Theorem ovolval5lem3 40868

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		ssrab2 3687	. . . . 5 ⊢ {𝑧 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))} ⊆ ℝ*
3	1, 2	eqsstri 3635	. . . 4 ⊢ 𝑄 ⊆ ℝ*
4		infxrcl 12163	. . . 4 ⊢ (𝑄 ⊆ ℝ* → inf(𝑄, ℝ*, < ) ∈ ℝ*)
5	3, 4	mp1i 13	. . 3 ⊢ (⊤ → inf(𝑄, ℝ*, < ) ∈ ℝ*)
6		ovolval5lem3.m	. . . . 5 ⊢ 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))}
7		ssrab2 3687	. . . . 5 ⊢ {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))} ⊆ ℝ*
8	6, 7	eqsstri 3635	. . . 4 ⊢ 𝑀 ⊆ ℝ*
9		infxrcl 12163	. . . 4 ⊢ (𝑀 ⊆ ℝ* → inf(𝑀, ℝ*, < ) ∈ ℝ*)
10	8, 9	mp1i 13	. . 3 ⊢ (⊤ → inf(𝑀, ℝ*, < ) ∈ ℝ*)
11	3	a1i 11	. . . 4 ⊢ (⊤ → 𝑄 ⊆ ℝ*)
12	8	a1i 11	. . . 4 ⊢ (⊤ → 𝑀 ⊆ ℝ*)
13		simpr 477	. . . . . 6 ⊢ ((𝑦 ∈ 𝑀 ∧ 𝑤 ∈ ℝ+) → 𝑤 ∈ ℝ+)
14	6	rabeq2i 3197	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑀 ↔ (𝑦 ∈ ℝ* ∧ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))))
15	14	biimpi 206	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑀 → (𝑦 ∈ ℝ* ∧ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))))
16	15	simprd 479	. . . . . . 7 ⊢ (𝑦 ∈ 𝑀 → ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))))
17	16	adantr 481	. . . . . 6 ⊢ ((𝑦 ∈ 𝑀 ∧ 𝑤 ∈ ℝ+) → ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))))
18		coeq2 5280	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔 = 𝑓 → ((,) ∘ 𝑔) = ((,) ∘ 𝑓))
19	18	rneqd 5353	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = 𝑓 → ran ((,) ∘ 𝑔) = ran ((,) ∘ 𝑓))
20	19	unieqd 4446	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = 𝑓 → ∪ ran ((,) ∘ 𝑔) = ∪ ran ((,) ∘ 𝑓))
21	20	sseq2d 3633	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑓 → (𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ↔ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑓)))
22		coeq2 5280	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = 𝑓 → ((vol ∘ (,)) ∘ 𝑔) = ((vol ∘ (,)) ∘ 𝑓))
23	22	fveq2d 6195	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = 𝑓 → (Σ^‘((vol ∘ (,)) ∘ 𝑔)) = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))
24	23	eqeq2d 2632	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑓 → (𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑔)) ↔ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))))
25	21, 24	anbi12d 747	. . . . . . . . . . . . . 14 ⊢ (𝑔 = 𝑓 → ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑔))) ↔ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))))
26	25	cbvrexv 3172	. . . . . . . . . . . . 13 ⊢ (∃𝑔 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑔))) ↔ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))))
27	26	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℝ* → (∃𝑔 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑔))) ↔ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))))
28	27	rabbiia 3185	. . . . . . . . . . 11 ⊢ {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)))}
29	1, 28	eqtr4i 2647	. . . . . . . . . 10 ⊢ 𝑄 = {𝑧 ∈ ℝ* ∣ ∃𝑔 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑔)))}
30		simp3r 1090	. . . . . . . . . 10 ⊢ ((𝑤 ∈ ℝ+ ∧ 𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ (𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))
31		eqid 2622	. . . . . . . . . 10 ⊢ (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))⟩)))
