hspmbllem3 - Mathbox for Glauco Siliprandi

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Theorem hspmbllem3 47162

Description: Any half-space of the n-dimensional Real numbers is Lebesgue measurable. Lemma 115F of [Fremlin1] p. 31. This proof handles the non-trivial cases (nonzero dimension and finite outer measure). (Contributed by Glauco Siliprandi, 24-Dec-2020.)

**Hypotheses**
Ref	Expression
hspmbllem3.h	⊢ 𝐻 = (𝑥 ∈ Fin ↦ (𝑙 ∈ 𝑥, 𝑦 ∈ ℝ ↦ X𝑘 ∈ 𝑥 if(𝑘 = 𝑙, (-∞(,)𝑦), ℝ)))
hspmbllem3.x	⊢ (𝜑 → 𝑋 ∈ Fin)
hspmbllem3.i	⊢ (𝜑 → 𝐾 ∈ 𝑋)
hspmbllem3.y	⊢ (𝜑 → 𝑌 ∈ ℝ)
hspmbllem3.a	⊢ (𝜑 → ((voln*‘𝑋)‘𝐴) ∈ ℝ)
hspmbllem3.s	⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑_m 𝑋))
hspmbllem3.c	⊢ 𝐶 = (𝑎 ∈ 𝒫 (ℝ ↑_m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})
hspmbllem3.l	⊢ 𝐿 = (ℎ ∈ ((ℝ × ℝ) ↑_m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)))
hspmbllem3.d	⊢ 𝐷 = (𝑎 ∈ 𝒫 (ℝ ↑_m 𝑋) ↦ (𝑟 ∈ ℝ⁺ ↦ {𝑖 ∈ (𝐶‘𝑎) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +_𝑒 𝑟)}))
hspmbllem3.10	⊢ 𝐵 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑗)‘𝑘))))
hspmbllem3.11	⊢ 𝑇 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑗)‘𝑘))))

**Assertion**
Ref	Expression
hspmbllem3	⊢ (𝜑 → (((voln‘𝑋)‘(𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) +_𝑒 ((voln‘𝑋)‘(𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌)))) ≤ ((voln*‘𝑋)‘𝐴))

Distinct variable groups: 𝐴,𝑎,ℎ,𝑖,𝑗,𝑘,𝑙,𝑥,𝑦 𝐴,𝑟,𝑎,ℎ,𝑖,𝑗 𝐵,𝑎,ℎ,𝑘,𝑙 𝐶,𝑎,ℎ,𝑖,𝑟 𝐷,𝑎,ℎ,𝑗,𝑘,𝑙,𝑥,𝑦 𝐷,𝑟 𝑖,𝐻,𝑗,𝑘 𝐾,𝑎,ℎ,𝑖,𝑗,𝑘,𝑙,𝑥,𝑦 𝐿,𝑎,ℎ,𝑖,𝑟 𝑇,𝑎,ℎ,𝑗,𝑘,𝑙 𝑋,𝑎,ℎ,𝑖,𝑗,𝑘,𝑙,𝑥,𝑦 𝑋,𝑟 𝑌,𝑎,ℎ,𝑖,𝑗,𝑘,𝑙,𝑥,𝑦 𝜑,𝑎,ℎ,𝑖,𝑗,𝑘,𝑙,𝑥,𝑦 𝜑,𝑟 Allowed substitution hints: 𝐵(𝑥,𝑦,𝑖,𝑗,𝑟) 𝐶(𝑥,𝑦,𝑗,𝑘,𝑙) 𝐷(𝑖) 𝑇(𝑥,𝑦,𝑖,𝑟) 𝐻(𝑥,𝑦,ℎ,𝑟,𝑎,𝑙) 𝐾(𝑟) 𝐿(𝑥,𝑦,𝑗,𝑘,𝑙) 𝑌(𝑟)

