ovolval4lem2 - Mathbox for Glauco Siliprandi

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Theorem ovolval4lem2 47291

Description: The value of the Lebesgue outer measure for subsets of the reals. Similar to ovolval3 47288, 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 4496	. . . . . . . . . . . . . . 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 4847	. . . . . . . . . . . . . 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 486	. . . . . . . . . . . . 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 5112	. . . . . . . . . . . . . 14 ⊢ ((1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛)) ↔ ⟨(1^st ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛))⟩ ∈ ≤ )
7	6	bilani 509	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) ∧ (1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛))) → ⟨(1^st ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛))⟩ ∈ ≤ )
8	5, 7	eqeltrd 2869	. . . . . . . . . . . 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)))⟩ ∈ ≤ )
9		iffalse 4499	. . . . . . . . . . . . . . 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)))
10	9	opeq2d 4847	. . . . . . . . . . . . . 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))⟩)
11	10	adantl 486	. . . . . . . . . . . . 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))⟩)
12		elmapi 8846	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → 𝑓:ℕ⟶(ℝ × ℝ))
13	12	ffvelcdmda 7080	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑓‘𝑛) ∈ (ℝ × ℝ))
14		xp1st 8018	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓‘𝑛) ∈ (ℝ × ℝ) → (1^st ‘(𝑓‘𝑛)) ∈ ℝ)
15	13, 14	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (1^st ‘(𝑓‘𝑛)) ∈ ℝ)
16	15	leidd 11780	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (1^st ‘(𝑓‘𝑛)) ≤ (1^st ‘(𝑓‘𝑛)))
17		df-br 5112	. . . . . . . . . . . . . . 15 ⊢ ((1^st ‘(𝑓‘𝑛)) ≤ (1^st ‘(𝑓‘𝑛)) ↔ ⟨(1^st ‘(𝑓‘𝑛)), (1^st ‘(𝑓‘𝑛))⟩ ∈ ≤ )
18	16, 17	sylib 221	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → ⟨(1^st ‘(𝑓‘𝑛)), (1^st ‘(𝑓‘𝑛))⟩ ∈ ≤ )
19	18	adantr 485	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) ∧ ¬ (1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛))) → ⟨(1^st ‘(𝑓‘𝑛)), (1^st ‘(𝑓‘𝑛))⟩ ∈ ≤ )
20	11, 19	eqeltrd 2869	. . . . . . . . . . . 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)))⟩ ∈ ≤ )
21	8, 20	pm2.61dan 824	. . . . . . . . . . 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)))⟩ ∈ ≤ )
22		xp2nd 8019	. . . . . . . . . . . . . 14 ⊢ ((𝑓‘𝑛) ∈ (ℝ × ℝ) → (2^nd ‘(𝑓‘𝑛)) ∈ ℝ)
23	13, 22	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (2^nd ‘(𝑓‘𝑛)) ∈ ℝ)
24	23, 15	ifcld 4537	. . . . . . . . . . . 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)
25		opelxpi 5699	. . . . . . . . . . . 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))
26	15, 24, 25	syl2anc 595	. . . . . . . . . . 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))
27	21, 26	elind 4159	. . . . . . . . . 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)))
28		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)))⟩)
29	27, 28	fmptd 7110	. . . . . . . . 9 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
30		reex 11191	. . . . . . . . . . . . 13 ⊢ ℝ ∈ V
31	30, 30	xpex 7752	. . . . . . . . . . . 12 ⊢ (ℝ × ℝ) ∈ V
32	31	inex2 5289	. . . . . . . . . . 11 ⊢ ( ≤ ∩ (ℝ × ℝ)) ∈ V
33	32	a1i 11	. . . . . . . . . 10 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → ( ≤ ∩ (ℝ × ℝ)) ∈ V)
34		nnex 12239	. . . . . . . . . . 11 ⊢ ℕ ∈ V
35	34	a1i 11	. . . . . . . . . 10 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → ℕ ∈ V)
36	33, 35	elmapd 8837	. . . . . . . . 9 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → (𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ↔ 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ))))
37	29, 36	mpbird 260	. . . . . . . 8 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → 𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ))
38	37	adantr 485	. . . . . . 7 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)))) → 𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ))
39		simpr 489	. . . . . . . . . 10 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑓)) → 𝐴 ⊆ ∪ ran ((,) ∘ 𝑓))
40		rexpssxrxp 11254	. . . . . . . . . . . . . . 15 ⊢ (ℝ × ℝ) ⊆ (ℝ^* × ℝ^*)
41	40	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → (ℝ × ℝ) ⊆ (ℝ^* × ℝ^*))
42	12, 41	fssd 6724	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → 𝑓:ℕ⟶(ℝ^* × ℝ^*))
43		2fveq3 6887	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑛 → (1^st ‘(𝑓‘𝑘)) = (1^st ‘(𝑓‘𝑛)))
44		2fveq3 6887	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑛 → (2^nd ‘(𝑓‘𝑘)) = (2^nd ‘(𝑓‘𝑛)))
45	43, 44	breq12d 5124	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑛 → ((1^st ‘(𝑓‘𝑘)) ≤ (2^nd ‘(𝑓‘𝑘)) ↔ (1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛))))
46	45	cbvrabv 3432	. . . . . . . . . . . . 13 ⊢ {𝑘 ∈ ℕ ∣ (1^st ‘(𝑓‘𝑘)) ≤ (2^nd ‘(𝑓‘𝑘))} = {𝑛 ∈ ℕ ∣ (1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛))}
47	42, 28, 46	ovolval4lem1 47290	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → (∪ ran ((,) ∘ 𝑓) = ∪ ran ((,) ∘ 𝐺) ∧ (vol ∘ ((,) ∘ 𝑓)) = (vol ∘ ((,) ∘ 𝐺))))
48	47	simpld 499	. . . . . . . . . . 11 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → ∪ ran ((,) ∘ 𝑓) = ∪ ran ((,) ∘ 𝐺))
49	48	adantr 485	. . . . . . . . . 10 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑓)) → ∪ ran ((,) ∘ 𝑓) = ∪ ran ((,) ∘ 𝐺))
50	39, 49	sseqtrd 3979	. . . . . . . . 9 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑓)) → 𝐴 ⊆ ∪ ran ((,) ∘ 𝐺))
51	50	adantrr 729	. . . . . . . 8 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)))) → 𝐴 ⊆ ∪ ran ((,) ∘ 𝐺))
