ovolval4lem2 - Mathbox for Glauco Siliprandi

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Theorem ovolval4lem2 45356

Description: The value of the Lebesgue outer measure for subsets of the reals. Similar to ovolval3 45353, but here 𝑓 is may represent unordered interval bounds. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

**Hypotheses**
Ref	Expression
ovolval4lem2.a	⊢ (𝜑 → 𝐴 ⊆ ℝ)
ovolval4lem2.m	⊢ 𝑀 = {𝑦 ∈ ℝ^* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)))}
ovolval4lem2.g	⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⟨(1^st ‘(𝑓‘𝑛)), if((1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛)), (1^st ‘(𝑓‘𝑛)))⟩)

**Assertion**
Ref	Expression
ovolval4lem2	⊢ (𝜑 → (vol‘𝐴) = inf(𝑀, ℝ^, < ))

Distinct variable groups: 𝐴,𝑓,𝑦 𝑛,𝐺 𝑓,𝑛 𝜑,𝑦 Allowed substitution hints: 𝜑(𝑓,𝑛) 𝐴(𝑛) 𝐺(𝑦,𝑓) 𝑀(𝑦,𝑓,𝑛)

Proof of Theorem ovolval4lem2 Dummy variables 𝑔 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovolval4lem2.a	. 2 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
2		ovolval4lem2.m	. . 3 ⊢ 𝑀 = {𝑦 ∈ ℝ^* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)))}
3		iftrue 4534	. . . . . . . . . . . . . . 15 ⊢ ((1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛)) → if((1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛)), (1^st ‘(𝑓‘𝑛))) = (2^nd ‘(𝑓‘𝑛)))
4	3	opeq2d 4880	. . . . . . . . . . . . . 14 ⊢ ((1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛)) → ⟨(1^st ‘(𝑓‘𝑛)), if((1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛)), (1^st ‘(𝑓‘𝑛)))⟩ = ⟨(1^st ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛))⟩)
5	4	adantl 482	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) ∧ (1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛))) → ⟨(1^st ‘(𝑓‘𝑛)), if((1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛)), (1^st ‘(𝑓‘𝑛)))⟩ = ⟨(1^st ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛))⟩)
6		df-br 5149	. . . . . . . . . . . . . . 15 ⊢ ((1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛)) ↔ ⟨(1^st ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛))⟩ ∈ ≤ )
7	6	biimpi 215	. . . . . . . . . . . . . 14 ⊢ ((1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛)) → ⟨(1^st ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛))⟩ ∈ ≤ )
8	7	adantl 482	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) ∧ (1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛))) → ⟨(1^st ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛))⟩ ∈ ≤ )
9	5, 8	eqeltrd 2833	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) ∧ (1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛))) → ⟨(1^st ‘(𝑓‘𝑛)), if((1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛)), (1^st ‘(𝑓‘𝑛)))⟩ ∈ ≤ )
10		iffalse 4537	. . . . . . . . . . . . . . 15 ⊢ (¬ (1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛)) → if((1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛)), (1^st ‘(𝑓‘𝑛))) = (1^st ‘(𝑓‘𝑛)))
11	10	opeq2d 4880	. . . . . . . . . . . . . 14 ⊢ (¬ (1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛)) → ⟨(1^st ‘(𝑓‘𝑛)), if((1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛)), (1^st ‘(𝑓‘𝑛)))⟩ = ⟨(1^st ‘(𝑓‘𝑛)), (1^st ‘(𝑓‘𝑛))⟩)
12	11	adantl 482	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) ∧ ¬ (1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛))) → ⟨(1^st ‘(𝑓‘𝑛)), if((1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛)), (1^st ‘(𝑓‘𝑛)))⟩ = ⟨(1^st ‘(𝑓‘𝑛)), (1^st ‘(𝑓‘𝑛))⟩)
13		elmapi 8842	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → 𝑓:ℕ⟶(ℝ × ℝ))
14	13	ffvelcdmda 7086	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑓‘𝑛) ∈ (ℝ × ℝ))
15		xp1st 8006	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓‘𝑛) ∈ (ℝ × ℝ) → (1^st ‘(𝑓‘𝑛)) ∈ ℝ)
16	14, 15	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (1^st ‘(𝑓‘𝑛)) ∈ ℝ)
17	16	leidd 11779	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (1^st ‘(𝑓‘𝑛)) ≤ (1^st ‘(𝑓‘𝑛)))
18		df-br 5149	. . . . . . . . . . . . . . 15 ⊢ ((1^st ‘(𝑓‘𝑛)) ≤ (1^st ‘(𝑓‘𝑛)) ↔ ⟨(1^st ‘(𝑓‘𝑛)), (1^st ‘(𝑓‘𝑛))⟩ ∈ ≤ )
