Theorem ovolval5lem3 47228

Description: The value of the Lebesgue outer measure for subsets of the reals, using covers of left-closed right-open intervals are used, instead of open intervals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Hypotheses

Ref	Expression
ovolval5lem3.m	⊢ 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))}
ovolval5lem3.q	⊢ 𝑄 = {𝑧 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))}

Assertion

Ref	Expression
ovolval5lem3	⊢ inf(𝑄, ℝ*, < ) = inf(𝑀, ℝ*, < )

Distinct variable groups:   𝐴,𝑓,𝑧,𝑦   𝑦,𝑀,𝑧   𝑄,𝑓,𝑦,𝑧

Allowed substitution hint:   𝑀(𝑓)

Proof of Theorem ovolval5lem3

Dummy variables 𝑔 𝑛 𝑤 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovolval5lem3.q	. . . . 5 ⊢ 𝑄 = {𝑧 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))}
2	1	ssrab3 4035	. . . 4 ⊢ 𝑄 ⊆ ℝ*
3		infxrcl 13337	. . . 4 ⊢ (𝑄 ⊆ ℝ* → inf(𝑄, ℝ*, < ) ∈ ℝ*)
4	2, 3	mp1i 13	. . 3 ⊢ (⊤ → inf(𝑄, ℝ*, < ) ∈ ℝ*)
5		ovolval5lem3.m	. . . . 5 ⊢ 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))}
6	5	ssrab3 4035	. . . 4 ⊢ 𝑀 ⊆ ℝ*
7		infxrcl 13337	. . . 4 ⊢ (𝑀 ⊆ ℝ* → inf(𝑀, ℝ*, < ) ∈ ℝ*)
8	6, 7	mp1i 13	. . 3 ⊢ (⊤ → inf(𝑀, ℝ*, < ) ∈ ℝ*)
9	2	a1i 11	. . . 4 ⊢ (⊤ → 𝑄 ⊆ ℝ*)
10	6	a1i 11	. . . 4 ⊢ (⊤ → 𝑀 ⊆ ℝ*)
11	5	reqabi 3437	. . . . . . 7 ⊢ (𝑦 ∈ 𝑀 ↔ (𝑦 ∈ ℝ* ∧ ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))))
12	11	simprbi 501	. . . . . 6 ⊢ (𝑦 ∈ 𝑀 → ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))))
13		coeq2 5830	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑓 → ((,) ∘ 𝑔) = ((,) ∘ 𝑓))
14	13	rneqd 5914	. . . . . . . . . . . . . 14 ⊢ (𝑔 = 𝑓 → ran ((,) ∘ 𝑔) = ran ((,) ∘ 𝑓))
15	14	unieqd 4878	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝑓 → ∪ ran ((,) ∘ 𝑔) = ∪ ran ((,) ∘ 𝑓))
16	15	sseq2d 3968	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑓 → (𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ↔ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑓)))
17		coeq2 5830	. . . . . . . . . . . . . 14 ⊢ (𝑔 = 𝑓 → ((vol ∘ (,)) ∘ 𝑔) = ((vol ∘ (,)) ∘ 𝑓))
18	17	fveq2d 6871	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝑓 → (Σ^‘((vol ∘ (,)) ∘ 𝑔)) = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))
19	18	eqeq2d 2773	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑓 → (𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑔)) ↔ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))))
20	16, 19	anbi12d 641	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑓 → ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑔))) ↔ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))))
21	20	cbvrexvw 3241	. . . . . . . . . 10 ⊢ (∃𝑔 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑔))) ↔ ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))))
22	21	rabbii 3419	. . . . . . . . 9 ⊢ {𝑧 ∈ ℝ* ∣ ∃𝑔 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑔)))} = {𝑧 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))}
23	1, 22	eqtr4i 2788	. . . . . . . 8 ⊢ 𝑄 = {𝑧 ∈ ℝ* ∣ ∃𝑔 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑔)))}
24		simp3r 1216	. . . . . . . 8 ⊢ ((𝑤 ∈ ℝ+ ∧ 𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ (𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))
25		eqid 2762	. . . . . . . 8 ⊢ (Σ^‘((vol ∘ (,)) ∘ (𝑚 ∈ ℕ ↦ ⟨((1st ‘(𝑓‘𝑚)) − (𝑤 / (2↑𝑚))), (2nd ‘(𝑓‘𝑚))⟩))) = (Σ^‘((vol ∘ (,)) ∘ (𝑚 ∈ ℕ ↦ ⟨((1st ‘(𝑓‘𝑚)) − (𝑤 / (2↑𝑚))), (2nd ‘(𝑓‘𝑚))⟩)))
26		elmapi 8830	. . . . . . . . 9 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → 𝑓:ℕ⟶(ℝ × ℝ))
27	26	3ad2ant2 1147	. . . . . . . 8 ⊢ ((𝑤 ∈ ℝ+ ∧ 𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ (𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝑓:ℕ⟶(ℝ × ℝ))