52		simpr 489	. . . . . . . . . 10 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓))) → 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)))
53	47	simprd 500	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → (vol ∘ ((,) ∘ 𝑓)) = (vol ∘ ((,) ∘ 𝐺)))
54		coass 6268	. . . . . . . . . . . . . 14 ⊢ ((vol ∘ (,)) ∘ 𝑓) = (vol ∘ ((,) ∘ 𝑓))
55	54	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → ((vol ∘ (,)) ∘ 𝑓) = (vol ∘ ((,) ∘ 𝑓)))
56		coass 6268	. . . . . . . . . . . . . 14 ⊢ ((vol ∘ (,)) ∘ 𝐺) = (vol ∘ ((,) ∘ 𝐺))
57	56	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → ((vol ∘ (,)) ∘ 𝐺) = (vol ∘ ((,) ∘ 𝐺)))
58	53, 55, 57	3eqtr4d 2814	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → ((vol ∘ (,)) ∘ 𝑓) = ((vol ∘ (,)) ∘ 𝐺))
59	58	fveq2d 6886	. . . . . . . . . . 11 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^{^}‘((vol ∘ (,)) ∘ 𝐺)))
60	59	adantr 485	. . . . . . . . . 10 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓))) → (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^{^}‘((vol ∘ (,)) ∘ 𝐺)))
61	52, 60	eqtrd 2804	. . . . . . . . 9 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓))) → 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝐺)))
62	61	adantrl 728	. . . . . . . 8 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)))) → 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝐺)))
63	51, 62	jca 520	. . . . . . 7 ⊢ ((𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)))) → (𝐴 ⊆ ∪ ran ((,) ∘ 𝐺) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝐺))))
64		coeq2 5845	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝐺 → ((,) ∘ 𝑔) = ((,) ∘ 𝐺))
65	64	rneqd 5929	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → ran ((,) ∘ 𝑔) = ran ((,) ∘ 𝐺))
66	65	unieqd 4887	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → ∪ ran ((,) ∘ 𝑔) = ∪ ran ((,) ∘ 𝐺))
67	66	sseq2d 3975	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ↔ 𝐴 ⊆ ∪ ran ((,) ∘ 𝐺)))
68		coeq2 5845	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → ((vol ∘ (,)) ∘ 𝑔) = ((vol ∘ (,)) ∘ 𝐺))
69	68	fveq2d 6886	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (Σ^{^}‘((vol ∘ (,)) ∘ 𝑔)) = (Σ^{^}‘((vol ∘ (,)) ∘ 𝐺)))
70	69	eqeq2d 2780	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑔)) ↔ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝐺))))
71	67, 70	anbi12d 643	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑔))) ↔ (𝐴 ⊆ ∪ ran ((,) ∘ 𝐺) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝐺)))))
72	71	rspcev 3588	. . . . . . 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))))
73	38, 63, 72	syl2anc 595	. . . . . 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))))
74	73	rexlimiva 3164	. . . . 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))))
75		inss2 4196	. . . . . . . . . 10 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
76		mapss 8887	. . . . . . . . . 10 ⊢ (((ℝ × ℝ) ∈ V ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)) → (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ⊆ ((ℝ × ℝ) ↑_m ℕ))
77	31, 75, 76	mp2an 704	. . . . . . . . 9 ⊢ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ⊆ ((ℝ × ℝ) ↑_m ℕ)
78	77	sseli 3939	. . . . . . . 8 ⊢ (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → 𝑔 ∈ ((ℝ × ℝ) ↑_m ℕ))
79	78	adantr 485	. . . . . . 7 ⊢ ((𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑔)))) → 𝑔 ∈ ((ℝ × ℝ) ↑_m ℕ))
80		simpr 489	. . . . . . 7 ⊢ ((𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑔)))) → (𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑔))))
81		coeq2 5845	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → ((,) ∘ 𝑓) = ((,) ∘ 𝑔))
82	81	rneqd 5929	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → ran ((,) ∘ 𝑓) = ran ((,) ∘ 𝑔))
83	82	unieqd 4887	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → ∪ ran ((,) ∘ 𝑓) = ∪ ran ((,) ∘ 𝑔))
84	83	sseq2d 3975	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ↔ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑔)))
85		coeq2 5845	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → ((vol ∘ (,)) ∘ 𝑓) = ((vol ∘ (,)) ∘ 𝑔))
86	85	fveq2d 6886	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑔)))
87	86	eqeq2d 2780	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → (𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)) ↔ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑔))))
88	84, 87	anbi12d 643	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓))) ↔ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑔)))))
89	88	rspcev 3588	. . . . . . 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))))
90	79, 80, 89	syl2anc 595	. . . . . 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))))
91	90	rexlimiva 3164	. . . . 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))))
92	74, 91	impbii 212	. . . 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))))
93	92	rabbii 3427	. . 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)))}
94	2, 93	eqtri 2792	. 2 ⊢ 𝑀 = {𝑦 ∈ ℝ^* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑔)))}
95	1, 94	ovolval3 47288	1 ⊢ (𝜑 → (vol‘𝐴) = inf(𝑀, ℝ^, < ))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 400 = wceq 1567 ∈ wcel 2149 ∃wrex 3095 {crab 3422 Vcvv 3461 ∩ cin 3910 ⊆ wss 3911 ifcif 4490 ⟨cop 4598 ∪ cuni 4874 class class class wbr 5111 ↦ cmpt 5194 × cxp 5660 ran crn 5663 ∘ ccom 5666 ⟶wf 6533 ‘cfv 6537 (class class class)co 7411 1^st c1st 7984 2^nd c2nd 7985 ↑_m cmap 8824 infcinf 9401 ℝcr 11099 ℝ^*cxr 11242 < clt 11243 ≤ cle 11244 ℕcn 12233 (,)cioo 13372 vol*covol 25590 volcvol 25591 Σ^{^}csumge0 47003

