Theorem uniioovol 25560

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 25535.) 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 13395	. . . . . 6 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ
2		uniioombl.1	. . . . . . 7 ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
3		inss2 4179	. . . . . . . 8 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
4		rexpssxrxp 11185	. . . . . . . 8 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
5	3, 4	sstri 3932	. . . . . . 7 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
6		fss 6680	. . . . . . 7 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)) → 𝐹:ℕ⟶(ℝ* × ℝ*))
7	2, 5, 6	sylancl 587	. . . . . 6 ⊢ (𝜑 → 𝐹:ℕ⟶(ℝ* × ℝ*))
8		fco 6688	. . . . . 6 ⊢ (((,):(ℝ* × ℝ*)⟶𝒫 ℝ ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
9	1, 7, 8	sylancr 588	. . . . 5 ⊢ (𝜑 → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
10	9	frnd 6672	. . . 4 ⊢ (𝜑 → ran ((,) ∘ 𝐹) ⊆ 𝒫 ℝ)
11		sspwuni 5043	. . . 4 ⊢ (ran ((,) ∘ 𝐹) ⊆ 𝒫 ℝ ↔ ∪ ran ((,) ∘ 𝐹) ⊆ ℝ)
12	10, 11	sylib 218	. . 3 ⊢ (𝜑 → ∪ ran ((,) ∘ 𝐹) ⊆ ℝ)
13		ovolcl 25459	. . 3 ⊢ (∪ ran ((,) ∘ 𝐹) ⊆ ℝ → (vol*‘∪ ran ((,) ∘ 𝐹)) ∈ ℝ*)
14	12, 13	syl 17	. 2 ⊢ (𝜑 → (vol*‘∪ ran ((,) ∘ 𝐹)) ∈ ℝ*)
15		eqid 2737	. . . . . 6 ⊢ ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
16		uniioombl.3	. . . . . 6 ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
17	15, 16	ovolsf 25453	. . . . 5 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
18		frn 6671	. . . . 5 ⊢ (𝑆:ℕ⟶(0[,)+∞) → ran 𝑆 ⊆ (0[,)+∞))
19	2, 17, 18	3syl 18	. . . 4 ⊢ (𝜑 → ran 𝑆 ⊆ (0[,)+∞))
20		icossxr 13380	. . . 4 ⊢ (0[,)+∞) ⊆ ℝ*
21	19, 20	sstrdi 3935	. . 3 ⊢ (𝜑 → ran 𝑆 ⊆ ℝ*)
22		supxrcl 13262	. . 3 ⊢ (ran 𝑆 ⊆ ℝ* → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
23	21, 22	syl 17	. 2 ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
24		ssid 3945	. . 3 ⊢ ∪ ran ((,) ∘ 𝐹) ⊆ ∪ ran ((,) ∘ 𝐹)
25	16	ovollb 25460	. . 3 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ∪ ran ((,) ∘ 𝐹) ⊆ ∪ ran ((,) ∘ 𝐹)) → (vol*‘∪ ran ((,) ∘ 𝐹)) ≤ sup(ran 𝑆, ℝ*, < ))
26	2, 24, 25	sylancl 587	. 2 ⊢ (𝜑 → (vol*‘∪ ran ((,) ∘ 𝐹)) ≤ sup(ran 𝑆, ℝ*, < ))
27	16	fveq1i 6837	. . . . . . . 8 ⊢ (𝑆‘𝑛) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑛)
28	2	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
29		elfznn 13502	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (1...𝑛) → 𝑥 ∈ ℕ)
30	15	ovolfsval 25451	. . . . . . . . . . 11 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑥) = ((2nd ‘(𝐹‘𝑥)) − (1st ‘(𝐹‘𝑥))))
31	28, 29, 30	syl2an 597	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (((abs ∘ − ) ∘ 𝐹)‘𝑥) = ((2nd ‘(𝐹‘𝑥)) − (1st ‘(𝐹‘𝑥))))
32		fvco3 6935	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹‘𝑥)))
33	28, 29, 32	syl2an 597	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹‘𝑥)))
34		ffvelcdm 7029	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (𝐹‘𝑥) ∈ ( ≤ ∩ (ℝ × ℝ)))
35	28, 29, 34	syl2an 597	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (𝐹‘𝑥) ∈ ( ≤ ∩ (ℝ × ℝ)))
36	35	elin2d 4146	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (𝐹‘𝑥) ∈ (ℝ × ℝ))
37		1st2nd2 7976	. . . . . . . . . . . . . . . . 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 6840	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → ((,)‘(𝐹‘𝑥)) = ((,)‘⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩))
40		df-ov 7365	. . . . . . . . . . . . . . 15 ⊢ ((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥))) = ((,)‘⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩)
