Theorem ovolval5lem3 45243

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 4078	. . . 4 ⊢ 𝑄 ⊆ ℝ*
3		infxrcl 13299	. . . 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 4078	. . . 4 ⊢ 𝑀 ⊆ ℝ*
7		infxrcl 13299	. . . 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 5853	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑓 → ((,) ∘ 𝑔) = ((,) ∘ 𝑓))
14	13	rneqd 5932	. . . . . . . . . . . . . 14 ⊢ (𝑔 = 𝑓 → ran ((,) ∘ 𝑔) = ran ((,) ∘ 𝑓))
15	14	unieqd 4918	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝑓 → ∪ ran ((,) ∘ 𝑔) = ∪ ran ((,) ∘ 𝑓))
16	15	sseq2d 4012	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑓 → (𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ↔ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑓)))
17		coeq2 5853	. . . . . . . . . . . . . 14 ⊢ (𝑔 = 𝑓 → ((vol ∘ (,)) ∘ 𝑔) = ((vol ∘ (,)) ∘ 𝑓))
18	17	fveq2d 6885	. . . . . . . . . . . . 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 ↦ ⟨((1st ‘(𝑓‘𝑚)) − (𝑤 / (2↑𝑚))), (2nd ‘(𝑓‘𝑚))⟩))) = (Σ^‘((vol ∘ (,)) ∘ (𝑚 ∈ ℕ ↦ ⟨((1st ‘(𝑓‘𝑚)) − (𝑤 / (2↑𝑚))), (2nd ‘(𝑓‘𝑚))⟩)))
26		elmapi 8831	. . . . . . . . 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 6886	. . . . . . . . . . 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 6886	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (2nd ‘(𝑓‘𝑚)) = (2nd ‘(𝑓‘𝑛)))
35	33, 34	opeq12d 4877	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ⟨((1st ‘(𝑓‘𝑚)) − (𝑤 / (2↑𝑚))), (2nd ‘(𝑓‘𝑚))⟩ = ⟨((1st ‘(𝑓‘𝑛)) − (𝑤 / (2↑𝑛))), (2nd ‘(𝑓‘𝑛))⟩)
36	35	cbvmptv 5257	. . . . . . . 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 45242	. . . . . . 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 43955	. . 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 13402	. . . . . . . . . . . . . . . . . . 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 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 43763	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝑓)‘𝑛) = ((1st ‘(𝑓‘𝑛))(,)(2nd ‘(𝑓‘𝑛))))
52	49, 50	fvovco 43763	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (([,) ∘ 𝑓)‘𝑛) = ((1st ‘(𝑓‘𝑛))[,)(2nd ‘(𝑓‘𝑛))))
53	48, 51, 52	3sstr4d 4027	. . . . . . . . . . . . . . . . 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 5011	. . . . . . . . . . . . . . . 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 13411	. . . . . . . . . . . . . . . . . . 19 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ
58	57	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → (,):(ℝ* × ℝ*)⟶𝒫 ℝ)
59		rexpssxrxp 11246	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
60	59	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → (ℝ × ℝ) ⊆ (ℝ* × ℝ*))
61	58, 60, 26	fcoss 43780	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ((,) ∘ 𝑓):ℕ⟶𝒫 ℝ)
62	61	ffnd 6708	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ((,) ∘ 𝑓) Fn ℕ)
63		fniunfv 7233	. . . . . . . . . . . . . . . 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 43789	. . . . . . . . . . . . . . . . . . 19 ⊢ [,):(ℝ* × ℝ*)⟶𝒫 ℝ*
66	65	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → [,):(ℝ* × ℝ*)⟶𝒫 ℝ*)
67	66, 60, 26	fcoss 43780	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ([,) ∘ 𝑓):ℕ⟶𝒫 ℝ*)
68	67	ffnd 6708	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ([,) ∘ 𝑓) Fn ℕ)
69		fniunfv 7233	. . . . . . . . . . . . . . . 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 4026	. . . . . . . . . . . . . 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 3990	. . . . . . . . . . . 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 44585	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ((vol ∘ (,)) ∘ 𝑓) = ((vol ∘ [,)) ∘ 𝑓))
76	75	fveq2d 6885	. . . . . . . . . . . . . 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 4072	. . . . . . 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 4014	. . . . . 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 4023	. . . . 5 ⊢ 𝑄 ⊆ 𝑀
86		infxrss 13305	. . . . 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 13121	. 2 ⊢ (⊤ → inf(𝑄, ℝ*, < ) = inf(𝑀, ℝ*, < ))
90	89	mptru 1549	1 ⊢ inf(𝑄, ℝ*, < ) = inf(𝑀, ℝ*, < )

