Theorem ovolval5lem3 46625

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 4041	. . . 4 ⊢ 𝑄 ⊆ ℝ*
3		infxrcl 13270	. . . 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 4041	. . . 4 ⊢ 𝑀 ⊆ ℝ*
7		infxrcl 13270	. . . 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 3426	. . . . . . 7 ⊢ (𝑦 ∈ 𝑀 ↔ (𝑦 ∈ ℝ* ∧ ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))))
12	11	simprbi 496	. . . . . 6 ⊢ (𝑦 ∈ 𝑀 → ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))))
13		coeq2 5812	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑓 → ((,) ∘ 𝑔) = ((,) ∘ 𝑓))
14	13	rneqd 5891	. . . . . . . . . . . . . 14 ⊢ (𝑔 = 𝑓 → ran ((,) ∘ 𝑔) = ran ((,) ∘ 𝑓))
15	14	unieqd 4880	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝑓 → ∪ ran ((,) ∘ 𝑔) = ∪ ran ((,) ∘ 𝑓))
16	15	sseq2d 3976	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑓 → (𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ↔ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑓)))
17		coeq2 5812	. . . . . . . . . . . . . 14 ⊢ (𝑔 = 𝑓 → ((vol ∘ (,)) ∘ 𝑔) = ((vol ∘ (,)) ∘ 𝑓))
18	17	fveq2d 6844	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝑓 → (Σ^‘((vol ∘ (,)) ∘ 𝑔)) = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))
19	18	eqeq2d 2740	. . . . . . . . . . . 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 3214	. . . . . . . . . 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 3408	. . . . . . . . 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 2755	. . . . . . . 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 2729	. . . . . . . 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 8799	. . . . . . . . 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 1202	. . . . . . . 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 6845	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (1st ‘(𝑓‘𝑚)) = (1st ‘(𝑓‘𝑛)))
31		oveq2 7377	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (2↑𝑚) = (2↑𝑛))
32	31	oveq2d 7385	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (𝑤 / (2↑𝑚)) = (𝑤 / (2↑𝑛)))
33	30, 32	oveq12d 7387	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → ((1st ‘(𝑓‘𝑚)) − (𝑤 / (2↑𝑚))) = ((1st ‘(𝑓‘𝑛)) − (𝑤 / (2↑𝑛))))
34		2fveq3 6845	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (2nd ‘(𝑓‘𝑚)) = (2nd ‘(𝑓‘𝑛)))
35	33, 34	opeq12d 4841	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ⟨((1st ‘(𝑓‘𝑚)) − (𝑤 / (2↑𝑚))), (2nd ‘(𝑓‘𝑚))⟩ = ⟨((1st ‘(𝑓‘𝑛)) − (𝑤 / (2↑𝑛))), (2nd ‘(𝑓‘𝑛))⟩)
36	35	cbvmptv 5206	. . . . . . . 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 46624	. . . . . . 7 ⊢ ((𝑤 ∈ ℝ+ ∧ 𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ (𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → ∃𝑧 ∈ 𝑄 𝑧 ≤ (𝑦 +𝑒 𝑤))
38	37	rexlimdv3a 3138	. . . . . 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 506	. . . . 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 45341	. . 3 ⊢ (⊤ → inf(𝑄, ℝ*, < ) ≤ inf(𝑀, ℝ*, < ))
42		eqeq1 2733	. . . . . . . . . 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 3157	. . . . . . . 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 3413	. . . . . . 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 484	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑓)) → 𝐴 ⊆ ∪ ran ((,) ∘ 𝑓))
47		ioossico 13375	. . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → 𝑓:ℕ⟶(ℝ × ℝ))
50		simpr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
51	49, 50	fvovco 45160	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝑓)‘𝑛) = ((1st ‘(𝑓‘𝑛))(,)(2nd ‘(𝑓‘𝑛))))
52	49, 50	fvovco 45160	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (([,) ∘ 𝑓)‘𝑛) = ((1st ‘(𝑓‘𝑛))[,)(2nd ‘(𝑓‘𝑛))))
53	48, 51, 52	3sstr4d 3999	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝑓)‘𝑛) ⊆ (([,) ∘ 𝑓)‘𝑛))
54	53	ralrimiva 3125	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ∀𝑛 ∈ ℕ (((,) ∘ 𝑓)‘𝑛) ⊆ (([,) ∘ 𝑓)‘𝑛))
55		ss2iun 4970	. . . . . . . . . . . . . . . 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 13384	. . . . . . . . . . . . . . . . . . 19 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ
58	57	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → (,):(ℝ* × ℝ*)⟶𝒫 ℝ)
59		rexpssxrxp 11195	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
60	59	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → (ℝ × ℝ) ⊆ (ℝ* × ℝ*))
61	58, 60, 26	fcoss 45177	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ((,) ∘ 𝑓):ℕ⟶𝒫 ℝ)
62	61	ffnd 6671	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ((,) ∘ 𝑓) Fn ℕ)
63		fniunfv 7203	. . . . . . . . . . . . . . . 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 45186	. . . . . . . . . . . . . . . . . . 19 ⊢ [,):(ℝ* × ℝ*)⟶𝒫 ℝ*
66	65	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → [,):(ℝ* × ℝ*)⟶𝒫 ℝ*)
67	66, 60, 26	fcoss 45177	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ([,) ∘ 𝑓):ℕ⟶𝒫 ℝ*)
68	67	ffnd 6671	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ([,) ∘ 𝑓) Fn ℕ)
69		fniunfv 7203	. . . . . . . . . . . . . . . 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 3998	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ∪ ran ((,) ∘ 𝑓) ⊆ ∪ ran ([,) ∘ 𝑓))
72	71	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑓)) → ∪ ran ((,) ∘ 𝑓) ⊆ ∪ ran ([,) ∘ 𝑓))
73	46, 72	sstrd 3954	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑓)) → 𝐴 ⊆ ∪ ran ([,) ∘ 𝑓))
74		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) → 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))
75	26	voliooicof 45967	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ((vol ∘ (,)) ∘ 𝑓) = ((vol ∘ [,)) ∘ 𝑓))
76	75	fveq2d 6844	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → (Σ^‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))
77	76	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) → (Σ^‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))
78	74, 77	eqtrd 2764	. . . . . . . . . . . 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 412	. . . . . . . . . 10 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) → (𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))))
81	80	reximia 3064	. . . . . . . . 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 4036	. . . . . . 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 3990	. . . . . 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 3995	. . . . 5 ⊢ 𝑄 ⊆ 𝑀
86		infxrss 13276	. . . . 5 ⊢ ((𝑄 ⊆ 𝑀 ∧ 𝑀 ⊆ ℝ*) → inf(𝑀, ℝ*, < ) ≤ inf(𝑄, ℝ*, < ))
87	85, 6, 86	mp2an 692	. . . 4 ⊢ inf(𝑀, ℝ*, < ) ≤ inf(𝑄, ℝ*, < )
88	87	a1i 11	. . 3 ⊢ (⊤ → inf(𝑀, ℝ*, < ) ≤ inf(𝑄, ℝ*, < ))
89	4, 8, 41, 88	xrletrid 13091	. 2 ⊢ (⊤ → inf(𝑄, ℝ*, < ) = inf(𝑀, ℝ*, < ))
90	89	mptru 1547	1 ⊢ inf(𝑄, ℝ*, < ) = inf(𝑀, ℝ*, < )

