Theorem uniioovol 25487

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 25462.) 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 13415	. . . . . 6 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ
2		uniioombl.1	. . . . . . 7 ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
3		inss2 4204	. . . . . . . 8 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
4		rexpssxrxp 11226	. . . . . . . 8 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
5	3, 4	sstri 3959	. . . . . . 7 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
6		fss 6707	. . . . . . 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 6715	. . . . . 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 6699	. . . 4 ⊢ (𝜑 → ran ((,) ∘ 𝐹) ⊆ 𝒫 ℝ)
11		sspwuni 5067	. . . 4 ⊢ (ran ((,) ∘ 𝐹) ⊆ 𝒫 ℝ ↔ ∪ ran ((,) ∘ 𝐹) ⊆ ℝ)
12	10, 11	sylib 218	. . 3 ⊢ (𝜑 → ∪ ran ((,) ∘ 𝐹) ⊆ ℝ)
13		ovolcl 25386	. . 3 ⊢ (∪ ran ((,) ∘ 𝐹) ⊆ ℝ → (vol*‘∪ ran ((,) ∘ 𝐹)) ∈ ℝ*)
14	12, 13	syl 17	. 2 ⊢ (𝜑 → (vol*‘∪ ran ((,) ∘ 𝐹)) ∈ ℝ*)
15		eqid 2730	. . . . . 6 ⊢ ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
16		uniioombl.3	. . . . . 6 ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
17	15, 16	ovolsf 25380	. . . . 5 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
18		frn 6698	. . . . 5 ⊢ (𝑆:ℕ⟶(0[,)+∞) → ran 𝑆 ⊆ (0[,)+∞))
19	2, 17, 18	3syl 18	. . . 4 ⊢ (𝜑 → ran 𝑆 ⊆ (0[,)+∞))
20		icossxr 13400	. . . 4 ⊢ (0[,)+∞) ⊆ ℝ*
21	19, 20	sstrdi 3962	. . 3 ⊢ (𝜑 → ran 𝑆 ⊆ ℝ*)
22		supxrcl 13282	. . 3 ⊢ (ran 𝑆 ⊆ ℝ* → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
23	21, 22	syl 17	. 2 ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
24		ssid 3972	. . 3 ⊢ ∪ ran ((,) ∘ 𝐹) ⊆ ∪ ran ((,) ∘ 𝐹)
25	16	ovollb 25387	. . 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 6862	. . . . . . . 8 ⊢ (𝑆‘𝑛) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑛)
28	2	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
29		elfznn 13521	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (1...𝑛) → 𝑥 ∈ ℕ)
30	15	ovolfsval 25378	. . . . . . . . . . 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 6963	. . . . . . . . . . . . . . 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 7056	. . . . . . . . . . . . . . . . . . 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 4171	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (𝐹‘𝑥) ∈ (ℝ × ℝ))
37		1st2nd2 8010	. . . . . . . . . . . . . . . . 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 6865	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → ((,)‘(𝐹‘𝑥)) = ((,)‘⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩))
40		df-ov 7393	. . . . . . . . . . . . . . 15 ⊢ ((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥))) = ((,)‘⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩)
41	39, 40	eqtr4di 2783	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → ((,)‘(𝐹‘𝑥)) = ((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥))))
42	33, 41	eqtrd 2765	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (((,) ∘ 𝐹)‘𝑥) = ((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥))))
43		ioombl 25473	. . . . . . . . . . . . 13 ⊢ ((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥))) ∈ dom vol
44	42, 43	eqeltrdi 2837	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (((,) ∘ 𝐹)‘𝑥) ∈ dom vol)
45		mblvol 25438	. . . . . . . . . . . 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 6865	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (vol*‘(((,) ∘ 𝐹)‘𝑥)) = (vol*‘((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥)))))
48		ovolfcl 25374	. . . . . . . . . . . . 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 25476	. . . . . . . . . . . 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 2769	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (vol‘(((,) ∘ 𝐹)‘𝑥)) = ((2nd ‘(𝐹‘𝑥)) − (1st ‘(𝐹‘𝑥))))
53	31, 52	eqtr4d 2768	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (((abs ∘ − ) ∘ 𝐹)‘𝑥) = (vol‘(((,) ∘ 𝐹)‘𝑥)))
54		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
55		nnuz 12843	. . . . . . . . . 10 ⊢ ℕ = (ℤ≥‘1)
56	54, 55	eleqtrdi 2839	. . . . . . . . 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 11613	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → ((2nd ‘(𝐹‘𝑥)) − (1st ‘(𝐹‘𝑥))) ∈ ℝ)
60	52, 59	eqeltrd 2829	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (vol‘(((,) ∘ 𝐹)‘𝑥)) ∈ ℝ)
