ovolval5lem3 - Mathbox for Glauco Siliprandi

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Theorem ovolval5lem3 45370

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 4081	. . . 4 ⊢ 𝑄 ⊆ ℝ^*
3		infxrcl 13312	. . . 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 4081	. . . 4 ⊢ 𝑀 ⊆ ℝ^*
7		infxrcl 13312	. . . 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 3455	. . . . . . 7 ⊢ (𝑦 ∈ 𝑀 ↔ (𝑦 ∈ ℝ^* ∧ ∃𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ)(𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ [,)) ∘ 𝑓)))))
12	11	simprbi 498	. . . . . 6 ⊢ (𝑦 ∈ 𝑀 → ∃𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ)(𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ [,)) ∘ 𝑓))))
13		coeq2 5859	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑓 → ((,) ∘ 𝑔) = ((,) ∘ 𝑓))
14	13	rneqd 5938	. . . . . . . . . . . . . 14 ⊢ (𝑔 = 𝑓 → ran ((,) ∘ 𝑔) = ran ((,) ∘ 𝑓))
15	14	unieqd 4923	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝑓 → ∪ ran ((,) ∘ 𝑔) = ∪ ran ((,) ∘ 𝑓))
16	15	sseq2d 4015	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑓 → (𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ↔ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑓)))
17		coeq2 5859	. . . . . . . . . . . . . 14 ⊢ (𝑔 = 𝑓 → ((vol ∘ (,)) ∘ 𝑔) = ((vol ∘ (,)) ∘ 𝑓))
18	17	fveq2d 6896	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝑓 → (Σ^{^}‘((vol ∘ (,)) ∘ 𝑔)) = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)))
19	18	eqeq2d 2744	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑓 → (𝑧 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑔)) ↔ 𝑧 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓))))
20	16, 19	anbi12d 632	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑓 → ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑧 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑔))) ↔ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)))))
21	20	cbvrexvw 3236	. . . . . . . . . 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 3439	. . . . . . . . 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 2764	. . . . . . . 8 ⊢ 𝑄 = {𝑧 ∈ ℝ^* ∣ ∃𝑔 ∈ ((ℝ × ℝ) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑧 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑔)))}
24		simp3r 1203	. . . . . . . 8 ⊢ ((𝑤 ∈ ℝ⁺ ∧ 𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ (𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ [,)) ∘ 𝑓)))) → 𝑦 = (Σ^{^}‘((vol ∘ [,)) ∘ 𝑓)))
25		eqid 2733	. . . . . . . 8 ⊢ (Σ^{^}‘((vol ∘ (,)) ∘ (𝑚 ∈ ℕ ↦ ⟨((1^st ‘(𝑓‘𝑚)) − (𝑤 / (2↑𝑚))), (2^nd ‘(𝑓‘𝑚))⟩))) = (Σ^{^}‘((vol ∘ (,)) ∘ (𝑚 ∈ ℕ ↦ ⟨((1^st ‘(𝑓‘𝑚)) − (𝑤 / (2↑𝑚))), (2^nd ‘(𝑓‘𝑚))⟩)))
26		elmapi 8843	. . . . . . . . 9 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → 𝑓:ℕ⟶(ℝ × ℝ))
27	26	3ad2ant2 1135	. . . . . . . 8 ⊢ ((𝑤 ∈ ℝ⁺ ∧ 𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ (𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ [,)) ∘ 𝑓)))) → 𝑓:ℕ⟶(ℝ × ℝ))
28		simp3l 1202	. . . . . . . 8 ⊢ ((𝑤 ∈ ℝ⁺ ∧ 𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ (𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ [,)) ∘ 𝑓)))) → 𝐴 ⊆ ∪ ran ([,) ∘ 𝑓))
29		simp1 1137	. . . . . . . 8 ⊢ ((𝑤 ∈ ℝ⁺ ∧ 𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ (𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ [,)) ∘ 𝑓)))) → 𝑤 ∈ ℝ⁺)
30		2fveq3 6897	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (1^st ‘(𝑓‘𝑚)) = (1^st ‘(𝑓‘𝑛)))
31		oveq2 7417	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (2↑𝑚) = (2↑𝑛))