32		elmapi 7879	. . . . . . . . . . 11 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → 𝑓:ℕ⟶(ℝ × ℝ))
33	32	3ad2ant2 1083	. . . . . . . . . 10 ⊢ ((𝑤 ∈ ℝ+ ∧ 𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ (𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝑓:ℕ⟶(ℝ × ℝ))
34		simp3l 1089	. . . . . . . . . 10 ⊢ ((𝑤 ∈ ℝ+ ∧ 𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ (𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝐴 ⊆ ∪ ran ([,) ∘ 𝑓))
35		simp1 1061	. . . . . . . . . 10 ⊢ ((𝑤 ∈ ℝ+ ∧ 𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ (𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝑤 ∈ ℝ+)
36		fveq2 6191	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑛 → (𝑓‘𝑚) = (𝑓‘𝑛))
37	36	fveq2d 6195	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → (1st ‘(𝑓‘𝑚)) = (1st ‘(𝑓‘𝑛)))
38		oveq2 6658	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑛 → (2↑𝑚) = (2↑𝑛))
39	38	oveq2d 6666	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → (𝑤 / (2↑𝑚)) = (𝑤 / (2↑𝑛)))
40	37, 39	oveq12d 6668	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → ((1st ‘(𝑓‘𝑚)) − (𝑤 / (2↑𝑚))) = ((1st ‘(𝑓‘𝑛)) − (𝑤 / (2↑𝑛))))
41	36	fveq2d 6195	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (2nd ‘(𝑓‘𝑚)) = (2nd ‘(𝑓‘𝑛)))
42	40, 41	opeq12d 4410	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → ⟨((1st ‘(𝑓‘𝑚)) − (𝑤 / (2↑𝑚))), (2nd ‘(𝑓‘𝑚))⟩ = ⟨((1st ‘(𝑓‘𝑛)) − (𝑤 / (2↑𝑛))), (2nd ‘(𝑓‘𝑛))⟩)
43	42	cbvmptv 4750	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ ↦ ⟨((1st ‘(𝑓‘𝑚)) − (𝑤 / (2↑𝑚))), (2nd ‘(𝑓‘𝑚))⟩) = (𝑛 ∈ ℕ ↦ ⟨((1st ‘(𝑓‘𝑛)) − (𝑤 / (2↑𝑛))), (2nd ‘(𝑓‘𝑛))⟩)
44	29, 30, 31, 33, 34, 35, 43	ovolval5lem2 40867	. . . . . . . . 9 ⊢ ((𝑤 ∈ ℝ+ ∧ 𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ (𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → ∃𝑧 ∈ 𝑄 𝑧 ≤ (𝑦 +𝑒 𝑤))
45	44	3exp 1264	. . . . . . . 8 ⊢ (𝑤 ∈ ℝ+ → (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ((𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))) → ∃𝑧 ∈ 𝑄 𝑧 ≤ (𝑦 +𝑒 𝑤))))
46	45	rexlimdv 3030	. . . . . . 7 ⊢ (𝑤 ∈ ℝ+ → (∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))) → ∃𝑧 ∈ 𝑄 𝑧 ≤ (𝑦 +𝑒 𝑤)))
47	46	imp 445	. . . . . 6 ⊢ ((𝑤 ∈ ℝ+ ∧ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → ∃𝑧 ∈ 𝑄 𝑧 ≤ (𝑦 +𝑒 𝑤))
48	13, 17, 47	syl2anc 693	. . . . 5 ⊢ ((𝑦 ∈ 𝑀 ∧ 𝑤 ∈ ℝ+) → ∃𝑧 ∈ 𝑄 𝑧 ≤ (𝑦 +𝑒 𝑤))
49	48	3adant1 1079	. . . 4 ⊢ ((⊤ ∧ 𝑦 ∈ 𝑀 ∧ 𝑤 ∈ ℝ+) → ∃𝑧 ∈ 𝑄 𝑧 ≤ (𝑦 +𝑒 𝑤))
50	11, 12, 49	infleinf 39588	. . 3 ⊢ (⊤ → inf(𝑄, ℝ*, < ) ≤ inf(𝑀, ℝ*, < ))
51		eqeq1 2626	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → (𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)) ↔ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))))
52	51	anbi2d 740	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) ↔ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))))
53	52	rexbidv 3052	. . . . . . . 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)))))
54	53	cbvrabv 3199	. . . . . . 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)))}
55		simpr 477	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑓)) → 𝐴 ⊆ ∪ ran ((,) ∘ 𝑓))
56		ioossico 12262	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1st ‘(𝑓‘𝑛))(,)(2nd ‘(𝑓‘𝑛))) ⊆ ((1st ‘(𝑓‘𝑛))[,)(2nd ‘(𝑓‘𝑛)))