41	39, 40	eqtr4di 2790	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → ((,)‘(𝐹‘𝑥)) = ((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥))))
42	33, 41	eqtrd 2772	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (((,) ∘ 𝐹)‘𝑥) = ((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥))))
43		ioombl 25546	. . . . . . . . . . . . 13 ⊢ ((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥))) ∈ dom vol
44	42, 43	eqeltrdi 2845	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (((,) ∘ 𝐹)‘𝑥) ∈ dom vol)
45		mblvol 25511	. . . . . . . . . . . 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 6840	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (vol*‘(((,) ∘ 𝐹)‘𝑥)) = (vol*‘((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥)))))
48		ovolfcl 25447	. . . . . . . . . . . . 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 597	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → ((1st ‘(𝐹‘𝑥)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑥)) ∈ ℝ ∧ (1st ‘(𝐹‘𝑥)) ≤ (2nd ‘(𝐹‘𝑥))))
50		ovolioo 25549	. . . . . . . . . . . 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 2776	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (vol‘(((,) ∘ 𝐹)‘𝑥)) = ((2nd ‘(𝐹‘𝑥)) − (1st ‘(𝐹‘𝑥))))
53	31, 52	eqtr4d 2775	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (((abs ∘ − ) ∘ 𝐹)‘𝑥) = (vol‘(((,) ∘ 𝐹)‘𝑥)))
54		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
55		nnuz 12822	. . . . . . . . . 10 ⊢ ℕ = (ℤ≥‘1)
56	54, 55	eleqtrdi 2847	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ≥‘1))
57	49	simp2d 1144	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (2nd ‘(𝐹‘𝑥)) ∈ ℝ)
58	49	simp1d 1143	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (1st ‘(𝐹‘𝑥)) ∈ ℝ)
59	57, 58	resubcld 11573	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → ((2nd ‘(𝐹‘𝑥)) − (1st ‘(𝐹‘𝑥))) ∈ ℝ)
60	52, 59	eqeltrd 2837	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (vol‘(((,) ∘ 𝐹)‘𝑥)) ∈ ℝ)
61	60	recnd 11168	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (vol‘(((,) ∘ 𝐹)‘𝑥)) ∈ ℂ)
62	53, 56, 61	fsumser 15687	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Σ𝑥 ∈ (1...𝑛)(vol‘(((,) ∘ 𝐹)‘𝑥)) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑛))
63	27, 62	eqtr4id 2791	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑆‘𝑛) = Σ𝑥 ∈ (1...𝑛)(vol‘(((,) ∘ 𝐹)‘𝑥)))
64		fzfid 13930	. . . . . . . 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 3130	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑥 ∈ (1...𝑛)((((,) ∘ 𝐹)‘𝑥) ∈ dom vol ∧ (vol‘(((,) ∘ 𝐹)‘𝑥)) ∈ ℝ))
67		fz1ssnn 13504	. . . . . . . . 9 ⊢ (1...𝑛) ⊆ ℕ
68		uniioombl.2	. . . . . . . . . . 11 ⊢ (𝜑 → Disj 𝑥 ∈ ℕ ((,)‘(𝐹‘𝑥)))
69	2, 32	sylan 581	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹‘𝑥)))
70	69	disjeq2dv 5058	. . . . . . . . . . 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 5059	. . . . . . . . 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 25528	. . . . . . . 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 1374	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol‘∪ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥)) = Σ𝑥 ∈ (1...𝑛)(vol‘(((,) ∘ 𝐹)‘𝑥)))
77	44	ralrimiva 3130	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ∈ dom vol)
78		finiunmbl 25525	. . . . . . . . 9 ⊢ (((1...𝑛) ∈ Fin ∧ ∀𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ∈ dom vol) → ∪ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ∈ dom vol)
79	64, 77, 78	syl2anc 585	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∪ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ∈ dom vol)
