Theorem uniioombl 25488

Description: A disjoint union of open intervals is measurable. (This proof does not use countable choice, unlike iunmbl 25452.) Lemma 565Ca of [Fremlin5] p. 214. (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
uniioombl	⊢ (𝜑 → ∪ ran ((,) ∘ 𝐹) ∈ dom vol)

Distinct variable groups:   𝑥,𝐹   𝜑,𝑥

Allowed substitution hint:   𝑆(𝑥)

Proof of Theorem uniioombl

Dummy variables 𝑓 𝑟 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ioof 13350	. . . . 5 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ
2		uniioombl.1	. . . . . 6 ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
3		inss2 4189	. . . . . . 7 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
4		rexpssxrxp 11160	. . . . . . 7 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
5	3, 4	sstri 3945	. . . . . 6 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
6		fss 6668	. . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)) → 𝐹:ℕ⟶(ℝ* × ℝ*))
7	2, 5, 6	sylancl 586	. . . . 5 ⊢ (𝜑 → 𝐹:ℕ⟶(ℝ* × ℝ*))
8		fco 6676	. . . . 5 ⊢ (((,):(ℝ* × ℝ*)⟶𝒫 ℝ ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
9	1, 7, 8	sylancr 587	. . . 4 ⊢ (𝜑 → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
10	9	frnd 6660	. . 3 ⊢ (𝜑 → ran ((,) ∘ 𝐹) ⊆ 𝒫 ℝ)
11		sspwuni 5049	. . 3 ⊢ (ran ((,) ∘ 𝐹) ⊆ 𝒫 ℝ ↔ ∪ ran ((,) ∘ 𝐹) ⊆ ℝ)
12	10, 11	sylib 218	. 2 ⊢ (𝜑 → ∪ ran ((,) ∘ 𝐹) ⊆ ℝ)
13		elpwi 4558	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝒫 ℝ → 𝑧 ⊆ ℝ)
14	13	ad2antrl 728	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) → 𝑧 ⊆ ℝ)
15		simprr 772	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) → (vol*‘𝑧) ∈ ℝ)
16		rphalfcl 12922	. . . . . . . . . 10 ⊢ (𝑟 ∈ ℝ+ → (𝑟 / 2) ∈ ℝ+)
17	16	rphalfcld 12949	. . . . . . . . 9 ⊢ (𝑟 ∈ ℝ+ → ((𝑟 / 2) / 2) ∈ ℝ+)
18		eqid 2729	. . . . . . . . . 10 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
19	18	ovolgelb 25379	. . . . . . . . 9 ⊢ ((𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ ∧ ((𝑟 / 2) / 2) ∈ ℝ+) → ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝑧 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝑧) + ((𝑟 / 2) / 2))))
20	14, 15, 17, 19	syl2an3an 1424	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) ∧ 𝑟 ∈ ℝ+) → ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝑧 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝑧) + ((𝑟 / 2) / 2))))
21	2	ad3antrrr 730	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ (𝑧 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝑧) + ((𝑟 / 2) / 2))))) → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
22		uniioombl.2	. . . . . . . . . 10 ⊢ (𝜑 → Disj 𝑥 ∈ ℕ ((,)‘(𝐹‘𝑥)))