Proof of Theorem hspmbllem3 Dummy variables 𝑏 𝑐 𝑒 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hspmbllem3.a	. . . 4 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐴) ∈ ℝ)
2		hspmbllem3.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ Fin)
3		inss1 4186	. . . . . 6 ⊢ (𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌)) ⊆ 𝐴
4		hspmbllem3.s	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑_m 𝑋))
5	3, 4	sstrid 3945	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌)) ⊆ (ℝ ↑_m 𝑋))
6	2, 5	ovncl 47101	. . . 4 ⊢ (𝜑 → ((voln*‘𝑋)‘(𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) ∈ (0[,]+∞))
7	3	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌)) ⊆ 𝐴)
8	2, 7, 4	ovnssle 47095	. . . 4 ⊢ (𝜑 → ((voln‘𝑋)‘(𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) ≤ ((voln‘𝑋)‘𝐴))
9	1, 6, 8	ge0lere 46068	. . 3 ⊢ (𝜑 → ((voln*‘𝑋)‘(𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) ∈ ℝ)
10	4	ssdifssd 4098	. . . . 5 ⊢ (𝜑 → (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌)) ⊆ (ℝ ↑_m 𝑋))
11	2, 10	ovncl 47101	. . . 4 ⊢ (𝜑 → ((voln*‘𝑋)‘(𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∈ (0[,]+∞))
12		difssd 4088	. . . . 5 ⊢ (𝜑 → (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌)) ⊆ 𝐴)
13	2, 12, 4	ovnssle 47095	. . . 4 ⊢ (𝜑 → ((voln‘𝑋)‘(𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ≤ ((voln‘𝑋)‘𝐴))
14	1, 11, 13	ge0lere 46068	. . 3 ⊢ (𝜑 → ((voln*‘𝑋)‘(𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∈ ℝ)
15		rexadd 13228	. . 3 ⊢ ((((voln‘𝑋)‘(𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) ∈ ℝ ∧ ((voln‘𝑋)‘(𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∈ ℝ) → (((voln‘𝑋)‘(𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) +_𝑒 ((voln‘𝑋)‘(𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌)))) = (((voln‘𝑋)‘(𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) + ((voln‘𝑋)‘(𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌)))))
16	9, 14, 15	syl2anc 593	. 2 ⊢ (𝜑 → (((voln‘𝑋)‘(𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) +_𝑒 ((voln‘𝑋)‘(𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌)))) = (((voln‘𝑋)‘(𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) + ((voln‘𝑋)‘(𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌)))))
17	2	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺) → 𝑋 ∈ Fin)
18		hspmbllem3.i	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ 𝑋)
19	18	ne0d 4292	. . . . . . 7 ⊢ (𝜑 → 𝑋 ≠ ∅)
20	19	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺) → 𝑋 ≠ ∅)
21	4	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺) → 𝐴 ⊆ (ℝ ↑_m 𝑋))
22		simpr 488	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺) → 𝑒 ∈ ℝ⁺)
23		hspmbllem3.c	. . . . . 6 ⊢ 𝐶 = (𝑎 ∈ 𝒫 (ℝ ↑_m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})
24		hspmbllem3.l	. . . . . 6 ⊢ 𝐿 = (ℎ ∈ ((ℝ × ℝ) ↑_m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)))
25		hspmbllem3.d	. . . . . 6 ⊢ 𝐷 = (𝑎 ∈ 𝒫 (ℝ ↑_m 𝑋) ↦ (𝑟 ∈ ℝ⁺ ↦ {𝑖 ∈ (𝐶‘𝑎) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +_𝑒 𝑟)}))
26	17, 20, 21, 22, 23, 24, 25	ovncvrrp 47098	. . . . 5 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺) → ∃𝑖 𝑖 ∈ ((𝐷‘𝐴)‘𝑒))
27		hspmbllem3.h	. . . . . . . 8 ⊢ 𝐻 = (𝑥 ∈ Fin ↦ (𝑙 ∈ 𝑥, 𝑦 ∈ ℝ ↦ X𝑘 ∈ 𝑥 if(𝑘 = 𝑙, (-∞(,)𝑦), ℝ)))
28	17	adantr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑖 ∈ ((𝐷‘𝐴)‘𝑒)) → 𝑋 ∈ Fin)
29	18	ad2antrr 736	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑖 ∈ ((𝐷‘𝐴)‘𝑒)) → 𝐾 ∈ 𝑋)
30		hspmbllem3.y	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ ℝ)
31	30	ad2antrr 736	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑖 ∈ ((𝐷‘𝐴)‘𝑒)) → 𝑌 ∈ ℝ)
32	22	adantr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑖 ∈ ((𝐷‘𝐴)‘𝑒)) → 𝑒 ∈ ℝ⁺)
33	21	adantr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑖 ∈ ((𝐷‘𝐴)‘𝑒)) → 𝐴 ⊆ (ℝ ↑_m 𝑋))
34		fveq1 6860	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = ℎ → (𝑖‘𝑗) = (ℎ‘𝑗))
35	34	fveq2d 6865	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = ℎ → (𝐿‘(𝑖‘𝑗)) = (𝐿‘(ℎ‘𝑗)))