28		simp3l 1215	. . . . . . . 8 ⊢ ((𝑤 ∈ ℝ+ ∧ 𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ (𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝐴 ⊆ ∪ ran ([,) ∘ 𝑓))
29		simp1 1149	. . . . . . . 8 ⊢ ((𝑤 ∈ ℝ+ ∧ 𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ (𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝑤 ∈ ℝ+)
30		2fveq3 6872	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (1st ‘(𝑓‘𝑚)) = (1st ‘(𝑓‘𝑛)))
31		oveq2 7404	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (2↑𝑚) = (2↑𝑛))
32	31	oveq2d 7412	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (𝑤 / (2↑𝑚)) = (𝑤 / (2↑𝑛)))
33	30, 32	oveq12d 7414	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → ((1st ‘(𝑓‘𝑚)) − (𝑤 / (2↑𝑚))) = ((1st ‘(𝑓‘𝑛)) − (𝑤 / (2↑𝑛))))
34		2fveq3 6872	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (2nd ‘(𝑓‘𝑚)) = (2nd ‘(𝑓‘𝑛)))
35	33, 34	opeq12d 4839	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ⟨((1st ‘(𝑓‘𝑚)) − (𝑤 / (2↑𝑚))), (2nd ‘(𝑓‘𝑚))⟩ = ⟨((1st ‘(𝑓‘𝑛)) − (𝑤 / (2↑𝑛))), (2nd ‘(𝑓‘𝑛))⟩)
36	35	cbvmptv 5204	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ ↦ ⟨((1st ‘(𝑓‘𝑚)) − (𝑤 / (2↑𝑚))), (2nd ‘(𝑓‘𝑚))⟩) = (𝑛 ∈ ℕ ↦ ⟨((1st ‘(𝑓‘𝑛)) − (𝑤 / (2↑𝑛))), (2nd ‘(𝑓‘𝑛))⟩)
37	23, 24, 25, 27, 28, 29, 36	ovolval5lem2 47227	. . . . . . 7 ⊢ ((𝑤 ∈ ℝ+ ∧ 𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ (𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → ∃𝑧 ∈ 𝑄 𝑧 ≤ (𝑦 +𝑒 𝑤))
38	37	rexlimdv3a 3167	. . . . . 6 ⊢ (𝑤 ∈ ℝ+ → (∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))) → ∃𝑧 ∈ 𝑄 𝑧 ≤ (𝑦 +𝑒 𝑤)))
39	12, 38	mpan9 514	. . . . 5 ⊢ ((𝑦 ∈ 𝑀 ∧ 𝑤 ∈ ℝ+) → ∃𝑧 ∈ 𝑄 𝑧 ≤ (𝑦 +𝑒 𝑤))
40	39	3adant1 1143	. . . 4 ⊢ ((⊤ ∧ 𝑦 ∈ 𝑀 ∧ 𝑤 ∈ ℝ+) → ∃𝑧 ∈ 𝑄 𝑧 ≤ (𝑦 +𝑒 𝑤))
41	9, 10, 40	infleinf 45947	. . 3 ⊢ (⊤ → inf(𝑄, ℝ*, < ) ≤ inf(𝑀, ℝ*, < ))
42		eqeq1 2766	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → (𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)) ↔ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))))
43	42	anbi2d 639	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) ↔ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))))
44	43	rexbidv 3186	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) ↔ ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))))
45	44	cbvrabv 3424	. . . . . . 7 ⊢ {𝑧 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))} = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))}
46		simpr 488	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑓)) → 𝐴 ⊆ ∪ ran ((,) ∘ 𝑓))
47		ioossico 13442	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1st ‘(𝑓‘𝑛))(,)(2nd ‘(𝑓‘𝑛))) ⊆ ((1st ‘(𝑓‘𝑛))[,)(2nd ‘(𝑓‘𝑛)))
48	47	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝑓‘𝑛))(,)(2nd ‘(𝑓‘𝑛))) ⊆ ((1st ‘(𝑓‘𝑛))[,)(2nd ‘(𝑓‘𝑛))))
49	26	adantr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → 𝑓:ℕ⟶(ℝ × ℝ))
50		simpr 488	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
51	49, 50	fvovco 45771	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝑓)‘𝑛) = ((1st ‘(𝑓‘𝑛))(,)(2nd ‘(𝑓‘𝑛))))
52	49, 50	fvovco 45771	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (([,) ∘ 𝑓)‘𝑛) = ((1st ‘(𝑓‘𝑛))[,)(2nd ‘(𝑓‘𝑛))))
53	48, 51, 52	3sstr4d 3991	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝑓)‘𝑛) ⊆ (([,) ∘ 𝑓)‘𝑛))
54	53	ralrimiva 3154	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ∀𝑛 ∈ ℕ (((,) ∘ 𝑓)‘𝑛) ⊆ (([,) ∘ 𝑓)‘𝑛))
55		ss2iun 4968	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑛 ∈ ℕ (((,) ∘ 𝑓)‘𝑛) ⊆ (([,) ∘ 𝑓)‘𝑛) → ∪ 𝑛 ∈ ℕ (((,) ∘ 𝑓)‘𝑛) ⊆ ∪ 𝑛 ∈ ℕ (([,) ∘ 𝑓)‘𝑛))
56	54, 55	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ∪ 𝑛 ∈ ℕ (((,) ∘ 𝑓)‘𝑛) ⊆ ∪ 𝑛 ∈ ℕ (([,) ∘ 𝑓)‘𝑛))
