hspmbllem3 - Mathbox for Glauco Siliprandi

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Theorem hspmbllem3 44056

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 4159	. . . . . 6 ⊢ (𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌)) ⊆ 𝐴
4		hspmbllem3.s	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑_m 𝑋))
5	3, 4	sstrid 3928	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌)) ⊆ (ℝ ↑_m 𝑋))
6	2, 5	ovncl 43995	. . . 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 43989	. . . 4 ⊢ (𝜑 → ((voln‘𝑋)‘(𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) ≤ ((voln‘𝑋)‘𝐴))
9	1, 6, 8	ge0lere 42960	. . 3 ⊢ (𝜑 → ((voln*‘𝑋)‘(𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) ∈ ℝ)
10	4	ssdifssd 4073	. . . . 5 ⊢ (𝜑 → (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌)) ⊆ (ℝ ↑_m 𝑋))
11	2, 10	ovncl 43995	. . . 4 ⊢ (𝜑 → ((voln*‘𝑋)‘(𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∈ (0[,]+∞))
12		difssd 4063	. . . . 5 ⊢ (𝜑 → (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌)) ⊆ 𝐴)
13	2, 12, 4	ovnssle 43989	. . . 4 ⊢ (𝜑 → ((voln‘𝑋)‘(𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ≤ ((voln‘𝑋)‘𝐴))
14	1, 11, 13	ge0lere 42960	. . 3 ⊢ (𝜑 → ((voln*‘𝑋)‘(𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∈ ℝ)
15		rexadd 12895	. . 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 583	. 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 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺) → 𝑋 ∈ Fin)
18		hspmbllem3.i	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ 𝑋)
19	18	ne0d 4266	. . . . . . 7 ⊢ (𝜑 → 𝑋 ≠ ∅)
20	19	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺) → 𝑋 ≠ ∅)
21	4	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺) → 𝐴 ⊆ (ℝ ↑_m 𝑋))
22		simpr 484	. . . . . 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 43992	. . . . 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 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑖 ∈ ((𝐷‘𝐴)‘𝑒)) → 𝑋 ∈ Fin)
29	18	ad2antrr 722	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑖 ∈ ((𝐷‘𝐴)‘𝑒)) → 𝐾 ∈ 𝑋)
30		hspmbllem3.y	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ ℝ)
31	30	ad2antrr 722	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑖 ∈ ((𝐷‘𝐴)‘𝑒)) → 𝑌 ∈ ℝ)
32	22	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑖 ∈ ((𝐷‘𝐴)‘𝑒)) → 𝑒 ∈ ℝ⁺)
33	21	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑖 ∈ ((𝐷‘𝐴)‘𝑒)) → 𝐴 ⊆ (ℝ ↑_m 𝑋))
34		fveq1 6755	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = ℎ → (𝑖‘𝑗) = (ℎ‘𝑗))
35	34	fveq2d 6760	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = ℎ → (𝐿‘(𝑖‘𝑗)) = (𝐿‘(ℎ‘𝑗)))
36	35	mpteq2dv 5172	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = ℎ → (𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗))) = (𝑗 ∈ ℕ ↦ (𝐿‘(ℎ‘𝑗))))
37	36	fveq2d 6760	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = ℎ → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) = (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(ℎ‘𝑗)))))
38	37	breq1d 5080	. . . . . . . . . . . . . . 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 3416	. . . . . . . . . . . . . 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 5175	. . . . . . . . . . . . 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 5175	. . . . . . . . . . . 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 2766	. . . . . . . . . . 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 484	. . . . . . . . . . 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 44039	. . . . . . . . . 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 764	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑖 ∈ ((𝐷‘𝐴)‘𝑒)) → (𝐵:ℕ⟶(ℝ ↑_m 𝑋) ∧ 𝑇:ℕ⟶(ℝ ↑_m 𝑋)))
48	47	simpld 494	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑖 ∈ ((𝐷‘𝐴)‘𝑒)) → 𝐵:ℕ⟶(ℝ ↑_m 𝑋))
