ℤ≥‘1)
56	54, 55	eleqtrdi 2844	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ≥‘1))
57	49	simp2d 1143	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (2nd ‘(𝐹‘𝑥)) ∈ ℝ)
58	49	simp1d 1142	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (1st ‘(𝐹‘𝑥)) ∈ ℝ)
59	57, 58	resubcld 11663	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → ((2nd ‘(𝐹‘𝑥)) − (1st ‘(𝐹‘𝑥))) ∈ ℝ)
60	52, 59	eqeltrd 2834	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (vol‘(((,) ∘ 𝐹)‘𝑥)) ∈ ℝ)
61	60	recnd 11261	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (vol‘(((,) ∘ 𝐹)‘𝑥)) ∈ ℂ)
62	53, 56, 61	fsumser 15744	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Σ𝑥 ∈ (1...𝑛)(vol‘(((,) ∘ 𝐹)‘𝑥)) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑛))
63	27, 62	eqtr4id 2789	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑆‘𝑛) = Σ𝑥 ∈ (1...𝑛)(vol‘(((,) ∘ 𝐹)‘𝑥)))
64		fzfid 13989	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1...𝑛) ∈ Fin)
65	44, 60	jca 511	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → ((((,) ∘ 𝐹)‘𝑥) ∈ dom vol ∧ (vol‘(((,) ∘ 𝐹)‘𝑥)) ∈ ℝ))
66	65	ralrimiva 3132	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑥 ∈ (1...𝑛)((((,) ∘ 𝐹)‘𝑥) ∈ dom vol ∧ (vol‘(((,) ∘ 𝐹)‘𝑥)) ∈ ℝ))
67		fz1ssnn 13570	. . . . . . . . 9 ⊢ (1...𝑛) ⊆ ℕ
68		uniioombl.2	. . . . . . . . . . 11 ⊢ (𝜑 → Disj 𝑥 ∈ ℕ ((,)‘(𝐹‘𝑥)))
69	2, 32	sylan 580	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹‘𝑥)))
70	69	disjeq2dv 5091	. . . . . . . . . . 11 ⊢ (𝜑 → (Disj 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) ↔ Disj 𝑥 ∈ ℕ ((,)‘(𝐹‘𝑥))))
71	68, 70	mpbird 257	. . . . . . . . . 10 ⊢ (𝜑 → Disj 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥))
72	71	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Disj 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥))
73		disjss1 5092	. . . . . . . . 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 25498	. . . . . . . 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 1373	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol‘∪ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥)) = Σ𝑥 ∈ (1...𝑛)(vol‘(((,) ∘ 𝐹)‘𝑥)))
77	44	ralrimiva 3132	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ∈ dom vol)
78		finiunmbl 25495	. . . . . . . . 9 ⊢ (((1...𝑛) ∈ Fin ∧ ∀𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ∈ dom vol) → ∪ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ∈ dom vol)
79	64, 77, 78	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∪ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ∈ dom vol)
80		mblvol 25481	. . . . . . . 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 2776	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑆‘𝑛) = (vol*‘∪ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥)))
83		iunss1 4982	. . . . . . . . 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 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
86		ffn 6705	. . . . . . . . 9 ⊢ (((,) ∘ 𝐹):ℕ⟶𝒫 ℝ → ((,) ∘ 𝐹) Fn ℕ)
87		fniunfv 7238	. . . . . . . . 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 3995	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∪ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ⊆ ∪ ran ((,) ∘ 𝐹))