61	60	recnd 11209	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑛)) → (vol‘(((,) ∘ 𝐹)‘𝑥)) ∈ ℂ)
62	53, 56, 61	fsumser 15703	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Σ𝑥 ∈ (1...𝑛)(vol‘(((,) ∘ 𝐹)‘𝑥)) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑛))
63	27, 62	eqtr4id 2784	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑆‘𝑛) = Σ𝑥 ∈ (1...𝑛)(vol‘(((,) ∘ 𝐹)‘𝑥)))
64		fzfid 13945	. . . . . . . 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 3126	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑥 ∈ (1...𝑛)((((,) ∘ 𝐹)‘𝑥) ∈ dom vol ∧ (vol‘(((,) ∘ 𝐹)‘𝑥)) ∈ ℝ))
67		fz1ssnn 13523	. . . . . . . . 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 5082	. . . . . . . . . . 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 5083	. . . . . . . . 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 25455	. . . . . . . 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 3126	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ∈ dom vol)
78		finiunmbl 25452	. . . . . . . . 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 25438	. . . . . . . 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 2771	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑆‘𝑛) = (vol*‘∪ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥)))
83		iunss1 4973	. . . . . . . . 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 6691	. . . . . . . . 9 ⊢ (((,) ∘ 𝐹):ℕ⟶𝒫 ℝ → ((,) ∘ 𝐹) Fn ℕ)
87		fniunfv 7224	. . . . . . . . 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 3986	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∪ 𝑥 ∈ (1...𝑛)(((,) ∘ 𝐹)‘𝑥) ⊆ ∪ ran ((,) ∘ 𝐹))
90	12	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∪ ran ((,) ∘ 𝐹) ⊆ ℝ)
91		ovolss 25393	. . . . . . 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 5132	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑆‘𝑛) ≤ (vol*‘∪ ran ((,) ∘ 𝐹)))
94	93	ralrimiva 3126	. . . 4 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝑆‘𝑛) ≤ (vol*‘∪ ran ((,) ∘ 𝐹)))
95	2, 17	syl 17	. . . . 5 ⊢ (𝜑 → 𝑆:ℕ⟶(0[,)+∞))
96		ffn 6691	. . . . 5 ⊢ (𝑆:ℕ⟶(0[,)+∞) → 𝑆 Fn ℕ)
97		breq1 5113	. . . . . 6 ⊢ (𝑦 = (𝑆‘𝑛) → (𝑦 ≤ (vol*‘∪ ran ((,) ∘ 𝐹)) ↔ (𝑆‘𝑛) ≤ (vol*‘∪ ran ((,) ∘ 𝐹))))
98	97	ralrn 7063	. . . . 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 13293	. . . 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 13122	1 ⊢ (𝜑 → (vol*‘∪ ran ((,) ∘ 𝐹)) = sup(ran 𝑆, ℝ*, < ))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3045   ∩ cin 3916   ⊆ wss 3917  𝒫 cpw 4566  ⟨cop 4598  ∪ cuni 4874  ∪ ciun 4958  Disj wdisj 5077   class class class wbr 5110   × cxp 5639  dom cdm 5641  ran crn 5642   ∘ ccom 5645   Fn wfn 6509  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390  1^st c1st 7969  2^nd c2nd 7970  Fincfn 8921  supcsup 9398  ℝcr 11074  0cc0 11075  1c1 11076   + caddc 11078  +∞cpnf 11212  ℝ^*cxr 11214   < clt 11215   ≤ cle 11216   − cmin 11412  ℕcn 12193  ℤ_≥cuz 12800  (,)cioo 13313  [,)cico 13315  ...cfz 13475  seqcseq 13973  abscabs 15207  Σcsu 15659  vol*covol 25370  volcvol 25371

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-inf2 9601  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 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 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-disj 5078  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-isom 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-of 7656  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-er 8674  df-map 8804  df-pm 8805  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-fi 9369  df-sup 9400  df-inf 9401  df-oi 9470  df-dju 9861  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-div 11843  df-nn 12194  df-2 12256  df-3 12257  df-n0 12450  df-z 12537  df-uz 12801  df-q 12915  df-rp 12959  df-xneg 13079  df-xadd 13080  df-xmul 13081  df-ioo 13317  df-ico 13319  df-icc 13320  df-fz 13476  df-fzo 13623  df-fl 13761  df-seq 13974  df-exp 14034  df-hash 14303  df-cj 15072  df-re 15073  df-im 15074  df-sqrt 15208  df-abs 15209  df-clim 15461  df-rlim 15462  df-sum 15660  df-rest 17392  df-topgen 17413  df-psmet 21263  df-xmet 21264  df-met 21265  df-bl 21266  df-mopn 21267  df-top 22788  df-topon 22805  df-bases 22840  df-cmp 23281  df-ovol 25372  df-vol 25373

This theorem is referenced by:  uniiccvol  25488  uniioombllem2  25491