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ⊤wtru 1541   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  {crab 3402   ⊆ wss 3911  𝒫 cpw 4559  ⟨cop 4591  ∪ cuni 4867  ∪ ciun 4951   class class class wbr 5102   ↦ cmpt 5183   × cxp 5629  ran crn 5632   ∘ ccom 5635   Fn wfn 6494  ⟶wf 6495  ‘cfv 6499  (class class class)co 7369  1^st c1st 7945  2^nd c2nd 7946   ↑_m cmap 8776  infcinf 9368  ℝcr 11043  ℝ^*cxr 11183   < clt 11184   ≤ cle 11185   − cmin 11381   / cdiv 11811  ℕcn 12162  2c2 12217  ℝ⁺crp 12927   +_𝑒 cxad 13046  (,)cioo 13282  [,)cico 13284  ↑cexp 14002  volcvol 25340  Σ^{^}csumge0 46333

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-inf2 9570  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121  ax-pre-sup 11122

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-isom 6508  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-of 7633  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-2o 8412  df-er 8648  df-map 8778  df-pm 8779  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-fi 9338  df-sup 9369  df-inf 9370  df-oi 9439  df-dju 9830  df-card 9868  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-div 11812  df-nn 12163  df-2 12225  df-3 12226  df-n0 12419  df-z 12506  df-uz 12770  df-q 12884  df-rp 12928  df-xneg 13048  df-xadd 13049  df-xmul 13050  df-ioo 13286  df-ico 13288  df-icc 13289  df-fz 13445  df-fzo 13592  df-fl 13730  df-seq 13943  df-exp 14003  df-hash 14272  df-cj 15041  df-re 15042  df-im 15043  df-sqrt 15177  df-abs 15178  df-clim 15430  df-rlim 15431  df-sum 15629  df-rest 17361  df-topgen 17382  df-psmet 21232  df-xmet 21233  df-met 21234  df-bl 21235  df-mopn 21236  df-top 22757  df-topon 22774  df-bases 22809  df-cmp 23250  df-ovol 25341  df-vol 25342  df-sumge0 46334

This theorem is referenced by:  ovolval5  46626