57		ioof 13451	. . . . . . . . . . . . . . . . . . 19 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ
58	57	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → (,):(ℝ* × ℝ*)⟶𝒫 ℝ)
59		rexpssxrxp 11227	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
60	59	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → (ℝ × ℝ) ⊆ (ℝ* × ℝ*))
61	58, 60, 26	fcoss 45786	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ((,) ∘ 𝑓):ℕ⟶𝒫 ℝ)
62	61	ffnd 6692	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ((,) ∘ 𝑓) Fn ℕ)
63		fniunfv 7231	. . . . . . . . . . . . . . . 16 ⊢ (((,) ∘ 𝑓) Fn ℕ → ∪ 𝑛 ∈ ℕ (((,) ∘ 𝑓)‘𝑛) = ∪ ran ((,) ∘ 𝑓))
64	62, 63	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ∪ 𝑛 ∈ ℕ (((,) ∘ 𝑓)‘𝑛) = ∪ ran ((,) ∘ 𝑓))
65		icof 45795	. . . . . . . . . . . . . . . . . . 19 ⊢ [,):(ℝ* × ℝ*)⟶𝒫 ℝ*
66	65	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → [,):(ℝ* × ℝ*)⟶𝒫 ℝ*)
67	66, 60, 26	fcoss 45786	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ([,) ∘ 𝑓):ℕ⟶𝒫 ℝ*)
68	67	ffnd 6692	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ([,) ∘ 𝑓) Fn ℕ)
69		fniunfv 7231	. . . . . . . . . . . . . . . 16 ⊢ (([,) ∘ 𝑓) Fn ℕ → ∪ 𝑛 ∈ ℕ (([,) ∘ 𝑓)‘𝑛) = ∪ ran ([,) ∘ 𝑓))
70	68, 69	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ∪ 𝑛 ∈ ℕ (([,) ∘ 𝑓)‘𝑛) = ∪ ran ([,) ∘ 𝑓))
71	56, 64, 70	3sstr3d 3990	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ∪ ran ((,) ∘ 𝑓) ⊆ ∪ ran ([,) ∘ 𝑓))
72	71	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑓)) → ∪ ran ((,) ∘ 𝑓) ⊆ ∪ ran ([,) ∘ 𝑓))
73	46, 72	sstrd 3946	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑓)) → 𝐴 ⊆ ∪ ran ([,) ∘ 𝑓))
74		simpr 488	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) → 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))
75	26	voliooicof 46570	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ((vol ∘ (,)) ∘ 𝑓) = ((vol ∘ [,)) ∘ 𝑓))
76	75	fveq2d 6871	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → (Σ^‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))
77	76	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) → (Σ^‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))
78	74, 77	eqtrd 2797	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) → 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))
79	73, 78	anim12dan 628	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))) → (𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))))
80	79	ex 416	. . . . . . . . . 10 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) → (𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))))
81	80	reximia 3097	. . . . . . . . 9 ⊢ (∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) → ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))))
82	81	a1i 11	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ* → (∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) → ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))))
83	82	ss2rabi 4029	. . . . . . 7 ⊢ {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))} ⊆ {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))}
84	45, 83	eqsstri 3982	. . . . . 6 ⊢ {𝑧 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))} ⊆ {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))}
85	84, 1, 5	3sstr4i 3987	. . . . 5 ⊢ 𝑄 ⊆ 𝑀
86		infxrss 13343	. . . . 5 ⊢ ((𝑄 ⊆ 𝑀 ∧ 𝑀 ⊆ ℝ*) → inf(𝑀, ℝ*, < ) ≤ inf(𝑄, ℝ*, < ))
87	85, 6, 86	mp2an 702	. . . 4 ⊢ inf(𝑀, ℝ*, < ) ≤ inf(𝑄, ℝ*, < )
88	87	a1i 11	. . 3 ⊢ (⊤ → inf(𝑀, ℝ*, < ) ≤ inf(𝑄, ℝ*, < ))
89	4, 8, 41, 88	xrletrid 13157	. 2 ⊢ (⊤ → inf(𝑄, ℝ*, < ) = inf(𝑀, ℝ*, < ))
90	89	mptru 1567	1 ⊢ inf(𝑄, ℝ*, < ) = inf(𝑀, ℝ*, < )