19	17, 18	sylib 217	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → ⟨(1^st ‘(𝑓‘𝑛)), (1^st ‘(𝑓‘𝑛))⟩ ∈ ≤ )
20	19	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) ∧ ¬ (1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛))) → ⟨(1^st ‘(𝑓‘𝑛)), (1^st ‘(𝑓‘𝑛))⟩ ∈ ≤ )
21	12, 20	eqeltrd 2833	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) ∧ ¬ (1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛))) → ⟨(1^st ‘(𝑓‘𝑛)), if((1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛)), (1^st ‘(𝑓‘𝑛)))⟩ ∈ ≤ )
22	9, 21	pm2.61dan 811	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → ⟨(1^st ‘(𝑓‘𝑛)), if((1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛)), (1^st ‘(𝑓‘𝑛)))⟩ ∈ ≤ )
23		xp2nd 8007	. . . . . . . . . . . . . 14 ⊢ ((𝑓‘𝑛) ∈ (ℝ × ℝ) → (2^nd ‘(𝑓‘𝑛)) ∈ ℝ)
24	14, 23	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (2^nd ‘(𝑓‘𝑛)) ∈ ℝ)
25	24, 16	ifcld 4574	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → if((1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛)), (1^st ‘(𝑓‘𝑛))) ∈ ℝ)
26		opelxpi 5713	. . . . . . . . . . . 12 ⊢ (((1^st ‘(𝑓‘𝑛)) ∈ ℝ ∧ if((1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛)), (1^st ‘(𝑓‘𝑛))) ∈ ℝ) → ⟨(1^st ‘(𝑓‘𝑛)), if((1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛)), (1^st ‘(𝑓‘𝑛)))⟩ ∈ (ℝ × ℝ))
27	16, 25, 26	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → ⟨(1^st ‘(𝑓‘𝑛)), if((1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛)), (1^st ‘(𝑓‘𝑛)))⟩ ∈ (ℝ × ℝ))
28	22, 27	elind 4194	. . . . . . . . . 10 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → ⟨(1^st ‘(𝑓‘𝑛)), if((1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛)), (1^st ‘(𝑓‘𝑛)))⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
29		ovolval4lem2.g	. . . . . . . . . 10 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⟨(1^st ‘(𝑓‘𝑛)), if((1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛)), (1^st ‘(𝑓‘𝑛)))⟩)
30	28, 29	fmptd 7113	. . . . . . . . 9 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
31		reex 11200	. . . . . . . . . . . . 13 ⊢ ℝ ∈ V
32	31, 31	xpex 7739	. . . . . . . . . . . 12 ⊢ (ℝ × ℝ) ∈ V
33	32	inex2 5318	. . . . . . . . . . 11 ⊢ ( ≤ ∩ (ℝ × ℝ)) ∈ V
34	33	a1i 11	. . . . . . . . . 10 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → ( ≤ ∩ (ℝ × ℝ)) ∈ V)
35		nnex 12217	. . . . . . . . . . 11 ⊢ ℕ ∈ V
36	35	a1i 11	. . . . . . . . . 10 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → ℕ ∈ V)
37	34, 36	elmapd 8833	. . . . . . . . 9 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → (𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ↔ 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ))))
38	30, 37	mpbird 256	. . . . . . . 8 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → 𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ))
39	38	adantr 481	. . . . . . 7 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)))) → 𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ))
40		simpr 485	. . . . . . . . . 10 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑓)) → 𝐴 ⊆ ∪ ran ((,) ∘ 𝑓))
41		rexpssxrxp 11258	. . . . . . . . . . . . . . 15 ⊢ (ℝ × ℝ) ⊆ (ℝ^* × ℝ^*)
42	41	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → (ℝ × ℝ) ⊆ (ℝ^* × ℝ^*))
43	13, 42	fssd 6735	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → 𝑓:ℕ⟶(ℝ^* × ℝ^*))
44		2fveq3 6896	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑛 → (1^st ‘(𝑓‘𝑘)) = (1^st ‘(𝑓‘𝑛)))
45		2fveq3 6896	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑛 → (2^nd ‘(𝑓‘𝑘)) = (2^nd ‘(𝑓‘𝑛)))
46	44, 45	breq12d 5161	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑛 → ((1^st ‘(𝑓‘𝑘)) ≤ (2^nd ‘(𝑓‘𝑘)) ↔ (1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛))))
47	46	cbvrabv 3442	. . . . . . . . . . . . 13 ⊢ {𝑘 ∈ ℕ ∣ (1^st ‘(𝑓‘𝑘)) ≤ (2^nd ‘(𝑓‘𝑘))} = {𝑛 ∈ ℕ ∣ (1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛))}