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542  ⊤wtru 1543   ∈ wcel 2107  ∀wral 3062  ∃wrex 3071  {crab 3433   ⊆ wss 3946  𝒫 cpw 4598  ⟨cop 4630  ∪ cuni 4904  ∪ ciun 4993   class class class wbr 5144   ↦ cmpt 5227   × cxp 5670  ran crn 5673   ∘ ccom 5676   Fn wfn 6530  ⟶wf 6531  ‘cfv 6535  (class class class)co 7396  1^st c1st 7960  2^nd c2nd 7961   ↑_m cmap 8808  infcinf 9423  ℝcr 11096  ℝ^*cxr 11234   < clt 11235   ≤ cle 11236   − cmin 11431   / cdiv 11858  ℕcn 12199  2c2 12254  ℝ⁺crp 12961   +_𝑒 cxad 13077  (,)cioo 13311  [,)cico 13313  ↑cexp 14014  volcvol 24949  Σ^{^}csumge0 44951

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 5281  ax-sep 5295  ax-nul 5302  ax-pow 5359  ax-pr 5423  ax-un 7712  ax-inf2 9623  ax-cnex 11153  ax-resscn 11154  ax-1cn 11155  ax-icn 11156  ax-addcl 11157  ax-addrcl 11158  ax-mulcl 11159  ax-mulrcl 11160  ax-mulcom 11161  ax-addass 11162  ax-mulass 11163  ax-distr 11164  ax-i2m1 11165  ax-1ne0 11166  ax-1rid 11167  ax-rnegex 11168  ax-rrecex 11169  ax-cnre 11170  ax-pre-lttri 11171  ax-pre-lttrn 11172  ax-pre-ltadd 11173  ax-pre-mulgt0 11174  ax-pre-sup 11175

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 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3965  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-int 4947  df-iun 4995  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6292  df-ord 6359  df-on 6360  df-lim 6361  df-suc 6362  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-isom 6544  df-riota 7352  df-ov 7399  df-oprab 7400  df-mpo 7401  df-of 7657  df-om 7843  df-1st 7962  df-2nd 7963  df-frecs 8253  df-wrecs 8284  df-recs 8358  df-rdg 8397  df-1o 8453  df-2o 8454  df-er 8691  df-map 8810  df-pm 8811  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-fi 9393  df-sup 9424  df-inf 9425  df-oi 9492  df-dju 9883  df-card 9921  df-pnf 11237  df-mnf 11238  df-xr 11239  df-ltxr 11240  df-le 11241  df-sub 11433  df-neg 11434  df-div 11859  df-nn 12200  df-2 12262  df-3 12263  df-n0 12460  df-z 12546  df-uz 12810  df-q 12920  df-rp 12962  df-xneg 13079  df-xadd 13080  df-xmul 13081  df-ioo 13315  df-ico 13317  df-icc 13318  df-fz 13472  df-fzo 13615  df-fl 13744  df-seq 13954  df-exp 14015  df-hash 14278  df-cj 15033  df-re 15034  df-im 15035  df-sqrt 15169  df-abs 15170  df-clim 15419  df-rlim 15420  df-sum 15620  df-rest 17355  df-topgen 17376  df-psmet 20910  df-xmet 20911  df-met 20912  df-bl 20913  df-mopn 20914  df-top 22365  df-topon 22382  df-bases 22418  df-cmp 22860  df-ovol 24950  df-vol 24951  df-sumge0 44952

This theorem is referenced by:  ovolval5  45244