ovolval4lem2 - Mathbox for Glauco Siliprandi

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Theorem ovolval4lem2 46836

Description: The value of the Lebesgue outer measure for subsets of the reals. Similar to ovolval3 46833, 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 4483	. . . . . . . . . . . . . . 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 4834	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . . 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 5097	. . . . . . . . . . . . . . 15 ⊢ ((1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛)) ↔ ⟨(1^st ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛))⟩ ∈ ≤ )
7	6	biimpi 216	. . . . . . . . . . . . . 14 ⊢ ((1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛)) → ⟨(1^st ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛))⟩ ∈ ≤ )
8	7	adantl 481	. . . . . . . . . . . . 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 2834	. . . . . . . . . . . 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 4486	. . . . . . . . . . . . . . 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 4834	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . . 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 8784	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → 𝑓:ℕ⟶(ℝ × ℝ))
14	13	ffvelcdmda 7027	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑓‘𝑛) ∈ (ℝ × ℝ))
15		xp1st 7963	. . . . . . . . . . . . . . . . 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 11701	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (1^st ‘(𝑓‘𝑛)) ≤ (1^st ‘(𝑓‘𝑛)))
18		df-br 5097	. . . . . . . . . . . . . . 15 ⊢ ((1^st ‘(𝑓‘𝑛)) ≤ (1^st ‘(𝑓‘𝑛)) ↔ ⟨(1^st ‘(𝑓‘𝑛)), (1^st ‘(𝑓‘𝑛))⟩ ∈ ≤ )
19	17, 18	sylib 218	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → ⟨(1^st ‘(𝑓‘𝑛)), (1^st ‘(𝑓‘𝑛))⟩ ∈ ≤ )
20	19	adantr 480	. . . . . . . . . . . . 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 2834	. . . . . . . . . . . 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 812	. . . . . . . . . . 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 7964	. . . . . . . . . . . . . 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 4524	. . . . . . . . . . . 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 5659	. . . . . . . . . . . 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 4150	. . . . . . . . . 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 7057	. . . . . . . . 9 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
31		reex 11115	. . . . . . . . . . . . 13 ⊢ ℝ ∈ V
32	31, 31	xpex 7696	. . . . . . . . . . . 12 ⊢ (ℝ × ℝ) ∈ V
33	32	inex2 5261	. . . . . . . . . . 11 ⊢ ( ≤ ∩ (ℝ × ℝ)) ∈ V
34	33	a1i 11	. . . . . . . . . 10 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → ( ≤ ∩ (ℝ × ℝ)) ∈ V)
35		nnex 12149	. . . . . . . . . . 11 ⊢ ℕ ∈ V
36	35	a1i 11	. . . . . . . . . 10 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → ℕ ∈ V)
37	34, 36	elmapd 8775	. . . . . . . . 9 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → (𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ↔ 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ))))
38	30, 37	mpbird 257	. . . . . . . 8 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → 𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ))
39	38	adantr 480	. . . . . . 7 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)))) → 𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ))
40		simpr 484	. . . . . . . . . 10 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑓)) → 𝐴 ⊆ ∪ ran ((,) ∘ 𝑓))
41		rexpssxrxp 11175	. . . . . . . . . . . . . . 15 ⊢ (ℝ × ℝ) ⊆ (ℝ^* × ℝ^*)
42	41	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → (ℝ × ℝ) ⊆ (ℝ^* × ℝ^*))