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1098   = wceq 1560  ⊤wtru 1561   ∈ wcel 2142  ∀wral 3076  ∃wrex 3086  {crab 3414   ⊆ wss 3904  𝒫 cpw 4555  ⟨cop 4588  ∪ cuni 4865  ∪ ciun 4949   class class class wbr 5100   ↦ cmpt 5181   × cxp 5645  ran crn 5648   ∘ ccom 5651   Fn wfn 6516  ⟶wf 6517  ‘cfv 6521  (class class class)co 7396  1^st c1st 7968  2^nd c2nd 7969   ↑_m cmap 8808  infcinf 9387  ℝcr 11072  ℝ^*cxr 11215   < clt 11216   ≤ cle 11217   − cmin 11414   / cdiv 11844  ℕcn 12210  2c2 12272  ℝ⁺crp 12993   +_𝑒 cxad 13112  (,)cioo 13349  [,)cico 13351  ↑cexp 14074  volcvol 25525  Σ^{^}csumge0 46936

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718  ax-inf2 9596  ax-cnex 11129  ax-resscn 11130  ax-1cn 11131  ax-icn 11132  ax-addcl 11133  ax-addrcl 11134  ax-mulcl 11135  ax-mulrcl 11136  ax-mulcom 11137  ax-addass 11138  ax-mulass 11139  ax-distr 11140  ax-i2m1 11141  ax-1ne0 11142  ax-1rid 11143  ax-rnegex 11144  ax-rrecex 11145  ax-cnre 11146  ax-pre-lttri 11147  ax-pre-lttrn 11148  ax-pre-ltadd 11149  ax-pre-mulgt0 11150  ax-pre-sup 11151

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4906  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-isom 6530  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-of 7660  df-om 7847  df-1st 7970  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-1o 8437  df-2o 8438  df-er 8678  df-map 8810  df-pm 8811  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-fi 9357  df-sup 9388  df-inf 9389  df-oi 9458  df-dju 9859  df-card 9897  df-pnf 11218  df-mnf 11219  df-xr 11220  df-ltxr 11221  df-le 11222  df-sub 11416  df-neg 11417  df-div 11845  df-nn 12211  df-2 12280  df-3 12281  df-n0 12482  df-z 12569  df-uz 12840  df-q 12950  df-rp 12994  df-xneg 13114  df-xadd 13115  df-xmul 13116  df-ioo 13353  df-ico 13355  df-icc 13356  df-fz 13513  df-fzo 13660  df-fl 13802  df-seq 14015  df-exp 14075  df-hash 14344  df-cj 15126  df-re 15127  df-im 15128  df-sqrt 15262  df-abs 15263  df-clim 15515  df-rlim 15516  df-sum 15714  df-rest 17451  df-topgen 17472  df-psmet 21416  df-xmet 21417  df-met 21418  df-bl 21419  df-mopn 21420  df-top 22954  df-topon 22971  df-bases 23006  df-cmp 23447  df-ovol 25526  df-vol 25527  df-sumge0 46937

This theorem is referenced by:  ovolval5  47229