32	31	oveq2d 7425	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (𝑤 / (2↑𝑚)) = (𝑤 / (2↑𝑛)))
33	30, 32	oveq12d 7427	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → ((1^st ‘(𝑓‘𝑚)) − (𝑤 / (2↑𝑚))) = ((1^st ‘(𝑓‘𝑛)) − (𝑤 / (2↑𝑛))))
34		2fveq3 6897	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (2^nd ‘(𝑓‘𝑚)) = (2^nd ‘(𝑓‘𝑛)))
35	33, 34	opeq12d 4882	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ⟨((1^st ‘(𝑓‘𝑚)) − (𝑤 / (2↑𝑚))), (2^nd ‘(𝑓‘𝑚))⟩ = ⟨((1^st ‘(𝑓‘𝑛)) − (𝑤 / (2↑𝑛))), (2^nd ‘(𝑓‘𝑛))⟩)
36	35	cbvmptv 5262	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ ↦ ⟨((1^st ‘(𝑓‘𝑚)) − (𝑤 / (2↑𝑚))), (2^nd ‘(𝑓‘𝑚))⟩) = (𝑛 ∈ ℕ ↦ ⟨((1^st ‘(𝑓‘𝑛)) − (𝑤 / (2↑𝑛))), (2^nd ‘(𝑓‘𝑛))⟩)
37	23, 24, 25, 27, 28, 29, 36	ovolval5lem2 45369	. . . . . . 7 ⊢ ((𝑤 ∈ ℝ⁺ ∧ 𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ (𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ [,)) ∘ 𝑓)))) → ∃𝑧 ∈ 𝑄 𝑧 ≤ (𝑦 +_𝑒 𝑤))
38	37	rexlimdv3a 3160	. . . . . 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 508	. . . . 5 ⊢ ((𝑦 ∈ 𝑀 ∧ 𝑤 ∈ ℝ⁺) → ∃𝑧 ∈ 𝑄 𝑧 ≤ (𝑦 +_𝑒 𝑤))
40	39	3adant1 1131	. . . 4 ⊢ ((⊤ ∧ 𝑦 ∈ 𝑀 ∧ 𝑤 ∈ ℝ⁺) → ∃𝑧 ∈ 𝑄 𝑧 ≤ (𝑦 +_𝑒 𝑤))
41	9, 10, 40	infleinf 44082	. . 3 ⊢ (⊤ → inf(𝑄, ℝ^, < ) ≤ inf(𝑀, ℝ^, < ))
42		eqeq1 2737	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → (𝑧 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)) ↔ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓))))
43	42	anbi2d 630	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓))) ↔ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)))))
44	43	rexbidv 3179	. . . . . . . 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 3443	. . . . . . 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 486	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑓)) → 𝐴 ⊆ ∪ ran ((,) ∘ 𝑓))
47		ioossico 13415	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1^st ‘(𝑓‘𝑛))(,)(2^nd ‘(𝑓‘𝑛))) ⊆ ((1^st ‘(𝑓‘𝑛))[,)(2^nd ‘(𝑓‘𝑛)))
48	47	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → ((1^st ‘(𝑓‘𝑛))(,)(2^nd ‘(𝑓‘𝑛))) ⊆ ((1^st ‘(𝑓‘𝑛))[,)(2^nd ‘(𝑓‘𝑛))))
49	26	adantr 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → 𝑓:ℕ⟶(ℝ × ℝ))
50		simpr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
51	49, 50	fvovco 43892	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝑓)‘𝑛) = ((1^st ‘(𝑓‘𝑛))(,)(2^nd ‘(𝑓‘𝑛))))
52	49, 50	fvovco 43892	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (([,) ∘ 𝑓)‘𝑛) = ((1^st ‘(𝑓‘𝑛))[,)(2^nd ‘(𝑓‘𝑛))))
53	48, 51, 52	3sstr4d 4030	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝑓)‘𝑛) ⊆ (([,) ∘ 𝑓)‘𝑛))
54	53	ralrimiva 3147	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → ∀𝑛 ∈ ℕ (((,) ∘ 𝑓)‘𝑛) ⊆ (([,) ∘ 𝑓)‘𝑛))
55		ss2iun 5016	. . . . . . . . . . . . . . . 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 13424	. . . . . . . . . . . . . . . . . . 19 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
58	57	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ)
59		rexpssxrxp 11259	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℝ × ℝ) ⊆ (ℝ^* × ℝ^*)
60	59	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → (ℝ × ℝ) ⊆ (ℝ^* × ℝ^*))
61	58, 60, 26	fcoss 43909	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → ((,) ∘ 𝑓):ℕ⟶𝒫 ℝ)
62	61	ffnd 6719	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → ((,) ∘ 𝑓) Fn ℕ)
63		fniunfv 7246	. . . . . . . . . . . . . . . 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 43918	. . . . . . . . . . . . . . . . . . 19 ⊢ [,):(ℝ^* × ℝ^)⟶𝒫 ℝ^