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-inf2 9610 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177 ax-pre-sup 11178

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3375 df-reu 3376 df-rab 3423 df-v 3463 df-sbc 3752 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4491 df-pw 4567 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-int 4915 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-se 5616 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7675 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8453 df-2o 8454 df-er 8694 df-map 8826 df-pm 8827 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fi 9371 df-sup 9402 df-inf 9403 df-oi 9472 df-dju 9887 df-card 9925 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-div 11872 df-nn 12234 df-2 12303 df-3 12304 df-n0 12505 df-z 12592 df-uz 12863 df-q 12973 df-rp 13017 df-xneg 13137 df-xadd 13138 df-xmul 13139 df-ioo 13376 df-ico 13378 df-icc 13379 df-fz 13536 df-fzo 13683 df-fl 13825 df-seq 14038 df-exp 14098 df-hash 14367 df-cj 15150 df-re 15151 df-im 15152 df-sqrt 15286 df-abs 15287 df-clim 15539 df-rlim 15540 df-sum 15738 df-rest 17475 df-topgen 17496 df-psmet 21483 df-xmet 21484 df-met 21485 df-bl 21486 df-mopn 21487 df-top 23020 df-topon 23037 df-bases 23072 df-cmp 23513 df-ovol 25592 df-vol 25593 df-sumge0 47004

This theorem is referenced by: ovolval4 47292

W3C validator