90	12	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∪ ran ((,) ∘ 𝐹) ⊆ ℝ)
91		ovolss 25436	. . . . . . 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 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol*‘∪ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥)) ≤ (vol*‘∪ ran ((,) ∘ 𝐹)))
93	82, 92	eqbrtrd 5141	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑆‘𝑛) ≤ (vol*‘∪ ran ((,) ∘ 𝐹)))
94	93	ralrimiva 3132	. . . 4 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝑆‘𝑛) ≤ (vol*‘∪ ran ((,) ∘ 𝐹)))
95	2, 17	syl 17	. . . . 5 ⊢ (𝜑 → 𝑆:ℕ⟶(0[,)+∞))
96		ffn 6705	. . . . 5 ⊢ (𝑆:ℕ⟶(0[,)+∞) → 𝑆 Fn ℕ)
97		breq1 5122	. . . . . 6 ⊢ (𝑦 = (𝑆‘𝑛) → (𝑦 ≤ (vol*‘∪ ran ((,) ∘ 𝐹)) ↔ (𝑆‘𝑛) ≤ (vol*‘∪ ran ((,) ∘ 𝐹))))
98	97	ralrn 7077	. . . . 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 257	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ ran 𝑆 𝑦 ≤ (vol*‘∪ ran ((,) ∘ 𝐹)))
101		supxrleub 13340	. . . 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 584	. . 3 ⊢ (𝜑 → (sup(ran 𝑆, ℝ*, < ) ≤ (vol*‘∪ ran ((,) ∘ 𝐹)) ↔ ∀𝑦 ∈ ran 𝑆 𝑦 ≤ (vol*‘∪ ran ((,) ∘ 𝐹))))
103	100, 102	mpbird 257	. 2 ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ (vol*‘∪ ran ((,) ∘ 𝐹)))
104	14, 23, 26, 103	xrletrid 13169	1 ⊢ (𝜑 → (vol*‘∪ ran ((,) ∘ 𝐹)) = sup(ran 𝑆, ℝ*, < ))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108  ∀wral 3051   ∩ cin 3925   ⊆ wss 3926  𝒫 cpw 4575  ⟨cop 4607  ∪ cuni 4883  ∪ ciun 4967  Disj wdisj 5086   class class class wbr 5119   × cxp 5652  dom cdm 5654  ran crn 5655   ∘ ccom 5658   Fn wfn 6525  ⟶wf 6526  ‘cfv 6530  (class class class)co 7403  1^st c1st 7984  2^nd c2nd 7985  Fincfn 8957  supcsup 9450  ℝcr 11126  0cc0 11127  1c1 11128   + caddc 11130  +∞cpnf 11264  ℝ^*cxr 11266   < clt 11267   ≤ cle 11268   − cmin 11464  ℕcn 12238  ℤ_≥cuz 12850  (,)cioo 13360  [,)cico 13362  ...cfz 13522  seqcseq 14017  abscabs 15251  Σcsu 15700  vol*covol 25413  volcvol 25414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727  ax-inf2 9653  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204  ax-pre-sup 11205

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-disj 5087  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-isom 6539  df-riota 7360  df-ov 7406  df-oprab 7407  df-mpo 7408  df-of 7669  df-om 7860  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-2o 8479  df-er 8717  df-map 8840  df-pm 8841  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-fi 9421  df-sup 9452  df-inf 9453  df-oi 9522  df-dju 9913  df-card 9951  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11466  df-neg 11467  df-div 11893  df-nn 12239  df-2 12301  df-3 12302  df-n0 12500  df-z 12587  df-uz 12851  df-q 12963  df-rp 13007  df-xneg 13126  df-xadd 13127  df-xmul 13128  df-ioo 13364  df-ico 13366  df-icc 13367  df-fz 13523  df-fzo 13670  df-fl 13807  df-seq 14018  df-exp 14078  df-hash 14347  df-cj 15116  df-re 15117  df-im 15118  df-sqrt 15252  df-abs 15253  df-clim 15502  df-rlim 15503  df-sum 15701  df-rest 17434  df-topgen 17455  df-psmet 21305  df-xmet 21306  df-met 21307  df-bl 21308  df-mopn 21309  df-top 22830  df-topon 22847  df-bases 22882  df-cmp 23323  df-ovol 25415  df-vol 25416

This theorem is referenced by:  uniiccvol  25531  uniioombllem2  25534