80		mblvol 25511	. . . . . . . 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 2778	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑆‘𝑛) = (vol*‘∪ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥)))
83		iunss1 4949	. . . . . . . . 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 6664	. . . . . . . . 9 ⊢ (((,) ∘ 𝐹):ℕ⟶𝒫 ℝ → ((,) ∘ 𝐹) Fn ℕ)
87		fniunfv 7197	. . . . . . . . 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 3959	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∪ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ⊆ ∪ ran ((,) ∘ 𝐹))
90	12	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∪ ran ((,) ∘ 𝐹) ⊆ ℝ)
91		ovolss 25466	. . . . . . 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 585	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol*‘∪ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥)) ≤ (vol*‘∪ ran ((,) ∘ 𝐹)))
93	82, 92	eqbrtrd 5108	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑆‘𝑛) ≤ (vol*‘∪ ran ((,) ∘ 𝐹)))
94	93	ralrimiva 3130	. . . 4 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝑆‘𝑛) ≤ (vol*‘∪ ran ((,) ∘ 𝐹)))
95	2, 17	syl 17	. . . . 5 ⊢ (𝜑 → 𝑆:ℕ⟶(0[,)+∞))
96		ffn 6664	. . . . 5 ⊢ (𝑆:ℕ⟶(0[,)+∞) → 𝑆 Fn ℕ)
97		breq1 5089	. . . . . 6 ⊢ (𝑦 = (𝑆‘𝑛) → (𝑦 ≤ (vol*‘∪ ran ((,) ∘ 𝐹)) ↔ (𝑆‘𝑛) ≤ (vol*‘∪ ran ((,) ∘ 𝐹))))
98	97	ralrn 7036	. . . . 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 13273	. . . 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 585	. . 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 13101	1 ⊢ (𝜑 → (vol*‘∪ ran ((,) ∘ 𝐹)) = sup(ran 𝑆, ℝ*, < ))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ∩ cin 3889   ⊆ wss 3890  𝒫 cpw 4542  ⟨cop 4574  ∪ cuni 4851  ∪ ciun 4934  Disj wdisj 5053   class class class wbr 5086   × cxp 5624  dom cdm 5626  ran crn 5627   ∘ ccom 5630   Fn wfn 6489  ⟶wf 6490  ‘cfv 6494  (class class class)co 7362  1^st c1st 7935  2^nd c2nd 7936  Fincfn 8888  supcsup 9348  ℝcr 11032  0cc0 11033  1c1 11034   + caddc 11036  +∞cpnf 11171  ℝ^*cxr 11173   < clt 11174   ≤ cle 11175   − cmin 11372  ℕcn 12169  ℤ_≥cuz 12783  (,)cioo 13293  [,)cico 13295  ...cfz 13456  seqcseq 13958  abscabs 15191  Σcsu 15643  vol*covol 25443  volcvol 25444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684  ax-inf2 9557  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110  ax-pre-sup 11111

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-disj 5054  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-se 5580  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-isom 6503  df-riota 7319  df-ov 7365  df-oprab 7366  df-mpo 7367  df-of 7626  df-om 7813  df-1st 7937  df-2nd 7938  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-1o 8400  df-2o 8401  df-er 8638  df-map 8770  df-pm 8771  df-en 8889  df-dom 8890  df-sdom 8891  df-fin 8892  df-fi 9319  df-sup 9350  df-inf 9351  df-oi 9420  df-dju 9820  df-card 9858  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-div 11803  df-nn 12170  df-2 12239  df-3 12240  df-n0 12433  df-z 12520  df-uz 12784  df-q 12894  df-rp 12938  df-xneg 13058  df-xadd 13059  df-xmul 13060  df-ioo 13297  df-ico 13299  df-icc 13300  df-fz 13457  df-fzo 13604  df-fl 13746  df-seq 13959  df-exp 14019  df-hash 14288  df-cj 15056  df-re 15057  df-im 15058  df-sqrt 15192  df-abs 15193  df-clim 15445  df-rlim 15446  df-sum 15644  df-rest 17380  df-topgen 17401  df-psmet 21340  df-xmet 21341  df-met 21342  df-bl 21343  df-mopn 21344  df-top 22873  df-topon 22890  df-bases 22925  df-cmp 23366  df-ovol 25445  df-vol 25446

This theorem is referenced by:  uniiccvol  25561  uniioombllem2  25564