43	13, 42	fssd 6677	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → 𝑓:ℕ⟶(ℝ^* × ℝ^*))
44		2fveq3 6837	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑛 → (1^st ‘(𝑓‘𝑘)) = (1^st ‘(𝑓‘𝑛)))
45		2fveq3 6837	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑛 → (2^nd ‘(𝑓‘𝑘)) = (2^nd ‘(𝑓‘𝑛)))
46	44, 45	breq12d 5109	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑛 → ((1^st ‘(𝑓‘𝑘)) ≤ (2^nd ‘(𝑓‘𝑘)) ↔ (1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛))))
47	46	cbvrabv 3407	. . . . . . . . . . . . 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 46835	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → (∪ ran ((,) ∘ 𝑓) = ∪ ran ((,) ∘ 𝐺) ∧ (vol ∘ ((,) ∘ 𝑓)) = (vol ∘ ((,) ∘ 𝐺))))
49	48	simpld 494	. . . . . . . . . . 11 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → ∪ ran ((,) ∘ 𝑓) = ∪ ran ((,) ∘ 𝐺))
50	49	adantr 480	. . . . . . . . . 10 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑓)) → ∪ ran ((,) ∘ 𝑓) = ∪ ran ((,) ∘ 𝐺))
51	40, 50	sseqtrd 3968	. . . . . . . . 9 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑓)) → 𝐴 ⊆ ∪ ran ((,) ∘ 𝐺))
52	51	adantrr 717	. . . . . . . 8 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)))) → 𝐴 ⊆ ∪ ran ((,) ∘ 𝐺))
53		simpr 484	. . . . . . . . . 10 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓))) → 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)))
54	48	simprd 495	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → (vol ∘ ((,) ∘ 𝑓)) = (vol ∘ ((,) ∘ 𝐺)))
55		coass 6222	. . . . . . . . . . . . . 14 ⊢ ((vol ∘ (,)) ∘ 𝑓) = (vol ∘ ((,) ∘ 𝑓))
56	55	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → ((vol ∘ (,)) ∘ 𝑓) = (vol ∘ ((,) ∘ 𝑓)))
57		coass 6222	. . . . . . . . . . . . . 14 ⊢ ((vol ∘ (,)) ∘ 𝐺) = (vol ∘ ((,) ∘ 𝐺))
58	57	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → ((vol ∘ (,)) ∘ 𝐺) = (vol ∘ ((,) ∘ 𝐺)))
59	54, 56, 58	3eqtr4d 2779	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → ((vol ∘ (,)) ∘ 𝑓) = ((vol ∘ (,)) ∘ 𝐺))
60	59	fveq2d 6836	. . . . . . . . . . 11 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^{^}‘((vol ∘ (,)) ∘ 𝐺)))
61	60	adantr 480	. . . . . . . . . 10 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓))) → (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^{^}‘((vol ∘ (,)) ∘ 𝐺)))
62	53, 61	eqtrd 2769	. . . . . . . . 9 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓))) → 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝐺)))
63	62	adantrl 716	. . . . . . . 8 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)))) → 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝐺)))
64	52, 63	jca 511	. . . . . . 7 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)))) → (𝐴 ⊆ ∪ ran ((,) ∘ 𝐺) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝐺))))
65		coeq2 5805	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝐺 → ((,) ∘ 𝑔) = ((,) ∘ 𝐺))
66	65	rneqd 5885	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → ran ((,) ∘ 𝑔) = ran ((,) ∘ 𝐺))
67	66	unieqd 4874	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → ∪ ran ((,) ∘ 𝑔) = ∪ ran ((,) ∘ 𝐺))
68	67	sseq2d 3964	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ↔ 𝐴 ⊆ ∪ ran ((,) ∘ 𝐺)))
69		coeq2 5805	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → ((vol ∘ (,)) ∘ 𝑔) = ((vol ∘ (,)) ∘ 𝐺))
70	69	fveq2d 6836	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (Σ^{^}‘((vol ∘ (,)) ∘ 𝑔)) = (Σ^{^}‘((vol ∘ (,)) ∘ 𝐺)))
71	70	eqeq2d 2745	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑔)) ↔ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝐺))))
72	68, 71	anbi12d 632	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑔))) ↔ (𝐴 ⊆ ∪ ran ((,) ∘ 𝐺) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝐺)))))