48	43, 29, 47	ovolval4lem1 45355	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → (∪ ran ((,) ∘ 𝑓) = ∪ ran ((,) ∘ 𝐺) ∧ (vol ∘ ((,) ∘ 𝑓)) = (vol ∘ ((,) ∘ 𝐺))))
49	48	simpld 495	. . . . . . . . . . 11 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → ∪ ran ((,) ∘ 𝑓) = ∪ ran ((,) ∘ 𝐺))
50	49	adantr 481	. . . . . . . . . 10 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑓)) → ∪ ran ((,) ∘ 𝑓) = ∪ ran ((,) ∘ 𝐺))
51	40, 50	sseqtrd 4022	. . . . . . . . 9 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑓)) → 𝐴 ⊆ ∪ ran ((,) ∘ 𝐺))
52	51	adantrr 715	. . . . . . . 8 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)))) → 𝐴 ⊆ ∪ ran ((,) ∘ 𝐺))
53		simpr 485	. . . . . . . . . 10 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓))) → 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)))
54	48	simprd 496	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → (vol ∘ ((,) ∘ 𝑓)) = (vol ∘ ((,) ∘ 𝐺)))
55		coass 6264	. . . . . . . . . . . . . 14 ⊢ ((vol ∘ (,)) ∘ 𝑓) = (vol ∘ ((,) ∘ 𝑓))
56	55	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → ((vol ∘ (,)) ∘ 𝑓) = (vol ∘ ((,) ∘ 𝑓)))
57		coass 6264	. . . . . . . . . . . . . 14 ⊢ ((vol ∘ (,)) ∘ 𝐺) = (vol ∘ ((,) ∘ 𝐺))
58	57	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → ((vol ∘ (,)) ∘ 𝐺) = (vol ∘ ((,) ∘ 𝐺)))
59	54, 56, 58	3eqtr4d 2782	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → ((vol ∘ (,)) ∘ 𝑓) = ((vol ∘ (,)) ∘ 𝐺))
60	59	fveq2d 6895	. . . . . . . . . . 11 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^{^}‘((vol ∘ (,)) ∘ 𝐺)))
61	60	adantr 481	. . . . . . . . . 10 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓))) → (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^{^}‘((vol ∘ (,)) ∘ 𝐺)))
62	53, 61	eqtrd 2772	. . . . . . . . 9 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓))) → 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝐺)))
63	62	adantrl 714	. . . . . . . 8 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)))) → 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝐺)))
64	52, 63	jca 512	. . . . . . 7 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)))) → (𝐴 ⊆ ∪ ran ((,) ∘ 𝐺) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝐺))))
65		coeq2 5858	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝐺 → ((,) ∘ 𝑔) = ((,) ∘ 𝐺))
66	65	rneqd 5937	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → ran ((,) ∘ 𝑔) = ran ((,) ∘ 𝐺))
67	66	unieqd 4922	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → ∪ ran ((,) ∘ 𝑔) = ∪ ran ((,) ∘ 𝐺))
68	67	sseq2d 4014	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ↔ 𝐴 ⊆ ∪ ran ((,) ∘ 𝐺)))
69		coeq2 5858	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → ((vol ∘ (,)) ∘ 𝑔) = ((vol ∘ (,)) ∘ 𝐺))
70	69	fveq2d 6895	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (Σ^{^}‘((vol ∘ (,)) ∘ 𝑔)) = (Σ^{^}‘((vol ∘ (,)) ∘ 𝐺)))
71	70	eqeq2d 2743	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑔)) ↔ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝐺))))
72	68, 71	anbi12d 631	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑔))) ↔ (𝐴 ⊆ ∪ ran ((,) ∘ 𝐺) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝐺)))))
73	72	rspcev 3612	. . . . . . 7 ⊢ ((𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝐺) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝐺)))) → ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑔))))
74	39, 64, 73	syl2anc 584	. . . . . 6 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)))) → ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑔))))
75	74	rexlimiva 3147	. . . . 5 ⊢ (∃𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓))) → ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑔))))
76		inss2 4229	. . . . . . . . . 10 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
77		mapss 8882	. . . . . . . . . 10 ⊢ (((ℝ × ℝ) ∈ V ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)) → (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ⊆ ((ℝ × ℝ) ↑_m ℕ))
