hspmbllem3 - Mathbox for Glauco Siliprandi

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Theorem hspmbllem3 45829

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 4220	. . . . . 6 ⊢ (𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌)) ⊆ 𝐴
4		hspmbllem3.s	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑_m 𝑋))
5	3, 4	sstrid 3985	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌)) ⊆ (ℝ ↑_m 𝑋))
6	2, 5	ovncl 45768	. . . 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 45762	. . . 4 ⊢ (𝜑 → ((voln‘𝑋)‘(𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) ≤ ((voln‘𝑋)‘𝐴))
9	1, 6, 8	ge0lere 44730	. . 3 ⊢ (𝜑 → ((voln*‘𝑋)‘(𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) ∈ ℝ)
10	4	ssdifssd 4134	. . . . 5 ⊢ (𝜑 → (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌)) ⊆ (ℝ ↑_m 𝑋))
11	2, 10	ovncl 45768	. . . 4 ⊢ (𝜑 → ((voln*‘𝑋)‘(𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∈ (0[,]+∞))
12		difssd 4124	. . . . 5 ⊢ (𝜑 → (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌)) ⊆ 𝐴)
13	2, 12, 4	ovnssle 45762	. . . 4 ⊢ (𝜑 → ((voln‘𝑋)‘(𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ≤ ((voln‘𝑋)‘𝐴))
14	1, 11, 13	ge0lere 44730	. . 3 ⊢ (𝜑 → ((voln*‘𝑋)‘(𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∈ ℝ)
15		rexadd 13208	. . 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 4327	. . . . . . 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 45765	. . . . 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 723	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑖 ∈ ((𝐷‘𝐴)‘𝑒)) → 𝐾 ∈ 𝑋)
30		hspmbllem3.y	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ ℝ)
31	30	ad2antrr 723	. . . . . . . 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 6880	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = ℎ → (𝑖‘𝑗) = (ℎ‘𝑗))
35	34	fveq2d 6885	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = ℎ → (𝐿‘(𝑖‘𝑗)) = (𝐿‘(ℎ‘𝑗)))
36	35	mpteq2dv 5240	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = ℎ → (𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗))) = (𝑗 ∈ ℕ ↦ (𝐿‘(ℎ‘𝑗))))
37	36	fveq2d 6885	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = ℎ → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) = (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(ℎ‘𝑗)))))
38	37	breq1d 5148	. . . . . . . . . . . . . . 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 3434	. . . . . . . . . . . . . 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 5243	. . . . . . . . . . . . 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 5243	. . . . . . . . . . . 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 2752	. . . . . . . . . . 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 45812	. . . . . . . . . 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 765	. . . . . . . . 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 767	. . . . . . . 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 13013	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺) → 𝑒 ∈ ℝ)
54	52, 53	rexaddd 13210	. . . . . . . . . 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 5164	. . . . . . . 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 723	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑖 ∈ ((𝐷‘𝐴)‘𝑒)) → ((voln*‘𝑋)‘𝐴) ∈ ℝ)
58	9	ad2antrr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑖 ∈ ((𝐷‘𝐴)‘𝑒)) → ((voln*‘𝑋)‘(𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) ∈ ℝ)
59	14	ad2antrr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑖 ∈ ((𝐷‘𝐴)‘𝑒)) → ((voln*‘𝑋)‘(𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∈ ℝ)
60		eqid 2724	. . . . . . . 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 2724	. . . . . . . 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 2724	. . . . . . . 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 45828	. . . . . . 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 1928	. . . . 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 3138	. . 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 11240	. . . 4 ⊢ (𝜑 → (((voln‘𝑋)‘(𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) + ((voln‘𝑋)‘(𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌)))) ∈ ℝ)
69		alrple 13182	. . . 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 257	. 2 ⊢ (𝜑 → (((voln‘𝑋)‘(𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) + ((voln‘𝑋)‘(𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌)))) ≤ ((voln*‘𝑋)‘𝐴))
72	16, 71	eqbrtrd 5160	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 1533 ∃wex 1773 ∈ wcel 2098 ≠ wne 2932 ∀wral 3053 {crab 3424 ∖ cdif 3937 ∩ cin 3939 ⊆ wss 3940 ∅c0 4314 ifcif 4520 𝒫 cpw 4594 {csn 4620 ∪ ciun 4987 class class class wbr 5138 ↦ cmpt 5221 × cxp 5664 ∘ ccom 5670 ⟶wf 6529 ‘cfv 6533 (class class class)co 7401 ∈ cmpo 7403 1^st c1st 7966 2^nd c2nd 7967 ↑_m cmap 8816 Xcixp 8887 Fincfn 8935 ℝcr 11105 0cc0 11106 + caddc 11109 -∞cmnf 11243 ≤ cle 11246 ℕcn 12209 ℝ⁺crp 12971 +_𝑒 cxad 13087 (,)cioo 13321 [,)cico 13323 ∏cprod 15846 volcvol 25314 Σ^{^}csumge0 45563 voln*covoln 45737

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-om 7849 df-1st 7968 df-2nd 7969 df-tpos 8206 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-dju 9892 df-card 9930 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-q 12930 df-rp 12972 df-xneg 13089 df-xadd 13090 df-xmul 13091 df-ioo 13325 df-ico 13327 df-icc 13328 df-fz 13482 df-fzo 13625 df-fl 13754 df-seq 13964 df-exp 14025 df-hash 14288 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-clim 15429 df-rlim 15430 df-sum 15630 df-prod 15847 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-starv 17211 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-rest 17367 df-0g 17386 df-topgen 17388 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-grp 18856 df-minusg 18857 df-subg 19040 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-cring 20131 df-oppr 20226 df-dvdsr 20249 df-unit 20250 df-invr 20280 df-dvr 20293 df-drng 20579 df-psmet 21220 df-xmet 21221 df-met 21222 df-bl 21223 df-mopn 21224 df-cnfld 21229 df-top 22718 df-topon 22735 df-bases 22771 df-cmp 23213 df-ovol 25315 df-vol 25316 df-sumge0 45564 df-ovoln 45738

This theorem is referenced by: hspmbl 45830

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