57	56	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝑓‘𝑛))(,)(2nd ‘(𝑓‘𝑛))) ⊆ ((1st ‘(𝑓‘𝑛))[,)(2nd ‘(𝑓‘𝑛))))
58	32	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → 𝑓:ℕ⟶(ℝ × ℝ))
59		simpr 477	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
60	58, 59	fvovco 39381	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝑓)‘𝑛) = ((1st ‘(𝑓‘𝑛))(,)(2nd ‘(𝑓‘𝑛))))
61	58, 59	fvovco 39381	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (([,) ∘ 𝑓)‘𝑛) = ((1st ‘(𝑓‘𝑛))[,)(2nd ‘(𝑓‘𝑛))))
62	60, 61	sseq12d 3634	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((((,) ∘ 𝑓)‘𝑛) ⊆ (([,) ∘ 𝑓)‘𝑛) ↔ ((1st ‘(𝑓‘𝑛))(,)(2nd ‘(𝑓‘𝑛))) ⊆ ((1st ‘(𝑓‘𝑛))[,)(2nd ‘(𝑓‘𝑛)))))
63	57, 62	mpbird 247	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝑓)‘𝑛) ⊆ (([,) ∘ 𝑓)‘𝑛))
64	63	ralrimiva 2966	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ∀𝑛 ∈ ℕ (((,) ∘ 𝑓)‘𝑛) ⊆ (([,) ∘ 𝑓)‘𝑛))
65		ss2iun 4536	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑛 ∈ ℕ (((,) ∘ 𝑓)‘𝑛) ⊆ (([,) ∘ 𝑓)‘𝑛) → ∪ 𝑛 ∈ ℕ (((,) ∘ 𝑓)‘𝑛) ⊆ ∪ 𝑛 ∈ ℕ (([,) ∘ 𝑓)‘𝑛))
66	64, 65	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ∪ 𝑛 ∈ ℕ (((,) ∘ 𝑓)‘𝑛) ⊆ ∪ 𝑛 ∈ ℕ (([,) ∘ 𝑓)‘𝑛))
67		ioof 12271	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ
68	67	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → (,):(ℝ* × ℝ*)⟶𝒫 ℝ)
69		rexpssxrxp 10084	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
70	69	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → (ℝ × ℝ) ⊆ (ℝ* × ℝ*))
71	32, 70	fssd 6057	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → 𝑓:ℕ⟶(ℝ* × ℝ*))
72		fco 6058	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((,):(ℝ* × ℝ*)⟶𝒫 ℝ ∧ 𝑓:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝑓):ℕ⟶𝒫 ℝ)
73	68, 71, 72	syl2anc 693	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ((,) ∘ 𝑓):ℕ⟶𝒫 ℝ)
74		ffn 6045	. . . . . . . . . . . . . . . . . . 19 ⊢ (((,) ∘ 𝑓):ℕ⟶𝒫 ℝ → ((,) ∘ 𝑓) Fn ℕ)
75	73, 74	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ((,) ∘ 𝑓) Fn ℕ)
76		fniunfv 6505	. . . . . . . . . . . . . . . . . 18 ⊢ (((,) ∘ 𝑓) Fn ℕ → ∪ 𝑛 ∈ ℕ (((,) ∘ 𝑓)‘𝑛) = ∪ ran ((,) ∘ 𝑓))
77	75, 76	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ∪ 𝑛 ∈ ℕ (((,) ∘ 𝑓)‘𝑛) = ∪ ran ((,) ∘ 𝑓))
78		icof 39411	. . . . . . . . . . . . . . . . . . . . 21 ⊢ [,):(ℝ* × ℝ*)⟶𝒫 ℝ*
79	78	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → [,):(ℝ* × ℝ*)⟶𝒫 ℝ*)
80		fco 6058	. . . . . . . . . . . . . . . . . . . 20 ⊢ (([,):(ℝ* × ℝ*)⟶𝒫 ℝ* ∧ 𝑓:ℕ⟶(ℝ* × ℝ*)) → ([,) ∘ 𝑓):ℕ⟶𝒫 ℝ*)
81	79, 71, 80	syl2anc 693	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ([,) ∘ 𝑓):ℕ⟶𝒫 ℝ*)
82		ffn 6045	. . . . . . . . . . . . . . . . . . 19 ⊢ (([,) ∘ 𝑓):ℕ⟶𝒫 ℝ* → ([,) ∘ 𝑓) Fn ℕ)
83	81, 82	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ([,) ∘ 𝑓) Fn ℕ)
84		fniunfv 6505	. . . . . . . . . . . . . . . . . 18 ⊢ (([,) ∘ 𝑓) Fn ℕ → ∪ 𝑛 ∈ ℕ (([,) ∘ 𝑓)‘𝑛) = ∪ ran ([,) ∘ 𝑓))
85	83, 84	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ∪ 𝑛 ∈ ℕ (([,) ∘ 𝑓)‘𝑛) = ∪ ran ([,) ∘ 𝑓))
86	77, 85	sseq12d 3634	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → (∪ 𝑛 ∈ ℕ (((,) ∘ 𝑓)‘𝑛) ⊆ ∪ 𝑛 ∈ ℕ (([,) ∘ 𝑓)‘𝑛) ↔ ∪ ran ((,) ∘ 𝑓) ⊆ ∪ ran ([,) ∘ 𝑓)))
87	66, 86	mpbid 222	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ∪ ran ((,) ∘ 𝑓) ⊆ ∪ ran ([,) ∘ 𝑓))