73	72	rspcev 3574	. . . . . . 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 3127	. . . . 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 4188	. . . . . . . . . 10 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
77		mapss 8825	. . . . . . . . . 10 ⊢ (((ℝ × ℝ) ∈ V ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)) → (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ⊆ ((ℝ × ℝ) ↑_m ℕ))
78	32, 76, 77	mp2an 692	. . . . . . . . 9 ⊢ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ⊆ ((ℝ × ℝ) ↑_m ℕ)
79	78	sseli 3927	. . . . . . . 8 ⊢ (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → 𝑔 ∈ ((ℝ × ℝ) ↑_m ℕ))
80	79	adantr 480	. . . . . . 7 ⊢ ((𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑔)))) → 𝑔 ∈ ((ℝ × ℝ) ↑_m ℕ))
81		simpr 484	. . . . . . 7 ⊢ ((𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑔)))) → (𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑔))))
82		coeq2 5805	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → ((,) ∘ 𝑓) = ((,) ∘ 𝑔))
83	82	rneqd 5885	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → ran ((,) ∘ 𝑓) = ran ((,) ∘ 𝑔))
84	83	unieqd 4874	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → ∪ ran ((,) ∘ 𝑓) = ∪ ran ((,) ∘ 𝑔))
85	84	sseq2d 3964	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ↔ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑔)))
86		coeq2 5805	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → ((vol ∘ (,)) ∘ 𝑓) = ((vol ∘ (,)) ∘ 𝑔))
87	86	fveq2d 6836	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑔)))
88	87	eqeq2d 2745	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → (𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)) ↔ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑔))))
89	85, 88	anbi12d 632	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓))) ↔ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑔)))))
90	89	rspcev 3574	. . . . . . 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 3127	. . . . 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 209	. . . 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 3402	. . 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 2757	. 2 ⊢ 𝑀 = {𝑦 ∈ ℝ^* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑔)))}
96	1, 95	ovolval3 46833	1 ⊢ (𝜑 → (vol‘𝐴) = inf(𝑀, ℝ^, < ))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ∃wrex 3058 {crab 3397 Vcvv 3438 ∩ cin 3898 ⊆ wss 3899 ifcif 4477 ⟨cop 4584 ∪ cuni 4861 class class class wbr 5096 ↦ cmpt 5177 × cxp 5620 ran crn 5623 ∘ ccom 5626 ⟶wf 6486 ‘cfv 6490 (class class class)co 7356 1^st c1st 7929 2^nd c2nd 7930 ↑_m cmap 8761 infcinf 9342 ℝcr 11023 ℝ^*cxr 11163 < clt 11164 ≤ cle 11165 ℕcn 12143 (,)cioo 13259 vol*covol 25417 volcvol 25418 Σ^{^}csumge0 46548

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-inf2 9548 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-pre-sup 11102

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8633 df-map 8763 df-pm 8764 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-fi 9312 df-sup 9343 df-inf 9344 df-oi 9413 df-dju 9811 df-card 9849 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-div 11793 df-nn 12144 df-2 12206 df-3 12207 df-n0 12400 df-z 12487 df-uz 12750 df-q 12860 df-rp 12904 df-xneg 13024 df-xadd 13025 df-xmul 13026 df-ioo 13263 df-ico 13265 df-icc 13266 df-fz 13422 df-fzo 13569 df-fl 13710 df-seq 13923 df-exp 13983 df-hash 14252 df-cj 15020 df-re 15021 df-im 15022 df-sqrt 15156 df-abs 15157 df-clim 15409 df-rlim 15410 df-sum 15608 df-rest 17340 df-topgen 17361 df-psmet 21299 df-xmet 21300 df-met 21301 df-bl 21302 df-mopn 21303 df-top 22836 df-topon 22853 df-bases 22888 df-cmp 23329 df-ovol 25419 df-vol 25420 df-sumge0 46549

This theorem is referenced by: ovolval4 46837

W3C validator