23	22	ad3antrrr 730	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ (𝑧 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝑧) + ((𝑟 / 2) / 2))))) → Disj 𝑥 ∈ ℕ ((,)‘(𝐹‘𝑥)))
24		uniioombl.3	. . . . . . . . 9 ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
25		eqid 2729	. . . . . . . . 9 ⊢ ∪ ran ((,) ∘ 𝐹) = ∪ ran ((,) ∘ 𝐹)
26	15	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) ∧ 𝑟 ∈ ℝ+) → (vol*‘𝑧) ∈ ℝ)
27	26	adantr 480	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ (𝑧 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝑧) + ((𝑟 / 2) / 2))))) → (vol*‘𝑧) ∈ ℝ)
28	16	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) ∧ 𝑟 ∈ ℝ+) → (𝑟 / 2) ∈ ℝ+)
29	28	adantr 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ (𝑧 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝑧) + ((𝑟 / 2) / 2))))) → (𝑟 / 2) ∈ ℝ+)
30	29	rphalfcld 12949	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ (𝑧 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝑧) + ((𝑟 / 2) / 2))))) → ((𝑟 / 2) / 2) ∈ ℝ+)
31		elmapi 8776	. . . . . . . . . 10 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) → 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
32	31	ad2antrl 728	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ (𝑧 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝑧) + ((𝑟 / 2) / 2))))) → 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
33		simprrl 780	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ (𝑧 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝑧) + ((𝑟 / 2) / 2))))) → 𝑧 ⊆ ∪ ran ((,) ∘ 𝑓))
34		simprrr 781	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ (𝑧 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝑧) + ((𝑟 / 2) / 2))))) → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝑧) + ((𝑟 / 2) / 2)))
35	21, 23, 24, 25, 27, 30, 32, 33, 18, 34	uniioombllem6 25487	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ (𝑧 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝑧) + ((𝑟 / 2) / 2))))) → ((vol*‘(𝑧 ∩ ∪ ran ((,) ∘ 𝐹))) + (vol*‘(𝑧 ∖ ∪ ran ((,) ∘ 𝐹)))) ≤ ((vol*‘𝑧) + (4 · ((𝑟 / 2) / 2))))
36	20, 35	rexlimddv 3136	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) ∧ 𝑟 ∈ ℝ+) → ((vol*‘(𝑧 ∩ ∪ ran ((,) ∘ 𝐹))) + (vol*‘(𝑧 ∖ ∪ ran ((,) ∘ 𝐹)))) ≤ ((vol*‘𝑧) + (4 · ((𝑟 / 2) / 2))))
37		rpcn 12904	. . . . . . . . . . . . 13 ⊢ (𝑟 ∈ ℝ+ → 𝑟 ∈ ℂ)
38	37	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) ∧ 𝑟 ∈ ℝ+) → 𝑟 ∈ ℂ)
39		2cnd 12206	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) ∧ 𝑟 ∈ ℝ+) → 2 ∈ ℂ)
40		2ne0 12232	. . . . . . . . . . . . 13 ⊢ 2 ≠ 0
41	40	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) ∧ 𝑟 ∈ ℝ+) → 2 ≠ 0)
42	38, 39, 39, 41, 41	divdiv1d 11931	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) ∧ 𝑟 ∈ ℝ+) → ((𝑟 / 2) / 2) = (𝑟 / (2 · 2)))
43		2t2e4 12287	. . . . . . . . . . . 12 ⊢ (2 · 2) = 4