88	87	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑓)) → ∪ ran ((,) ∘ 𝑓) ⊆ ∪ ran ([,) ∘ 𝑓))
89	55, 88	sstrd 3613	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑓)) → 𝐴 ⊆ ∪ ran ([,) ∘ 𝑓))
90	89	adantrr 753	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))) → 𝐴 ⊆ ∪ ran ([,) ∘ 𝑓))
91		simpr 477	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) → 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))
92	32	voliooicof 40213	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ((vol ∘ (,)) ∘ 𝑓) = ((vol ∘ [,)) ∘ 𝑓))
93	92	fveq2d 6195	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → (Σ^‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))
94	93	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) → (Σ^‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))
95	91, 94	eqtrd 2656	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) → 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))
96	95	adantrl 752	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))) → 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))
97	90, 96	jca 554	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))) → (𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))))
98	97	ex 450	. . . . . . . . . 10 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) → (𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))))
99	98	reximia 3009	. . . . . . . . 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))))
100	99	rgenw 2924	. . . . . . . 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))))
101		ss2rab 3678	. . . . . . . 8 ⊢ ({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)))} ↔ ∀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)))))
102	100, 101	mpbir 221	. . . . . . 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)))}
103	54, 102	eqsstri 3635	. . . . . 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)))}
104	1, 6	sseq12i 3631	. . . . . 6 ⊢ (𝑄 ⊆ 𝑀 ↔ {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)))})
105	103, 104	mpbir 221	. . . . 5 ⊢ 𝑄 ⊆ 𝑀
106		infxrss 12169	. . . . 5 ⊢ ((𝑄 ⊆ 𝑀 ∧ 𝑀 ⊆ ℝ*) → inf(𝑀, ℝ*, < ) ≤ inf(𝑄, ℝ*, < ))
107	105, 8, 106	mp2an 708	. . . 4 ⊢ inf(𝑀, ℝ*, < ) ≤ inf(𝑄, ℝ*, < )
108	107	a1i 11	. . 3 ⊢ (⊤ → inf(𝑀, ℝ*, < ) ≤ inf(𝑄, ℝ*, < ))
109	5, 10, 50, 108	xrletrid 11986	. 2 ⊢ (⊤ → inf(𝑄, ℝ*, < ) = inf(𝑀, ℝ*, < ))
110	109	trud 1493	1 ⊢ inf(𝑄, ℝ*, < ) = inf(𝑀, ℝ*, < )

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ⊤wtru 1484   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  {crab 2916   ⊆ wss 3574  𝒫 cpw 4158  ⟨cop 4183  ∪ cuni 4436  ∪ ciun 4520   class class class wbr 4653   ↦ cmpt 4729   × cxp 5112  ran crn 5115   ∘ ccom 5118   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  1^st c1st 7166  2^nd c2nd 7167   ↑_𝑚 cmap 7857  infcinf 8347  ℝcr 9935  ℝ^*cxr 10073   < clt 10074   ≤ cle 10075   − cmin 10266   / cdiv 10684  ℕcn 11020  2c2 11070  ℝ⁺crp 11832   +_𝑒 cxad 11944  (,)cioo 12175  [,)cico 12177  ↑cexp 12860  volcvol 23232  Σ^{^}csumge0 40579

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417  df-rest 16083  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-bases 20750  df-cmp 21190  df-ovol 23233  df-vol 23234  df-sumge0 40580

This theorem is referenced by:  ovolval5  40869