66	65	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → [,):(ℝ^* × ℝ^)⟶𝒫 ℝ^)
67	66, 60, 26	fcoss 43909	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → ([,) ∘ 𝑓):ℕ⟶𝒫 ℝ^*)
68	67	ffnd 6719	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → ([,) ∘ 𝑓) Fn ℕ)
69		fniunfv 7246	. . . . . . . . . . . . . . . 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 4029	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → ∪ ran ((,) ∘ 𝑓) ⊆ ∪ ran ([,) ∘ 𝑓))
72	71	adantr 482	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑓)) → ∪ ran ((,) ∘ 𝑓) ⊆ ∪ ran ([,) ∘ 𝑓))
73	46, 72	sstrd 3993	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑓)) → 𝐴 ⊆ ∪ ran ([,) ∘ 𝑓))
74		simpr 486	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓))) → 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)))
75	26	voliooicof 44712	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → ((vol ∘ (,)) ∘ 𝑓) = ((vol ∘ [,)) ∘ 𝑓))
76	75	fveq2d 6896	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^{^}‘((vol ∘ [,)) ∘ 𝑓)))
77	76	adantr 482	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓))) → (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^{^}‘((vol ∘ [,)) ∘ 𝑓)))
78	74, 77	eqtrd 2773	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓))) → 𝑦 = (Σ^{^}‘((vol ∘ [,)) ∘ 𝑓)))
79	73, 78	anim12dan 620	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)))) → (𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ [,)) ∘ 𝑓))))
80	79	ex 414	. . . . . . . . . 10 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓))) → (𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ [,)) ∘ 𝑓)))))
81	80	reximia 3082	. . . . . . . . 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 4075	. . . . . . 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 4017	. . . . . 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 4026	. . . . 5 ⊢ 𝑄 ⊆ 𝑀
86		infxrss 13318	. . . . 5 ⊢ ((𝑄 ⊆ 𝑀 ∧ 𝑀 ⊆ ℝ^) → inf(𝑀, ℝ^, < ) ≤ inf(𝑄, ℝ^*, < ))
87	85, 6, 86	mp2an 691	. . . 4 ⊢ inf(𝑀, ℝ^, < ) ≤ inf(𝑄, ℝ^, < )
88	87	a1i 11	. . 3 ⊢ (⊤ → inf(𝑀, ℝ^, < ) ≤ inf(𝑄, ℝ^, < ))
89	4, 8, 41, 88	xrletrid 13134	. 2 ⊢ (⊤ → inf(𝑄, ℝ^, < ) = inf(𝑀, ℝ^, < ))
90	89	mptru 1549	1 ⊢ inf(𝑄, ℝ^, < ) = inf(𝑀, ℝ^, < )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ⊤wtru 1543 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 {crab 3433 ⊆ wss 3949 𝒫 cpw 4603 ⟨cop 4635 ∪ cuni 4909 ∪ ciun 4998 class class class wbr 5149 ↦ cmpt 5232 × cxp 5675 ran crn 5678 ∘ ccom 5681 Fn wfn 6539 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 1^st c1st 7973 2^nd c2nd 7974 ↑_m cmap 8820 infcinf 9436 ℝcr 11109 ℝ^*cxr 11247 < clt 11248 ≤ cle 11249 − cmin 11444 / cdiv 11871 ℕcn 12212 2c2 12267 ℝ⁺crp 12974 +_𝑒 cxad 13090 (,)cioo 13324 [,)cico 13326 ↑cexp 14027 volcvol 24980 Σ^{^}csumge0 45078

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-er 8703 df-map 8822 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9505 df-dju 9896 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-z 12559 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ioo 13328 df-ico 13330 df-icc 13331 df-fz 13485 df-fzo 13628 df-fl 13757 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-clim 15432 df-rlim 15433 df-sum 15633 df-rest 17368 df-topgen 17389 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-top 22396 df-topon 22413 df-bases 22449 df-cmp 22891 df-ovol 24981 df-vol 24982 df-sumge0 45079

This theorem is referenced by: ovolval5 45371

W3C validator