44	43	oveq2i 7360	. . . . . . . . . . 11 ⊢ (𝑟 / (2 · 2)) = (𝑟 / 4)
45	42, 44	eqtrdi 2780	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) ∧ 𝑟 ∈ ℝ+) → ((𝑟 / 2) / 2) = (𝑟 / 4))
46	45	oveq2d 7365	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) ∧ 𝑟 ∈ ℝ+) → (4 · ((𝑟 / 2) / 2)) = (4 · (𝑟 / 4)))
47		4cn 12213	. . . . . . . . . . 11 ⊢ 4 ∈ ℂ
48	47	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) ∧ 𝑟 ∈ ℝ+) → 4 ∈ ℂ)
49		4ne0 12236	. . . . . . . . . . 11 ⊢ 4 ≠ 0
50	49	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) ∧ 𝑟 ∈ ℝ+) → 4 ≠ 0)
51	38, 48, 50	divcan2d 11902	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) ∧ 𝑟 ∈ ℝ+) → (4 · (𝑟 / 4)) = 𝑟)
52	46, 51	eqtrd 2764	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) ∧ 𝑟 ∈ ℝ+) → (4 · ((𝑟 / 2) / 2)) = 𝑟)
53	52	oveq2d 7365	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) ∧ 𝑟 ∈ ℝ+) → ((vol*‘𝑧) + (4 · ((𝑟 / 2) / 2))) = ((vol*‘𝑧) + 𝑟))
54	36, 53	breqtrd 5118	. . . . . 6 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) ∧ 𝑟 ∈ ℝ+) → ((vol*‘(𝑧 ∩ ∪ ran ((,) ∘ 𝐹))) + (vol*‘(𝑧 ∖ ∪ ran ((,) ∘ 𝐹)))) ≤ ((vol*‘𝑧) + 𝑟))
55	54	ralrimiva 3121	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) → ∀𝑟 ∈ ℝ+ ((vol*‘(𝑧 ∩ ∪ ran ((,) ∘ 𝐹))) + (vol*‘(𝑧 ∖ ∪ ran ((,) ∘ 𝐹)))) ≤ ((vol*‘𝑧) + 𝑟))
56		inss1 4188	. . . . . . . . 9 ⊢ (𝑧 ∩ ∪ ran ((,) ∘ 𝐹)) ⊆ 𝑧
57	56	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) → (𝑧 ∩ ∪ ran ((,) ∘ 𝐹)) ⊆ 𝑧)
58		ovolsscl 25385	. . . . . . . 8 ⊢ (((𝑧 ∩ ∪ ran ((,) ∘ 𝐹)) ⊆ 𝑧 ∧ 𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧 ∩ ∪ ran ((,) ∘ 𝐹))) ∈ ℝ)
59	57, 14, 15, 58	syl3anc 1373	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) → (vol*‘(𝑧 ∩ ∪ ran ((,) ∘ 𝐹))) ∈ ℝ)
60		difssd 4088	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) → (𝑧 ∖ ∪ ran ((,) ∘ 𝐹)) ⊆ 𝑧)
61		ovolsscl 25385	. . . . . . . 8 ⊢ (((𝑧 ∖ ∪ ran ((,) ∘ 𝐹)) ⊆ 𝑧 ∧ 𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧 ∖ ∪ ran ((,) ∘ 𝐹))) ∈ ℝ)
62	60, 14, 15, 61	syl3anc 1373	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) → (vol*‘(𝑧 ∖ ∪ ran ((,) ∘ 𝐹))) ∈ ℝ)
63	59, 62	readdcld 11144	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) → ((vol*‘(𝑧 ∩ ∪ ran ((,) ∘ 𝐹))) + (vol*‘(𝑧 ∖ ∪ ran ((,) ∘ 𝐹)))) ∈ ℝ)
64		alrple 13108	. . . . . 6 ⊢ ((((vol*‘(𝑧 ∩ ∪ ran ((,) ∘ 𝐹))) + (vol*‘(𝑧 ∖ ∪ ran ((,) ∘ 𝐹)))) ∈ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (((vol*‘(𝑧 ∩ ∪ ran ((,) ∘ 𝐹))) + (vol*‘(𝑧 ∖ ∪ ran ((,) ∘ 𝐹)))) ≤ (vol*‘𝑧) ↔ ∀𝑟 ∈ ℝ+ ((vol*‘(𝑧 ∩ ∪ ran ((,) ∘ 𝐹))) + (vol*‘(𝑧 ∖ ∪ ran ((,) ∘ 𝐹)))) ≤ ((vol*‘𝑧) + 𝑟)))
65	63, 15, 64	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) → (((vol*‘(𝑧 ∩ ∪ ran ((,) ∘ 𝐹))) + (vol*‘(𝑧 ∖ ∪ ran ((,) ∘ 𝐹)))) ≤ (vol*‘𝑧) ↔ ∀𝑟 ∈ ℝ+ ((vol*‘(𝑧 ∩ ∪ ran ((,) ∘ 𝐹))) + (vol*‘(𝑧 ∖ ∪ ran ((,) ∘ 𝐹)))) ≤ ((vol*‘𝑧) + 𝑟)))