36	35	mpteq2dv 5191	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = ℎ → (𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗))) = (𝑗 ∈ ℕ ↦ (𝐿‘(ℎ‘𝑗))))
37	36	fveq2d 6865	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = ℎ → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) = (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(ℎ‘𝑗)))))
38	37	breq1d 5107	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = ℎ → ((Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln‘𝑋)‘𝑎) +_𝑒 𝑟) ↔ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(ℎ‘𝑗)))) ≤ (((voln‘𝑋)‘𝑎) +_𝑒 𝑟)))
39	38	cbvrabv 3423	. . . . . . . . . . . . . 14 ⊢ {𝑖 ∈ (𝐶‘𝑎) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln‘𝑋)‘𝑎) +_𝑒 𝑟)} = {ℎ ∈ (𝐶‘𝑎) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(ℎ‘𝑗)))) ≤ (((voln‘𝑋)‘𝑎) +_𝑒 𝑟)}
40	39	mpteq2i 5193	. . . . . . . . . . . . 13 ⊢ (𝑟 ∈ ℝ⁺ ↦ {𝑖 ∈ (𝐶‘𝑎) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln‘𝑋)‘𝑎) +_𝑒 𝑟)}) = (𝑟 ∈ ℝ⁺ ↦ {ℎ ∈ (𝐶‘𝑎) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(ℎ‘𝑗)))) ≤ (((voln‘𝑋)‘𝑎) +_𝑒 𝑟)})
41	40	mpteq2i 5193	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ 𝒫 (ℝ ↑_m 𝑋) ↦ (𝑟 ∈ ℝ⁺ ↦ {𝑖 ∈ (𝐶‘𝑎) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln‘𝑋)‘𝑎) +_𝑒 𝑟)})) = (𝑎 ∈ 𝒫 (ℝ ↑_m 𝑋) ↦ (𝑟 ∈ ℝ⁺ ↦ {ℎ ∈ (𝐶‘𝑎) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(ℎ‘𝑗)))) ≤ (((voln‘𝑋)‘𝑎) +_𝑒 𝑟)}))
42	25, 41	eqtri 2784	. . . . . . . . . . 11 ⊢ 𝐷 = (𝑎 ∈ 𝒫 (ℝ ↑_m 𝑋) ↦ (𝑟 ∈ ℝ⁺ ↦ {ℎ ∈ (𝐶‘𝑎) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(ℎ‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +_𝑒 𝑟)}))
43		simpr 488	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑖 ∈ ((𝐷‘𝐴)‘𝑒)) → 𝑖 ∈ ((𝐷‘𝐴)‘𝑒))
44		hspmbllem3.10	. . . . . . . . . . 11 ⊢ 𝐵 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑗)‘𝑘))))
45		hspmbllem3.11	. . . . . . . . . . 11 ⊢ 𝑇 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑗)‘𝑘))))
46	28, 33, 32, 23, 24, 42, 43, 44, 45	ovncvr2 47145	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑖 ∈ ((𝐷‘𝐴)‘𝑒)) → (((𝐵:ℕ⟶(ℝ ↑_m 𝑋) ∧ 𝑇:ℕ⟶(ℝ ↑_m 𝑋)) ∧ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘))) ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘))))) ≤ (((voln*‘𝑋)‘𝐴) +_𝑒 𝑒)))
47	46	simplld 777	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑖 ∈ ((𝐷‘𝐴)‘𝑒)) → (𝐵:ℕ⟶(ℝ ↑_m 𝑋) ∧ 𝑇:ℕ⟶(ℝ ↑_m 𝑋)))
48	47	simpld 498	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑖 ∈ ((𝐷‘𝐴)‘𝑒)) → 𝐵:ℕ⟶(ℝ ↑_m 𝑋))
49	47	simprd 499	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑖 ∈ ((𝐷‘𝐴)‘𝑒)) → 𝑇:ℕ⟶(ℝ ↑_m 𝑋))
50	46	simplrd 779	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑖 ∈ ((𝐷‘𝐴)‘𝑒)) → 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘)))
51	46	simprd 499	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑖 ∈ ((𝐷‘𝐴)‘𝑒)) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘))))) ≤ (((voln*‘𝑋)‘𝐴) +_𝑒 𝑒))
52	1	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺) → ((voln*‘𝑋)‘𝐴) ∈ ℝ)
53	22	rpred 13030	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺) → 𝑒 ∈ ℝ)
54	52, 53	rexaddd 13230	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺) → (((voln‘𝑋)‘𝐴) +_𝑒 𝑒) = (((voln‘𝑋)‘𝐴) + 𝑒))
55	54	adantr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑖 ∈ ((𝐷‘𝐴)‘𝑒)) → (((voln‘𝑋)‘𝐴) +_𝑒 𝑒) = (((voln‘𝑋)‘𝐴) + 𝑒))
56	51, 55	breqtrd 5123	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑖 ∈ ((𝐷‘𝐴)‘𝑒)) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘))))) ≤ (((voln*‘𝑋)‘𝐴) + 𝑒))
57	1	ad2antrr 736	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑖 ∈ ((𝐷‘𝐴)‘𝑒)) → ((voln*‘𝑋)‘𝐴) ∈ ℝ)
58	9	ad2antrr 736	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑖 ∈ ((𝐷‘𝐴)‘𝑒)) → ((voln*‘𝑋)‘(𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) ∈ ℝ)
59	14	ad2antrr 736	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑖 ∈ ((𝐷‘𝐴)‘𝑒)) → ((voln*‘𝑋)‘(𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∈ ℝ)
60		eqid 2761	. . . . . . . 8 ⊢ (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑_m 𝑥), 𝑏 ∈ (ℝ ↑_m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘)))))) = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑_m 𝑥), 𝑏 ∈ (ℝ ↑_m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))