49	47	simprd 495	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑖 ∈ ((𝐷‘𝐴)‘𝑒)) → 𝑇:ℕ⟶(ℝ ↑_m 𝑋))
50	46	simplrd 766	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑖 ∈ ((𝐷‘𝐴)‘𝑒)) → 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘)))
51	46	simprd 495	. . . . . . . . 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 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺) → ((voln*‘𝑋)‘𝐴) ∈ ℝ)
53	22	rpred 12701	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺) → 𝑒 ∈ ℝ)
54	52, 53	rexaddd 12897	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺) → (((voln‘𝑋)‘𝐴) +_𝑒 𝑒) = (((voln‘𝑋)‘𝐴) + 𝑒))
55	54	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑖 ∈ ((𝐷‘𝐴)‘𝑒)) → (((voln‘𝑋)‘𝐴) +_𝑒 𝑒) = (((voln‘𝑋)‘𝐴) + 𝑒))
56	51, 55	breqtrd 5096	. . . . . . . 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 722	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑖 ∈ ((𝐷‘𝐴)‘𝑒)) → ((voln*‘𝑋)‘𝐴) ∈ ℝ)
58	9	ad2antrr 722	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑖 ∈ ((𝐷‘𝐴)‘𝑒)) → ((voln*‘𝑋)‘(𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) ∈ ℝ)
59	14	ad2antrr 722	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑖 ∈ ((𝐷‘𝐴)‘𝑒)) → ((voln*‘𝑋)‘(𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∈ ℝ)
60		eqid 2738	. . . . . . . 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 2738	. . . . . . . 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 2738	. . . . . . . 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 44055	. . . . . . 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 412	. . . . . 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 1937	. . . . 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 3107	. . 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 10935	. . . 4 ⊢ (𝜑 → (((voln‘𝑋)‘(𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) + ((voln‘𝑋)‘(𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌)))) ∈ ℝ)
69		alrple 12869	. . . 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 583	. . 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 256	. 2 ⊢ (𝜑 → (((voln‘𝑋)‘(𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) + ((voln‘𝑋)‘(𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌)))) ≤ ((voln*‘𝑋)‘𝐴))
72	16, 71	eqbrtrd 5092	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 205 ∧ wa 395 = wceq 1539 ∃wex 1783 ∈ wcel 2108 ≠ wne 2942 ∀wral 3063 {crab 3067 ∖ cdif 3880 ∩ cin 3882 ⊆ wss 3883 ∅c0 4253 ifcif 4456 𝒫 cpw 4530 {csn 4558 ∪ ciun 4921 class class class wbr 5070 ↦ cmpt 5153 × cxp 5578 ∘ ccom 5584 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ∈ cmpo 7257 1^st c1st 7802 2^nd c2nd 7803 ↑_m cmap 8573 Xcixp 8643 Fincfn 8691 ℝcr 10801 0cc0 10802 + caddc 10805 -∞cmnf 10938 ≤ cle 10941 ℕcn 11903 ℝ⁺crp 12659 +_𝑒 cxad 12775 (,)cioo 13008 [,)cico 13010 ∏cprod 15543 volcvol 24532 Σ^{^}csumge0 43790 voln*covoln 43964

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-tpos 8013 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-dju 9590 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-seq 13650 df-exp 13711 df-hash 13973 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-clim 15125 df-rlim 15126 df-sum 15326 df-prod 15544 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-rest 17050 df-0g 17069 df-topgen 17071 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-grp 18495 df-minusg 18496 df-subg 18667 df-cmn 19303 df-abl 19304 df-mgp 19636 df-ur 19653 df-ring 19700 df-cring 19701 df-oppr 19777 df-dvdsr 19798 df-unit 19799 df-invr 19829 df-dvr 19840 df-drng 19908 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-cnfld 20511 df-top 21951 df-topon 21968 df-bases 22004 df-cmp 22446 df-ovol 24533 df-vol 24534 df-sumge0 43791 df-ovoln 43965

This theorem is referenced by: hspmbl 44057

W3C validator