ovolval5lem3 - Mathbox for Glauco Siliprandi

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Theorem ovolval5lem3 45356

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 4079	. . . 4 ⊢ 𝑄 ⊆ ℝ^*
3		infxrcl 13308	. . . 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 4079	. . . 4 ⊢ 𝑀 ⊆ ℝ^*
7		infxrcl 13308	. . . 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 3454	. . . . . . 7 ⊢ (𝑦 ∈ 𝑀 ↔ (𝑦 ∈ ℝ^* ∧ ∃𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ)(𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ [,)) ∘ 𝑓)))))
12	11	simprbi 497	. . . . . 6 ⊢ (𝑦 ∈ 𝑀 → ∃𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ)(𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ [,)) ∘ 𝑓))))
13		coeq2 5856	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑓 → ((,) ∘ 𝑔) = ((,) ∘ 𝑓))
14	13	rneqd 5935	. . . . . . . . . . . . . 14 ⊢ (𝑔 = 𝑓 → ran ((,) ∘ 𝑔) = ran ((,) ∘ 𝑓))
15	14	unieqd 4921	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝑓 → ∪ ran ((,) ∘ 𝑔) = ∪ ran ((,) ∘ 𝑓))
16	15	sseq2d 4013	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑓 → (𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ↔ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑓)))
17		coeq2 5856	. . . . . . . . . . . . . 14 ⊢ (𝑔 = 𝑓 → ((vol ∘ (,)) ∘ 𝑔) = ((vol ∘ (,)) ∘ 𝑓))
18	17	fveq2d 6892	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝑓 → (Σ^{^}‘((vol ∘ (,)) ∘ 𝑔)) = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)))
19	18	eqeq2d 2743	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑓 → (𝑧 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑔)) ↔ 𝑧 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓))))
20	16, 19	anbi12d 631	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑓 → ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑧 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑔))) ↔ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)))))
21	20	cbvrexvw 3235	. . . . . . . . . 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 3438	. . . . . . . . 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 2763	. . . . . . . 8 ⊢ 𝑄 = {𝑧 ∈ ℝ^* ∣ ∃𝑔 ∈ ((ℝ × ℝ) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑧 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑔)))}
24		simp3r 1202	. . . . . . . 8 ⊢ ((𝑤 ∈ ℝ⁺ ∧ 𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ (𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ [,)) ∘ 𝑓)))) → 𝑦 = (Σ^{^}‘((vol ∘ [,)) ∘ 𝑓)))
25		eqid 2732	. . . . . . . 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 8839	. . . . . . . . 9 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → 𝑓:ℕ⟶(ℝ × ℝ))
27	26	3ad2ant2 1134	. . . . . . . 8 ⊢ ((𝑤 ∈ ℝ⁺ ∧ 𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ (𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ [,)) ∘ 𝑓)))) → 𝑓:ℕ⟶(ℝ × ℝ))
28		simp3l 1201	. . . . . . . 8 ⊢ ((𝑤 ∈ ℝ⁺ ∧ 𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ (𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ [,)) ∘ 𝑓)))) → 𝐴 ⊆ ∪ ran ([,) ∘ 𝑓))
29		simp1 1136	. . . . . . . 8 ⊢ ((𝑤 ∈ ℝ⁺ ∧ 𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ (𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ [,)) ∘ 𝑓)))) → 𝑤 ∈ ℝ⁺)
30		2fveq3 6893	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (1^st ‘(𝑓‘𝑚)) = (1^st ‘(𝑓‘𝑛)))
31		oveq2 7413	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (2↑𝑚) = (2↑𝑛))
32	31	oveq2d 7421	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (𝑤 / (2↑𝑚)) = (𝑤 / (2↑𝑛)))
33	30, 32	oveq12d 7423	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → ((1^st ‘(𝑓‘𝑚)) − (𝑤 / (2↑𝑚))) = ((1^st ‘(𝑓‘𝑛)) − (𝑤 / (2↑𝑛))))
34		2fveq3 6893	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (2^nd ‘(𝑓‘𝑚)) = (2^nd ‘(𝑓‘𝑛)))
35	33, 34	opeq12d 4880	. . . . . . . . 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 5260	. . . . . . . 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 45355	. . . . . . 7 ⊢ ((𝑤 ∈ ℝ⁺ ∧ 𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ (𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ [,)) ∘ 𝑓)))) → ∃𝑧 ∈ 𝑄 𝑧 ≤ (𝑦 +_𝑒 𝑤))
38	37	rexlimdv3a 3159	. . . . . 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 507	. . . . 5 ⊢ ((𝑦 ∈ 𝑀 ∧ 𝑤 ∈ ℝ⁺) → ∃𝑧 ∈ 𝑄 𝑧 ≤ (𝑦 +_𝑒 𝑤))
40	39	3adant1 1130	. . . 4 ⊢ ((⊤ ∧ 𝑦 ∈ 𝑀 ∧ 𝑤 ∈ ℝ⁺) → ∃𝑧 ∈ 𝑄 𝑧 ≤ (𝑦 +_𝑒 𝑤))
41	9, 10, 40	infleinf 44068	. . 3 ⊢ (⊤ → inf(𝑄, ℝ^, < ) ≤ inf(𝑀, ℝ^, < ))
42		eqeq1 2736	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → (𝑧 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)) ↔ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓))))
43	42	anbi2d 629	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓))) ↔ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)))))