61		eqid 2761	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑_m 𝑋) ↦ (ℎ ∈ 𝑋 ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), (𝑐‘ℎ), if((𝑐‘ℎ) ≤ 𝑦, (𝑐‘ℎ), 𝑦))))) = (𝑦 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑_m 𝑋) ↦ (ℎ ∈ 𝑋 ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), (𝑐‘ℎ), if((𝑐‘ℎ) ≤ 𝑦, (𝑐‘ℎ), 𝑦)))))
62		eqid 2761	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑_m 𝑋) ↦ (ℎ ∈ 𝑋 ↦ if(ℎ = 𝐾, if(𝑥 ≤ (𝑐‘ℎ), (𝑐‘ℎ), 𝑥), (𝑐‘ℎ))))) = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑_m 𝑋) ↦ (ℎ ∈ 𝑋 ↦ if(ℎ = 𝐾, if(𝑥 ≤ (𝑐‘ℎ), (𝑐‘ℎ), 𝑥), (𝑐‘ℎ)))))
63	27, 28, 29, 31, 32, 48, 49, 50, 56, 57, 58, 59, 60, 61, 62	hspmbllem2 47161	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑖 ∈ ((𝐷‘𝐴)‘𝑒)) → (((voln‘𝑋)‘(𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) + ((voln‘𝑋)‘(𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌)))) ≤ (((voln*‘𝑋)‘𝐴) + 𝑒))
64	63	ex 416	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺) → (𝑖 ∈ ((𝐷‘𝐴)‘𝑒) → (((voln‘𝑋)‘(𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) + ((voln‘𝑋)‘(𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌)))) ≤ (((voln*‘𝑋)‘𝐴) + 𝑒)))
65	64	exlimdv 1952	. . . . 5 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺) → (∃𝑖 𝑖 ∈ ((𝐷‘𝐴)‘𝑒) → (((voln‘𝑋)‘(𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) + ((voln‘𝑋)‘(𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌)))) ≤ (((voln*‘𝑋)‘𝐴) + 𝑒)))
66	26, 65	mpd 15	. . . 4 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺) → (((voln‘𝑋)‘(𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) + ((voln‘𝑋)‘(𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌)))) ≤ (((voln*‘𝑋)‘𝐴) + 𝑒))
67	66	ralrimiva 3153	. . 3 ⊢ (𝜑 → ∀𝑒 ∈ ℝ⁺ (((voln‘𝑋)‘(𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) + ((voln‘𝑋)‘(𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌)))) ≤ (((voln*‘𝑋)‘𝐴) + 𝑒))
68	9, 14	readdcld 11204	. . . 4 ⊢ (𝜑 → (((voln‘𝑋)‘(𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) + ((voln‘𝑋)‘(𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌)))) ∈ ℝ)
69		alrple 13202	. . . 4 ⊢ (((((voln‘𝑋)‘(𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) + ((voln‘𝑋)‘(𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌)))) ∈ ℝ ∧ ((voln‘𝑋)‘𝐴) ∈ ℝ) → ((((voln‘𝑋)‘(𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) + ((voln‘𝑋)‘(𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌)))) ≤ ((voln‘𝑋)‘𝐴) ↔ ∀𝑒 ∈ ℝ⁺ (((voln‘𝑋)‘(𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) + ((voln‘𝑋)‘(𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌)))) ≤ (((voln*‘𝑋)‘𝐴) + 𝑒)))
70	68, 1, 69	syl2anc 593	. . 3 ⊢ (𝜑 → ((((voln‘𝑋)‘(𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) + ((voln‘𝑋)‘(𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌)))) ≤ ((voln‘𝑋)‘𝐴) ↔ ∀𝑒 ∈ ℝ⁺ (((voln‘𝑋)‘(𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) + ((voln‘𝑋)‘(𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌)))) ≤ (((voln‘𝑋)‘𝐴) + 𝑒)))
71	67, 70	mpbird 259	. 2 ⊢ (𝜑 → (((voln‘𝑋)‘(𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) + ((voln‘𝑋)‘(𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌)))) ≤ ((voln*‘𝑋)‘𝐴))
72	16, 71	eqbrtrd 5119	1 ⊢ (𝜑 → (((voln‘𝑋)‘(𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) +_𝑒 ((voln‘𝑋)‘(𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌)))) ≤ ((voln*‘𝑋)‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 = wceq 1559 ∃wex 1798 ∈ wcel 2141 ≠ wne 2956 ∀wral 3075 {crab 3413 ∖ cdif 3899 ∩ cin 3901 ⊆ wss 3902 ∅c0 4283 ifcif 4477 𝒫 cpw 4552 {csn 4579 ∪ ciun 4946 class class class wbr 5097 ↦ cmpt 5178 × cxp 5641 ∘ ccom 5647 ⟶wf 6511 ‘cfv 6515 (class class class)co 7390 ∈ cmpo 7392 1^st c1st 7962 2^nd c2nd 7963 ↑_m cmap 8801 Xcixp 8872 Fincfn 8920 ℝcr 11065 0cc0 11066 + caddc 11069 -∞cmnf 11207 ≤ cle 11210 ℕcn 12203 ℝ⁺crp 12986 +_𝑒 cxad 13105 (,)cioo 13342 [,)cico 13344 ∏cprod 15923 volcvol 25512 Σ^{^}csumge0 46896 voln*covoln 47070