66	55, 65	mpbird 257	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) → ((vol*‘(𝑧 ∩ ∪ ran ((,) ∘ 𝐹))) + (vol*‘(𝑧 ∖ ∪ ran ((,) ∘ 𝐹)))) ≤ (vol*‘𝑧))
67	66	expr 456	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝒫 ℝ) → ((vol*‘𝑧) ∈ ℝ → ((vol*‘(𝑧 ∩ ∪ ran ((,) ∘ 𝐹))) + (vol*‘(𝑧 ∖ ∪ ran ((,) ∘ 𝐹)))) ≤ (vol*‘𝑧)))
68	67	ralrimiva 3121	. 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝒫 ℝ((vol*‘𝑧) ∈ ℝ → ((vol*‘(𝑧 ∩ ∪ ran ((,) ∘ 𝐹))) + (vol*‘(𝑧 ∖ ∪ ran ((,) ∘ 𝐹)))) ≤ (vol*‘𝑧)))
69		ismbl2 25426	. 2 ⊢ (∪ ran ((,) ∘ 𝐹) ∈ dom vol ↔ (∪ ran ((,) ∘ 𝐹) ⊆ ℝ ∧ ∀𝑧 ∈ 𝒫 ℝ((vol*‘𝑧) ∈ ℝ → ((vol*‘(𝑧 ∩ ∪ ran ((,) ∘ 𝐹))) + (vol*‘(𝑧 ∖ ∪ ran ((,) ∘ 𝐹)))) ≤ (vol*‘𝑧))))
70	12, 68, 69	sylanbrc 583	1 ⊢ (𝜑 → ∪ ran ((,) ∘ 𝐹) ∈ dom vol)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053   ∖ cdif 3900   ∩ cin 3902   ⊆ wss 3903  𝒫 cpw 4551  ∪ cuni 4858  Disj wdisj 5059   class class class wbr 5092   × cxp 5617  dom cdm 5619  ran crn 5620   ∘ ccom 5623  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349   ↑_m cmap 8753  supcsup 9330  ℂcc 11007  ℝcr 11008  0cc0 11009  1c1 11010   + caddc 11012   · cmul 11014  ℝ^*cxr 11148   < clt 11149   ≤ cle 11150   − cmin 11347   / cdiv 11777  ℕcn 12128  2c2 12183  4c4 12185  ℝ⁺crp 12893  (,)cioo 13248  seqcseq 13908  abscabs 15141  vol*covol 25361  volcvol 25362

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 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-inf2 9537  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086  ax-pre-sup 11087

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-disj 5060  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-isom 6491  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-of 7613  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-er 8625  df-map 8755  df-pm 8756  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-fi 9301  df-sup 9332  df-inf 9333  df-oi 9402  df-dju 9797  df-card 9835  df-acn 9838  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-div 11778  df-nn 12129  df-2 12191  df-3 12192  df-4 12193  df-n0 12385  df-z 12472  df-uz 12736  df-q 12850  df-rp 12894  df-xneg 13014  df-xadd 13015  df-xmul 13016  df-ioo 13252  df-ico 13254  df-icc 13255  df-fz 13411  df-fzo 13558  df-fl 13696  df-seq 13909  df-exp 13969  df-hash 14238  df-cj 15006  df-re 15007  df-im 15008  df-sqrt 15142  df-abs 15143  df-clim 15395  df-rlim 15396  df-sum 15594  df-rest 17326  df-topgen 17347  df-psmet 21253  df-xmet 21254  df-met 21255  df-bl 21256  df-mopn 21257  df-top 22779  df-topon 22796  df-bases 22831  df-cmp 23272  df-ovol 25363  df-vol 25364

This theorem is referenced by:  uniiccmbl  25489