78	32, 76, 77	mp2an 690	. . . . . . . . 9 ⊢ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ⊆ ((ℝ × ℝ) ↑_m ℕ)
79	78	sseli 3978	. . . . . . . 8 ⊢ (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → 𝑔 ∈ ((ℝ × ℝ) ↑_m ℕ))
80	79	adantr 481	. . . . . . 7 ⊢ ((𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑔)))) → 𝑔 ∈ ((ℝ × ℝ) ↑_m ℕ))
81		simpr 485	. . . . . . 7 ⊢ ((𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑔)))) → (𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑔))))
82		coeq2 5858	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → ((,) ∘ 𝑓) = ((,) ∘ 𝑔))
83	82	rneqd 5937	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → ran ((,) ∘ 𝑓) = ran ((,) ∘ 𝑔))
84	83	unieqd 4922	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → ∪ ran ((,) ∘ 𝑓) = ∪ ran ((,) ∘ 𝑔))
85	84	sseq2d 4014	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ↔ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑔)))
86		coeq2 5858	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → ((vol ∘ (,)) ∘ 𝑓) = ((vol ∘ (,)) ∘ 𝑔))
87	86	fveq2d 6895	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑔)))
88	87	eqeq2d 2743	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → (𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)) ↔ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑔))))
89	85, 88	anbi12d 631	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓))) ↔ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑔)))))
90	89	rspcev 3612	. . . . . . 7 ⊢ ((𝑔 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑔)))) → ∃𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓))))
91	80, 81, 90	syl2anc 584	. . . . . 6 ⊢ ((𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑔)))) → ∃𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓))))
92	91	rexlimiva 3147	. . . . 5 ⊢ (∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑔))) → ∃𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓))))
93	75, 92	impbii 208	. . . 4 ⊢ (∃𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓))) ↔ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑔))))
94	93	rabbii 3438	. . 3 ⊢ {𝑦 ∈ ℝ^* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)))} = {𝑦 ∈ ℝ^* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑔)))}
95	2, 94	eqtri 2760	. 2 ⊢ 𝑀 = {𝑦 ∈ ℝ^* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑔)))}
96	1, 95	ovolval3 45353	1 ⊢ (𝜑 → (vol‘𝐴) = inf(𝑀, ℝ^, < ))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∃wrex 3070 {crab 3432 Vcvv 3474 ∩ cin 3947 ⊆ wss 3948 ifcif 4528 ⟨cop 4634 ∪ cuni 4908 class class class wbr 5148 ↦ cmpt 5231 × cxp 5674 ran crn 5677 ∘ ccom 5680 ⟶wf 6539 ‘cfv 6543 (class class class)co 7408 1^st c1st 7972 2^nd c2nd 7973 ↑_m cmap 8819 infcinf 9435 ℝcr 11108 ℝ^*cxr 11246 < clt 11247 ≤ cle 11248 ℕcn 12211 (,)cioo 13323 vol*covol 24978 volcvol 24979 Σ^{^}csumge0 45068

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 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-inf2 9635 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187

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 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7669 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-2o 8466 df-er 8702 df-map 8821 df-pm 8822 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fi 9405 df-sup 9436 df-inf 9437 df-oi 9504 df-dju 9895 df-card 9933 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-n0 12472 df-z 12558 df-uz 12822 df-q 12932 df-rp 12974 df-xneg 13091 df-xadd 13092 df-xmul 13093 df-ioo 13327 df-ico 13329 df-icc 13330 df-fz 13484 df-fzo 13627 df-fl 13756 df-seq 13966 df-exp 14027 df-hash 14290 df-cj 15045 df-re 15046 df-im 15047 df-sqrt 15181 df-abs 15182 df-clim 15431 df-rlim 15432 df-sum 15632 df-rest 17367 df-topgen 17388 df-psmet 20935 df-xmet 20936 df-met 20937 df-bl 20938 df-mopn 20939 df-top 22395 df-topon 22412 df-bases 22448 df-cmp 22890 df-ovol 24980 df-vol 24981 df-sumge0 45069

This theorem is referenced by: ovolval4 45357

W3C validator