44	43	rexbidv 3178	. . . . . . . 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 3442	. . . . . . 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 485	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑓)) → 𝐴 ⊆ ∪ ran ((,) ∘ 𝑓))
47		ioossico 13411	. . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → 𝑓:ℕ⟶(ℝ × ℝ))
50		simpr 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
51	49, 50	fvovco 43877	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝑓)‘𝑛) = ((1^st ‘(𝑓‘𝑛))(,)(2^nd ‘(𝑓‘𝑛))))
52	49, 50	fvovco 43877	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (([,) ∘ 𝑓)‘𝑛) = ((1^st ‘(𝑓‘𝑛))[,)(2^nd ‘(𝑓‘𝑛))))
53	48, 51, 52	3sstr4d 4028	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝑓)‘𝑛) ⊆ (([,) ∘ 𝑓)‘𝑛))
54	53	ralrimiva 3146	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → ∀𝑛 ∈ ℕ (((,) ∘ 𝑓)‘𝑛) ⊆ (([,) ∘ 𝑓)‘𝑛))
55		ss2iun 5014	. . . . . . . . . . . . . . . 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 13420	. . . . . . . . . . . . . . . . . . 19 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
58	57	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ)
59		rexpssxrxp 11255	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℝ × ℝ) ⊆ (ℝ^* × ℝ^*)
60	59	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → (ℝ × ℝ) ⊆ (ℝ^* × ℝ^*))
61	58, 60, 26	fcoss 43894	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → ((,) ∘ 𝑓):ℕ⟶𝒫 ℝ)
62	61	ffnd 6715	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → ((,) ∘ 𝑓) Fn ℕ)
63		fniunfv 7242	. . . . . . . . . . . . . . . 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 43903	. . . . . . . . . . . . . . . . . . 19 ⊢ [,):(ℝ^* × ℝ^)⟶𝒫 ℝ^
66	65	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → [,):(ℝ^* × ℝ^)⟶𝒫 ℝ^)
67	66, 60, 26	fcoss 43894	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → ([,) ∘ 𝑓):ℕ⟶𝒫 ℝ^*)
68	67	ffnd 6715	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → ([,) ∘ 𝑓) Fn ℕ)
69		fniunfv 7242	. . . . . . . . . . . . . . . 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 4027	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → ∪ ran ((,) ∘ 𝑓) ⊆ ∪ ran ([,) ∘ 𝑓))
72	71	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑓)) → ∪ ran ((,) ∘ 𝑓) ⊆ ∪ ran ([,) ∘ 𝑓))
73	46, 72	sstrd 3991	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑓)) → 𝐴 ⊆ ∪ ran ([,) ∘ 𝑓))
74		simpr 485	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓))) → 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)))
75	26	voliooicof 44698	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → ((vol ∘ (,)) ∘ 𝑓) = ((vol ∘ [,)) ∘ 𝑓))
76	75	fveq2d 6892	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^{^}‘((vol ∘ [,)) ∘ 𝑓)))
77	76	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓))) → (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^{^}‘((vol ∘ [,)) ∘ 𝑓)))
78	74, 77	eqtrd 2772	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓))) → 𝑦 = (Σ^{^}‘((vol ∘ [,)) ∘ 𝑓)))
79	73, 78	anim12dan 619	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)))) → (𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ [,)) ∘ 𝑓))))
80	79	ex 413	. . . . . . . . . 10 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓))) → (𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ [,)) ∘ 𝑓)))))
81	80	reximia 3081	. . . . . . . . 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 4073	. . . . . . 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 4015	. . . . . 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 4024	. . . . 5 ⊢ 𝑄 ⊆ 𝑀
86		infxrss 13314	. . . . 5 ⊢ ((𝑄 ⊆ 𝑀 ∧ 𝑀 ⊆ ℝ^) → inf(𝑀, ℝ^, < ) ≤ inf(𝑄, ℝ^*, < ))
87	85, 6, 86	mp2an 690	. . . 4 ⊢ inf(𝑀, ℝ^, < ) ≤ inf(𝑄, ℝ^, < )
88	87	a1i 11	. . 3 ⊢ (⊤ → inf(𝑀, ℝ^, < ) ≤ inf(𝑄, ℝ^, < ))
89	4, 8, 41, 88	xrletrid 13130	. 2 ⊢ (⊤ → inf(𝑄, ℝ^, < ) = inf(𝑀, ℝ^, < ))
90	89	mptru 1548	1 ⊢ inf(𝑄, ℝ^, < ) = inf(𝑀, ℝ^, < )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ⊤wtru 1542 ∈ wcel 2106 ∀wral 3061 ∃wrex 3070 {crab 3432 ⊆ wss 3947 𝒫 cpw 4601 ⟨cop 4633 ∪ cuni 4907 ∪ ciun 4996 class class class wbr 5147 ↦ cmpt 5230 × cxp 5673 ran crn 5676 ∘ ccom 5679 Fn wfn 6535 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 1^st c1st 7969 2^nd c2nd 7970 ↑_m cmap 8816 infcinf 9432 ℝcr 11105 ℝ^*cxr 11243 < clt 11244 ≤ cle 11245 − cmin 11440 / cdiv 11867 ℕcn 12208 2c2 12263 ℝ⁺crp 12970 +_𝑒 cxad 13086 (,)cioo 13320 [,)cico 13322 ↑cexp 14023 volcvol 24971 Σ^{^}csumge0 45064

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8699 df-map 8818 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-dju 9892 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-rlim 15429 df-sum 15629 df-rest 17364 df-topgen 17385 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-top 22387 df-topon 22404 df-bases 22440 df-cmp 22882 df-ovol 24972 df-vol 24973 df-sumge0 45065

This theorem is referenced by: ovolval5 45357

W3C validator