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7712 ax-inf2 9589 ax-cnex 11122 ax-resscn 11123 ax-1cn 11124 ax-icn 11125 ax-addcl 11126 ax-addrcl 11127 ax-mulcl 11128 ax-mulrcl 11129 ax-mulcom 11130 ax-addass 11131 ax-mulass 11132 ax-distr 11133 ax-i2m1 11134 ax-1ne0 11135 ax-1rid 11136 ax-rnegex 11137 ax-rrecex 11138 ax-cnre 11139 ax-pre-lttri 11140 ax-pre-lttrn 11141 ax-pre-ltadd 11142 ax-pre-mulgt0 11143 ax-pre-sup 11144

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-int 4903 df-iun 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-se 5597 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6282 df-ord 6343 df-on 6344 df-lim 6345 df-suc 6346 df-iota 6471 df-fun 6517 df-fn 6518 df-f 6519 df-f1 6520 df-fo 6521 df-f1o 6522 df-fv 6523 df-isom 6524 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7654 df-om 7841 df-1st 7964 df-2nd 7965 df-frecs 8255 df-wrecs 8286 df-recs 8335 df-rdg 8374 df-1o 8430 df-2o 8431 df-er 8671 df-map 8803 df-pm 8804 df-ixp 8873 df-en 8921 df-dom 8922 df-sdom 8923 df-fin 8924 df-fi 9350 df-sup 9381 df-inf 9382 df-oi 9451 df-dju 9852 df-card 9890 df-pnf 11211 df-mnf 11212 df-xr 11213 df-ltxr 11214 df-le 11215 df-sub 11409 df-neg 11410 df-div 11838 df-nn 12204 df-2 12273 df-3 12274 df-n0 12475 df-z 12562 df-uz 12833 df-q 12943 df-rp 12987 df-xneg 13107 df-xadd 13108 df-xmul 13109 df-ioo 13346 df-ico 13348 df-icc 13349 df-fz 13506 df-fzo 13653 df-fl 13795 df-seq 14008 df-exp 14068 df-hash 14337 df-cj 15116 df-re 15117 df-im 15118 df-sqrt 15252 df-abs 15253 df-clim 15505 df-rlim 15506 df-sum 15704 df-prod 15924 df-rest 17441 df-topgen 17462 df-psmet 21403 df-xmet 21404 df-met 21405 df-bl 21406 df-mopn 21407 df-top 22941 df-topon 22958 df-bases 22993 df-cmp 23434 df-ovol 25513 df-vol 25514 df-sumge0 46897 df-ovoln 47071

This theorem is referenced by: hspmbl 47163

W3C validator